Is the knowledge of the nature and properties of the heavenly bodies; their magnitudes, distances, and motions, both real and apparent; together with the natural causes by which their revolutions are performed.

History of Astronomy.

It is probable that astronomy has existed almost from the beginning of the world. As there is nothing more surprising than the regularity of those great luminous bodies, that seem to turn incessantly round the earth, it is easy to judge, that one of the first curiosities of mankind was to consider their courses, and observe their periods. But it was not curiosity only that induced men to apply themselves to astronomical speculations; necessity itself may be said to have obliged them to it; for if the seasons are not observed, which are distinguished by the motion of the sun, it is impossible to succeed in agriculture. If the times proper for making voyages were not previously known, commerce could not be carried on. If the duration of the month and year were not determined, a certain order could not be established in civil affairs, nor the days allotted to the exercise of religion be fixed. Thus, as neither agriculture, commerce, polity, nor religion, could dispense with the want of astronomy, it is evident that mankind were obliged to apply themselves to that science from the beginning of the world.

What Ptolemy relates of the observations of the heavens, by which Hipparchus reformed astronomy almost 2000 years ago, proves sufficiently, that, in the most ancient times, this science was much studied.

It is agreed that astronomy was cultivated in a particular manner by the Chaldeans. The height of the tower of Babel, which the vanity of men erected about 150 years after the flood, the level and extensive plains of that country, the nights in which they breathed the fresh air after the troublesome heats of the day, an unbroken horizon, a pure and serene sky, all conspired to engage that people to contemplate the vast extent of the heavens, and the motions of the stars. From Chaldea astronomy passed into Egypt, and soon after was carried into Phenicia, where they began to apply its speculative observations to the uses of navigation, by which the Phenicians soon became masters of the sea and of commerce.

What made them bold in undertaking long voyages, was their custom of steering their ships by the observation of one of the stars of the little bear, which, being near the immovable point of the heavens called the pole, is the most proper to serve as a guide in navigation. Other nations, less skillful in astronomy, observed only the great bear in their voyages; but as that constellation is too far from the pole to be capable of serving as a certain guide in long voyages, they did not dare to stand out so far to sea as to lose sight of the coasts; and if a storm happened to drive them into the ocean, or upon some unknown shore, it was impossible for them to know by the heavens into what part of the world the tempest had carried them.

Thales, having at length brought the science of the stars from Phenicia into Greece, taught the Greeks to know the constellation of the little bear, and to make use of it as their guide in navigation. He also taught them the theory of the motion of the sun and moon, by which he accounted for the length and shortness of the days, determined the number of the days of the solar year, and not only explained the cause of eclipses, but showed the art of predicting them, which he even reduced to practice, foretelling an eclipse which happened soon after. The merit of a knowledge so uncommon in those days, made him pass for the oracle of his times, and occasioned his being reckoned the first of the seven sages of Greece.

Anaximander was his disciple, to whom Pliny and Diogenes Laertius ascribed the invention of the terrestrial globe; or, according to Strabo, geographical maps. Anaximander is said also to have erected a gnomon at Sparta, by the means of which he observed the equinoxes and solstices, and to have determined the obliquity of the ecliptic more exactly than had ever been done before; which was necessary for dividing the terrestrial globe into five zones, and for distinguishing the climates, that were afterwards used by geographers for showing the situation of all the places of the earth. The Greeks, assisted by the instructions they had received from Thales and Anaximander, ventured to make considerable voyages, and planted several colonies in remote countries.

Commerce having induced the learned men of Greece to visit other nations, they greatly increased their astronomical knowledge from conversing with the Egyptian priests, who had long made the science of the stars their profession. They also learned many things from the Pythagorean philosophers in Italy, who by some are said to have made so considerable a progress in this science, that they had rejected the common opinions, and and asserted that the earth and planets moved round the sun, which was at rest in the centre of the system. But others affirm that Pythagoras only mentioned this as a conjecture, which he did not pretend to establish as a system.

Meton greatly distinguished himself at Athens by his profound knowledge in astronomy. He lived in the time of the Peloponnesian war; and was the inventor of the golden number, still placed in the calendar.

The Greeks also improved their knowledge from conversing with the druids, who, according to Julius Cæsar, instructed their pupils in the knowledge of the stars, and of the magnitudes of the heavenly bodies.

This species of learning was more ancient in Gaul than is generally imagined. Strabo has preserved a famous observation, made by Pytheas at Marseilles about 2000 years ago, with regard to the proportion of the sun's shadow to the height of a gnomon at the time of the solstice. Were the circumstances of this observation exactly known, it would be sufficient to resolve the important question, Whether the obliquity of the ecliptic be or be not subject to variation?

Pytheas was not contented with making observations in his own country. His passion for astronomy and geography induced him to travel through Europe, from the pillars of Hercules to the mouths of the Tanais. He also advanced along the shore of the western ocean, towards the north pole, and observed that the days grew longer about the summer solstice, in proportion as he travelled; so that, in the island of Thule, the sun rose almost as soon as it set, the tropic continuing entirely above the horizon. By this means he proved the fallacy of what some philosophers had advanced, namely, that those climates were not habitable; and at the same time showed the method of distinguishing the climates by the length of the days and nights.

About the time of Pytheas, several of the Greeks applied themselves to astronomy in emulation of each other. Eudoxus, the disciple of Plato, not being satisfied with what was taught on that subject in the schools of Athens, repaired to Egypt, to cultivate astronomy at its source; and having a letter of recommendation from Agelaius king of Sparta, to Nechanebus king of Egypt, he remained 16 months with the astronomers of that country. At his return, he composed several books upon astronomy; and, among others, a description of the constellations, which Aratus, some time after, turned into verse, by order of Antigonus.

Aristotle, the disciple of Plato, and the contemporary of Eudoxus, made use of astronomy for improving physics and geography. He attempted to determine, by means of astronomical observations, both the figure and magnitude of the earth. He demonstrated, that it was of a spherical form, by the circular appearance of its shadow on the disk of the moon in eclipses; and from the inequality of the meridian altitudes of the sun, which are different in different latitudes.

Callisthenes, who attended Alexander the Great, having been sent to Babylon, found there astronomical observations made by the Babylonians during the space of 1903 years, and sent them to Aristotle.

The princes who succeeded Alexander in the kingdom of Egypt were very careful to draw the most famous astronomers to their courts by their liberality; so that Alexandria soon became the seat of astronomy.

The famous Conon made a vast number of observations; but they have not reached our hands. Aristyllus and Timocharis observed the places of the fixed stars, in order to improve navigation and geography. Eratosthenes measured a degree of the meridian, in order to determine the magnitude of the earth. Hipparchus, who also resided at Alexandria, laid the foundation for a methodical system of astronomy; for a new star happening to appear, he made a catalogue of the fixed stars, consisting of 1022. He also described their motion round the poles of the ecliptic, and at the same time applied himself to establish a theory of the solar and lunar motion.

The Romans, who aspired to the empire of the world, encouraged astronomy, and endeavoured to carry it nearer to perfection; and in the reign of Antoninus it began to assume a new face: for Ptolemy, who may be called the restorer of this science, improving from the lights of his predecessors, and adding the observations of Hipparchus, Timocharis, and those of the Babylonians, to his own, composed a system of astronomy, entitled, "The Great Syntaxis." It contained the theory and tables of the motion of the sun, moon, and other planets, and of the fixed stars.

But as the beginning of great works are never perfect, it is no wonder that Ptolemy's work was not free from errors and defects. Many ages, however, elapsed without any one's presuming either to correct or complete it. At last, the Arabian princes, having conquered the countries where astronomy had long flourished, procured the work of Ptolemy to be translated into their own language, and called it the Almagest. Nor did they stop here; they caused many observations to be made, by which it appeared, that the greatest declination of the sun was one-third of a degree less than what Ptolemy had made it; and that the motion of the fixed stars was not so slow as he believed it. They also ordered a large extent of country under the same meridian to be measured, in order to determine the length of a degree.

This example of the khalifs excited the princes of Europe to promote the improvement of astronomy. The emperor Frederic the second, willing that the Christians should understand astronomy as well as the Barbarians, caused the Almagest of Ptolemy to be translated from the Arabic into Latin; and Alphonso king of Castile assembled the most able astronomers from all parts. By his orders they applied themselves to reform astronomy, and compose new tables, which from him were called the Alphonsine Tables.

This work awakened the curiosity of the learned of Europe: they applied themselves to invent instruments for facilitating the observations of the heavenly bodies; they calculated ephemerides, and composed tables for finding the declinations of the planets; and laboured successfully to facilitate the calculation of eclipses. The noble Dane Tycho Brahe was a far more accurate observer than any that preceded him. He published from his own observations, a catalogue of 770 fixed stars; and Nicholas Copernicus revived the ancient Pythagorean system.

John Kepler, a most excellent astronomer, discovered, by the help of Tycho's labours, the true system of the world, and the laws that regulate the motion of the celestial bodies. Galileo, the Florentine philosopher, The first and most obvious phenomenon is the daily rising of the sun in the east; and his setting in the west; after which the moon and stars appear, still keeping the same westerly course, till we lose sight of them altogether. This cannot be long taken notice of, before we must likewise perceive that neither the sun nor moon always rise exactly in the same point of the heavens. If we begin to observe the sun, for instance, in the beginning of March, we will find that he seems to rise almost every day sensibly more to the northward than he did the day before, to continue longer above the horizon, and to be more vertical at mid-day. This continues till towards the end of June, when he is observed to move backward in the same manner; and this retrograde motion continues to the end of December, or near it, when he begins again to move forwards, and so on.

The motion of the moon through the heavens, as well as her appearance at different times, are still more remarkable than those of the sun. When she first becomes visible at the time she is called the new-moon, she appears in the western part of the heavens, and seems to be at no great distance from the sun himself. Every night she not only increases in size, but removes to a greater distance from the sun, till at last she appears in the eastern part of the horizon, just at the time the sun disappears in the western. After this she gradually moves farther and farther eastward, and therefore rises every night later and later, till at last she seems to approach the sun as nearly in the east as she did in the west, and rises only a little before him in the morning, as in the first part of her course she set in the west not long after him. All these different appearances are completed in the space of a month, after which they begin in the same order as before. They are not, however, at all times regular; for at some seasons of the year, particularly in harvest, the moon appears for several days to be stationary in the heavens, and to recede no farther from the sun, in consequence of which she rises for that time nearly at the same hour every night.

In contemplating the stars, it is observed that some of them have the singular property of neither rising in the east, nor setting in the west; but seem to turn round one immovable point, near which is placed a single star called the pole or pole-star. This point is more or less elevated according to the different parts of the earth from which we take our view. The inhabitants of Lapland, for instance, see it much more elevated above the horizon, or more vertical, than we do; we see it more vertical than it appears to the inhabitants of France and Spain; and they, again, see it more elevated than the inhabitants of Barbary. By continually travelling south, this star would at length seem depressed in the horizon, and another point would appear in the south part of the horizon, round which the stars in that quarter would seem to turn. In this part of the heavens, however, there is no star so near the pole as there is in the northern part, neither is the number of stars in the southern part of the heavens so great as in the northern. Supposing us still to travel southward, the north-pole would then entirely disappear, and the whole hemisphere would appear to turn round a single point in the south, as the northern hemisphere appears to us to turn round the pole-star.—The general appearance When we further consider the stars, we will find the greatest part of them to keep their places with respect to one another; that is, if we observe two stars having a certain apparent distance from each other this night, we will observe them to have the same tomorrow, and every other succeeding night; but we will by no means observe them to have the same places either with respect to the sun or moon, as must be readily understood from what we have already said. Neither do all the stars in the heavens appear to be of this fixed kind. Some of them, on the contrary, change their places very remarkably with regard to the fixed stars, and with regard to one another. Of these there are only five, distinguished by the name of planets, (from ῥάπτω, to err or wander), and called by the names of Mercury, Venus, Mars, Jupiter, and Saturn.

Mercury is a small star, but emits a very bright white light; though, by reason of his always keeping near the sun, he is seldom to be seen; and when he does make his appearance, his motion towards the sun is so swift, that he can only be discerned for a short time. He appears a little after sunset, and again a little before sunrise.

Venus, the most beautiful star in the heavens, known by the names of the morning and evening star, likewise keeps near the sun, though she recedes from him almost double the distance of Mercury. In consequence of this property she is never seen in the eastern quarter of the heavens when the sun is in the western; but always seems to attend him in the evening, or to give notice of his approach in the morning.

Mars is of a red fiery colour, and always gives a much duller light than Venus, though sometimes he equals her in size. He is not subject to the same limitation in his motions as Mercury or Venus; but appears sometimes very near the sun, and sometimes at a great distance from him; sometimes rising when the sun sets, or setting when he rises. Of this planet it is remarkable, that when he approaches any of the fixed stars, which all the planets frequently do, these stars change their colour, grow dim, and often become totally invisible, though at some little distance from the body of the planet.

Jupiter and Saturn likewise often appear at great distances from the sun. The former shines with a bright light somewhat reddish, and the latter with a pale faint one; and the motion of Saturn among the fixed stars is so slow, that, unless carefully observed, he will not be thought to move at all.

Besides the motions which we observe in all these planets, their apparent magnitudes are very different at different times. Every person must have observed that Venus, though she constantly appears with great splendour, is not always equally big. But this apparent difference of magnitude is most remarkable in the planet Mars, which sometimes appears no less than 25 times larger than at others. This increase of magnitude is likewise very remarkable in Jupiter, but less so in Saturn and Mercury.

Though we have thus described the motions of the planets, with respect to their apparent distances from the sun, they by no means appear to us to move regularly in the heavens, but, on the contrary, in the most complex and confused manner that can be imagined, sometimes going forward, sometimes backward, and sometimes seeming to be stationary. Plate XLIV. fig. 2 represents the apparent paths of Mercury and Venus, as traced by Cassini and Mr Ferguson. They all seem to describe looped curves; but it is not known when any of these curves would return into themselves, except that of Venus, which returns into itself every eighth year. In the figure referred to, that which has the fewest loops is the apparent path of Venus, the other that of Mercury. On each side of the loops they appear stationary; in that part of each loop near the earth, retrograde; and in every other part of their path, direct.

These, however, are not the only moving bodies which are to be observed in the celestial regions. The five above mentioned are indeed the only ones which appear almost constantly, or disappear only at certain intervals, and then as certainly return. But there are others which appear at uncertain intervals, and with a very different aspect from the planets. These are called Comets, from their having a long tail, somewhat resembling the appearance of hair. This, however, is not always the case; for some comets have appeared which were as well defined and as round as planets; but in general they have a luminous matter diffused around them, or projecting out from them, which to appearance very much resembles the Aurora Borealis. When these appear, they come in a direct line towards the sun, as if they were going to fall into his body; and after having disappeared for some time in consequence of their proximity to that luminary, they fly off again on the other side as fast as they came, projecting a tail much greater and brighter in their recess from him than when they advanced towards him; but, getting daily at a farther distance from us in the heavens, they continually lose their splendour, and at last totally disappear. Their apparent magnitude is very different: sometimes they appear only of the brightness of the fixed stars; at other times they will equal the diameter of Venus, and sometimes even of the sun or moon themselves. So, in 1652, Hevelius observed a comet which seemed not inferior to the moon in size, though it had not so bright a splendour, but appeared with a pale and dim light, and had a dismal aspect. These bodies will also sometimes lose their splendour suddenly, while their apparent bulk remains unaltered. With respect to their apparent motions, they have all the inequalities of the planets; sometimes seeming to go forwards, sometimes backwards, and sometimes to be stationary.

Though the fixed stars are the only marks by which astronomers are enabled to judge of the courses of the seemingly moveable ones, and though they have never been observed to change their places, yet they seem not to be endowed with the permanency even of the earth and planets, but to be perishable or destructible by accident, and likewise generable by some natural cause. Several stars observed by the ancients are now no more to be seen, but are destroyed; and new ones have appeared, which were unknown to the ancients. Some of them have also disappeared for some time, and again become visible. Of these, a very remarkable one is mentioned by Dr Keil, in that part of the heavens called... called the neck of the whale; which for eight or nine months of the year withdrew itself from the sight, and for the other three or four months was constantly changing its lustre and bigness. Its appearances were attended with the greatest irregularities; sometimes it appeared much smaller than at others; sometimes it disappeared in three months, and sometimes appeared for four months; nor did the increase or decrease of its magnitude answer to the difference of the times of its appearance.

We are also assured from the observations of astronomers, that some stars have been observed which never were seen before, and for a certain time they have distinguished themselves by their superlative lustre; but afterwards decreasing, they vanished by degrees, and were no more to be seen. One of these stars being first seen and observed by Hipparchus, the chief of the ancient astronomers, set him upon composing a catalogue of the fixed stars, that by it posterity might learn whether any of the stars perish, and others are produced afresh.

After several ages, another new star appeared to Tycho Brahe and the astronomers that were contemporary with him; which put him on the same design with Hipparchus, namely, the making a catalogue of the fixed stars. Of this, and of the other stars which have appeared since that time, we have the following history by Dr Halley:

"The first new star in the chair of Cassiopeia, was not seen by Cornelius Gemma on the eighth of November 1572, who says, he that night considered that part of the heaven in a very serene sky, and saw it not: but that the next night, November 9th, it appeared with a splendour surpassing all the fixed stars, and scarce less bright than Venus. This was not seen by Tycho Brahe before the 14th of the same month: but from thence he assures us that it gradually decreased and died away, so as in March 1574, after fifteen months, to be no longer visible; and at this day no signs of it remain. The place thereof in the sphere of fixed stars, by the accurate observations of the same Tycho, was $9^\circ 17' \text{a} 1^\text{ma} * \gamma$, with $53^\circ 45'$ north latitude.

Such another star was seen and observed by the scholars of Kepler, to begin to appear on Sept. 30th, anno 1604, which was not to be seen the day before: but it broke out at once with a lustre surpassing that of Jupiter; and like the former, it died away gradually, and in much about the same time disappeared totally, there remaining no footsteps thereof in January 1605. This was near the ecliptic, following the right leg of Serpentarius; and by the observations of Kepler and others, was in $7^\circ 20' \text{co} \text{a} 1^\text{ma} * \gamma$, with north latitude $1^\circ 56'$. These two seem to be of a distinct species from the rest, and nothing like them has appeared since.

But between them, viz. in the year 1596, we have the first account of the wonderful star in Collo Ceti, seen by David Fabricius on the third of August, a. v. as bright as a star of the third magnitude, which has been since found to appear and disappear periodically, its period being precisely enough seven revolutions in six years, tho' it returns not always with the same lustre. Nor is it ever totally extinguished, but may at all times be seen with a fix-foot tube. This was singular in its kind, till that in Collo Cygni was discovered. It preceeds the first star of Aries $1^\circ 40'$, with $15^\circ 57'$ south latitude.

Another new star was first discovered by William Janfonius in the year 1600, in pellor, or rather in reductione, Colli Cygni, which exceeded not the third magnitude. This having continued some years, became at length so small, as to be thought by some to have disappeared entirely: but in the years 1657, 58, and 59, it again arose to the third magnitude; though soon after it decayed by degrees to the fifth or sixth magnitude, and at this day is to be seen as such in $9^\circ 18' 38'' \text{a} 1^\text{ma} * \gamma$, with $55^\circ 29'$ north latitude.

A fifth new star was first seen by Hevelius in the year 1670, on July 15th, a. v. as a star of the third magnitude, but by the beginning of October was scarce to be perceived by the naked eye. In April following it was again as bright as before, or rather greater than of the third magnitude, yet wholly disappeared about the middle of August. The next year, in March 1672, it was seen again, but not exceeding the fifth magnitude: since when, it has been no further visible, though we have frequently sought for its return; its place is $9^\circ 3' 17'' \text{a} 1^\text{ma} * \gamma$, and has lat. north $47^\circ 28'$.

The sixth and last is that discovered by Mr G. Kirch in the year 1686, and its period determined to be of $404\frac{1}{2}$ days; and though it rarely exceeds the fifth magnitude, yet it is very regular in its returns, as we found in the year 1714. Since then we have watched, as the absence of the moon and clearness of the weather would permit, to catch the first beginning of its appearance in a fix-foot tube, that, bearing a very great aperture, discovers most minute stars. And on June 15th, last, it was first perceived like one of the very least telescopical stars: but in the rest of that month and July, it gradually increased, so as to become in August visible to the naked eye; and so continued all the month of September. After that, it again died away by degrees; and on the eighth of December, at night, was scarce discernible by the tube; and, as near as could be guessed, equal to what it was at its first appearance on June 25th: so that this year it has been seen in all near six months, which is but little less than half its period: and the middle, and consequently the greatest brightness, falls about the 16th of September."

Concerning the changes which happen among the fixed stars, Mr Montanere, professor of mathematics at Bononia, gave the following account, in a letter to the Royal Society, dated April 30th, 1670. "There are now wanting in the heavens two stars of the second magnitude in the stern of the ship Argo, and its yard; Bayerus marked them with the letters $\beta$ and $\gamma$. I, and others, observed them in the year 1664, upon the occasion of the comet that appeared that year: when they disappeared first, I know not: only I am sure that in the year 1668, upon the 10th of April, there was not the least glimpse of them to be seen; and yet the rest about them, even of the third and fourth magnitudes, remained the same. I have observed many more changes among the fixed stars, even to the number of an hundred, though none of them are so great as those I have shewed."

A very remarkable appearance in the heavens is Galaxy, or that called the galaxy or milky-way. This is a broad milky-way circle, sometimes double, but for the most part single; surrounding the whole celestial concave. It is of a whitish colour, somewhat resembling a faint Aurora Borealis; but Mr Brydone, in his journey to the top of mount Etna, found that phenomenon to make a glorious appearance, being, as he expresses it, like a pure flame that shot across the heavens.

The only appearances, besides those already mentioned, which are very observable by the unassisted eye, are those unexpected obscurations of the sun and moon, commonly called eclipses. These are too well known, and attract the attention too much, to need any particular description. We have, however, accounts very well authenticated, of obscurations of the sun continuing for a much longer time than a common eclipse possibly can do, and likewise of the darkness being much greater than it usually is on such occasions; and that these accounts are probably true, we shall afterwards have occasion to observe.

Sect. II. Of the Appearances of the Celestial Bodies through Telescopes.

By means of this instrument we are enabled in some measure to ascend into the celestial regions, and view the sun, moon, and stars, as they would appear to us if we were brought as many times nearer to them as the telescope magnifies, provided the light proceeding from the luminary we view was diminished in the same proportion. Thus, supposing we view the moon through a telescope magnifying 1000 times, her face will appear 1000 times bigger to us than it does to our naked eye, and we will perceive vast numbers of spots in it which are imperceptible to our naked eye; but then she will also appear 1000 times more dim through the telescope than she does to the naked eye; so that those who look at her through a telescope for the first time, will be greatly disappointed, if they are not warned of this diminution of light. The reason of this is, that the telescope cannot increase the quantity of light which falls upon itself from the moon, and by which only she is or can be visible to us. This quantity of light, however, is by the magnifying powers of the telescope spread over a proportionably large surface; and therefore the moon or other body appears magnified indeed, but with a splendor vastly inferior to that with which she appears to the naked eye.

Hence we may see, that the advantages arising from telescope, with regard to the giving us a near view of the celestial bodies, are not near so great as one would at first imagine; for though we should suppose a telescope capable of magnifying 240,000 times, by which we could see the moon at the distance of only a single mile, we could only have this view with a light 240,000 times less than the moon at present affords us; and how imperfectly objects at the distance of a mile could be seen by such a light, any one may easily imagine. We are not, however, to imagine, that the light here spoken of would be 240,000 times less than moon-light; it would only be 240,000 times less than that wherewith she appears illuminated at present. The moon, we know, is enlightened by the sun, as well as the earth is; and could the telescope increase the quantity of light as much as it does the apparent surface, we would, with a telescope magnifying 240,000 times, see all objects in the moon as well as we do terrestrial objects at a mile's distance when the sun shines bright-

eft: but by reason of this incapacity of the telescope to increase the quantity of light, we could only see the lunar objects as well as we would do terrestrial ones a mile distant, with a light 240,000 times less than the light of the sun, or with a light little better than moonlight; for the light of the sun is computed to be 300,000 times stronger than that of the moon.

What we have here said of the moon is applicable to all the other celestial bodies, but not in the same degree. If we look at the sun, for instance, whose lustre is too great for our naked eye to bear, supposing the telescope to magnify as many millions of times as there are miles between us and the sun, we would perceive objects in him as distinctly as could be done at the distance of a mile, not with the light of sunshine upon this earth, but with that intense light which is emitted from the sun at a mile's distance from his body. This is on the supposition that the telescope could increase the quantity of the light as well as magnify the apparent surface; but, being destitute of this power, the light with which the sun would be seen through such a telescope is as many millions of times less than the abovementioned intense light, as the number of times the telescope magnifies, and which, according to the latest calculations, behoved to be upwards of 95 millions. It must be observed, however, that no telescopes ever have been, or probably ever will be invented, whose magnifying powers are so great as either of those we have mentioned. The greatest magnifying power we have yet heard of in any telescope is one made by the late Mr James Short, which magnifies 12,000 times; and, by having another applied to it, is said to magnify 60,000 times.

From these considerations it will be apparent, that our telescopic views of the celestial objects can be but imperfect, and that conjectures drawn from them must be very vague and uncertain. It is also plain, that, ceteris paribus, our views of Venus and Mercury, the planets nearest the sun, ought to be more distinct than of the more remote ones, Mars, Jupiter, and Saturn; because Venus and Mercury are much more strongly illuminated than the others; but this is not found to hold in fact; which is very surprising, as a strong light ought to be much farther and more strongly reflected from any object than a weak one. The general appearances of each of the celestial bodies, when viewed with the best telescopes, the large one by Mr Short abovementioned excepted, are as follow.

1. The sun, when viewed with but an ordinary instrument, and sometimes through a piece of smoked glass without any telescope, discovers on his surface numbers of black, or rather less bright, spots, of various shapes and sizes. Sometimes these spots will vanish in a very short time from their first appearance; sometimes they travel over his whole disk, or visible surface, from west to east, when they disappear; and in 12 or 13 days they appear again, so as to be known, by their size and shape, to be the same that formerly disappeared. Those, however, which are of the longest continuance, never appear to have any durability or solidity of existence, but soon vanish and become bright like the rest of the surface.

The most remarkable phenomena of these spots have been remarked by Scheiner and Hevelius, and are as follow. 1. Every spot, which hath a nucleus, or considerably... siderably dark part, hath also an umbra, or fainter shade, surrounding it. 2. The boundary betwixt the nucleus and umbra is always distinct and well defined.

3. The increase of a spot is gradual, the breadth of the nucleus and umbra dilating at the same time. 4. In like manner, the decrease of a spot is gradual, the breadth of the nucleus and umbra contracting at the same time. 5. The exterior boundary of the umbra never consists of sharp angles; but is always curvilinear, how irregular soever the outline of the nucleus may be.

6. The nucleus of a spot, whilst on the decrease, often changes its figure by the umbra encroaching irregularly upon it, inasmuch that in a small space of time new encroachments are discernible, whereby the boundary betwixt the nucleus and umbra is perpetually varying.

7. It often happens, by these encroachments, that the nucleus of a spot is divided into two or more nuclei.

8. The nuclei of the spots vanish sooner than the umbra. 9. Small umbrae are often seen without nuclei.

10. An umbra of any considerable size is seldom seen without a nucleus in the middle of it. 11. When a spot which consisted of a nucleus and umbra is about to disappear, if it is not succeeded by a facula, or spot brighter than the rest of the disk, the place where it was is soon after not distinguishable from the rest.

In the Philosophical Transactions, Vol. LXIV. Dr Willson, professor of astronomy at Glasgow, hath given a dissertation on the nature of the solar spots, and mentions the following appearances. 1. When the spot is about to disappear on the western edge of the sun's limb, the eastern part of the umbra first contracts, then vanishes, the nucleus and western part of the umbra remaining; then the nucleus gradually contracts and vanishes, while the western part of the umbra remains. At last this disappears also; and if the spot remains long enough to become again visible, the eastern part of the umbra first becomes visible, then the nucleus; and when the spot approaches the middle of the disk, the nucleus appears environed by the umbra on all sides, as already mentioned. 2. When two spots lie very near to one another, the umbra is deficient on that side which lies next the other spot; and this will be the case, though a large spot should be contiguous to one much smaller; the umbra of the large spot will be totally wanting on that side next the small one. If there are little spots on each side of the large one, the umbra does not totally vanish; but appears flattened, or pressed in towards the nucleus on each side. When the little spots disappear, the umbra of the large one extends itself as usual. This circumstance, he observes, may sometimes prevent the disappearance of the umbra in the manner above-mentioned; so that the western umbra may disappear before the nucleus, if a small spot happens to break out on that side.

In the same volume, p. 337. Mr Wolfstien observes, that the appearances mentioned by Dr Willson are not constant. He positively affirms, that the facula or bright spots on the sun are often converted into dark ones. "I have many times (says he) observed, near the eastern limb, a bright facula just come on, which has the next day shewn itself as a spot, though I do not recollect to have seen such a facula near the western one after a spot's disappearance. Yet I believe, both these circumstances have been observed by others; and perhaps not only near the limbs. The circumstance of the facula being converted into spots I think I may be sure of. That there is generally (perhaps always) a mottled appearance over the face of the sun, when carefully attended to, I think I may be as certain. It is most visible towards the limbs, but I have undoubtedly seen it in the centre; yet I do not recollect to have observed this appearance, or indeed any spots, towards his poles. Once I saw, with a 12 inch reflector, a spot burst to pieces while I was looking at it. I could not expect such an event, and therefore cannot be certain of the exact particulars; but the appearance as it struck me at the time was like that of a piece of ice when dashed on a frozen pond, which breaks to pieces and slides in various directions." He also acquaints us, that the nuclei of the spots are not always in the middle of the umbrae; and gives the figure of one seen November 13th 1773, which is a remarkable instance to the contrary. Mr Dunn, however, in his new Atlas of the mundane system, gives some particulars very different from the above. "The face of the sun (says he) has frequently many large black spots, of various forms and dimensions, which move from east to west, and round the sun, according to some observations in 25 days, according to others in 26, and according to some in 27 days. The black or central part of each spot is in the middle of a great number of very small ones, which permit the light to pass between them. The small spots are scarce ever in contact with the central ones: but what is most remarkable, when the whole spot is near the limb of the sun, the surrounding small ones form nearly a straight line, and the central part projects a little over it, like Saturn in his ring."

The spots are by no means confined to one part of the sun's disk; though we have not heard of any being observed about his polar regions; and though their direction is from east to west, yet the paths they describe in their course over the disk are exceedingly different; sometimes being straight lines, sometimes curves, sometimes descending from the northern to the southern part of the disk, sometimes ascending from the southern to the northern, &c. This was observed by Mr Derham, (Philos. Trans. No. 330.) who hath given figures of the apparent paths of many different spots, wherein the months in which they appeared, and their particular progress each day, are marked.

Besides these spots, there are others which sometimes appear very round and black, travelling over the disk of the sun in a few hours. They are totally unlike the others, and will be shown to proceed from an interposition of the planets Mercury and Venus between the earth and the sun. Excepting the two kinds of spots above-mentioned, however, no kind of object is discoverable on the surface of the sun, but he appears like an immense ocean of elementary fire or light.

2. With the moon the case is very different. Many darkish spots appear in her to the naked eye; and, view through a telescope, their number is prodigiously increased; she also appears very plainly to be more protuberant in the middle than at the edges, or to have the figure of a globe, and not a flat circle. This protuberance, or globular figure, may also be perceived in some degree by the naked eye, when she does not shine very bright. When the moon is horned or gibbous, the one side appears very ragged and uneven, but but the other always exactly defined and circular. The spots in the moon always keep their places exactly; never vanishing, or going from one side to the other, as those of the sun do.

The astronomers Florentius, Langrenus, John Hevelius of Dantzig, Grimaldus, and Ricciolus, have drawn the face of the moon as she is seen through telescopes magnifying between two and 300 times. Particular care has been taken to note all the shining parts in her surface; and, for the better distinguishing them, each has been marked with a proper name. Langrenus and Ricciolus have divided the lunar regions among the philosophers, astronomers, and other eminent men; but Hevelius, fearing lest the philosophers should quarrel about the division of their lands, has spoiled them of their property, and given the names belonging to different countries, islands, and seas on earth, to different parts of the moon's surface, without regard to situation or figure.

We have already observed, that when the planet Mars approaches any of the fixed stars, they lose their light, and sometimes totally disappear before he seems to touch them; but it is not so with the moon; for though the very often comes in betwixt us and the stars, they preserve their lustre till immediately in seeming contact with her, when they suddenly disappear, and as suddenly re-appear on the opposite side. When Saturn, however, was hid by the moon in June 1762, Mr Dunn, who watched his appearance at the meridian, observed a kind of faint shadow to follow him for a little from the edge of the moon's disk. This appearance is represented 6th Plate XLII. fig. 2.

3. Mercury, when looked at through telescopes magnifying about 200 or 300 times, appears equally luminous throughout his whole surface, without the least dark spot. He appears indeed to have the same difference of phases with the moon, being sometimes horned, sometimes gibbous, and sometimes shining with a full round face; but at all times perfectly well defined, without any ragged edge, and very bright.

4. Venus puts on the same appearances with the Moon, or with Mercury; only she is not so bright as the latter. Spots have sometimes been seen in her, which appeared of the nature of those in the sun; others have been observed more permanent. In 1666, October 14th, Mr Cassini observed, towards the middle of the disk of this planet, a part more luminous than the rest, and two dark spots to the westward of it. He could make nothing of his observations at that time, nor had he any opportunity of observing the spots again till April 28th 1667. At this time Venus had an horned appearance, and a quarter of an hour before sun-rising; the bright spot was distant from the southern horn little more than \( \frac{1}{4} \) of its diameter. Near the eastern part of the circumference he saw an oblong spot, which was nearer to the northern than the southern horn. At the rising of the sun, he perceived the bright part distant from the southern horn by \( \frac{1}{4} \) of its diameter; so that it was plainly moving from south to north. He was very much surprised, however, to find, that though this spot appeared to move from south to north while it remained on the southern part of the disk, yet, when he saw it on the northern part, it was plainly moving from north to south. From frequent observation, he imagined that this bright spot finished its motion, and returned to the same place of the disk from whence it let out, in about 23 hours, though not without some irregularity. The same irregular motion he observed in the obscure spots. In 1672 and 1686, with a telescope 54 feet long, Mr Cassini thought he saw a moon, or satellite, belonging to this planet, and having the same phases with Venus herself, but not so well defined. It was not above \( \frac{1}{4} \) of the diameter of Venus distant from her body. Mr Short also made the like observation some years ago.

5. Mars viewed through a telescope, sometimes appears gibbous, but never horned. He has one very large spot in his disk, which at different times changes its figure, as well as shifts its place. Dr Hook observed this spot carefully in 1665, and Mr Cassini in 1666; and the latter, by diligently continuing his observations, found it to return to the same place in the space of 24 hours and 40 minutes. Through a 36 foot telescope, made use of by Dr Hook, this planet appeared almost as big as the full moon. The phases are represented 4th Plate XLII. fig. 3.

Besides these spots, Mars sometimes appears to have darkish fillets or belts along his disk, which are called fasciae, and appear parallel to his equator. The fixed stars which he approaches, suffer the same diminution of their light when observed through a telescope, as when seen by the naked eye.

6. Jupiter has the same general appearance with Mars; only that the fixed stars never suffer any diminution of light by his approaching them. In 1666, Mr Hook observed the body of Jupiter through a telescope 60 feet long, and found the apparent diameter \( \frac{1}{4} \) to be about four times as great as that of the full moon. This planet has more remarkable and more distinct fasciae than Mars, which are terminated by parallel lines. They do not, however, appear to be permanent substances, being often interrupted and broken, and sometimes vanishing entirely. In some of these belts large black spots have appeared, which moved swiftly over the disk from east to west, and returned again to the same place; those nearest the planet's equator in nine hours 56 minutes; and those nearer the poles, according to Dr Smith, in 9 hours and 50 minutes. In 1665, Mr Cassini saw a very large spot in one of Jupiter's belts, which he observed continually for the space of two years, and determined exactly the time of its appearance and disappearance. In 1672, he observed it to return to the same place from whence it had set out in one night, having had an opportunity of viewing Jupiter during 10 hours that night, and thus thoroughly ascertained himself of the exact time it took. In 1677, this spot vanished, and was not seen again till 1679; afterwards, for the space of almost three years, it continually showed itself, and then gradually vanished; and since that time has frequently appeared and disappeared. From the year 1665, in which it first appeared, to the year 1678, it appeared and vanished no fewer than eight times. The apparent motion of these spots over the disk is very unequal when they first appear; on the eastern limb of the disk, their motion is slow, and they appear narrow; but, as they advance towards the middle, they grow broader, and their motion becomes much quicker; and when they approach the western limb, they again change their figure, and move more slowly. The belts undergo several changes without out being broken; sometimes becoming narrower, sometimes increasing in breadth, sometimes advancing towards each other, sometimes receding, &c.

What is most remarkable of this planet is, that it is attended by four satellites or moons, which evidently circulate round it; as they sometimes recede a little from, and then come nearer to, the disk; sometimes are hid behind the body of the planet, and then appear again on the other side, receding to a little distance, then come near and pass over it; and it is observed that when they pass between us and the disk of Jupiter, they often resemble black round spots, like Mercury and Venus passing over the disk of the sun; their shadows in the mean time travelling over the disk like other spots, having the same appearance, and marching either before or behind the satellites according to the situation of the earth and Jupiter with respect to the sun. With good telescopes, black spots have also been observed on the disks of these secondary planets themselves.

7. Saturn, the most remote of all the planets, makes a still more remarkable appearance; being encompassed with a luminous ring, represented at Plate XLII. fig. 1, and Plate XLIII. fig. 5. This ring appears double when seen thro' good telescopes, as represented in the figures, and keeps always parallel to itself; by which means it disappears once every fifteen years, according to the situation of the earth and Saturn with regard to the Sun. The reason of its disappearance is, that it is turned edge-wise to us, when its thinness renders it invisible. When it begins to turn its edge towards us, it appears somewhat thicker on one side than another, and the thickest edge has been observed on different sides of the planet. On the body of Saturn, spots have likewise been sometimes seen, but much more obscurely than those of Mars or Jupiter; and no dark spots have ever been observed on its surface. This planet is attended by five moons, whose general appearances resemble those of Jupiter; only they are more obscure, and require better telescopes to discover them on account of their great distance.

8. The Comets, viewed through a telescope, have a very different appearance from any of the planets. The nucleus, or star, seems much more dim. Sturmius tells us, that observing the comet of 1680 with a telescope, it appeared like a coal dimly glowing; or a rude mass of matter illuminated with a dusky fumid light, less sensible at the extremes than in the middle; and not at all like a star, which appears with a round disk and a vivid light.

Hevelius observed of the comet in 1661, that its body was of a yellowish colour, bright and conspicuous, but without any glittering light. In the middle was a dense ruddy nucleus, almost equal to Jupiter, encompassed with a much fainter, thinner matter.—Feb. 5th. The nucleus was somewhat bigger and brighter, of a gold colour, but its light more dusky than the rest of the stars; it appeared also divided into a number of parts.—Feb. 6th. The nuclei still appeared, though less than before. One of them on the left side of the lower part of the disk appeared to be much denser and brighter than the rest; its body round, and representing a little lucid star; the nuclei still encompassed with another kind of matter.—Feb. 10th. The nuclei more obscure and confused, but brighter at top than at bottom.—Feb. 13th. The head diminished much both in brightness and in magnitude.—March 24th. Its roundness a little impaired, and the edges lacquered.—March 28th. Its matter much dispersed; and no distinct nucleus at all appearing.

Weigelius, who saw through a telescope the comet of 1664, the moon, and a little cloud illuminated by the sun, at the same time; observed that the moon appeared of a continued luminous surface, but the comet very different, being perfectly like the little cloud enlightened by the sun's beams.

The comets, too, are to appearance surrounded with their atmospheres of a prodigious size, often rising ten times higher than the nucleus. They have often likewise different phases, like the moon. Those of 1744 and 1769 had both of them this appearance, at Plate XLII. fig. 6. The latter also, when viewed through a telescope, seemed to turn swiftly round on its axis, and to emit flashes or sparks of electric light from all parts of it; which sparks were instantly impelled with great violence towards the tail.

As for the tails of comets, they resemble the streams of electric light more than anything else. That of 1769 seemed perpetually to float out in straight lines of a pale silver hue, lengthening and shortening at each instant, and forming frequently some of the configurations assumed by the Aurora Borealis. Dr Halley, about the year 1716, hath observed the same thing. Speaking of a remarkable Aurora Borealis, he says, "That the great streams of light so much resembled the tails of comets, that at first sight they might well be taken for such;" and afterwards adds, "This light seems to have a great affinity with that which electric bodies emit in the dark."

9. The fixed stars, when viewed through the best telescopes, appear not at all magnified, but rather diminished in bulk, by reason that the telescope takes off that twinkling appearance they make to the naked eye, and which increases their apparent magnitude. Their number, however, appears increased to prodigiously, that 70 stars have been counted in the constellation called the Pleiades, and no fewer than 2000 in that of Orion. The galaxy, or milky-way, appears in a manner made up of stars so small and so close, that they cannot be discerned singly. To these stars it owes a good part of its light, though not the whole, as some small specks called nebulae are discovered in the heavens, having much the same appearance, and which have not always stars within them. Of these we have the following account in the Philosophical Transactions, No. 347. "Some of these bright spots discover no sign of a star in the middle of them; and the irregular form of those that the nebulae have, shews them not to proceed from the illumination of a central body. These are six in number, all which we will describe in the order of time, as they were discovered; giving their places in the sphere of fixed stars, to enable the curious, who are furnished with good telescopes, to take the satisfaction of contemplating them.

"The first and most considerable is that in the middle of Orion's sword, marked with δ by Bayer in his Uranometria, as a single star of the third magnitude; and is so accounted by Ptolemy, Tycho Brahe, and Hevelius; but it is in reality two very contiguous stars, environed with a very large transparent bright spot, thro' which they appear with several others. There are curiously... riously described by Huygenius in his *Systema Saturnium*, p. 8, who there calls this brightness Portentum cui certe simile aliud nusquam apud reliquas fixas potuit animadvertere; affirming, that he found it by chance in the year 1656. The middle of this is at present in 11° 19' with south lat. 28° 4'.

"About the year 1661, another of this sort was discovered by Bullialdus, in *Cingula Andromedae*. This is neither in Tycho nor Bayer, having been omitted, as are many others, because of its smallness: but it is inserted in the catalogue of Hevelius, who has improperly called it *Nebulosa* instead of *Nebula*; it has no sign of a star in it, but appears like a pale cloud, and seems to emit a radiant beam into the north-east, as that in Orion does into the south-east. It precedes in right ascension the northern in the girdle, or *Bayero*, about a degree and three quarters, and has longitude at this time 9° 24', with north lat. 33° 1'.

"The third is near the ecliptic, between the head and bow of Sagittary, not far from the point of the winter solstice. This, it seems, was found in the year 1665 by a German gentleman M. J. Abraham Ible, whilst he attended the motion of Saturn then near his aphelion. This is small, but very luminous, and emits a ray like the former. Its place at this time is 19° 4', with north lat. 35° 4'.

"A fourth was found by Dr Edm. Halley in the year 1677, when he was making the catalogue of the southern stars. It is in the Centaur, that which Ptolemy calls *Centaurus*; which he names in *dorfo Equino Nebula*, and is *Bayero*; it is in appearance between the fourth and fifth magnitude, and emits but a small light for its breadth, and is without a radiant beam; this never rises in England, but at this time its place is 11° 5', with 35° 4' south lat.

"A fifth was discovered by Mr G. Kirch in the year 1681, preceding the right foot of Antinous: it is of itself but a small obscure spot, but has a star that shines through it, which makes it the more luminous. The longitude of this is at present 19° 9' circiter, with 17° 5' north latitude.

"The sixth and last was accidentally hit upon by Dr Edm. Halley in the constellation of Hercules, in the year 1714. It is nearly in a right line with 5° and 6° of Bayer, somewhat nearer to 5° than 6°; and by comparing its situation among the stars, its place is sufficiently near in 11° 26', with 57° north lat. This is but a little patch; but it shows itself to the naked eye, when the sky is serene and the moon absent."

Sect. III. Conclusions from the foregoing Appearances, and Conjectures concerning them.

From the abovementioned appearances, either with the naked eye, or through telescopes, the conclusions that can be drawn with certainty are very few, and may be reduced to the following:

1. Mercury and Venus are between the earth and the sun, circulating round him as the moon does round the earth: Mars, Jupiter, and Saturn, are at a greater distance, including the earth also in their orbit. This appears from these two planets putting on the phases of the moon; and from Mars, Jupiter, and Saturn, never doing so: from which, and from the apparent irregularity of the planet motions, the motion of the earth necessarily follows, as shall be afterwards more fully explained.

2. The planets Jupiter and Mars revolve on their axis, or turn round, the one in about 9 hours 56 minutes, the other in 24 hours 40 minutes. With regard to Venus, the case is doubtful. From Mr Cassini's observations, it might be inferred, that, if she revolves at all, it is in somewhat more than 23 hours; but, according to the later observations of M. Bianchini, she revolves only in 24 days 8 hours. As to Mercury and Saturn, as no spots have ever been observed on their disk, nothing can be advanced concerning their revolutions.

3. From the motion of the solar spots from east to west, it hath been concluded that he revolves round his axis in the same direction. This, however, we humbly apprehend, may be justly disputed, for the following reason. If the apparent motion of the solar spots depended on his rotation round his axis, then all their apparent paths behoved to be parallel to one another; which they are not, as must be evident from the figures we have given. We acknowledge, that an inclination of the sun's axis to the ecliptic must make them describe paths, at some times of the year, which must deviate from parallelism to those described at other times, in proportion to the quantity of the angle of inclination: but still, at the same times of the year, their paths ought to be parallel to one another; and this, it is manifest from looking at the figures, they are not. Among the paths of those marked by Mr Derham the greatest inequalities in this respect are to be observed, whether we compare them with one another, or with that of the spots in 1769 by Dr Wilson; (see 2nd Plate XLII. fig. 1.)—How far this may deserve attention, we submit to the consideration of the learned.

4. The moon and planets are all opaque bodies, of a globular figure, which shine not by their own light, but by the reflection of that of the sun.

The conjectures which have been formed concerning the nature of the celestial bodies are so numerous, that a recital of them would fill a volume; while at the same time many of them are so ridiculous, that absurdity itself would seem almost to have been exhausted on this subject.

As a specimen of what were the opinions of the ancient philosophers concerning the nature of the sun, it may suffice to mention, that Anaximander and Anaximenes held, that there was a circle of fire all along the heavens, which they called the circle of the sun; between the earth and this fiery circle was placed another circle of some opaque matter, in which there was a hole like the mouth of a German flute. Through this hole the light was transmitted, and appeared to the inhabitants of this earth as a round and distinct body of fire. The eclipses of the sun were occasioned by stopping this hole.

We must not, however, imagine, that the opinions of all the ancients were equally absurd with those of Anaximander and Anaximenes. Many of them had more just notions, tho' very imperfect and obscure. Anaxagoras held the sun to be a fiery globe of some solid substance, bigger than Peloponnesus; and many of the moderns have adopted this notion, only increasing the magnitude of the globe prodigiously. Sir Isaac Newton has proposed it as a query, Whether the sun and fixed stars are not great Earths made vehemently hot, whose... whose parts are kept from fuming away by the vast weight and density of their superincumbent atmospheres, and whose heat is preserved by the prodigious action and reaction of their parts upon one another? Agreeable to this idea, Mr Derham and many others have formed conjectures concerning the solar spots, taking them for the smoke of new volcanoes breaking out in that fiery body; which smoke, being more dense towards the middle, and more rare towards the edges, gives the appearance of a nucleus and umbra. Many, however, are of opinion that they are only exhalations raised by the intense heat of the sun, and consequently a kind of clouds rising in his atmosphere. Some have taken them for planets nearer the sun than Mercury; but this their perishableness will by no means allow us to believe. Others have imagined, that they were new and unformed worlds in a chaotic state, as our earth originally was. In the abovementioned dissertation, Dr Wilton's opinion is of opinion, that the solar spots are vast cavities in the body of the sun himself, and even gives the following method for measuring their depth. "All the foregoing appearances, when taken together, and when duly considered, seem to prove in the most convincing manner, that the nucleus of this spot (December 1769) was considerably beneath the level of the sun's spherical surface.

"The next thing which I took into consideration, was to think of some means whereby I could form an estimate of its depth. At the time of the observation I had on Dec. 12th, I had remarked that the breadth of the side of the umbra next the limb was about 1°4"; but, for determining the point in question, it was also requisite to know the inclination of the shelving side of the umbra to the sun's spherical surface. And here it occurred, that, in the case of a large spot, this would in some measure be deduced from observation. For, at the time when the side of the umbra is just hid, or begins first to come in view, it is evident, that a line joining the eye and its observed edge, or uppermost limit, coincides with the plane of its declivity. By measuring therefore the distance of the edge from the limb, when this change takes place, and by representing it by a projection, the inclination or declivity may in some measure be ascertained. For in fig. 2, let ILDK be a portion of the sun's limb, and ABCD a section of the spot, SL the sun's semidiameter, LG the observed distance from the limb, when the side of the umbra changes, then will the plane of the umbra CD coincide with the line EDG drawn perpendicular to SL at the point G. Let FH be a tangent to the limb at the point D, and join SD.

"Since GL, the verified sine of the angle LSD, is given by observation, that angle is given, which by the figure is equal to FDE, or GDH; which angle is therefore given, and is the angle of inclination of the plane of the umbra to the sun's spherical surface. In the small triangle therefore CMD, which may be considered as rectangular, the angle MDC is given, and the side DC equal to AB is given nearly by observation; therefore the side MC is given, which may be regarded as the depth of the nucleus without any material error.

"I had not an opportunity, in the course of the foregoing observations, to measure the distance GL, not having seen the spot at the time when either of the sides of the umbra changed. It is, however, certain, that when the spot came upon the disk for the second time, this change happened some time in the night between the 11th and 12th of December, and I judge that the distance of the plane of the umbra, when in a line with the eye, must have been about 1°55" from the sun's eastern limb; from which we may safely conclude, that the nucleus of the spot was, at that time, not less than a semidiameter of the earth below the level of the sun's spherical surface, and made the bottom of an amazing cavity, from the surface downwards, whose other dimensions were of much greater extent."

Having thus established it, as an absolute certainty, that the solar spots are vast cavities in the sun, the Doctor next proceeds to offer some queries and conjectures concerning the nature of the sun himself, and to answer some objections to his hypothesis, which it must be confessed is somewhat uncommon. He begins with asking, Whether it is not reasonable to think, that the vast body of the sun is made up of two kinds of matter very different in their qualities; that by far the greatest part is solid and dark; and that this dark globe is encompassed with a thin covering of that resplendent substance, from which the sun would seem to derive the whole of his vivifying heat and energy.—This, if granted, will afford a satisfactory solution of the appearance of spots; because, if any part of this resplendent surface shall be by any means displaced, the dark globe must necessarily appear; the bottom of the cavity corresponding to the nucleus, and the shelving sides to the umbra. The shining substance, he thinks, may be displaced by the action of some elastic vapour generated within the substance of the dark globe. This vapour, dwelling into such a volume as to reach up to the surface of the luminous matter, would thereby throw it aside in all directions: and as we cannot expect any regularity in the production of such a vapour, the irregular appearance and disappearance of the spots is by that means accounted for; as the reflux of the luminous matter must always occasion the dark nucleus gradually to decrease, till at last it becomes indistinguishable from the rest of the surface.

Here an objection occurs, viz. That, on this supposition, the nucleus of a spot whilst on the decrease should always appear nearly circular, by the gradual descent of the luminous matter from all sides to cover it. But to this the Doctor replies, that in all probability the surface of the dark globe is very uneven and mountainous, which prevents the regular reflux of the shining matter. This, he thinks, is rendered very probable by the enormous mountains and cavities which are observed in the moon; and why, says he, may there not be the same on the surface of the sun? He thinks his hypothesis also confirmed by the dividing of the nucleus into several parts, which might arise from the luminous matter flowing in different channels in the bottom of the hollow.—The appearance of the umbra after the nucleus is gone, he thinks, may be owing to a cavity remaining in the luminous matter, tho' the dark globe is entirely covered.

As to a motion of the spots, distinct from what they are supposed to receive from the rotation of the sun round his axis, he says he never could observe any, except what might be attributed to the enlargement or diminution of them when in the neighbourhood of one another. another. "But," says he, "what would further contribute towards forming a judgement of this kind in, the apparent alteration of the relative place, which must arise from the motion across the disk on a spherical surface, a circumstance which I am uncertain if it has been sufficiently attended to." This is the circumstance from which we deduced the objection against the sun's motion; and from comparing the figures given by Mr Derham and others, we cannot help thinking that it hath never been attended to as it ought.

The abovementioned hypothesis, the Doctor thinks, is further confirmed by the disappearance of the umbra on the sides of spots contiguous to one another; as the action of the elastic vapour must necessarily drive the luminous matter away from each, and thus as it were accumulate it between them, so that no umbra can be perceived. As to the luminous matter itself, he conjectures, that it cannot be any very ponderous fluid, but that it rather resembles a dense fog which broods on the surface of the sun's dark body. His general conclusion being somewhat extraordinary, we shall give it in his own words.

"According to the view of things given in the foregoing queries, there would seem to be something very extraordinary in the dark and unignited state of the great internal globe of the sun. Does not this seem to indicate that the luminous matter that encompasses it derives not its splendour from any intensity of heat? For, if this were the case, would not the parts underneath, which would be perpetually in contact with that glowing matter, be heated to such a degree, as to become luminous and bright? At the same time it must be confessed, that although the internal globe was in reality much ignited, yet when any part of it forming the nucleus of a spot is exposed to our view, and is seen in competition with a substance of such amazing splendour, it is no wonder that an inferior degree of light should, in these cases, be unperceivable.

"In order to obtain some knowledge of this point, I think an experiment might be tried, if we had an opportunity of a very large spot, by making a contrivance in the eye-piece of a telescope, whereby an observer could look at the nucleus alone with the naked eye, without being in danger of light coming from any other part of the sun. In this case, if the observer found no greater splendour than what might be expected from a planet very near the sun, and illumined by as much of his surface as corresponds to the spot's umbra, we might reasonably conclude, that the solar matter, at the depth of the nucleus, is in reality not ignited. But from the nature of the thing, doth there seem any necessity for thinking that there prevails such a raging and fervent heat as many have imagined? It is proper here to attend to the distinction betwixt this shining matter of the sun, and the rays of light which proceed from it. It may perhaps be thought, that the reaction of the rays upon the matter, at their emission, may be productive of a violent degree of heat. But whoever would urge this argument in favour of the sun being intensely heated, as arising from the nature of the thing, ought to consider that all polished bodies are less and less disposed to be heated by the action of the rays of light, in proportion as their surfaces are more polished, and as their powers of reflection are brought to a greater degree of perfection. And is there not a strong analogy betwixt the reaction of light upon matter in cases where it is reflected, and in cases where it is emitted?"

With regard to this hypothesis, a decisive argument against it is what we have already taken notice of from Mr Wollaston, (Phil. Trans. Vol. LXIV. p. 337), if his assertion be just, that the appearance of an hollow in the body of the sun, on which the Doctor founds his arguments, is not constant. To himself, he says, they had the appearance of hollows in the tops of volcanoes. Mr Marshal of Pennsylvania too, in a letter to Dr Franklin, gives it as his opinion, that the spots are near, if not closely adhering to, the sun's surface; and tells us, that he never observed the same spot appear again on the eastern limb, in the same figure and position. At any rate, Dr Wilson's hypothesis seems exceptionable for the following reasons.

1. The notion of a shelving declivity formed in a fluid substance by an elastic vapour, is utterly irreconcilable with the nature of any fluid with which we are acquainted. If we suppose the luminous matter of the sun to have any kind of fluidity analogous to that of water, melted metals, &c. we know that a discharge of elastic vapour through them could occasion nothing but a prodigious bubbling; or if we suppose the vapour to issue with immense force, perhaps a round spot might be formed, but without any umbra, as the expansive power must decrease every moment after getting vent, and the fluid would contract the orifice on all sides, till a round mouth was left just sufficient to allow the emission of the quantity of vapour generated. If the luminous matter is supposed to be endowed with any degree of viscosity, the same effect must happen in a greater degree; and it is impossible that any shelving cavity could be produced.

With regard to a fog, as it is not a fluid per se, but a multitude of aqueous particles floating in our atmosphere, if we compare the luminous matter of the sun to these aqueous particles, we must also suppose them swimming in some other fluid; and, at any rate, the shelving sides of a cavity of fog appear inconsistent, and also its running in channels in the bottom of a hollow, which last the Doctor gives as a reason for the breaking of the nucleus of a spot in pieces.

2. The vulgar prejudice in favour of the immense heat at the body of the sun seems to be extremely well founded, because we know of no substance that emits a bright white light like the sun, but what is also capable of burning very violently. We see the effect of the rays of the sun himself in burning mirrors*, which exceed any degree of heat we can raise by other means. No plausible reason can therefore be given why the heat ingulfs should not increase in proportion as we come near the sun. Sir Isaac Newton hath clearly been of this opinion; and we apprehend it must be the opinion of every person who is resolved not to reject the evidence of sense entirely.

At the outer surface of the luminous matter of the sun, therefore, any solid body must be heated beyond all imagination, by the intensity of the light proceeding from that matter. Now, we know that light is emitted from luminous bodies in all directions, down as well as up, and backward as well as forward. If a most violent heat behoved to take place at the outer surface of the luminous matter, the same must take place at the inner surface also, which is contiguous to the dark globe of the sun according to the Doctor's supposition. It is impossible then to imagine, that a globe of any solid matter could have resisted for near 6000 years such a violent heat, without being in a state of ignition more violent than we can have any idea of.

On the whole, we cannot help thinking, that, with regard to the solar spots, no reasonable hypothesis hath been yet formed. Those who think they are the smoke of volcanoes, or clouds exhaled from the body of the sun, ought first to prove that the sun hath a solid body like our earth. It is true, we cannot produce fire on this earth without terrestrial fuel: but this is no reason why it cannot be done in the celestial spaces; on the contrary, we have the most convincing proofs that it really can be done. The explosions of electrical batteries, flashes of lightening, and some kinds of meteors, resemble the bright light of the sun much more than any fire we can make with our fuel. It is true, these are momentary, and it is proper they should be so for the safety of this world and its inhabitants: but this is no reason for supposing that there is not a natural cause sufficient for the preservation of a pure flame of that kind where it is proper it always should exist; and if this should be granted, the body of the sun may reasonably enough be supposed only to consist of elementary fire or light.

Concerning the moon, it is allowed on all hands, that there are prodigious inequalities on her surface. This is proved by looking at her through a telescope, at any other time than when she is full: for then there is no regular line bounding light and darkness; but the confines of these parts appear as it were toothed and cut with innumerable notches and breaks; and even in the dark part, near the borders of the lucid surface, there are seen some small spaces enlightened by the sun's beams. Upon the fourth day after new moon, there may be perceived some shining points like rocks or small islands within the dark body of the moon; but not far from the confines of light and darkness there are observed other little spaces which join to the enlightened surface, but run out into the dark side, which by degrees change their figure, till at last they come wholly within the illuminated face, and have no dark parts round them at all. Afterwards many more shining spaces are observed to arise by degrees, and to appear within the dark side of the moon, which before they drew near to the confines of light and darkness were invisible, being without any light, and totally immersed in the shadow. The contrary is observed in the decreasing phases, where the lucid spaces which joined the illuminated surface by degrees recede from it, and, after they are quite separated from the confines of light and darkness, remain for some time visible, till at last they also disappear. Now it is impossible that this should be the case, unless these shining points were higher than the rest of the surface, so that the light of the sun may reach them.

Not content with perceiving the bare existence of these lunar mountains, astronomers have endeavoured to measure their height in the following manner. Let EGD be the hemisphere of the moon illuminated by the sun, ECD the diameter of the circle bounding light and darkness, and A the top of a hill within the dark part when it first begins to be illuminated. Observe with a telescope the proportion of the right line AE, or the distance of the point A from the lucid surface to the diameter of the moon ED; and, because in this case the ray of light ES touches the globe of the moon, AEC will be a right angle by 16 prop. of Euclid's third book; and therefore in the triangle AEC having the two sides AE and EC, we can find out the third side AC; from which subducting BC or EC, there will remain AB the height of the mountain. Ricciolus affirms, that upon the fourth day after new moon he has observed the top of the hill called St Catherine's to be illuminated, and that it was distant from the confines of the lucid surface about a sixteenth part of the moon's diameter. Therefore, if CE = 8, AE will be 1, and AC² = CE² + AE² by prop. 47. of Euclid's first book. Now, the square of CE being 64, and the square of AE being 1, the square of AC will be 65, whose square root is 8.062, which expresses the length of AC. From which deducting BC = 8, there will remain AB = 0.062. So that CB or CE is therefore to AB, as 8 is to 0.062, that is, as 8000 is to 62. If the diameter of the moon therefore was known, the height of this mountain would also be known. This demonstration is taken from Dr Keil, who supposes the semidiameter of the moon to be 1182 miles; according to which, the mountain must be somewhat more than nine miles of perpendicular height: but astronomers having now determined the moon's semidiameter to be only 1090 miles, the height of the mountain will be nearly 8½ miles.

Here we cannot help remarking, that it is extremely improbable such enormous mountains should exist in so small a planet. The abovementioned height is almost three times what is allowed to the highest mountains on earth; and it is certainly altogether contrary to the proportions and analogy wisely observed throughout nature, to imagine that the moon should have hills 8½ miles high, while the earth, which is more than 40 times as large, should have none exceeding three miles in height. At any rate, the extreme disagreement and contradiction among geometers concerning the height of terrestrial mountains, must make all their calculations concerning the heights of lunar ones appear still less worthy of credit.

Concerning the nature of the moon's substance there have been many conjectures formed. Some have imagined, that, besides the light reflected from the sun, the moon hath also some obscure light of her own, by which she would be visible without being illuminated by the sun-beams. In proof of this it is urged, that during the time of even total eclipses the moon is still visible, appearing of a dull red colour, as if obscured by a great deal of smoke. In reply to this it hath been advanced, that this is not always the case; the moon sometimes disappearing totally in the time of an eclipse, so as not to be discernible by the best glasses, while little stars of the fifth and sixth magnitudes were distinctly seen as usual. This phenomenon was observed by Kepler twice, in the year 1580 and 1583; and by Hevelius in 1620. Ricciolus and other Jesuits at Bologna, and many people throughout Holland, observed the same on April 14th, 1643; yet at Venice and Vienna, she was all the time conspicuous. In the year 1793, Dec 23, there was another total obscuration. At Arles, she appeared of a yellowish brown; at Avignon, ruddy and transparent, as if the sun had shone through her; at Marseilles, one part was reddish and the other very dusky; and at length, though in a clear sky, she totally disappeared. The general reason for her appearance at all during the time of eclipses shall be given afterwards; but as for these particular phenomena, they have not yet, as far as we know, been satisfactorily accounted for; and indeed astronomers in general seem industriously to avoid speaking of them.

Different conjectures have also been formed concerning the spots on the moon's surface. Some philosophers have been so taken with the beauty of the brightest places observed in her disk, that they have imagined them to be rocks of diamonds; and others have compared them to pearls and precious stones. Dr Keil and the greatest part of astronomers now are of opinion that these are only the tops of mountains, which by reason of their elevation are more capable of reflecting the sun's light than others which are lower. The dullish spots, he says, cannot be seas, nor any thing of a liquid substance; because when examined by the telescope, they appear to consist of an infinity of caverns and empty pits, whose shadows fall within them, which can never be the case with seas, or any liquid substance; but, even within these spots, brighter places are also to be observed; which, according to his hypothesis, ought to be the points of rocks standing up within the cavities.

On the other side, it has been urged, that if all the dark spots observed in the moon's surface were really the shadows of mountains, or of the sides of deep pits, they could not possibly be so permanent as they are found to be, but behaved to vary according to the position of the moon with regard to the sun, as we find shadows on earth are varied according as the earth is turned towards or from the sun. Accordingly it is pretended, that variable spots are actually discovered on the moon's disk, and that the direction of these is always opposite to the sun. Hence they are found among those parts which are soonest illuminated in the increasing moon, and in the decreasing moon lose their light sooner than the intermediate ones; running round, and appearing sometimes longer, and sometimes shorter. The permanent dark spots, therefore, it is said, must be some matter which is not fitted for reflecting the rays of the sun so much as the bright parts do; and this property, we know by experience, belongs to water rather than land: whence these philosophers conclude, that the moon, as well as our earth, is made up of land and seas.

It hath been much disputed whether there is about the moon any kind of atmosphere similar to what we breathe. Against the existence of this it hath been urged, that the moon constantly appears with the same lustre when there are no clouds in our air; which could not be expected, were she surrounded with an atmosphere like ours, the different changes of which behoved sometimes to diminish and at other times to increase her lustre. The strongest argument, however, is drawn from the refractive power of our atmosphere, which is well known to have a great influence on the rays of light proceeding either from celestial or terrestrial bodies, so as to cause them deviate from a straight line, and of consequence behoved to have the same effect on those which passed through the atmosphere of the moon, if any such there was. But no such effect has ever been perceived. The smallest fixed stars, as hath been already observed, preserve their lustre undiminished till they are suddenly covered by the moon's limb, and as suddenly appear on the other side, without being at all affected by their approach to her, as they are by the planet Mars. For this reason, too, the same philosophers maintain, that there can neither be clouds nor rain in the lunar regions, but that she enjoys a perpetual and uninterrupted serenity.

To these arguments it hath been replied, 1. That the appearances on which they are founded are not constant. Hevelius writes, that he has several times found in skies perfectly clear, when even stars of the fifth and seventh magnitude were visible, that at the same altitude of the moon, and the same elongation from the earth, and with one and the same telescope, the moon and its macule do not appear equally lucid, clear, and conspicuous, at all times; but are much brighter and more distinct at some times than at others. From the circumstances of this observation, say they, it is evident that the reason of this phenomenon is not either in our air, in the tube, in the moon, or in the spectator's eye; but must be looked for in something existing about the moon. Along with this, the phenomena already mentioned of the different appearances of the moon in the total eclipses are also urged, and are derived from the different constitutions of the lunar atmosphere at that time. Cassini frequently observed Saturn, Jupiter, and the fixed stars, to have their circular figure changed into an elliptical one, when they approached either the illuminated or dark edge of the moon's limb, and in other occultations found no change of figure at all. Mr Dunn particularly viewed Saturn at his emergence from behind the moon, in order to determine this question; and the appearance which he then observed, inclined him to think there was an atmosphere about the moon. 2. In the total eclipses of the sun, we find the moon encompassed with a lucid ring parallel to her periphery. Of this we have too many observations to doubt: In the great eclipse of 1713, the ring was very visible at London and elsewhere. The same was observed by Kepler in 1605, at Naples and Antwerp. Wolfius relates the same of an eclipse in 1666 at Leipzig, which is described at large in the Acta Eruditorum, with this notable circumstance, that the part next the moon was visibly the most illuminated, being considerably brighter than that further from it; which is also confirmed by the observations of the French astronomers in 1706.

For these reasons, it is concluded by many, that there is about the moon some fluid, which conforms itself to her figure, and is more dense near her surface than at a distance from it; and that it both reflects and refracts the rays of the sun. It is also concluded, that the lunar atmosphere is not always in the same state, as it sometimes changes the figures of the stars, and sometimes not: and in the several eclipses just mentioned, there was observed a trembling of the moon's limb immediately before immersion, with an appearance of thin smoke flying over it during immersion, very visible in England. Hence, as these phenomena are observed to happen in our atmosphere when full of vapours, it is concluded that the lunar atmosphere has been in a similar state at these times: and since at other times these appearances are not to be observed, it is thought that then the lunar air has been clear and transparent, by reason of rain, dew, or snow, having fallen.

The strongest argument for a lunar atmosphere is that drawn from the luminous ring around her in solar eclipses; eclipses; and this seems so conclusive, that Dr Halley himself was almost convinced of the existence of such an atmosphere by it. He tells us, however, that several very great astronomers did not think such a thing at all probable. Other astronomers have imagined that it proceeded from a solar atmosphere, because it followed the centre of the sun, and not that of the moon. Some there are who ridicule both these opinions, and take this appearance to be an undeniable proof of the moon's being included within the atmosphere of the earth. We observe, say they, at all times, when the sky is clear, the body of the sun to be surrounded with a very bright and dazzling circle, which extends to a considerable distance, and whose centre always coincides with that of the sun. This circle we never ascribe to the atmosphere of the sun, but to that of the earth. If the sun is hid by a mountain, part of this circle continues visible after his body disappears; and that part of the luminous circle is brightest, which is apparently nearest the mountain: yet we never ascribe this to the atmosphere of the mountain, but to the common atmosphere of the earth which lies beyond the mountain. In the case of solar eclipses, therefore, why should we imagine the luminous circle to proceed from any thing else than that part of the earth's atmosphere which lies beyond the moon? In this case, it will also follow the centre of the sun; and the tremulations of that part of our atmosphere lying between us and the moon being observed by means of the sun's strong light, will occasion the apparent tremulation of the moon's limb already taken notice of at the time of immersion. To have recourse in this case, say they, to such an ens rationis as the solar or lunar atmosphere, is truly solving abscurrum per obscurum. As for the small refractions of the light of the planets and fixed stars which have sometimes been observed when they approach the moon's limb, the abovementioned persons think they may be reasonably accounted for on the same principles with the tides; namely, by an accumulation of the aqueous vapours floating in our atmosphere under that part of the heavens where the moon is; and consequently the light of the stars will be more or less refracted, according as it passes through a larger or lesser quantity of the accumulated vapour. This last hypothesis might give some encouragement to bishop Wilkins's scheme of flying to the moon, which the want of a continued atmosphere between that luminary and the earth would be an effectual bar against: but tho' we should suppose the art of flying possible, and likewise that the air all the way up were fit for supporting our life, the bishop's journey would be rather too long; for allowing him to fly 60 miles an hour, and to proceed day and night without intermission, he must be five months before he came to his journey's end, supposing the moon's mean distance 240,000 miles, as it is commonly thought to be.

With regard to the planets Mercury, Venus, Mars, Jupiter, and Saturn, few conjectures have been formed: only that the temperature of the two inferior ones, Mercury and Venus, must be much hotter, and that of the superior ones, Mars, Jupiter, and Saturn, considerably colder, than that of the earth. That the latter are much less enlightened than this earth, we are absolutely certain: but whether they are not encompassed with atmospheres, which prevent them from suffering any violent excess of cold, we are entirely ignorant; as well as whether those of Mercury and Venus may not diminish the heat which at first would appear to be so violent at the small distances from the sun at which they circulate. Mars, we are certain, has an atmosphere about his body which is capable of refracting and absorbing the light of other stars; but what its other properties are, we have no means of knowing. The spots on Venus, Mr Dunn informs us, are thought to be seas. The belts observable on the superior planets have occasioned much speculation. Some have thought that they were inherent on the surfaces of the planets; and thus some philosophers have said, that greater changes take place on the body of the planet Jupiter, than what would happen to this earth were the ocean and dry land to change places. Others, however, have imagined that they are similar to our clouds: but their constant appearance seems contradictory to such a supposition; for tho' it is true that they are but of a transitory nature, they are vastly more permanent than any clouds to be observed on earth. For this reason, some have imagined that they are only certain parts of the atmosphere of the planet of a different constitution, and less capable of transmitting the sun's light than others. These dark zones, they think, may have arisen from the superior planets enjoying a perpetual equinox, the reason of which shall be given when we come to speak of the causes of the different seasons on earth. The moons of Jupiter and Saturn, and the ring with which the latter is surrounded, are thought to be designed for reflecting the sun's light upon these planets; and their number is imagined to be sufficient to compensate in some measure for the great distance at which they are placed from the sun.

We cannot conclude this article without taking notice, that astronomers seem not to agree whether the belts of Jupiter are really darker or brighter than the rest of his disk. In Chambers's Dictionary, we have the following account of them, under the word Faecia. "Faecia, in astronomy, two rows of bright spots, observed on Jupiter's body; appearing like swaths, or belts. The faecia or belts of Jupiter are more lucid than the rest of his disk, and are terminated by parallel lines. They are sometimes broader, and sometimes narrower; nor do they always possess the same part of the disk. M. Huygens, likewise, observed a very large kind of faecia in Mars; but it was darker than the rest of the disk, and took up the middle part thereof." Most of the astronomical writers, at least the more common ones, seem to be pretty silent on the subject, but generally incline us to think that they are dark. In Mr Wollaston's paper already quoted from Phil. Trans. Vol. LXIV. he mentions both bright and dark belts.

With regard to comets, innumerable conjectures have been formed. The ancients in general were of opinion that they were meteors formed in our atmosphere, only of a more permanent substance than the common ones; and that they were signs of the wrath of God: which last opinion hath been preferred among the vulgar to this day. Some, however, thought otherwise; and ascribed them to be a kind of planets that revolved round the sun, but in more extensive circles than the others: but these were few in number; and the general opinion of their being signs of divine wrath, and prefigu- It was not, however, till some time after people began to throw off the fetters of superstition and ignorance which had so long held them, that any rational hypothesis was formed concerning comets. Kepler, in other respects a very great genius, indulged the most extravagant conjectures, not only concerning comets, but the whole system of nature in general. The planets he imagined to be huge animals who swam round the sun by means of certain fins acting upon the ethereal fluid, as those of fishes do on the water; and agreeable to this notion, he imagined the comets to be monstrous and uncommon animals generated in the celestial spaces; and he explained how the air engendered them by an animal faculty. A yet more ridiculous opinion, if possible, was that of John Bodin, a learned man of France in the 16th century. He maintained that comets "are spirits, which having lived on the earth innumerable ages, and being at last arrived on the confines of death, celebrate their last triumph, or are recalled to the firmament like shining stars! This is followed by famine, plague, &c., because the cities and people destroy the governors and chiefs who appease the wrath of God." This opinion, ridiculous as it is, he says he borrowed from the philosopher Democritus, who imagined them to be the souls of famous heroes; but this opinion being irreconcilable with Bodin's Christian sentiments, he was obliged to suppose them to be a kind of genii, or spirits subject to death, like those so much mentioned in the Mahometan fables.

Others, again, have denied the existence of comets, and maintained that they were only false appearances occasioned by the refraction or reflection of light; as if the light could refract or reflect of itself, without any subsistence from whence it was to be reflected or refracted.

The first rational conjecture we meet with is that of James Bernoulli, an Italian astronomer, who imagined them to be the satellites of some very distant planet, which was invisible to us on account of its distance, as were also the satellites unless when in a certain part of their course. But though we call this a rational conjecture, in comparison of the others, it is nevertheless very absurd; as it supposes a satellite to leave its primary planet in darkness, in order to enlighten other orbs with which it has nothing to do.

The first rate astronomers of England, as Newton, Flamsteed, Halley, &c., have been perfectly satisfied that the comets were a kind of planets which revolved round the sun in very eccentric ellipses; and have accordingly calculated the returns of some of them, and made conjectures concerning the use they may probably be of in the general system of nature. Cassini and some of the French philosophers thought this opinion highly probable; but De La Hire and others opposed it. The whole event of the dilute, however, it is plain, behoved to turn on the observation of the return of comets, and the calculation of their periodical times like those of other planets; which, whenever it was fully done, behoved to put the matter beyond a doubt; but until this was done, not at once or twice, but a great number of times, there behoved still to be an uncertainty, let the arguments for their return be supposed as probable as we please. This Sir Isaac Newton and Dr Halley attempted to do. Having observed the accounts of comets in history, and found some of them to appear at equal intervals of time, it was concluded that these were the same comets, and would appear again after an interval of equal length. Thus, Sir Isaac Newton having observed a comet in the year 1680, of great apparent magnitude; and found that such an one had appeared before, at an interval of 575 years between each appearance; he concluded that these were different appearances of the same comet, and that it would again appear 575 years afterwards. The same celebrated philosopher, along with Dr Halley and the others, calculated the period of the comet which appeared in 1682, and found that it ought to appear again in the year 1758, its periodical time being about 75½ years. The most important objection that arose to the return of this comet, was the inequality of its periods, which were as follows: "That from August 25th 1531, to the 26th of October 1607, was performed in 76 years and two months; that from October 26th 1607, to September 14th 1682, was rather less than 75 years; and its last period, from the 14th of September 1682, to the 13th of March 1759, which was the longest of all, was 76 years and six months, or 27,937 days, amounting to 583 days more than in the preceding period.

Dr Halley was aware of these differences, and at first confessed himself to be a little staggered by them; nor would he have had the courage to pronounce its return so positively, if history had not informed him, that comets had appeared in 1456, 1380, and 1305, which put their identity out of all doubt.

The appearances happening alternately in 75 and 76 years, and as the preceding period was only of 75 years, it was natural to suppose that the next would amount to 76. But as the difficulties arising from these inequalities in the periods were foreseen and obviated by Dr Halley, we cannot do better than to infer his own words.

Perhaps some may object to the diversity of their inclinations and periods, which is greater than what is observed in the revolutions of the same planet; seeing one period exceeded the other by more than the space of one year, and the inclination of the comet of the year 1682 exceeded that of the year 1607 by 22 entire minutes. But let it be considered what I mentioned at the end of the tables of Saturn, where it was proved that one period of that planet is sometimes longer than another by 13 days; and that is evidently occasioned by the force of gravity tending towards the centre of Jupiter, which force indeed in equal distances is only the 1000th part of that force tending to the sun itself, by which the planets are retained in their orbits. But by a more accurate computation, the force of Jupiter upon Saturn, for example, in the great conjunction as they call it, January 26, in the year 1683, was found to be to the force of the sun upon the same Saturn, as 1 to 186; the sum of the forces therefore is to the force of the sun, as 187 to 186. But at the same distance from the centre, the periodic times of bodies revolving in a circle are in the subduplicate ratio of the forces with which they are urged: wherefore the gravity being increased by the 186th part of itself, the periodic time will be shortened by about the 374th part, that is by a whole month in Saturn. How much more is a comet liable to these errors, which makes its excursion... near four times higher than Saturn; and whose velocity being increased by less than the 120th part of itself, would change its elliptic orbit into a parabolic trajectory?

But it happened in the summer of 1681, that the comet seen in the following year, in its descent towards the sun, was in conjunction with Jupiter in such a manner, and for several months so near him, that during all that time it must have been urged likewise towards the centre of Jupiter with near the 50th part of that force by which it tended towards the sun: whence, according to the theory of gravity, the arc of the elliptic orbit, which it would have described had Jupiter been absent, must be bent inwards towards Jupiter in an hyperbolic form winding, and have assumed a kind of curve very compounded, and as hitherto not to be managed by the geometers; in which the velocity and direction of the moving body, in proportion to the cause, would be very different from what it otherwise had been in the ellipses.

Hence a reason may be assigned for the change of its inclination: for as the comet in this part of its path had Jupiter on the north almost in a perpendicular direction to its path, that portion of its orbit must be bent towards that quarter; and therefore its tangent being inclined to a greater angle towards the plane of the ecliptic, the angle of the inclination of the plane itself must be necessarily increased. Besides the comet continuing long in the neighbourhood of Jupiter, after it had come towards him from parts much more remote from the sun with a slower motion, and now being urged with the joint central forces of both, must have acquired more accelerated velocity, than it could lose in its recess from Jupiter by forces acting a contrary way, its motion being more swift, and the time being less.

When the comet of 1682 descended towards the sun and became visible, Europe had scarce recovered from the terrible panic into which it had been thrown but 13 months before by the great comet. However, this was comparatively too inconsiderable to be much regarded; for it was little imagined then, that the least of the two would become the most interesting, and that it would be for ever celebrated by posterity for having taught mankind how to know all the rest. But however inferior to the other this comet may have appeared in vulgar eyes, astronomers observed it with the greatest attention. Hevelius at Dantzig, Kitch at Leipzig, Flamsteed and Halley in England, Zimmermann at Nuremberg, Baer at Toulon, Montanari at Padua, and Picard, Cassini, and la Hire, at Paris. This list of names will suffice to shew that there can be no scarcity of good observations upon this comet during that appearance.

In 1607 it was observed by the famous Kepler, who published his observations together with his general theory. The 16th of September old style, the sky being very clear, Kepler first saw this comet upon the bridge at Prague; and though it had no tail when he first discovered it, yet afterwards it had one of considerable length and splendour. It was likewise observed by Longomontanus, September 18. He says it appeared as large as Jupiter, though with a very obscure and pale light; that the tail was pretty long and more dense than the tails of comets usually are, but as pale in colour as the comet itself.

In the preceding revolution of 1531, we find our comet observed by the astronomer Appian at Ingolstadt, the same who first remarked that the tails of comets were always in an opposite direction to the sun; which to him was an evident proof that the sun was the cause of such eruptions.

In 1456, there was a very remarkable exhibition of the same comet. Cometa inaudita magnitudinis tota mensis Junii cum praelonga cauda, ita ut duo fere signa colli comprehenderit, (Theatrum Comet.)

It is difficult to comprehend how the comet, whose tail was so inconsiderable in its last appearance, should in this have one of sixty degrees: but M. de la Lande, in his Theory of Comets, p. 127, accounts for this difference in the following manner. "I find," says this active astronomer, that if the comet reached its perihelion in the beginning of June, it ought to have appeared at night towards the middle of the month with 60 degrees of elongation and a very northern latitude, its distance from the earth being less than the semidiameter of the sun: so that in this position, which of all others is the most favourable, it must have appeared in all the splendour allowed to it by the old chronicles. Perhaps by duo signa they only mean the extent of two constellations, which is often much less than two signs of the ecliptic."

In 1379 and 1380 we find two comets mentioned by Alfedius and Lubienietzki, but without any particulars as to the time or form of their appearance.

In 1505, our comet again appears, according to the historians of that time, in all its terrors. Cometa horrenda magnitudinis visus est circa ferias paschatis, quem fecit eft pefidentia maxima. It is very likely that the horror occasioned by the plague had augmented the terrible impression left by the comet; however, upon calculation, it does appear that the comet must this year have passed very near the earth.

The history of this comet might be traced much higher by consulting Ecksturmus, Riccioli, Alfedius, and Lubienietzki. Among the 415 comets mentioned by this last writer, we find one for the year 1230, which appears to be the very comet in question; another 1005, three periods before; it is found in 930, and higher up in the year 550, marked by the taking of Rome by Totila. All the historians of the empire speak of a great comet in the year 399, which may have been the same. Cometa fuit prodigiosae magnitudinis, horribilis aspectu, comam ad terram usque demittere visus.

In 323, that is to say, 76 years before, a comet also appeared in Virgo; and in short it would be easy to mount, without quitting the same periods, as high as 130 years before Christ, when, according to Justin, one appeared at the birth of Mithridates. But, in these early periods, there would be great danger of meeting with some of those fabulous comets with which it was thought necessary perhaps to embellish every famous reign."

As this comet engaged the attention of the most celebrated astronomers more than any other, and as its course was calculated by Newton, Halley, Maupertuis, Clairault, De Lisle, Le Monnier, La Caille, Messier, La Lande, Pingre, &c., who unanimously determined that it ought to appear in 1758, it is not to be wondered dered that those who respected these illustrious names, should expect its return in that year, with absolute certainty, and even with no small degree of fear, as Dr Halley himself had thought that these bodies might possibly strike upon the earth in some of their revolutions, and occasion its utter destruction, or come so near as at least to occasion terrible calamities to those parts which were most exposed to their malignant influences. A mortal fear accordingly seized the minds of great numbers; and this panic seems to have been kept up by some astronomers of inferior note probably with views not entirely disinterested. Of this the following sentence in a book entitled *The Theory of Comets Illustrated*, &c. by B. Martin, 1757, is a remarkable instance. "It is well known to astronomers how near that dreadful comet of 1680 approached to the earth's orbit. Also the comets of 1472, 1618, 1684, and the comet which we now expect, with many others, pass so near the orbit of the earth, that it will not be without reason if our fears and apprehensions are considerably raised thereat. However, the reader need not be under any needless terror about the return of this comet: for if it appears before the beginning of next May, it can do no harm; as he may be easily convinced by the view of the comet's orbit which I published some time ago."

The fatal period at length arrived; but no comet appeared. This was a prodigious disappointment to astronomers; and as they were now in danger of being turned into ridicule, it became absolutely necessary to find out some reasons for the retardation of this comet which had been so certainly expected. These were happily found out by Mr Clairaut; who accounted for its nonappearance, from retardations occasioned by the attraction of Jupiter and Saturn, which two planets are found to have an effect upon each other's motions, and must have the same upon any other body that comes near them, as shall be more fully explained in the next section. He found, "that the action of Jupiter upon the comet, during the whole revolution of 1531 to 1607, had occasioned a diminution of 19 days in its period, which would not have happened by the mere force of the sun; and at the same time had altered its elements so as to produce an acceleration of near 31 days in the following period.

Proceeding afterwards to the revolution from 1607 to 1682, the action of Jupiter turns out much more considerable: for it occasions an acceleration of about 420 days, which added to the 31 resulting from the action of the same planet during the preceding period, amounts in all to 451 days of diminution in the time of its period; which would not have happened merely by its inclination to the sun.

Now if we take the difference of these two accelerations, in order to know how much shorter the second period was than the first, it appears to be 432 days; which differs only 37 days from the time resulting from the observations.

And this period appears to be still diminished by the action of Saturn. Indeed this diminution is not much, because the effects of Saturn's force are almost reciprocally destroyed in the two first periods.

Hence we see that the theory gives within a month the difference so remarkable between the two known revolutions of this comet. Now if we consider the length of these periods, the complication of the two causes of their irregularity, and the nature of the problem by which they are measured; this new demonstration of the Newtonian system will perhaps be found as striking as any one that has hitherto been given.

By comparing, in like manner, the force of the action of Jupiter, during the second period of the comet, with that which will be terminated at its approaching return; I find the revolution about which we are at present interested will be 518 days longer than the preceding, occasioned by the action of Jupiter upon the comet, from its last mean distance to its perihelion; that is, for the last seven or eight years; an interval, during which there can hardly be more than 15 days alteration.

As to Saturn, the result of its action on the comet is much more considerable compared with the two first revolutions; for I find the present period protracted more than 100 days by it, independent likewise of its action since 1751, and another small object which I have not had time to determine. From these considerations, then, it appears to me, that the expected comet ought to arrive at its perihelion about the middle of the month of April next ensuing."

This period of M. Clairault was found to be somewhat too long; for on the 21st of January 1759, a January comet made its appearance, and was seen at different times, to the third of June the same year. The following is Mr Messier's description of it, as viewed April 1st. "When I saw this comet again on the first of April, I could very plainly discern its tail; but could not ascertain its length, because of the morning twilight, which was then beginning, and soon increased much: it filled the field of the telescope; and must have extended far beyond: according to what I have observed, the tail of the comet must have spread to more than 25 degrees: the nucleus was considerable, but not well terminated, and it apparently exceeded the size of stars of the first magnitude; it was of a pale whitish colour, not unlike that of Venus. The nebulosity which surrounded the nucleus, and went on lessening, shewed reddish colours; and these colours grew more vivid towards the brightest part of the tail. The morning twilight, which increased apace, soon put an end to these appearances, and afterwards made the comet itself disappear: however, I had been able to perceive it with the naked eye, when it was somewhat disengaged from the vapours of the horizon." On the first of May it appeared to the naked eye larger than stars of the first magnitude, the nucleus being surrounded with a great coma. Its light was but faint, like that of the planets seen thro' the thick vapours of the horizon. It would have appeared brighter but for the light of the moon. In this last appearance the comet was in the sextant, or 60° from the sun, and was observed by most of the astronomers of Europe.

The triumph of the astronomers seemed now to be complete; and accordingly Dr Bevis exultingly says, Phil. Trans. Vol. LI. "I think I may now venture to pronounce this to be the same as the comet of 1682; and am about making out its future tract. If I presume rightly, it will in a short time become in a manner stationary, but diminish very fast both in size and light, the earth and it receding from one another almost in a direct line. It is at this time about four times nearer..." Thus we have given the strength of the evidence for the return of comets; and the appearance of that in 1759 is commonly urged as an undeniable proof of their revolution round the sun, and consequently being a kind of planets: but, however strong this evidence may be reckoned, a regard for truth obliges us to take notice, that M. Clairault could not be mistaken, whether he had concluded the comet to have been accelerated or retarded by the action of Jupiter and Saturn; for a comet appeared in the year 1757, in the months of September and October. Had he determined the retardation of the comet to be twice as great as he actually did, still he would have been right by the event, for another comet appeared in 1760. It must be owned that these are perplexing circumstances. It is very singular, that out of four years, in which three comets appeared, the only one in which no comet was to be seen should be that very year in which the greatest astronomers that ever existed had foretold the appearance of one.

How far this consideration renders the real accomplishment of Dr Halley's prediction uncertain, we submit to the judgment of the impartial; but, as it is certain that small comets are very frequently to be seen; so, till some better marks for distinguishing one from another than any yet known are found out, perhaps this part of astronomy may not appear to be sufficiently established. Another comet is expected in the year 1789, when some further evidence either one way or another may be hoped for.

The revolution of comets round the sun, and their permanent existence as planets, being granted, we are naturally led to inquire what kind of substances they are, and for what use designed. Sir Isaac Newton, considering the near approach of that of 1680 to the sun, has computed the heat they sometimes undergo to be inconceivably great. That one, in particular, he thought to be heated to a degree 2000 times greater than red-hot iron. In consequence of this calculation he naturally imagined that they were bodies of extreme solidity, in order to sustain such an intensity of heat; and that, notwithstanding their running out into the vast regions of space, where they were exposed to the most intense degrees of cold, they would hardly be cool on their returning again to the sun. Indeed, according to his calculation, the comet of 1680 must be forever in a violent state of ignition. He hath computed that a globe of red-hot iron of the same dimensions with the earth would scarce be cool in 50,000 years. If then the comet be supposed to cool 100 times faster than red-hot iron, as its heat was 2000 times greater, it must require upwards of a million of years to cool it. In the short period of 575 years, therefore, its heat would be in a manner scarce diminished; and therefore, in its next and every succeeding revolution it must acquire an increase of heat; so that, since the creation, having received a proportional addition in every succeeding revolution, it must now be in a state of ignition very little inferior to that of the sun himself. Sir Isaac hath farther concluded, that this comet must be considerably retarded in every revolution by the atmosphere of the sun within which it enters; and thus must continually come nearer and nearer his body, till at last it falls into it. This, he thinks, may be one use of the comets: viz. to furnish fuel for the sun, which would otherwise be in danger of wasting, from the continual emission of its light.

As to the tails with which comets are almost constantly attended, Sir Isaac Newton shews, that the atmospheres of comets will furnish vapour sufficient to form their tails. This he argues from that wonderful rarefaction observed in our air, at a distance from the earth: a cubic inch of common air, at the distance of half the earth's diameter, or 4000 miles, would expand itself so as to fill a space larger than the whole region of the stars. Since then the coma or atmosphere of a comet is ten times higher than the surface of the nucleus, counting from the centre thereof; the tail, ascending much higher, must needs be immensely rare: so that it is no wonder the stars should be visible through it.

Now, the ascent of vapours into the tail of the comet, he supposes to be occasioned by the rarefaction of the matter of the atmosphere at the time of the perihelion. Smoke, it is observed, ascends a chimney by the impulse of the air wherein it floats; and air, rarefied by heat, ascends by the diminution of its specific gravity, taking up the smoke along with it: why then should not the tail of a comet be supposed to be raised after the same manner by the sun? for the sun-beams do not act on the mediums they pass through, any otherwise than by reflection and refraction.

The reflecting particles, then, being warmed by the action, will again warm the ether wherewith they are compounded; and this, rarefied by the heat, will have its specific gravity, whereby it before tended to descend, diminished by the rarefaction, so as to ascend, and to carry along with it those reflecting particles whereof the tail of the comet is composed.

This ascent of the vapours will be promoted by their circular motion round the sun; by means whereof they will endeavour to recede from the sun, while the sun's atmosphere, and the other matters in the celestial spaces, are either at rest, or nearly so, as having no motion but what they receive from the sun's circumrotation.

Thus are the vapours raised into the tails of comets in the neighbourhood of the sun, where the orbits are most curve; and where the comets, being within the denser atmosphere of the sun, have their tails of the greatest length.

The tails thus produced, by preserving that motion, and at the same time gravitating towards the sun, will move round his body in ellipses, in like manner as their heads; and by this means will ever accompany and freely adhere to their head. In effect, the gravitation of the vapours towards the sun will no more occasion the tails of comets to forfetake their heads, and fall down towards the sun, than the gravitation of their heads will occasion them to fall off from their tails; but, by their common gravitation, they will either fall down together to the sun, or be together suspended or retarded. This gravitation, therefore, does not at all hinder, but that the heads and tails of comets may receive and retain any position towards each other, which either the abovementioned causes, or any other, may occasion.

The tails therefore, thus produced in the perihelion of comets, will go off along with their head into remote regions; and either return thence, together with the comets, after a long series of years; or rather be there lost, and vanish by little and little, and the comet be left bare; till, at its return, descending towards the sun, some little short tails be gradually and slowly produced from the heads; which afterwards, in the perihelion, descending down into the sun's atmosphere, will be immensely increased.

The vapours thus dilated, rarefied, and diffused thro' all the celestial regions, the same author observes, may probably, by little and little, by means of their own gravity, be attracted down to the planets, and become intermingled with their atmospheres.

He adds, that for the conservation of the water and moisture of the planets, comets seem absolutely requisite; from whose condensed vapours and exhalations all that moisture which is spent in vegetations and putrefactions, and turned into dry earth, &c. may be refurnished and recruited. For all vegetables grow and increase wholly from fluids; and, again, as to their greatest part, turn by putrefaction into earth, an earthly slime being perpetually precipitated to the bottom of putrefying liquors. Hence, the quantity of dry earth must continually increase, and the moisture of the globe decrease, and at last be quite evaporated; if it have not a continual supply from some part or other of the universe. And I suspect, adds our great author, that the spirit, which makes the finest, subtlest, and best part of our air, and which is absolutely requisite for the life and being of all things, comes principally from the comets.

On this principle, there seems to be some foundation for the popular opinion of preludes from comets: since the tail of a comet, thus intermingled with our atmosphere, may produce changes very sensible in animal and vegetable bodies.

This account of the formation of the tails of comets, we find controverted by Dr Hamilton of Dublin, in a small treatise intitled Conjectures on the nature of the Aurora Borealis; and on the tails of Comets. His hypothesis is, that the comets are of use to bring back the electric fluid to the planets, which is continually discharged from the higher regions of their atmospheres. Having given at length the abovementioned opinion of Sir Isaac, "We find (says he) in this account, that Sir Isaac attributes the ascent of comets tails to their being rarer and lighter, and moving round the sun more swiftly, than the solar atmosphere, with which he supposes them to be surrounded whilst in the neighbourhood of the sun; he says also, that whatever position (in respect to each other) the head and tail of a comet then receive, they will keep the same afterwards most freely; and in another place he observes, "That the celestial spaces must be entirely void of any power of resisting, since not only the solid bodies of the planets and comets, but even the exceeding thin vapours of which comets tails are formed, move thro' those spaces with immense velocity, and yet with the greatest freedom." I cannot help thinking that this account is liable to many difficulties and objections, and that it seems not very consistent with itself or with the phenomena.

"I do not know that we have any proof of the existence of a solar atmosphere of any considerable extent, nor are we anywhere taught how to guess at the limits of it. It is evident that the existence of such an atmosphere cannot be proved merely by the ascent of comets tails from the sun, as that phenomenon may possibly arise from some other cause. However, let us suppose, for the present, that the ascent of comets tails is owing to an atmosphere surrounding the sun, and see how the effects arising from thence will agree with the phenomena. When a comet comes into the solar atmosphere, and is then descending almost directly to the sun, if the vapours which compose the tail are raised up from it by the superior density and weight of that atmosphere, they must rise into those parts that the comet has left, and therefore at that time they may appear in a direction opposite to the sun. But as soon as the comet comes near the sun, and moves in a direction nearly at right angles with the direction of its tail, the vapours which then arise, partaking of the great velocity of the comet, and being specifically lighter than the medium in which they move, and being vastly expanded through it, must necessarily suffer a resistance immensely greater than what the small and dense body of the comet meets with, and consequently cannot possibly keep up with it, but must be left behind, or, as it were, driven backwards by the resistance of that medium into a line directed towards the parts which the comet has left, and therefore can no longer appear in a direction opposite to the sun. And, in like manner, when a comet passes its perihelion, and begins to ascend from the sun, it certainly ought to appear ever after with its tail behind it, or in a direction pointed towards the sun; for if the tail of the comet be specifically lighter than the medium in which it moves with so great velocity, it must be just as impossible it should move foremost, as it is that a torch moved swiftly thro' the air should project its flame and smoke before it. Since therefore we find that the tail of a comet, even when it is ascending from the sun, moves foremost, and appears in a direction nearly opposite to the sun, I think we must conclude that the comet and its tail do not move in a medium heavier and denser than the matter of which the tail consists, and consequently that the constant ascent of the tail from the sun must be owing to some other cause. For that the solar atmosphere should have density and weight sufficient to raise up the vapours of a comet from the sun, and yet not be able to give any sensible resistance to these vapours in their rapid progress through it, are two things inconsistent with each other. And therefore, since the tail of a comet is found to move as freely as the body does, we ought rather to conclude that the celestial spaces are void of all resisting matter, than that they are filled with a solar atmosphere, be it ever so rare.

"But there is, I think, a further consideration which will shew, that the received opinion, as to the ascent of comets tails, is not agreeable to the phenomena, and may at the same time lead us to some knowledge of the matter of which these tails consist; which I suspect is of a very different nature from what it has been hitherto supposed to be. Sir Isaac says, the vapours, of which the tail of a comet consists, grow hot by reflecting the rays of the sun, and thereby warm and rarefy the medium which surrounds them; which must therefore ascend from the sun, and carry with it the reflecting particles of which the tail is formed; for he always speaks of the tail as shining by reflected light. But one would rather imagine, from the phenomena, that the matter which forms a comet's tail has not the least sensible power of reflecting the rays of light. light. For it appears from Sir Isaac's observation, which I have quoted already, that the light of the smallest stars, coming to us through the immense thickness of a comet's tail, does not suffer the least diminution. And yet, if the tail can reflect the light of the sun so copiously, as it must do if its great splendour be owing to such reflection, it must undoubtedly have the same effect on the light of the stars; that is, it must reflect back the light which comes from the stars behind it, and by so doing must intercept them from our sight, considering its vast thickness, and how exceedingly slender a ray is that comes from a small star; or if it did not intercept their whole light, it must at least increase their twinkling. But we do not find that it has even this small effect, for those stars that appear through the tail are not observed to twinkle more than others in their neighbourhood. Since therefore this fact is supported by observations, what can be a plainer proof that the matter of a comet's tail has no power of reflecting the rays of light? and consequently that it must be a self-shining substance. But the same thing will further appear, from considering that bodies reflect and refract light by one and the same power; and therefore if comets tails want the power of refracting the rays of light, they must also want the power of reflecting them. Now, that they want this refracting power appears from hence: If that great column of transparent matter which forms a comet's tail, and moves either in a vacuum, or in some medium of a different density from its own, had any power of refracting a ray of light coming through it from a star to us, that ray must be turned far out of its way in passing over the great distance between the comet and the earth; and, therefore, we should very sensibly perceive the smallest refraction that the light of the stars might suffer in passing through a comet's tail. The consequence of such a refraction must be very remarkable: the stars that lie near the tail would, in some cases, appear double; for they would appear in their proper places by their direct rays, and we should see their images behind the tail, by means of their rays which it might refract to our eyes; and those stars that were really behind the tail would disappear in some situations, their rays being turned aside from us by refraction. In short, it is easy to imagine what strange alterations would be made in the apparent places of the fixed stars by the tails of comets, if they had a power of refracting their light, which could not fail to be taken notice of, if any such ever happened. But since astronomers have not mentioned any such apparent changes of place among the stars, I take it for granted that the stars seen through all parts of a comet's tail appear in their proper places, and with their usual colours, and consequently I infer that the rays of light suffer no refraction in passing through a comet's tail. And thence I conclude (as before) that the matter of a comet's tail has not the power of refracting or reflecting the rays of light, and must therefore be a lucid or self-shining substance."

But, whatever probability the Doctor's conjecture concerning the materials whereof the tails are formed may have in it, his criticism on Sir Isaac Newton's account of them seems not to be just: for that great philosopher supposes the comets to have an atmosphere peculiar to themselves; and consequently, in their nearest approaches to the sun, both comet and atmosphere are immersed in the atmosphere of that luminary. In this case, the atmosphere of the comet being prodigiously heated on the side next to the sun, and consequently the equilibrium in it broken, the denser parts will continually pour in from the regions farther from the sun; for the same reason, the more rarefied part which is before, will continually fly off opposite to the sun, being displaced by that which comes from behind; for tho' we must suppose the comet and its atmosphere to be heated on all sides to an extreme degree, yet still that part which is fartherest from the sun will be less hot, and consequently more dense, than what is nearest to his body. The consequence of this is, that there must be a constant stream of dense atmosphere descending towards the sun, and another stream of rarefied vapours and atmosphere ascending on the contrary side; just as, in a common fire, there is a constant stream of dense air descending, which pushes up another of rarefied air, flame, and smoke. The resistance of the solar atmosphere may indeed be very well supposed to occasion the curvature observable in the tails of comets, and their being better defined in the fore part than behind; and this appearance we think Dr Hamilton's Dr Hamilton's hypothesis is incapable of solving. We grant, that there is the utmost probability that the tails of comets are streams of electric matter; but they who advance a theory of any kind ought to solve every phenomenon, otherwise their theory is insufficient. It was incumbent on Dr Hamilton, therefore, to have explained how this stream of electric matter comes to be bent into a curve, and also why it is better defined and brighter on the outer side of the arch than on the inner. This indeed he attempts, in the following manner: "But that this curvature was not owing to any resisting matter appears from hence, that the tail must be bent into a curve though it met with no resistance; for it could not be a right line, unless all its particles were projected in parallel directions, and with the same velocity, and unless the comet moved uniformly in a right line. But the comet moves in a curve, and each part of the tail is projected in a direction opposite to the sun, and at the same time partakes of the motion of the comet; so that the different parts of the tail must move on in lines which diverge from each other; and a line drawn from the head of a comet to the extremity of the tail will be parallel to a line drawn from the sun to the place where the comet was when that part of the tail began to ascend, as Sir Isaac observes; and so all the chords, or lines drawn from the head of the comet to the intermediate parts of the tail, will be respectively parallel to lines drawn from the sun to the places where the comet was when these parts of the tail began to ascend. And therefore, since these chords of the tail will be of different lengths, and parallel to different lines, they must make different angles, with a great circle passing through the sun and comet, and consequently a line passing through their extremities will be a curve.

"It is observed, that the convex side of the tail which is turned from the sun is better defined, and shines a little brighter, than the concave side. Sir Isaac accounts for this, by saying, that the vapour on the convex side is freer (that is, has ascended later) than that on the concave side; and yet I cannot see how the particles on the convex side can be thought to have ascended later than those on the concave side which may be nearer to the head of the comet. I think it rather looks as if the tail, in its rapid motion, met with some slight resistance just sufficient to cause a small condensation in that side of it which moves foremost, and which would occasion it to appear a little brighter and better defined than the other side; which slight resistance may arise from that subtle ether which is supposed to be dispersed through the celestial regions, or from this very electric matter dispersed in the same manner, if it be different from the ether."

On the last part of this observation we must remark, that though a slight resistance in the ethereal medium would have served Sir Isaac Newton's turn, it will by no means serve Dr Hamilton's; for though a stream of water or air may be easily destroyed or broken by resistance, yet a stream of electric matter seems to set every obstacle at defiance. If a sharp needle is placed on the conductor of an electric machine, and the machine set in motion, we will perceive a small stream of electric matter issuing from the point; but though we blow against this stream of fire with the utmost violence, it is impossible either to move it, or to brighten it on the side against which we blow. If the celestial spaces then are full of a subtle ether capable of thus affecting a stream of electric matter, we may be sure that it also will resist very violently; and we are then as much difficulted to account for the projectile motion continuing amidst such violent resistance; for if the ether resists the tail of the comet, it is impossible to prove that it doth not resist the head also.

This objection may appear to some to be but weakly founded, as we perceive the electric fluid to be endowed with such extreme subtilty, and to yield to the impression of solid bodies with such facility, that we easily imagine it to be of a very passive nature in all cases. But we are inclined to think, that this fluid only shews itself passive where it passes from one body into another, which it seems very much inclined to do of itself. We are also much mistaken, if it will not be found, on proper examination of all the phenomena, that the only way we can manage the electric fluid at all is by allowing it to direct its own motions. In all cases where we ourselves attempt to assume the government of it, it shews itself the most untractable and stubborn being in nature. But these things come more properly under the article Electricity, where they are fully considered. Here, it is sufficient to observe, that a stream of electric matter resists air, and from the phenomena of electric repulsion we are sure that one stream of electric matter resists another; from which we may be also certain, that if a stream of electric matter moves in an aerial fluid, such fluid will resist it, and we can only judge of the degree of resistance it meets with in the heavens from what we observe on earth. Here we see the most violent blast of air has no effect upon a stream of electric fluid; in the celestial regions, either air, or some other fluid, has an effect upon it according to Dr Hamilton. The resistance of that fluid, therefore, must be greater than that of the most violent blast of air we can imagine.

As to the Doctor's method of accounting for the curvature of the comet's tail, it might do very well on Sir Isaac Newton's principles, but cannot do so on his.

There is no comparison between the celerity with which rarefied vapour ascends in our atmosphere, and that whereby the electric fluid is discharged. The velocity of the latter seems to equal that of light; of consequence, supposing the velocity of the comet to be equal to that of the earth in its annual course, and its tail equal in length to the distance of the sun from the earth, the curvature of the tail could only be to a straight line, as the velocity of the comet in its orbit is to the velocity of light, which, according to the calculations of Dr Bradley, is as 10,201 to 1.—The apparent curvature of such a comet's tail, therefore, would at this rate only be $\frac{1}{10,201}$ part of its visible length, and thus behoved always to be imperceptible. The velocity of comets is indeed sometimes inconceivably great. Mr Brydone observed one at Palermo, in July 1770, which in 24 hours described an arch in the heavens upwards of 50 degrees in length; according to which he supposes, that, if it was as far distant as the sun, it must have moved at the rate of upwards of 60 millions of miles in a day. But this comet was attended with no tail, so that we cannot be certain whether the curvature of the tails of these bodies corresponds with their velocity or not.

On this occasion Mr Brydone observes, that the comets without tails seem to be of a very different species from those which have tails: to the latter, he says, they appear to bear a much less resemblance than they do even to planets. He tells us that comets with tails have seldom been visible but on their recess from the sun; that they are kindled up, and receive their alarming appearance, in their near approach to this glorious luminary; but that those without tails are seldom or never seen but on their way to the sun, and he does not recollect any whose return has been tolerably well ascertained. "I remember indeed (says he), a few years ago, a small one, that was said to have been discovered by a telescope after it had passed the sun, but never more became visible to the naked eye. This assertion is easily made, and nobody can contradict it; but it does not at all appear probable that it should have been so much less luminous after it had passed the sun, than before it approached him: and I will own to you, when I have heard that the return of these comets had escaped the eyes of the most acute astronomers, I have been tempted to think that they did not return at all, but were absorbed in the body of the sun, which their violent motion towards him seemed to indicate." He then attempts to account for the continual emission of the sun's light without waste, by supposing that there are numberless bodies throughout the universe that are attracted into the body of the sun, which serve to supply the waste of light, and which for some time remain obscure, and occasion spots on his surface, till at last they are perfectly dissolved and become bright like the rest. This hypothesis will account for the dark spots becoming as bright, or even brighter than the rest of the disk, but will by no means account for the brighter spots becoming dark. Of this comet too, Mr Brydone remarks, that it was evidently surrounded by an atmosphere which refracted the light of the fixed stars, and even seemed to cause them change their places as the comet came near them.

A very strange opinion we find set forth in a book Mr Cole's entitled hypothesis. entitled "Observations and Conjectures on the nature and properties of light, and on the theory of comets, by William Cole." This gentleman supposes that the comets belong to no particular system; but were originally projected in such directions as would successively expose them to the attraction of different centres, and thus they would describe various curves of the parabolic and the hyperbolic kind. This treatise is written in answer to some objections thrown out in Mr Brydone's tour, against the motions of the comets by means of the two forces of gravitation and projection, which were thought sufficient for that purpose by Sir Isaac Newton; of which we shall treat as fully as our limits will allow, in the next section.

With regard to the fixed stars, they are generally supposed to be of the same nature with our sun, each of them attended by planets as he is; and these planets, as well as those which attend our sun, are supposed like this earth to be inhabited by rational creatures.—The strongest argument for the fixed stars being suns is taken from the impossibility of magnifying their diameters by the best telescopes, which is thought to arise from the vast distance at which they are placed from the earth. As it is impossible that they can be seen by any reflection of light from the sun at such immense distances, we must therefore necessarily suppose them endowed with a power of emitting light from their own bodies; and by comparing the apparent diameters of objects at different distances, it is conjectured that our sun would appear but like a star, was he to be removed to the distance at which they are placed. Of consequence, the fixed stars are supposed to be equal if not superior in magnitude to that which is the centre of our system; and as it would be absurd to suppose the wise Author of nature to have made so many suns without anything to shine upon, it is thence concluded that they are attended by planets, which receive the same benefit from them that the earth does from our sun. In like manner, it would be absurd to suppose so many habitable worlds enlightened by suns without having any inhabitants; and therefore it is concluded, that all the planets of every system are inhabited; to corroborate which hypothesis the infinite benevolence of the Deity is urged, who would not, it is thought, suffer any part of the visible creation to want living creatures that might be spectators of his goodness. It is also asserted, that even the planets belonging to our own system are but of a very trifling use to this earth, their whole combined light being much less than that of the moon alone. Much less can we suppose the fixed stars to be made for the use of this earth: for many of them are utterly invisible without the assistance of telescopes; and when they are thus seen, only appear as so many shining points; and it would be absurd to the highest degree to think that they were created merely to be seen by astronomers.

The fixed stars are not supposed to be at equal distances from us, but to be more remote in proportion to their apparent smallness. This supposition is necessary to prevent any interference of their planets; and thus there may be as great a distance between a star of the first magnitude and one of the second apparently close to it, as between the earth and the fixed stars first mentioned.

Those who take the contrary side of the question, affirm that the disappearance of some of the fixed stars is a demonstration that they cannot be suns, as it would be to the highest degree absurd to think that God would create a sun which might disappear of a sudden and leave its planets and their inhabitants in endless night. Yet this opinion, we find adopted by Dr Keil, who tells us, "It is no ways improbable that these stars lost their brightness by a prodigious number of spots which entirely covered and overwhelmed them, &c."

In what dismal condition must their planets remain, who have nothing but the dim and twinkling light of the fixed stars to enlighten them?" Others, however, have made suppositions more favourable to the benevolent character of the Deity. Sir Isaac Newton thinks that the sudden blaze of some stars may have been occasioned by the falling of a comet into them, by which means they would be enabled to emit a prodigious light for a little time, after which they would gradually return to their former state. Others have thought that the variable ones, which disappear for a time, were planets, which were only visible during some part of their course; but this their apparent immobility, notwithstanding their decrease of lustre, will not allow us to think. Some have imagined, that one side of them might be naturally much darker than the other, and when by the revolution of the star upon its axis the dark side was turned towards us, the star became invisible, and, for the same reason, after some interval, resumed its former lustre. Lastly, Mr Dunn, (Phil. Mr Dunn's Transl. Vol. LII.) in a dissertation concerning the appearance increase of magnitude in the heavenly bodies when they approach the horizon, conjectures that the interpolation of some gross atmosphere may solve the phenomena both of nebulous and new stars. "The phenomena of nebulous and new stars (says he) have engaged the attention of curious astronomers; but none that I know of have given any reason for the appearance of nebulous stars. Possibly what has been before advanced may also be applicable for investigating reasons for those strange appearances in the remotest parts of the universe.

"From many instances which might be produced concerning the nature and properties of lights and illuminations on the earth's surface, concerning the nature and properties of the earth's atmosphere, and concerning the atmospheres and illuminations of comets, we may safely conclude, that the atmospheres of comets and of our earth are more gross in their nature than the ethereal medium which is generally diffused through the solar system. Possibly a more aqueous vapour in the one than the other makes the difference. Now, as the atmospheres of comets and of planets in our solar system are more gross than the ether which is generally diffused through our solar system, why may not the ethereal medium diffused throughout those other solar systems (whose centres are their respective fixed stars) be more gross than the ethereal medium diffused throughout our solar system? This indeed is an hypothesis, but such an one as agrees exactly with nature. For these nebulous stars appear so much like comets, both to the naked eye and through telescopes, that the one cannot always, by any difference of their extraneous light, be known from the other.

"Such orbs of gross ether reflecting light more copiously, or like the atmospheres of comets, may help..." us to judge of the magnitudes of the orbs illuminated by those remote suns, when all other means seem to fail.

"The appearance of new stars, and disappearance of others, possibly may be occasioned by the interposition of such an ethereal medium, within their respective orbs, as either admits light to pass freely, or wholly absorbs it at certain times, whilst light is constantly pursuing its journey through the vast regions of space."

The other arguments for the plurality of worlds, the opposers of this doctrine attempt to evade, by telling us that the not being able to magnify the fixed stars with telescopes is not a sufficient proof of their immense distance. We know, say they, that light may be reflected so as that no telescope whatever could magnify the object from which it is reflected; and this without supposing it at any great distance. Thus, the light of the sun reflected from a polished globe of metal, or of glass quicksilvered, throws an image of him like a small star, and which would be visible at a great distance, without a possibility of being magnified. If we then suppose the fixed stars to be polished globes reflecting the sun's light, we may account for the impossibility of magnifying them without attributing to them such an extravagant distance. An hypothesis something similar to this was published a few years ago at London. A formidable objection against it seems to be the apparent immobility of the stars with regard to one another, notwithstanding the continual motion of the earth, which one would naturally imagine behoved to cause a considerable variation of the apparent places of the fixed stars at different times of the year.

Sect. IV. Of the different Systems by which the Celestial Phenomena have been accounted for.

In treating of the various systems which have been invented in different ages, we do not mean to give an account of all, or even the greatest part, of the absurdities that have been broached by individuals on this subject; but shall confine ourselves to those systems which have been of considerable note, and been generally followed for a number of years. Concerning the opinions of the very first astronomers about the system of nature, we are necessarily as ignorant as we are of those astronomers themselves. Whatever opinions are handed down to us, must be of a vastly later date than the introduction of astronomy among mankind. If we may hazard a conjecture, however, we are inclined to think that the first opinions on this subject were much more just than those that were held afterwards for many ages.

We are told that Pythagoras maintained the motion of the earth, which is now universally believed, but at that time appears to have been the opinion of only a few detached individuals of Greece. As the Greeks borrowed many things from the Egyptians, and Pythagoras had travelled into Egypt and Phoenice, it is probable he might receive an account of this hypothesis from thence; but whether he did so or not, we have now no means of knowing, neither is it of any importance whether he did or not. Certain it is, however, that this opinion did not prevail in his days, nor for many ages after. In the 2nd century after Christ, the very name of the Pythagorean hypothesis was suppressed by a system erected by the famous geographer and astronomer Claudius Ptolemaeus. This system, which commonly goes by the name of the Ptolemaic, he seems not to have originally invented, but adopted as the prevailing one of that age; and perhaps made it somewhat more consistent than it was before. He supposed the earth at rest in the centre of the universe. Round the earth, and the nearest to it of all the heavenly bodies, the moon performed its monthly revolutions. Next to the moon, was placed the planet Mercury; then Venus; and above that the Sun, Mars, Jupiter, and Saturn, in their proper orbits; then the sphere of the fixed stars; above these, two spheres of what he called crystalline heavens; above these was the primum mobile, which by turning round once in 24 hours, by some unaccountable means or other, carried all the rest along with it. This primum mobile was encompassed by the empyrean heaven, which was of a cubic form, and the seat of angels and blessed spirits. Besides the motions of all the heavens round the earth once in 24 hours, each planet was supposed to have a particular motion of its own; the moon, for instance, once in a month, performed an additional revolution, the sun in a year, &c.

It is easy to see, that, on this supposition, the confused motions of the planets already described, could never be accounted for. Had they circulated uniformly round the earth, their apparent motion ought always to have been equal and uniform, without appearing either stationary, or retrograde, in any part of their courses. In consequence of this objection, Ptolemy was obliged to invent a great number of circles interfering with each other, which he called epicycles and excentrics. These proved a ready and effectual salvo for all the defects of his system; as whenever a planet was deviating from the course it ought, on his plan, to have followed, it was then only moving in an epicycle or an excentric, and would in due time fall into its proper path. As to the natural causes by which the planets were directed to move in these epicycles and excentrics, it is no wonder that he found himself much at a loss, and was obliged to have recourse to divine power for an explanation, or, in other words, to own that his system was unintelligible.

This system continued to be in vogue till the beginning of the 16th century, when Nicolaus Copernicus, a native of Thorn (a city of regal Prussia), and a man of great abilities, began to try whether a more satisfactory manner of accounting for the apparent motions of the heavenly bodies could not be obtained than was afforded by the Ptolemaic hypothesis. He had recourse to every author upon the subject, to see whether any had been more consistent in explaining the irregular motions of the stars, than the mathematical schools; but received no satisfaction, till he found, first from Cicero, that Nicetas the Syracusan had maintained the motion of the earth; and next, from Plutarch, that others of the ancients had been of the same opinion. From the small hints he could obtain from the ancients, Copernicus then deduced a most complete system capable of solving every phenomenon in a satisfactory manner. From him this system hath ever afterwards been called the Copernican, and represented Plate XLIII. fig. 1. Here the sun is supposed to be in the centre; next him revolves the planet Mercury; then Venus; next, the Earth, with the Moon; beyond these, Mars, Jupiter, and Saturn. turn; and far beyond the orbit of Saturn, he supposed the fixed stars to be placed, which formed the boundaries of the visible creation.

Though this hypothesis afforded the only natural and satisfactory solution of the phenomena which so much perplexed Ptolemy's system, it met with great opposition at first; which is not to be wondered at, considering the age in which he lived. Even the famous astronomer Tycho Brahe could never attest to the earth's motion, which was the foundation of Copernicus's scheme. He therefore invented another system, whereby he avoided the ascribing of motion to the earth, and at the same time got clear of the difficulties with which Ptolemy was embarrassed. In this system, the earth was supposed the centre of the orbits of the sun and moon; but the sun was supposed to be the centre of the orbits of the five planets; so that the sun with all the planets were by Tycho Brahe supposed to turn round the earth, in order to save the motion of the earth round its axis once in 24 hours. This system was never much followed, the superiority of the Copernican scheme being evident at first sight.

The system of Copernicus coming soon into universal credit, philosophers began to inquire into the causes of the planetary motions; and here, without entering upon what has been advanced by detached individuals, we shall content ourselves with giving an account of the three famous systems, the Cartesian, the Newtonian, and what is sometimes called the Mechanical system.

Des Cartes, the founder of that system which since his time has been called the Cartesian, flourished about the beginning of the 17th century. His system seems to have been borrowed from the philosophers Democritus and Epicurus; who held, that every thing was formed by a particular motion of very minute bodies called atoms, which could not be divided into smaller parts. But tho' the philosophy of Des Cartes resembled that of the Corpuscularians, in accounting for all the phenomena of nature merely from matter and motion; he differed from them in supposing the original parts of matter capable of being broken. To this property his Materia Subtilis owes its origin. To each of his atoms, or rather small masses of matter, Des Cartes attributed a motion on its axis, and likewise maintained that there was a general motion of the whole matter of the universe round like a vortex or whirlpool. From this complicated motion, those particles, which were of an angular form, would have their angles broke off; and the fragments which were broke off being smaller than the particles from which they were abraded, behoved to form a matter of a more subtile kind than that made of large particles; and as there was no end of the abrasion, different kinds of matter of all degrees of fineness would be produced. The finest sorts, he thought, would naturally separate themselves from the rest, and be accumulated in particular places. The finest of all would therefore be collected in the sun, which was the centre of the universe, whose vortex was the whole ethereal matter in the creation. As all the planets were immersed in this vortex, they behoved to be carried round by it, in different times, proportioned to their distances; those which were nearest the sun, circulating the most quickly; and those farthest off, more slowly; as those parts of a vortex which are farthest removed from the centre are observed to circulate more slowly than those which are nearest. Besides this general vortex of the sun, each of the planets had a particular vortex of their own by which their secondary planets were carried round, and any other body that happened to come within reach of it would likewise be carried away.

It is easy to see, from this short account of Des Cartes's system, that the whole of it was a mere petitio principii: for had he been required to prove the existence of his materia subtilis, he must undoubtedly have failed in the attempt; and hence, though his hypothesis was for some time followed for want of a better, yet it gave way to the Newtonian almost as soon as the latter was proposed.

When Sir Isaac Newton undertook the reformation of philosophy, he proceeded upon a very different plan from all those who had gone before him, as he professed to assume nothing as an hypothesis which was not deduced from what is obvious to our eyes; and thus, by arguing from those things which are within our reach, he thought we might come to know with certainty, what must happen in the celestial regions, to which access is denied us. The manner in which he was first led to form his system of gravitation, which hath since been so universally received, is said to have been as follows. He was sitting alone in a garden, when some apples, falling from a tree, led his thoughts very closely upon the subject of gravity; and reflecting on the law of gravity of that principle, he began to consider, that, as this power is not found to be sensibly diminished at the remotest distance from the centre of the earth to which we can rise, neither at the tops of the loftiest buildings, nor on the summits of the highest mountains, it appeared to him reasonable to conclude that this power must extend much further than was usually thought.

"Why not as far as the moon?" (said he to himself); and if so, her motion must be influenced by it; perhaps she is retained in her orbit thereby; however, though the power of gravity is not sensibly weakened in the little change of distance at which we can place ourselves from the centre of the earth, yet it is very possible, that, as high as the moon, this power may differ much in strength from what it is here." To make an estimate what might be the degree of this diminution, he considered with himself, that if the moon be retained in her orbit by the force of gravity, no doubt the primary planets are detained in their orbits by a similar gravitation towards the sun; and by comparing the periods of the several planets with their distances from the sun, he found, that if any power like gravity held them in their courses, its strength must decrease in the duplicate proportion of the increase of distance. This was concluded from a supposition that these bodies moved in perfect circles round the sun; which though they are not found to do exactly, yet the error was but of little consequence. The fact itself will be understood from the following considerations. Let BYU (Plate XLIV.fig.3.) be the orbit of a planet round the sun S. It is manifest from inspection, that during the time in which the planet moves from B to Y, it deviates in a perpendicular direction from the tangent line A.B.X., by the space X.Y, equal to B.Y which is the verified sine of the arch B.Y. Were the planet therefore retained in its orbit by a gravitating power towards the sun, it would require for this purpose a power of gra- gravitation sufficient to make it fall towards the sun, or, at least, to deviate from a tangent, by the whole length of the versed sine of the arch which it describes in the same time. Thus, suppose the planet is observed to move from B to Y in the space of a minute; we thence know, that, had it been left to the force of gravity alone, it would have fallen from B to y in the same time. In proportion as the radius is lengthened, the length of the arch B Y will become a proportionably smaller part of the circle, and thus approach more nearly to the right line A B X; so that, if we suppose the planet both to take longer time to describe the same space in its orbit, and likewise to deviate less from a straight line in describing that space than another does, it is plain that the force of gravity must be much greater in the latter than in the former. By comparing in this manner the distances of the several planets from the sun, and of the secondary planets from their primaries, Sir Isaac Newton observed the gravitating power towards each body to decrease in a duplicate proportion to the increase of distance from the sun, or from the primary planet, as already mentioned.

In the course of these observations a very singular circumstance occurred, namely, that the force of gravity was not at all proportioned to the apparent bulk of the planets, or even of the sun himself. In this respect the earth seems to have the advantage over all the other bodies in the system. This is discovered by comparing the bulk of the sun with that of the several planets, and the bulk of the earth with that of the moon, and her distance from the earth. From this comparison it appears, that our moon is vastly larger in proportion than any of those belonging to Jupiter or Saturn, and consequently more difficult to be retained in her orbit: her distance is also much greater than that of the most remote satellites of these planets, as is obvious from their figures.

This seemed to be a difficulty in Sir Isaac's hypothesis of gravity: for if such a property existed in all bodies, and was proportioned to their quantities of matter, it was naturally asked why it was not in proportion to the bulk of the sun or planets. To obviate this objection, Sir Isaac, observing the effect of fire on our earth to rarefy bodies and make them occupy more space than they did before, naturally enough concluded that the sun was of a more rare substance, or contained less matter in proportion to his bulk than the earth did; and having considered the matter mathematically, he concluded the sun to be four times rarer than the earth.

But though the difficulty was removed with regard to the sun, it still remained as to Jupiter and Saturn; both of these planets being found by observations of the distances and periodical times of their satellites, to be rarer than the earth; whereas, in the opinion of Des Cartes and other philosophers, they were more dense, as being placed at a greater distance from the sun, and in a colder region. Indeed, according to the above-mentioned cause of the sun's rarefaction, these planets ought to have been accounted more dense bodies than this earth; but whatever was the reason, Sir Isaac and all philosophers who followed him have considered them as much rarer. For this supposition, they have not assigned any natural cause; but chosen to refer it to the will of the Deity, and think his wisdom is manifested in placing the densest planets next the sun, that they may be able to resist his heat, and the rarest at a distance from him, that they may not be too much consolidated by the cold.

As for Des Cartes's method of accounting for the planetary motions, Sir Isaac Newton entirely overturned it, by shewing, that, if they were carried about in a vortex of fluid matter, their periodical times behoved to be directly as the squares of their distances from the sun; whereas from observation it is found, that the squares of the periodical times of the planets are as the cubes of their distances. He also shewed, that no motion could be continued in a fluid medium, because whatever body moved in such a medium behoved to communicate its motion to the fluid. In proportion as motion was communicated to the fluid, it behoved to be lost to the body; which, besides, would be resisted by the fluid itself, and this resistance would be in the proportion of the square of the velocity wherein the body moved: and for these reasons he concluded that the celestial regions were entirely void of matter, excepting perhaps some exceedingly rare exhalations from the planets and comets, and the rays of light, which were considered by him as substances of such extreme tenuity as to give no sensible resistance to any body whatever.

To account for the perpetual motions of the planets and comets in their orbs, Newton had recourse to the force of gravity already mentioned, and a projectile force compounded with it. These two forces had indeed been proposed by Mr Horrocks some time before with the same view, and the power of gravity as existing in the celestial regions had been hinted at by several philosophers; but before Sir Isaac, little notice had been taken of this scheme; owing, no doubt, to the inferiority of those geniuses who had proposed it, to that of our celebrated philosopher. As Sir Isaac was ignorant of any natural power by which the planets could be impelled in the direction of a tangent line to any part of their orbits, he was obliged to have recourse for one of his forces to the immediate action of the Deity himself. According to him, God having created this world, and impressed the universal law of attraction or gravitation upon matter, impelled each of the planets in the direction of a right line touching their orbits. Being immediately acted upon by the attraction of the sun, their courses were bent from a straight line into a circle; and the same causes still continuing to act, the original rectilinear direction was changed into one nearly circular, which has continued ever since. The same reason was given for the continued motion of the secondary planets round their primaries; but as for the motion of the earth, sun, and planets, round their axis, we have not heard that the Newtonian philosophers assign any natural cause.

Upon this plan Sir Isaac Newton accounted for the motion both of planets and comets; the latter of which had been, and still is, an unsupernatural objection to the system of Des Cartes: for, if such a vast whirlpool of fluid matter, as that philosopher supposed, had existed, it is absolutely impossible, without a continued miracle, but it must have carried along with it the comets as well as the planets. The manner in which Sir Isaac Newton demonstrates the operation of the projectile and gravitating forces upon the planets so as to direct them... them in circles round the sun, is by supposing the orbit they describe divided into a vast number of infinitely small parts, each of which will not differ from a right line, and consequently the whole curve may be considered as consisting of the diagonals of parallelograms infinitely small, one of whose sides is represented by the space the planet would have moved through by the projectile force alone, and the other by that which it would have moved through by the force of gravity alone in the same time. Those who want to enter deeply into this speculation will find themselves amply satisfied by looking into Newton's Principia; but for the sake of those who may be less learned, and yet want to have a general knowledge of these things, we subjoin the following explanation by Mr Ferguson.

From the uniform projectile motion of bodies in straight lines, and the universal power of attraction which draws them off from these lines, the curvilinear motions of all the planets arise. If the body A be projected along the right line A B X, in open space, where it meets with no resistance, and is not drawn aside by any other power, it will for ever go on with the same velocity, and in the same direction. For the force which moves it from A to B in any given time, will carry it from B to X in as much more time, and so on, there being nothing to obstruct or alter its motion. But if, when this projectile force has carried it, suppose to B, the body S begins to attract it, with a power duly adjusted, and perpendicular to its motion at B, it will then be drawn from the straight line A B X, and forced to revolve about S in the circle B Y T U. When the body A comes to U, or any other part of its orbit, if the small body s, within the sphere of U's attraction, be projected as in the right line Z, with a force perpendicular to the attraction of U, then s will go round U in the orbit W, and accompany it in its whole course round the body S. Here, S may represent the sun, U the earth, and u the moon.

If a planet at B gravitates, or is attracted, toward the sun, so as to fall from B to s, in the time that the projectile force would have carried it from B to X, it will describe the curve B Y by the combined action of these two forces, in the same time that the projectile force singly would have carried it from B to X, or the gravitating power singly have caused it to descend from B to y; and these two forces being duly proportioned, and perpendicular to each other, the planet obeying them both will move in the circle B Y U. To make the projectile force balance the gravitating power so exactly, as that the body may move in a circle, the projectile velocity of the body must be such as it would have acquired by gravity alone in falling through half the radius of the circle.

But if, whilst the projectile force would carry the planet from B to b, the sun's attraction (which constitutes the planet's gravitation) should bring it down from B to 1, the gravitating power would then be too strong for the projectile force, and would cause the planet to describe the curve B C. When the planet comes to C, the gravitating power (which always increases as the square of the distance from the sun diminishes) will be yet stronger for the projectile force; and, by conspiring in some degree therewith, will accelerate the planet's motion all the way from C to K; causing it to describe the arcs B C, C D, D E, E F, &c. all in equal times. Having its motion thus accelerated, it thereby gains so much centrifugal force, or tendency to fly off at K in the line K k, as overcomes the sun's attraction; and the centrifugal force being too great to allow the planet to be brought nearer the sun, or even to move round him in the circle K l m n, &c. it goes off, and ascends in the curve K L M N, &c. its motion decreasing as gradually from K to B, as it increased from B to K, because the sun's attraction now acts against the planet's projectile motion just as much as it acted with it before. When the planet has got round to B, its projectile force is as much diminished from its mean state about G or N, as it was augmented at K; and so the sun's attraction being more than sufficient to keep the planet from going off at B, it describes the same orbit over again, by virtue of the same forces or powers.

A double projectile force will always balance a quadruple power of gravity. Let the planet at B have twice as great an impulse from thence towards X, as it had before; that is, in the same length of time that it was projected from B to b, as in the last example, let it now be projected from B to c; and it will require four times as much gravity to retain it in its orbit; that is, it must fall as far as from B to 4 in the time that the projectile force would carry it from B to c; otherwise it could not describe the curve BD, as is evident by the figure. But, in as much time as the planet moves from B to C in the higher part of its orbit, it moves from I to K, or from K to L, in the lower part thereof; because, from the joint action of these two forces, it must always describe equal areas in equal times throughout its annual course. These areas are represented by the triangles BSC, CSD, DSE, ESF, &c. whose contents are equal to one another, quite round the figure.

As the planets approach nearer the sun, and recede farther from him, in every revolution; there may be some difficulty in conceiving the reason why the power of gravity, when it once gets the better of the projectile force, does not bring the planets nearer and nearer the sun in every revolution, till they fall upon and unite with him; or why the projectile force, when it once gets the better of gravity, does not carry the planets farther and farther from the sun, till it remove them quite out of the sphere of his attraction, and cause them to go on in straight lines for ever afterward. But by considering the effects of these powers as described in the two last articles, this difficulty will be removed. Suppose a planet at B to be carried by the projectile force as far as from B to b, in the time that gravity would have brought it down from B to 1; by these two forces it will describe the curve B C. When the planet comes down to K, it will be but half as far from the sun S as it was at B; and therefore, by gravitating four times as strongly towards him, it would fall from K to V in the same length of time that it would have fallen from B to 1 in the higher part of its orbit, that is, through four times as much space; but its projectile force is then so much increased at K, as would carry it from K to k in the same time; being double of what it was at B; and is therefore too strong for the gravitating power, either to draw the planet to the sun, or cause it go round him in the circle K l m n, &c. which would require its falling from K to u, thro' a greater space than gravity can draw it, whilst the projectile force is such as would carry it from K to k; and therefore the planet ascends in its orbit KLMN, decreasing in its velocity for the causes already assigned.

"By the above-mentioned law, bodies will move in all kinds of ellipses, whether long or short, if the spaces they move in be void of resistance. Only, those which move in the longer ellipses, have so much the less projectile force impressed upon them in the higher parts of their orbits; and their velocities, in coming down towards the sun, are so prodigiously increased by his attraction, that their centrifugal forces in the lower parts of their orbits are so great as to overcome the sun's attraction there, and cause them to ascend again towards the higher parts of their orbits; during which time, the sun's attraction acting contrary to the motions of those bodies, causes them to move slower and slower, until their projectile forces are diminished almost to nothing; and then they are brought back again by the sun's attraction, as before.

"The sun and planets mutually attract each other: the power by which they do so we call gravity. But whether this power be mechanical or not, is very much disputed. Observation proves, that the planets disturb one another's motions by it; and that it decreases according to the squares of the distances of the sun and planets; as light, which is known to be material, likewise does. Hence gravity should seem to arise from the agency of some subtle matter pressing towards the sun and planets, and acting, like all mechanical causes, by contact. But, on the other hand, when we consider that the degree or force of gravity is exactly in proportion to the quantities of matter in those bodies, without any regard to their bulks or quantities of surface, acting as freely on their internal as external parts; it seems to surpass the power of mechanism; and to be either the immediate agency of the Deity, or effected by a law originally established and impressed on all matter by him. But some affirm that matter, being altogether inert, cannot be impressed with any law even by almighty power; and that the Deity, or some subordinate intelligence, must therefore be constantly compelling the planets toward the sun, and moving them with the same irregularities and disturbances which gravity would cause, if it could be supposed to exist. But, if a man may venture to publish his own thoughts, it seems to me no more an absurdity, to suppose the Deity capable of infusing a law, or what laws he pleases, into matter, than to suppose him capable of giving it existence at first. The manner of both is equally inconceivable to us; but neither of them imply a contradiction in our ideas, and what implies no contradiction is within the power of Omnipotence.

"That the projectile force was at first given by the Deity, is evident. For, since matter can never put itself in motion, and all bodies may be moved in any direction whatsoever; and yet the planets, both primary and secondary, move from west to east, in planes nearly coincident; whilst the comets move in all directions, and in planes very different from one another; these motions can be owing to no mechanical cause or necessity, but to the free will and power of an intelligent Being.

"Whatever gravity be, it is plain that it acts every moment of time: for if its action should cease, the projectile force would instantly carry off the planets in straight lines from those parts of their orbits where gravity left them. But, the planets being once put into motion, there is no occasion for any new projectile force, unless they meet with some resistance in their orbits; nor for any amending hand, unless they disturb one another too much by their mutual attractions.

"It is found that there are disturbances among the planets in their motions, arising from their mutual attractions when they are in the same quarter of the heavens; and the best modern observers find that our years are not always precisely of the same length (a). Besides, there is reason to believe that the moon is somewhat nearer the earth now than she was formerly; her periodical month being shorter than it was in former ages. For our astronomical tables, which in the present age show the times of solar and lunar eclipses to great precision, do not answer so well for very ancient eclipses. Hence it appears, that the moon does not move in a medium void of all resistance; and therefore her projectile force being a little weakened, whilst there is nothing to diminish her gravity, she must be gradually approaching nearer the earth, describing smaller and smaller circles round it in every revolution, and finishing her period sooner, although her absolute motion with regard to space be not so quick now as it was formerly: and, therefore, she must come to the earth at last; unless that Being which gave her a sufficient projectile force at the beginning, adds a little more to it in due time. And, as all the planets move in spaces full of ether and light, which are material easily applicable, they too must meet with some resistance. And therefore, if their gravities are not diminished, nor their projectile forces increased, they must necessarily approach nearer and nearer the sun, and at length fall upon and unite with him.

"Here we have a strong philosophical argument against the eternity of the world. For, had it existed eternal, from eternity, and been left by the Deity to be governed by the combined actions of the above forces or powers, generally called Laws, it had been at an end long ago. And if it be left to them, it must come to an end. But we may be certain that it will last as long as was intended by its Author, who ought no more to be found fault with for framing so perishable a work, than for making man mortal (b)."

Though this system hath now obtained in a manner universal

(a) If the planets did not mutually attract one another, the areas described by them would be exactly proportional to the times of description. But observations prove, that these areas are not in such exact proportion, and are most varied when the greatest number of planets are in any particular quarter of the heavens. When any two planets are in conjunction, their mutual attractions, which tend to bring them nearer to one another, draws the inferior one a little farther from the sun, and the superior one a little nearer to him; by which means, the figure of their orbits is somewhat altered; but this alteration is too small to be discovered in several ages.

(b) A difficulty of this kind we find obviated in a pamphlet intitled Thoughts on General Gravitation, &c. &c. &c. The author of this performance, after considering how necessary a projectile force is to counteract the power of gravity acting universal credit, it was not without opposition at first; nor are there wanting at this day some who not only object to it, but charge the hypothesis of universal gravity or attraction itself with being an absurdity. As we reckon ourselves bound to strict impartiality in every dispute, and consequently to give the opinions and arguments of one party as well as another, that every reader may be enabled to judge for himself, we hope it will give no just cause of offence if we mention the principal arguments that have been made use of against our celebrated philosopher.

At the first appearance of the Newtonian system, Leibnitz, and other learned men abroad, objected to his principle of attraction, as being an unknown power of which Sir Isaac himself did not pretend to know the cause; and consequently that it was of the same nature with the occult qualities formerly made use of, and which were only another way of expressing ignorance. To avoid the force of this objection, it was answered on Sir Isaac's part, that by the word attraction was meant no more than the bare effect or action of some cause not yet discovered; and that philosophers ought to search for the cause. On the other hand, it was replied, that if attraction was the effect of any natural cause, such cause behoved to be material, and of consequence the matter of which that cause was formed behoved to be definite of attraction. If this was the case, say they, Sir Isaac Newton was in the wrong to make gravity or attraction an universal law of matter, when it is only partial: He ought to have contented himself with observing the phenomena of the celestial motions, without pretending to assign any cause for them at all.

This may suffice to give a short view of the dispute concerning attraction as an effect. With regard to it as a cause, (for to tell the truth, most Newtonian philosophers, and even Sir Isaac Newton himself, have spoken of it both as cause and effect (c): "If (say they) we affirm with Dr Cotes, that it is the most simple of causes, beyond which we cannot penetrate, it is undoubtedly right to call it an invisible, hidden, or occult cause, property, or quality; and therefore explaining phenomena by attraction, is only explaining them by occult qualities; or, in other words, owning our ignorance. If we believe what Sir John Pringle hath said in his discourse concerning the attraction of mountains, namely, that gravity hath been long acknowledged by the Royal Society to be "a quality impressed by the Creator on all matter, whether of the earth or of the heavens, whether at rest or in motion;" then there is no doubt that the Royal Society acknowledges it also to be an invisible quality, a hidden quality, or an occult quality, according as we please to express ourselves. If, with Mr Ferguson, we call it a law impressed by the Creator on all matter, there is no difference, except merely in the word, between an occult law and an occult quality. If, as Mr Baxter and others suppose, attraction is the immediate operation of God himself, then the occultation is not only infinitely increased, but we have this additional inconvenience, that the Deity is chargeable with doing whatever is done by means of attraction, whether good or evil."

These are the general objections that are made to the doctrine of attraction itself. In particular, it is said to be demonstrably insufficient to produce even the common phenomena of falling bodies; but these objections

adging on the planets, applies the same rule to the sun, and to the stars which are supposed to be other suns. As we cannot settle the boundaries of gravitation, he thinks it reasonable to suppose that all the suns in the universe have a projectile force as well as the planets, and gravitate towards a common centre, round which they describe orbits, carrying with them their systems of planets, as the primary planets carry their secondaries along with them in their revolutions round the sun. That something is necessary to counteract the gravitation of the sun towards the planets, is certain; for tho' the projectile power hinders the planets from coming to the sun, yet what hinders the sun from coming to the planets, as Sir Isaac Newton shews that he is continually moved from his place by their attraction? Difficulties of this kind probably occurred to Sir Isaac himself, when he propounded it as a query, "What hinders the fixed stars from falling upon one another?" This query, which is taken for the motto of the abovementioned pamphlet, is there answered in the manner just now related; and to shew the credibility of the hypothesis, it is urged, that the stars "Sirius, Canopus, Procyon, Regulus, alpha Aquilae, Pollux, Arcturus, and many others, are found to be all moving through absolute space with incredible velocity," and the whole stars in the firmament are suspected to do so by the first astronomers of this age." This assertion is supported by the testimonies of Dr Halley, Le Monier, Maclayne, and Mayer, who have observed changes in the places of the fixed stars; but till these changes are fully ascertained and determined, little can be said upon this subject.

(c) This will appear by the following quotations from the writings of Sir Isaac himself, and several of the most eminent of his followers.

Sir Isaac Newton. "Gravity exists and acts." Princip. p. ult.

Dr Friend. "In explaining gravity, Newton has demonstrated it to arise from an attractive force." Phil. Trans. No. 331.

M. Maupertuis. "It should be remembered in justice to Sir Isaac Newton, he has never considered attraction as an explanation of gravity. He considers it not as a cause, but as an effect." Astron. Phys. par. M. de Guendelhez, p. 348.

Dr Cotes. "Gravity is the most simple of causes." Princip. pref. p. 9.

Dr Clarke. It has often been distinctly declared, that "by the term attraction we do not mean to express the cause of bodies tending toward each other, but barely the effect, the effect itself, the phenomenon, or matter of fact." p. 335.

Dr Desaguliers. "Attraction seems to be settled by the great Creator as the first of second causes." Phil. Trans. No. 454.

Mr Rewing. "When we use the term attraction, we do not determine the physical cause of it, but use it to signify an effect." Vol. I. p. 17.

Sir Isaac Newton. "There are agents in nature able to make the particles of bodies stick together by very strong attractions, and it is the business of experimental philosophy to find them out."

Dr Desaguliers. "We are not solicitous about the cause of attraction." Phil. Trans. No. 454.

Dr Friend. "I believe attraction will always be occult." Phil. Trans. No. 331. objections come more properly to be noticed under the article Gravity. As to the particular phenomenon in question, namely, the revolutions of the celestial bodies in orbits nearly circular, the following objections are raised against the Newtonian scheme.

1. "It is highly absurd, not to say impious, to think of compounding a motion by a mixture of Divine and created power. If the Deity projects the planets in a right line, how can a created quality make them deviate from that line? If we are to suppose the Deity to project them at all, why do we not suppose him to project them in a circular or elliptical direction? Or can God only project bodies in a right line as men can do?"

2. "The Newtonian method of explaining the celestial motions by the diagonals of parallelograms infinitely small, involves us in a contradiction no less than that of supposing the radii of a circle to be parallel to one another. This is evident from an inspection of fig. 3, Plate XLIV. When the planet sets out from B, it is drawn by the force of gravity towards S in the direction SB. When it arrives at Y, gravity no longer draws it in the direction SB, or XY, but in that of a radius drawn from S to Y. Unless therefore we suppose these two radii to be parallel to one another, which is evidently absurd, we cannot say that the planet moves from B to Y by a force compounded of those represented by BX and BY; as the latter is every moment varying in its direction. By supposing the arches to be infinitely small indeed, we may think to get over the difficulty; but it is a mere deception. Perhaps it may be thought no absurdity to say, that one radius of a circle is parallel to another drawn at an infinitely small distance from it; but to a Being whose eyes are infinitely good, the absurdity will appear as great as it does to us to say that the lines SD and SB are parallel to one another; and whatever appears an absurdity to him, will also appear an absurdity to us when multiplied so as to become visible.

3. "Granting that there is no absurdity in the abovementioned supposition, it is absolutely impossible that the power of gravity, acting in the manner which it is found to do on earth, can be the means of retaining any body in a circular or any other kind of orbit, but must at last overcome the strongest projectile force we can imagine. Neither will any law, by which we can suppose this power to be regulated, answer our purpose better. If we suppose gravity to be a power acting uniformly, the planet will run off in a curve approaching every moment nearer to a straight line. If it is accelerated in any proportion whatever, the projectile force, however strong, must at last be overcome; and the planet for some time behoved to describe a strange kind of curve, very different either from a circle, parabola, or ellipse, and would at last come to the centre.

"To illustrate this, let S represent the sun, diffusing the attractive rays SB, SA, SB, &c. Let us next suppose a planet projected from B, in the direction BI, in such a manner that it would move through the spaces BA, BB, BC, &c. by the force of projection alone, in the same times that it would describe the spaces BI, B2, B3, &c. by the force of gravity alone. According to the laws of compound forces, therefore, it must describe the diagonal of the parallelogram BX, in the same time that it would have described the side BA by the projectile force alone. The diagonal of a right-angled parallelogram is always longer than its side; the planet therefore has, in the same time in which it would have described the space BA, described that of BX, which is larger; and consequently has received an addition of velocity. Let us now suppose the force of gravity to be annihilated, or to cease its action. It is plain that the planet, by reason of the acceleration it has already received, would move from x to y in the same time that it would have moved from a to b by the projectile force originally impressed upon it. The diagonal of the former parallelogram must therefore become the side of the next one, and if we suppose gravity again to begin its action, the planet will evidently acquire an additional acceleration during the next moment, and thus describe the line yz. For the same reason, it is plain, that the side of every succeeding parallelogram must be the diagonal of the former; and thus the planet will describe the curve Bxzyzm, &c., which, if traced out, would become a kind of increasing spiral, that would carry it, after the first half revolution, farther and farther off from the centre S.

"Let us now suppose the gravitating power to accelerate bodies in the celestial regions in the same proportion that it is found to do on earth, and the case will be still worse. We must always consider, that the power of gravity tends to bring bodies to a centre, and not merely away from the tangent line BI. In proportion, therefore, as the force of gravity is increased, the planet must every moment come nearer the centre than it would have been had it moved uniformly on in the straight line BI. By the time, therefore, it would have arrived at a by the original force of projection, it must be nearer the centre S by the space BI, and consequently be at x. By the time it would have arrived at b, it must be nearer the centre by the space B4, and consequently be at s. For the same reasons it must come continually nearer and nearer the centre, describing a kind of curve Bxzyzm, &c., till at last it falls into it altogether.

"This matter is equally capable of being illustrated from the figure by which Mr Ferguson illustrates the Newtonian hypothesis. For, supposing the planet to have moved from B to Y, by the united forces of projection and gravitation, in the same time that it would have moved from B to X by the force of projection, it is plain, that supposing the power of gravity to cease when the planet has arrived at Y, it would then fly off, in a line touching the point Y, with a velocity increased in the proportion of the length of the arch BY to the line BX. Thus the projectile velocity behoved to be for ever increased, if we suppose the gravitating power to act uniformly; and to be entirely destroyed, if it is supposed to increase in the proportion 1, 3, 5, 7, &c., as it is found to do on earth.

"To state this in a different manner: Whatever power, motion, or velocity, receives a continual addition, must ultimately become infinite, or greater than any assignable quantity. If the power of gravity, therefore, acts on a planet uniformly, it must continually increase its projectile force, because it obliges it to move in the diagonal of a parallelogram, of which the projectile velocity is one side, and the gravitating power another. The projectile force, therefore, receiving a continual addition, will be ultimately increased beyond all..." calculation, and consequently the planet will perpetually remove farther and farther from the centre. If the power of gravity increases every moment after it first begins to act, as it is found to do on earth, it is equally plain that it must ultimately become infinite, and cannot be overcome by any projectile force whatever.

We have already observed, that by the two forces of gravitation and projection, Sir Isaac Newton accounted for the motion of the comets as well as planets. The former he supposed to revolve in very eccentric ellipses; on account of their having got only a small degree of projection at first, by which means they are brought very near the sun by the force of gravity, from which they again acquire a prodigious degree of projectile force that carries them off on the other side, till being gradually weakened by the attractive power, they return; and so on. To this doctrine in the particular instance of comets, Mr Brydone hath made some objections, on the occasion of the comet observed at Palermo, which we shall here lay before our readers in his own words.

"The astronomy of comets, from what I can remember of it, appears to be clogged with very great difficulties, and even some seeming absurdities. It is difficult to conceive, that these immense bodies, after being drawn to the sun with the velocity of a million of miles in an hour; when they have at last come almost to touch him, should then fly off from his body with the same velocity they approach it; and that too, by the power of this very motion that his attraction has occasioned. The demonstration of this I remember is very curious and ingenious; but I with it may be entirely free from sophistry. No doubt, in bodies moving in curves round a fixed centre, as the centripetal motion increases, the centrifugal one increases likewise—but how this motion, which is only generated by the former, should at last get the better of the power that produces it; and that too, at the very time this power has acquired its utmost force and energy; seems somewhat difficult to conceive. It is the only instance I know, wherein, the effect increasing regularly with the cause, at last, whilst the cause is still acting with full vigour, the effect entirely gets the better of the cause, and leaves it in the lurch. For the body attracted is at last carried away with infinite velocity from the attracting body:—by what power is it carried away?—Why, say our philosophers, by the very power of this attraction, which has now produced a new power superior to itself, to wit, the centrifugal force. However, perhaps all this may be reconcilable to reason; far be it from me to presume attacking so glorious a system as that of attraction. The law that the heavenly bodies are said to observe, in describing equal areas in equal times, is supposed to be demonstrated; and by this it would appear, that the centripetal and centrifugal forces alternately get the mastery of one another.

"However, I cannot help thinking it somewhat hard to conceive, that gravity should always get the better of the centrifugal force, at the very time that its action is the smallest, when the comet is at its greatest distance from the sun; and that the centrifugal force should get the better of gravity, at the very time that its action is the greatest, when the comet is at its nearest point to the sun.

"To a common observer, it would rather appear, that the sun, like an electric body, after it had once charged the objects that it attracted with its own effluvia or atmosphere, by degrees loses its attraction, and at last even repels them; and that the attracting power, like what we likewise observe in electricity, does not return again till the effluvia imbibed from the attracting body is dispelled or dissipated; when it is again attracted, and so on alternately. For it appears (at least to an unphilosophical observer) somewhat repugnant to reason to say that a body flying off from another body some thousands of miles in a minute, should all the time be violently attracted by that body, and that it is even by virtue of this very attraction that it is flying off from it.—He would probably ask, What more could it do, pray, were it really to be repelled?

"Had the system of electricity, and of repulsion as well as attraction, been known and established in the last age, I have little doubt that the profound genius of Newton would have called it to his aid; and perhaps accounted in a more satisfactory manner for many of the great phenomena of the heavens. To the best of my remembrance, we know of no body that possesses, in any considerable degree, the power of attraction, that in certain circumstances does not likewise possess the power of repulsion; the magnet, the tourmaline, amber, glass, and every electrical substance. Now, from analogy, as we find the sun so powerfully endowed with attraction, why may we not likewise suppose him to be possessed of repulsion? Indeed, this very power seems to be confessed by the Newtonians to reside in the sun in a most wonderful degree; for they assure us he repels the rays of light with such amazing force, that they fly upwards of 80 millions of miles in seven minutes. Now why should we confine this repulsion to the rays of light only? as they are material, may not other matter brought near his body be affected in the same manner? Indeed one would imagine, that their motion alone would create the most violent repulsion; and that the force with which they are perpetually flowing from the sun, would most effectually prevent every other body from approaching him; for this we find is the constant effect of a rapid stream of any other matter. But let us examine a little more his effects on comets. The tails of these bodies are probably their atmospheres rendered highly electrical, either from the violence of their motion, or from their proximity to the sun. Of all the bodies we know, there is none in so constant and so violent an electrical state as the higher regions of our own atmosphere. Of this I have been long convinced; for, send up a kite with a small wire about its string, only to the height of 12 or 1300 feet, and at all times it will produce fire, as I have found by frequent experience; sometimes when the air was perfectly clear, without a cloud in the hemisphere; at other times, when it was thick and hazy, and totally unfit for electrical operations below. Now, as this is the case at so small a height, and as we find the effect still grows stronger in proportion as the kite advances, (for I have sometimes observed, that a little blast of wind, suddenly raising the kite about 100 feet, has more than doubled the effect) what must it be in very great elevations? Indeed we may often judge of it from the violence with which the clouds are agitated, from the meteors formed above the regions of the clouds, and particularly from the Aurora Borealis, which has been observed to have much the same colour and appearance as the matter that forms the tails of comets.

"Now what must be the effect of so vast a body as our atmosphere, made strongly electrical, when it happens to approach any other body? It must always be either violently attracted or repelled, according to the positive or negative quality (in the language of electricians) of the body that it approaches.

"It has ever been observed, that the tails of comets (just as we should expect from a very light fluid body attached to a solid heavy one) are drawn after the comets as long as they are at a distance from the sun; but as soon as the comet gets near his body, the tail veers about to that side of the comet that is in the opposite direction from the sun, and no longer follows the comet, but continues its motion sideways, opposing its whole length to the medium through which it passes, rather than allow it in any degree to approach the sun. Indeed, its tendency to follow the body of the comet is still observable, were it not prevented by some force superior to that tendency; for the tail is always observed to bend a little to that side from whence the comet is flying. This perhaps is some proof too, that it does not move in an absolute vacuum.

"When the comet reaches its perihelion, the tail is generally very much lengthened; perhaps by the rarefaction from the heat; perhaps by the increase of the sun's repulsion, or that of his atmosphere. It still continues projected, exactly in the opposite direction from the sun; and when the comet moves off again to the regions of space, the tail, instead of following it, as it did on its approach, is projected a vast way before it, and still keeps the body of the comet exactly opposed betwixt it and the sun; till by degrees, as the distance increases, the length of the tail is diminished; the repulsion probably becoming weaker and weaker.

"It has likewise been observed, that the length of these tails are commonly in proportion to the proximity of the comet to the sun. That of 1680 threw out a train that would almost have reached from the sun to the earth. If this had been attracted by the sun, would it not have fallen upon his body, when the comet at that time was not one fourth of his diameter distant from him? but, instead of this, it was darted away to the opposite side of the heavens, even with a greater velocity than that of the comet itself. Now what can this be owing to, if not to a repulsive power in the sun, or his atmosphere?

"And indeed it would at first appear but little less absurd, to say that the tail of the comet is all this time violently attracted by the sun, although it be driven away in an opposite direction from him, as to say the same of the comet itself. It is true, this repulsion seems to begin much sooner to affect the tail than the body of the comet; which is supposed always to pass the sun before it begins to fly away from him, which is by no means the case with the tail. The repulsive force, therefore, (if there is any such,) is in a much less proportion than the attractive one; and probably just only enough to counterbalance the latter, when these bodies are in their perihelions, and to turn them so much aside as to prevent their falling into the body of the sun. The projectile force they have acquired will then carry them out to the heavens, and repulsion probably diminishing as they recede from the sun's atmosphere, his attraction will again take place and retard their motion regularly, till they arrive at their aphelia, when they once more begin to return to him."

The only system which appears at present in opposition to the Newtonian is that which we formerly called Mechanical, and was first published by John Hutchinson Esq; with a pretence of its being extracted from the sacred writings. He was contemporary with Sir Isaac, and expressed no small inaccuracy against him, but without success; his unintelligible manner of writing rendering him in a great measure inaccessible even to his friends, and his malice disgusting every body else. The system, however, still continues, with some considerable alterations and improvements. The most confident account we have been able to procure of it as it stands at present, is what follows.

"The motions of the celestial bodies are all of them entirely dependent on the action of the sun, and this action consists in the emission of his light. As we see that no fire can be preserved on earth without the influx of air, so it is reasonable to think that the sun himself cannot be supported without the influx of a stream of air from every side, proportioned to his immense magnitude. This air, by which the sun's heat and light are preserved, is of a purer nature than what we breathe, as being perfectly destitute of aqueous and other vapours with which our air is always loaded. The matter of which the fire, the light, and the air, are composed, is ultimately the same: the only difference we perceive is, that when the ethereal matter appears to us as fire or light, it acts with violence from a centre to a circumference; when as air, it acts more mildly, as from a circumference to a centre. The reason of this apparent mildness of the action of air, in comparison with that of fire or light, is owing only to the greatness of the particles of the former which make its action less sensible; as a push with the head of a pin is much less sensible than one with the point of it, though the one be given with no more force than the other.

"The manner in which the gross air is formed, appears from the following consideration. The light being emitted from the sun, in particles inconceivably minute, must very soon strike against other particles of the same nature with itself, of which the universe is supposed to be perfectly full, which either are not moving at all, or with a slower motion, or in a contrary direction. When a particle has thus lost its motion in a direct line from the sun, it will move along with the particle by which it was stopped, as one body. The reason of this is the pressure of the circumambient fluid; for till the violent action of the light from the sun overcomes the pressure behind and on each side, the two particles cannot separate, having no motion but what is given them by the rest of the fluid. The consequence of this is, that such particles must every moment come nearer the sun; for, the universe being supposed an absolute plenum, if any particle of matter goes out from the centre, another must return to it; otherwise there could be no motion. The light, therefore, consisting of all those particles which have a tendency outward, cannot possibly be emitted without displacing those which have no such tendency, and consequently bringing them nearer to the centre; and thus they must always continue to approach, till at last Thus it appears that the light consists of the very finest particles, each of which moves by itself, and unconnected with any other. When two or more particles of light lose their motion while contiguous to one another, they cannot be separated till they come to the sun himself; or at least to some place where the action from the centre is nearly equal to that from the circumference. This, it is evident, can only take place perfectly in the sun himself; and hence, though our common fires do reduce the air to a greater degree of fineness than when it enters them, they are far from being sufficient to reduce it to the utmost degree of fineness possible; therefore their light is always weak and obscure, compared with that of the sun; and for the same reason, the sun-beams excite a stronger heat than can be raised by any furnace. Hence it is easy to see, that between the light of the sun, and the grossest air, there may, and necessarily will, be fluids of all degrees of grossestness or density, in which dense only the word density can be used on this plan. Each of these fluids will constitute a natural power, or secondary cause, which will act according to its degree of density in particular circumstances, and thus be subservient to the production of different natural phenomena, according to the original appointment of the Creator. A fluid of this intermediate degree of density we evidently see in electricity, which appears vastly more subtle than air, though not quite so much as light.

Thus we have considered the universe as consisting only of the ethereal matter which appears to us in the different modifications of fire, light, air, electric fluid, fixed air, &c. Let us now suppose a planet, or any other porous body, our earth for instance, introduced into it, or created out of nothing by Divine power. Immediately upon its immersion into this mixture of fluids, it is evident that the finer parts would be impelled by the pressure of the rest into the smallest pores of the body; while, by the pressure of the grossest fluids, its particles would be brought as close as those of the finest ethereal fluid which had already infusated themselves would permit. The consequence of this behoved to be the formation of an atmosphere denser than that of the common ethereal fluid with which the whole universe is filled; for the finer parts being as it were drained out from among that part of the common fluid, and infusing themselves into the pores of the earth or other planet, the more gross fluids must necessarily be driven towards the surface, where they will remain, without a possibility of their being separated, unless the planet should fall into the sun; because, in any other part of the creation, the pressure from without must be greater than that from within, and consequently let us suppose the atmosphere to be ever so much rarefied, it never could be destroyed, or leave the body to which it originally belonged.

Having now seen how, on the foregoing principles, every planet must be surrounded with a particular atmosphere of its own, distinct from that of the common ethereal fluid, which we shall henceforth call the atmosphere of the sun, we must now consider the consequences. In the first place, it is obvious, that by means of this atmosphere the violent action of the sun's light will be moderated, as well as of the particles of air that are continually returning towards him; so that, let their impulse be ever so strong, they cannot act on the planet or its inhabitants but through the medium of the atmosphere; and thus the earth becomes a comfortable habitation, when it would otherwise be utterly unfit for the residence of living creatures. The atmosphere of the sun is, as we have already seen, in a perpetual motion, the one part going out, and the other returning. The earth, with its atmosphere, or any other planet, therefore, cannot be supposed in any part of the creation where this action subsists, but that its force part, or that which is turned towards the sun, must be subject to the impulse of those particles of light which are issuing out from him, as the back-part is to the impulse of the particles of air returning towards him. By the one it is pushed out from, and by the other impelled towards, the sun; but as these two impulses are necessarily equal to one another, the planet must still continue at the same distance, without being able to approach or recede in the least. Nevertheless, as both impulses are inconceivably violent, the planet must make an effort to get away, proportionate to the strength of both of them. It must therefore fly off at the tide, because the resistance there is least (it being easier to cross a stream at right angles, than to go directly against it); and having once begun to do so, it must continue its motion, as long as the cause subsists by which it was originally produced; and as the impetus of the light ascending, and the air descending, are in all places equal, it is plain the planet can neither approach nearer to, nor recede farther from, the sun, but must continually circulate round him.

In this way it is easy to see how the earth or any other planet might continue to move round the sun in orbits, a perfect circle; but to explain the planetary motions in elliptic orbits, requires a farther consideration. We have hitherto supposed the planet to be perfectly impenetrable even by the finest part of the solar atmosphere; but it can scarce be supposed that any body in nature is perfectly so; and a little consideration will easily shew us, that, unless the body is perfectly impenetrable, it must describe an ellipse, more or less eccentric, according to its degree of penetrability. The reason of this is, that, in proportion to the transmission of the finer parts, the impulse of the grossest ones from behind becomes stronger, and consequently the planet must approach the sun, till, the impulse of the light becoming so violent that by its plentiful transmission the air behind is somewhat repelled, and of consequence the pressure from that quarter lessened, the planet gradually recedes to its former distance.

That this is really the case, is evident to our senses from the comets. Being formed in a different manner from the planets, they transmit great part of the light, and finer parts of the solar atmosphere; of consequence their lateral motion is but small, and they move almost directly towards the sun. The nearer they approach him, the greater must their velocity be; for it is demonstrable, that the quantity of light emitted from any luminous body, and of consequence the impulse of it, increases in a duplicate proportion to the decrease of distance from that body. If the impulse of light from the sun increases in this manner, so must the impulse of the air towards him; for as the quantity of light emitted is always equal to that of the air which flows in, the impetus of the one must always be equal to that of the other. As the comet gets nearer the sun, the finer parts of the solar atmosphere are transmitted in great quantity, and form its tail, which is nothing else than a stream of electric fluid with which the celestial regions everywhere abound, and which is driven with violence through the body of the comet by the impulse of the light before it. It is not to be supposed that such an immense stream of this matter can be discharged without affecting the air behind the comet. We see that this is the case by the bending of the tail, which indicates a very violent resistance; and this resistance, together with the impulse of the light before, regulates the motion of the comet, and prevents it from flying with the velocity of light itself.

When the comet draws near the sun, the tail increases prodigiously; and the body, being now heated, begins to repel the air on all sides; in consequence of which, a lateral motion is at first begun, as if it was to describe a circle round the sun at a very small distance from his body; but the heat still increasing, it is at last hurried off by the impulse of the light, while the electric stream goes before, as it were to clear its way, and keep off the too great prelude of the air which would retard its progress. As the comet gets farther from the sun into the more dense regions of his atmosphere, the heat begins to abate, and the grotto air to act with its full force; and then, most of the finer parts of his atmosphere being again transmitted imperceptibly, it begins to circulate for a little way, and soon descends again towards the sun.

With this hypothesis, which consists merely in reasoning analogically from what we observe on earth to what is observed in the celestial motions, all the phenomena of nature are in perfect conformity. By it the revolutions of the planets round their axes are easily accounted for, and for which the Newtonian philosophers have never assigned any reason. Such a revolution must be the necessary consequence of the resistance made to the planet's motion, while it flies off from between the ascending stream of light, and the descending one of air. According to this hypothesis too, the velocities of the planets nearest the sun ought to be greatest in their orbits, and their revolutions on their axes the slowest. This is confirmed by experience; Mercury moves swifter than Venus, and Venus swifter than the earth. If Mr Bianchini's observations also are to be credited, Venus moves on her axis 23 times slower than the earth. This slowness in the diurnal revolutions necessarily follows from the greater fluidity of the solar atmosphere near the body of the sun than at a distance; and consequently its being less able to make a lateral resistance. For the same reason, the superior planets, Mars, Jupiter, and Saturn, ought to revolve more slowly in their orbits, and more quickly round their axis. This is remarkably the case with Jupiter, who revolves round his axis with great rapidity, though his motion in his orbit is much slower than that of the earth. Mars seems to be an exception; for though he moves more slowly in his orbit, his motion round his axis is also a little slower than that of the earth. But here we must consider the size of the planet, which is greatly inferior to that of the earth; so that though he moves in a denser medium than the earth does, yet the fineness of his body, and slowness of motion in his orbit, lessen the lateral resistance in such a manner, that his diurnal revolution cannot be completed in such a short time as it could be were his body equal to the earth in size. The magnitude of Jupiter's body also, as well as the density of the medium he moves in, probably contribute to his quick diurnal revolution, which is vastly more swift than that of either the earth or Mars. As for Saturn and Mercury, no spots having ever been observed on them whereby their diurnal revolution could be ascertained, nothing can be determined with regard to them in this particular.

It remains now only to account for the motions of the secondary planets round their primaries; and here there are some appearances which make it probable that the secondary planets are retained in their orbits by the power of electricity. It is observed, that our moon keeps always the same side towards the earth; and this any small electrified body is constantly observed to do towards the body which electrifies it. It hath been observed, that the moons of Jupiter, when passing over his disk, appear to us like black spots; whence it is probable that only one side of the secondary planets is capable of reflecting the light, and therefore that all of them keep constantly the same side towards their primaries. That the combined powers of light and electricity are capable of producing a motion round a centre, may be proved by experiment, which in all cases is worth a thousand speculations.

Let a light hollow ball of cork covered over with electrical brafs or gold leaf be suspended by a pretty long silk thread, so as just to touch the knob of an electrified vial placed on a table. It will instantly be driven off to some distance, and, after a few vibrations, will remain at rest. If a lighted candle is now placed at some distance behind it, so that the flame of the candle may be nearly as high as the knob of the vial, the cork will instantly be agitated, and, after some irregular motions, will describe a curve round the knob of the vial, seemingly of the elliptic kind; and this it will continue to do, sometimes moving in one direction, sometimes in another, till the force of electricity in the vial is almost exhausted. It must be owned, that the circulation here is far from being regular; for sometimes the ellipsis is very eccentric, sometimes it hath very little eccentricity; very often the cork ball will strike upon the knob of the vial, &c. but these irregularities can only be attributed to want of skill in the operator to adjust the forces to one another in a proper manner. But if we, by a few sparks from an electric conductor, can make a cork perform some hundreds of revolutions in an irregular manner round the knob of a vial, what cannot the Deity do, who hath the whole powers of light and electricity at his command, who knows their nature perfectly, and whose mechanical skill hath no limits besides the nature of the materials he employs?

The electric power is by most philosophers allowed to have a principal share in all the natural operations.

This experiment we have repeated with success; but whether the consequences deduced from it by the author of this account can be justly drawn, we must submit to better judges. tions on this earth. Experience shews, that, so far from diminishing, it grows much stronger the higher we ascend. As we can, therefore, set no bounds to this increase of power, it seems most reasonable to suppose that the moon receives it from the earth, as the cork ball in the experiment receives it from the knob of the vial; and that, being continually drawn off by the sun, it occasions the circulation of the moon in a similar, though much more regular, manner."

Having now given an account of the principal systems that have appeared, and recounted a great number of arguments pro and con, on almost every particular, some general conclusion will naturally be expected from us; as otherwise many readers may think our intention has been to confound them, by advancing a multiplicity of opinions, and leading them into a chaos from which no real knowledge can be extracted.

Here we must observe, that all the arguments that ever have been brought against Sir Isaac Newton, only tend to invalidate what is called his Physical system, or that part of it which accounts for the phenomena of nature. As for that part of astronomy which consists in the knowledge of the phenomena themselves, what Sir Isaac hath advanced on that head may be looked upon as absolutely certain, and is controverted by nobody. With regard to the dispute concerning attraction, it is of little consequence whether it is cause or effect. The word attraction, or some other perhaps equally improper, must be made use of, though we even were already acquainted with the cause of gravity. But it must be remembered, that if attraction is ever discovered to be the effect of a material cause, the cause itself must be distinct of all attraction, or tendency of one part to another, and consequently have very different properties from other matter.

As to the two powers of gravitation and projection which Sir Isaac Newton assumed as the causes of the planetary revolutions, it is of the utmost importance to a physical astronomer to be ascertained whether these forces are capable of producing the effects ascribed to them or not. Objections similar to those above inserted have been published long ago, and we are surprised that no plain and direct answer hath yet been given to them. In 1762, a book entitled "The Principles of Natural Philosophy, &c." by William Jones," made its appearance, in which, among other things, the author undertook to prove, that by a combination of gravitation and projectile force no lasting motion could be produced. As far as we know, no answer has ever been published to this treatise; and upon looking into the Monthly Review, Vol. xxvii. p. 122, we were surprised to find the author censured, rather uncandidly, for controverting Sir Isaac's opinion, while not a single word is offered in answer to his objection, or a hint given where such a thing could be found. In other respects, the Reviewers own that Sir Isaac Newton himself has reasoned very weakly and inconclusively in physical matters. Such concessions as these will necessarily create doubts in the mind of every person that reads them; and therefore particular care ought to be taken in distinguishing where his reasoning is solid and invincible, and where it is not to be regarded. In 1764, another treatise of the same nature, entitled "Short Observations on the Principles and Moving Powers assumed by the present System of Philosophy," was published. In this the whole physical part of Sir Isaac's system was attacked and even ridiculed. The author asserted the insufficiency of the two forces of gravitation and projection to keep a planet in its orbit; and, if no other powers than these acted upon it, that it behoved to be hurried off in an eccentric curve. Being unacquainted with any answer to this treatise, we were again obliged to have recourse to the Reviewers; when, in the Review for May 1764, we found the following answer, viz. that the "argument is fallacious, because he doth not take into consideration the time in which gravity acts on moving bodies." Certainly an objection of such a capital nature as this merited a more particular answer, or a direction to some other treatise where such an answer might be found. It is in a manner incredible, that such an excellent mathematician as Sir Isaac Newton should have assumed two powers as first principles, which were utterly insufficient to produce the effect he ascribed to them; and, on the other hand, if they are sufficient, we are entirely at a loss to account for the want of replies to such objections, in the common astronomical treatises, when others, of at least as little consequence, are fully obviated.

But further, we are afraid, that most philosophers, even the most zealous advocates for Sir Isaac Newton, are inclined to admit the existence of a power in the celestial regions, which must either be the cause of the planetary revolutions, or will utterly destroy their motions. The power we mean is that of electricity. We have already quoted Dr Hamilton conjecturing the tails of comets are streams of electric matter; and indeed their resemblance to the Aurora Borealis is so great, that it is almost impossible to ascribe the one to electricity, and the other to any different cause. But let us attend to the consequences of this supposition. The tails of comets are immensely large. Sir Isaac Newton computed that of the comet in 1680 to be eighty millions of miles in length. What inconceivable power must not such a stream of electric matter be attended with? We are sure that by its means the comet would attract at the distance of 80,000,000 miles, and how much farther we cannot tell. If we suppose the sun to be the fountain of electricity as well as of heat and light, then undoubtedly he must attract and repel by means of his electric as well as his gravitating power; so that the law of gravity must either be an effect of the electrical power, or behaved to be perpetually interrupted by it. If, with Mr Henry, Cavallo, and others, we suppose the electric fluid to be a modification of the element of fire, there is an end of Sir Isaac Newton's physical system, to all intents and purposes, and downright Hutchinsonianism comes in place of it; for Hutchinson's very first and fundamental principle is, that elementary fire is sent forth as such from the sun, into the planetary regions and beyond them, where it is converted into a different substance, no matter whether air, or electric fluid.

These things we take notice of, in order to show how cautious philosophers ought to be in indulging conjectures; as, by so doing, it may often happen, that they will pull down with one hand what they build up with the other. For our own part, we cannot pretend to decide; but as the Newtonian system is so generally made use of for the solution of the celestial phenomena, nomena, we shall still suppose it to be the only true one, and proceed to give a particular explanation of these phenomena according to it.

Sect. V. The Copernican System particularly considered and explained, on the Newtonian Principles.

The sun, with the planets and comets which move round him as their centre, constitute the solar system. Those planets which are near the sun, not only finish their circuits sooner, but likewise move faster in their respective orbits, than those which are more remote from him. Their motions are all performed from west to east, in orbits nearly circular.

The sun, an immense globe of fire, is placed near the common centre, or rather in the lower focus, of the orbits of all the planets and comets (x); and turns round his axis in 25 days 6 hours. His diameter is computed to be 890,000 miles; and, by the various attractions of the circumvolving planets, he is agitated by a small motion round the centre of gravity of the system. All the planets, as seen from him, move the same way, and according to the order of signs in the graduated circle Ψ & Π & Σ &c., which represents the great ecliptic, or circle annually described by the earth round the sun; but, as seen from any one planet, they appear sometimes to go backward, sometimes forward, and sometimes to stand still; not in circles nor ellipses, but in looped curves which never return into themselves; as already observed n°6. Round the path of Venus and Mercury is marked the graduated circle representing the ecliptic. The dotted lines from the earth to the ecliptic are added for shewing Mercury's apparent or geocentric motion therein for one year; in which time his path makes three loops, and goes on a little farther; which shews that he has three inferior, and as many superior, conjunctions with the sun in that time; and also that he is six times stationary, and thrice retrograde.

Let us now trace his motion for one year in the figure.

Suppose Mercury to be setting out from A towards B (between the earth and left-hand corner of the plate), and as seen from the earth, his motion will then be direct, or according to the order of the signs. But when he comes to B, he appears to stand still in the 23rd degree of M at F, as shown by the line BF. Whilst he goes from B to C, the line BF, supposed to move with him, goes backward from F to E, or contrary to the order of signs; and when he is at C, he appears stationary at E, having gone back 11½ degrees. Now, suppose him stationary on the first of January at C, on the 10th thereof he will appear in the heavens as at 20, near F; on the 20th he will be seen as at G; on the 31st, at H; on the 10th of February, at I; on the 20th, at K; and on the 28th, at L; as the dotted lines show, which are drawn through every tenth day's motion in his looped path, and continued to the ecliptic. On the 10th of March he appears at M; on the 20th, at N; and on the 31st, at O. On the 10th of April he appears stationary at P; on the 20th he seems to have gone back again to O; and on the 30th he appears stationary at Q, having gone back 11½ degrees. Thus Mercury seems to go forward 4 signs 11 degrees, or 131 degrees; and to go back only 11 or 12 degrees, at a mean rate. From the 30th of April to the 10th of May he seems to move from Q to R; and on the 20th he is seen at S, going forward in the same manner again, according to the order of letters; and backward when they go back; which it is needless to explain any farther, as the reader can trace him out so easily thro' the rest of the year. The same appearances happen in Venus's motion; but as she moves slower than Mercury, there are longer intervals of time between them. The comets come from all parts of the heavens, and move in all sorts of directions.

The axis of a planet is a line conceived to be drawn Axes of the through its centre, about which it revolves as if on a planet's real axis. The extremities of this line, terminating in opposite points of the planet's surface, are called its poles. That which points towards the northern part of the heavens, is called the north pole; and the other, pointing towards the southern part, is called the south pole. A bowl whirled from one's hand into the open air turns round such a line within itself, whilst it moves forward; and such are the lines we mean when we speak of the axes of the heavenly bodies.

Let us suppose the earth's orbit to be a thin, even, solid plane; cutting the sun through the centre, and extended out as far as the starry heavens, where it will mark the great circle called the ecliptic. This circle we suppose to be divided into 12 equal parts, called signs; each sign into 30 equal parts, called degrees; each degree into 60 equal parts, called minutes; and every minute into 60 equal parts, called seconds; so that a second is the 60th part of a minute; a minute the 60th part of a degree; and a degree the 360th part of a circle, or 360th part of a sign. The planes of the orbits of all the other planets likewise cut the sun in halves; but, extended to the heavens, form circles different from one another, and from the ecliptic; one half of each being on the north side, and the other on the south side of it. Consequently the orbit of each planet crosses the ecliptic in two opposite points, which are called the planet's nodes. These nodes are all in Nodes of different parts of the ecliptic; and therefore, if the planetary traits remained visible in the heavens, they would in some measure resemble the different runs of waggon-wheels crossing one another in different parts, but never going far asunder. That node, or intersection of the orbit of any planet with the earth's orbit, from which the planet ascends northward above the ecliptic, is called the ascending node of the planet; and the other, which is directly opposite thereto, is called its descending node. Saturn's ascending node is in 21 deg. 13 min. of Cancer, Σ; Jupiter's in 7 deg. 29 min. of the same sign; Mars's in 17 deg. 17 min. of Taurus.

(x) Astronomers are not far from the truth, when they reckon the sun's centre to be in the lower focus of all the planetary orbits. Though, strictly speaking, if we consider the focus of Mercury's orbit to be in the sun's centre, the focus of Venus's orbit will be in the common centre of gravity of the sun and Mercury; the focus of the earth's orbit in the common centre of gravity of the sun, Mercury, and Venus; the focus of the orbit of Mars in the common centre of gravity of the sun, Mercury, Venus, and the earth; and so of the rest. Yet, the focuses of the orbits of all the planets, except Saturn, will not be sensibly removed from the centre of the sun; nor will the focus of Saturn's orbit recede sensibly from the common centre of gravity of the sun and Jupiter. Taurus; Venus's in 13 deg. 59 min. of Gemini; and Mercury's in 14 deg. 43 min. of Taurus. Here we consider the earth's orbit as the standard, and the orbits of all the other planets as oblique to it.

When we speak of the planets' orbits, all that is meant is their paths through the open and unresisting space in which they move; and are kept in, by the attractive power of the sun, and the projectile force impressed upon them at first; between which power and force there is so exact an adjustment, that they continue in the same tracts without any solid orbits to confine them.

Mercury, the nearest planet to the sun, goes round him (as in the circle marked §) in 87 days 23 hours of our time nearly; which is the length of his year. But being seldom seen, and no spots appearing on his surface or disk, the time of his rotation on his axis, or the length of his days and nights, is as yet unknown. His distance from the sun is computed to be 36,841,448 miles, and his diameter 3000. In his course round the sun, he moves at the rate of 109,699 miles every hour. His light and heat from the sun are almost seven times as great as ours, and the sun appears to him almost seven times as large as to us. The great heat on this planet is no argument against its being inhabited; since the Almighty could as easily suit the bodies and constitutions of its inhabitants to the heat of their dwelling, as he has done to the temperature of our earth. And it is very probable that the people there have such an opinion of us, as we have of the inhabitants of Jupiter and Saturn, namely, that we must be intolerably cold, and have very little light, at so great a distance from the sun.

The orbit of Mercury is inclined seven degrees to the ecliptic; and that node from which he ascends northward above the ecliptic, is in the 14th degree of Taurus; the opposite, in the 14th degree of Scorpio. The earth is in these points on the 6th of November, and 4th of May, new style; and when Mercury comes to either of his nodes at his inferior conjunction (v) about these times, he will appear to pass over the disk or face of the sun, like a dark round spot. But in all other parts of his orbit his conjunctions are invisible, because he either goes above or below the sun.

Mr Whiston has given us an account of several periods at which Mercury may be seen on the sun's disk, viz., in the year 1782, Nov. 12th, at 3 h. 44 m. in the afternoon; 1786, May 4th, at 6 h. 57 m. in the forenoon; 1789 Dec. 6th, at 3 h. 55 m. in the afternoon; and 1799, May 7th, at 2 hours 34 m. in the afternoon. There will be several intermediate transits, but none of them visible at London.

Venus, the next planet in order, is computed to be 68,891,486 miles from the sun; and by moving at the rate of 80,295 miles every hour in her orbit (as in the circle marked §), she goes round the sun in 224 days 17 hours of our time nearly; in which, though it be the full length of her year, she has only 9½ days according to Bianchini's observations: so that, to her, every day and night together is as long as 24½ days and nights with us. This odd quarter of a day in every year makes every fourth year a leap-year to Venus; as the like does to our earth. Her diameter is 9330 miles. Her orbit is included by the earth's; for if it were not, she might be seen as often in opposition to the sun, as she is in conjunction with him; but she was never seen 90 degrees, or a fourth part of a circle, from the sun.

When Venus appears west of the sun, she rises before him in the morning, and is called the morning star; when she appears east of the sun, she shines in the evening after he sets, and is then called the evening star; being each in its turn for 290 days. It may perhaps be surprising at first, that Venus should keep longer on the east or west of the sun, than the whole time of her period round him. But the difficulty vanishes when we consider that the earth is all the while going round the sun the same way, though not so quick as Venus; and therefore her relative motion to the earth must in every period be as much slower than her absolute motion in her orbit, as the earth during that time advances forward in the ecliptic; which is 220 degrees.

The axis of Venus is inclined 75 degrees to the axis of her orbit, which is 51½ degrees more than our earth's axis is inclined to the axis of the ecliptic; and therefore her seasons vary much more than ours do. The north pole of her axis inclines toward the 20th degree of Aquarius, our earth's to the beginning of Cancer; consequently the northern parts of Venus have summer in the signs where those of our earth have winter, and vice versa.

The artificial day at each pole of Venus is as long as 112½ natural days on our earth.

The sun's greatest declination on each side of her equator amounts to 75 degrees; therefore her tropics are only 15 degrees from her poles, and her polar circles as far from her equator. Consequently, the tropics of Venus are between her polar circles and her poles; contrary to what those of our earth are.

As her annual revolution contains only 9½ of her days, the sun will always appear to go through a whole sign, or twelfth part of her orbit, in little more than three quarters of her natural day, or nearly in 18½ of our days and nights.

Because her day is so great a part of her year, the sun changes his declination in one day to much, that if he passes vertically, or directly over head of any given place on the tropic, the next day he will be 26 degrees from it; and whatever place he passes vertically over when in the equator, one day's revolution will remove him 36½ degrees from it. So that the sun changes his declination every day in Venus about 14 degrees more, at a mean rate, than he does in a quarter of a year on our earth. This appears to be providentially ordered, for preventing the too great effects of the sun's heat (which is twice as great on Venus as on the earth), so that he cannot shine perpendicularly on the same places for two days together; and on that account, the heated places have time to cool.

If the inhabitants about the north pole of Venus fix their south, or meridian line, through that part of the heavens where the sun comes to his greatest height, or north declination, and call those the east and west points of their horizon, which are 90 degrees on each side from that point where the horizon is cut by the meridian line, these inhabitants will have the following remarkable appearances:

(r) When he is between the earth and the sun in the nearer part of his orbit. The sun will rise 22° degrees north of the east; and going on 112° degrees, as measured on the plane of the horizon, he will cross the meridian at an altitude of 12° degrees; then making an entire revolution without setting, he will cross it again at an altitude of 48° degrees; at the next revolution he will cross the meridian, as he comes to his greatest height and declination, at the altitude of 75° degrees; being then only 15 degrees from the zenith, or that point of the heavens which is directly over head; and thence he will descend in the like spiral manner; crossing the meridian first at the altitude of 48° degrees, next at the altitude of 12° degrees; and going on thence 112° degrees, he will set 22° degrees north of the west; so that, after having been 4 revolutions above the horizon, he descends below it to exhibit the like appearances at the south pole.

At each pole, the sun continues half a year without setting in summer, and as long without rising in winter; consequently the polar inhabitants of Venus have only one day and one night in the year; as it is at the poles of our earth. But the difference between the heat of summer and cold of winter, or of mid-day and mid-night, on Venus, is much greater than on the earth; because on Venus, as the sun is for half a year together above the horizon of each pole in its turn, so he is for a considerable part of that time near the zenith; and during the other half of the year always below the horizon, and for a great part of that time at least 70 degrees from it. Whereas, at the poles of our earth, although the sun is for half a year together above the horizon; yet he never ascends above, nor descends below it, more than 23° degrees. When the sun is in the equinoctial, or in that circle which divides the northern half of the heavens from the southern, he is seen with one half of his disk above the horizon of the north-pole, and the other half above the horizon of the south-pole; so that his centre is in the horizon of both poles; and then, descending below the horizon of one, he ascends gradually above that of the other. Hence, in a year, each pole has one spring, one harvest, a summer as long as them both, and a winter equal in length to the other three seasons.

At the polar circles of Venus, the seasons are much the same as at the equator, because there are only 15 degrees between them; only the winters are not quite so long, nor the summers so short; but the four seasons come twice round every year.

At Venus's tropics, the sun continues for about 15 of our weeks together without setting in summer, and as long without rising in winter. Whilst he is more than 15 degrees from the equator, he neither rises to the inhabitants of the one tropic, nor sets to those of the other; whereas, at our terrestrial tropics, he rises and sets every day of the year.

At Venus's tropics, the seasons are much the same as at her poles; only the summers are a little longer, and the winters a little shorter.

At her equator, the days and nights are always of the same length; and yet the diurnal and nocturnal arches are very different, especially when the sun's declination is about the greatest; for then, his meridian altitude may sometimes be twice as great as his midnight depression, and at other times the reverse. When the sun is at his greatest declination, either north or south, his rays are as oblique at Venus's equator, as they are at London on the shortest day of winter. Therefore, at her equator, there are two winters, two summers, two springs, and two autumns, every year. But because the sun stays for some time near the tropics, and passes so quickly over the equator, every winter there will be almost twice as long as summer; the four seasons returning twice in that time, which consists only of 9½ days.

Those parts of Venus which lie between the poles and tropics, and between the tropics and polar circles, and also between the polar circles and equator, partake more or less of the phenomena of these circles, as they are more or less distant from them.

From the quick change of the sun's declination it happens, that if he rises due east on any day, he will not set due west on that day, as with us; for if the place where he rises due east be on the equator, he will set on that day almost west north-west, or about 18° degrees north of the west. But if the place be in 45 degrees north latitude, then on the day that the sun rises due east he will set north-west by west, or 33 degrees north of the west. And in 62 degrees north latitude, when he rises in the east, he sets not in that revolution, but just touches the horizon 10 degrees to the west of the north point; and ascends again, continuing for 3½ revolutions above the horizon without setting. Therefore no place has the forenoon and afternoon of the same day equally long, unless it be on the equator, or at the poles.

The sun's altitude at noon, or any other time of the Longitude day, and his amplitude at rising and setting, being very different at places on the same parallel of latitude, according to the different longitudes of those places, the longitude will be almost as easily found on Venus, as the latitude is found on the earth; which is an advantage we can never have, because the daily change of the sun's declination is much too small for that important purpose.

On this planet, where the sun crosses the equator in any year, he will have 9 degrees of declination from that place on the same day and hour next year; and will cross the equator 90 degrees farther to the west; which makes the time of the equinox a quarter of a day (or about six of our days) later every year. Hence although the spiral in which the sun's motion is performed be of the same form every year, yet it will not be the very same, because the sun will not pass vertically over the same places till four annual revolutions are finished.

We may suppose that the inhabitants of Venus Every will be careful to add a day to some particular part of fourth year every fourth year; which will keep the same seasons to the same days. For, as the great annual change of the equinoxes and solstices shifts the seasons a quarter of a day every year, they would be shifted through all the days of the year in 35 years. But by means of this intercalary day, every fourth year will be a leap-year: which will bring her time to an even reckoning, and keep her calendar always right.

At the transits of Venus over the sun in 1761 and 1769, astronomers were very careful to observe whether any satellite belonging to this planet could be discovered; but as none was to be seen, it is now generally concluded that she has none, but that Cassini and Mr Short were mistaken. The earth is the next planet above Venus in the system. It is 95,173,000 miles from the sun, and goes round him (as in the circle ⊙) in 365 days 5 hours 49 minutes, from any equinox or solstice to the same again; but from any fixed star to the same again as seen from the sun, in 365 days 6 hours 9 minutes; the former being the length of the tropical year, and the latter the length of the sidereal. It travels at the rate of 68,000 miles every hour; which motion, though upwards of 140 times swifter than that of a cannon ball, is little more than half as swift as Mercury's motion in his orbit. The earth's diameter is 7970 miles; and by turning round its axis every 24 hours from west to east, it causes an apparent diurnal motion of all the heavenly bodies from east to west. By this rapid motion of the earth on its axis, the inhabitants about the equator are carried 1042 miles every hour, whilst those on the parallel of London are carried only about 580, besides the 68,000 miles by the annual motion above-mentioned, which is common to all places whatever.

The earth's axis makes an angle of 23½ degrees with the axis of its orbit, and keeps always the same oblique direction, inclining nearly to the same fixed stars (⊙) throughout its annual course, which causes the returns of spring, summer, autumn, and winter. That the sun, and not the earth, is the centre of our solar system, may be demonstrated beyond a possibility of doubt, from considering the forces of gravitation and projection, by which all the celestial bodies are retained in their orbits. For, if the sun moves about the earth, the earth's attractive power must draw the sun towards it from the line of projection so as to bend its motion into a curve. But the sun being at least 227,000 times as heavy as the earth, by being so much weightier as its quantity of matter is greater, it must move 227,000 times as slowly toward the earth, as the earth does toward the sun; and consequently the earth would fall to the sun in a short time, if it had not a very strong projectile motion to carry it off. The earth therefore, as well as every other planet in the system, must have a rectilinear impulse, to prevent its falling into the sun. To say, that gravitation retains all the other planets in their orbits without affecting the earth, which is placed between the orbits of Mars and Venus, is as absurd as to suppose that six cannon-bullets might be projected upward to different heights in the air, and that five of them should fall down to the ground; but the sixth, which is neither the highest nor the lowest, should remain suspended in the air without falling, and the earth move round about it.

There is no such thing in nature as a heavy body moving round a light one as its centre of motion. A pebble fastened to a mill-stone by a string, may by an easy impulse be made to circulate round the mill-stone; but no impulse can make a mill-stone circulate round a loose pebble; for the mill-stone would go off, and carry the pebble along with it.

The sun is so immensely bigger and heavier than the earth, that, if he was moved out of his place, not only the earth, but all the other planets, if they were united into one mass, would be carried along with the sun, as the pebble would be with the mill-stone.

By considering the law of gravitation, which takes place throughout the solar system, in another light, it will be evident that the earth moves round the sun in a year, and not the sun round the earth. It has been observed, that the power of gravity decreases as the square of the distance increases; and from this it follows with mathematical certainty, that when two or more bodies move round another as their centre of motion, the squares of their periodic times will be to one another in the same proportion, as the cubes of their distances from the central body. This holds precisely with regard to the planets round the sun, and the satellites round the planets; the relative distances of all which are well known. But, if we suppose the sun to move round the earth, and compare its period with the moon's by the above rule, it will be found that the sun would take no less than 173,510 days to move round the earth; in which case our year would be 475 times as long as it now is. To this we may add, that the aspects of increase and decrease of the planets, the times of their seeming to stand still, and to move direct and retrograde, answer precisely to the earth's motion; but not at all to the sun's, without introducing the most absurd and monstrous suppositions, which would destroy all harmony, order, and simplicity, in the system. Moreover, if the earth be supposed to stand still, and the stars to revolve in free spaces about the earth in 24 hours, it is certain that the forces by which the stars revolve in their orbits are not directed to the earth, but to the centres of the several orbits; that is, of the several parallel circles which the stars on different sides of the equator describe every day; and the like inferences may be drawn from the supposed diurnal motion of the planets, since they are never in the equinoctial but twice, in their courses with regard to the starry heavens. But that forces should be directed to no central body, on which they physically depend, but to innumerable imaginary points in the axis of the earth produced to the poles of the heavens, is an hypothesis too absurd to be allowed of by any rational creature. And it is still more absurd to imagine that these forces should increase exactly in proportion to the distances from this axis; for this is an indication of an increase to infinity; whereas the force of attraction is found to decrease in receding from the fountain from whence it flows. But the farther any star is from the quiescent pole, the greater must be the orbit which it describes; and yet it appears to go round in the same time as the nearest star to the pole does. And if we take into consideration the twofold motion observed in the stars, one diurnal round the axis of the earth in 24 hours, and the other round the axis of the ecliptic in 25,920 years, it would require an explication of such a perplexed composition of forces, as could by no means be reconciled with any physical theory.

The strongest objection that can be made against the earth's motion round the sun, is, that in opposite points of the earth's orbit, its axis, which always keeps a parallel direction, would point to different fixed stars; which is not found to be fact. But this objection is easily removed, by considering the immense distance

(c) This is not strictly true, as will appear when we come to treat of the recession of the equinoctial points in the heavens, which recession is equal to the deviation of the earth's axis from its parallelism: but this is rather too frail to be feasible in an age, except to those who make very nice observations. distance of the stars in respect of the diameter of the earth's orbit; the latter being no more than a point when compared to the former. If we lay a ruler on the side of a table, and along the edge of the ruler view the top of a spire at ten miles distance; then lay the ruler on the opposite of the table in a parallel situation to what it had before, and the spire will still appear along the edge of the ruler; because our eyes, even when assisted by the best instruments, are incapable of distinguishing so small a change at so great a distance.

Dr Bradley, our late astronomer royal, found by a long series of the most accurate observations, that there is a small apparent motion of the fixed stars, occasioned by the aberration of their light; and so exactly answering to an annual motion of the earth, as evinces the same, even to a mathematical demonstration. He considered this matter in the following manner: he imagined CA, fig. 5, to be a ray of light falling perpendicularly upon the line BD; that, if the eye is at rest at A, the object must appear in the direction AC, whether light be propagated in time or in an instant. But if the eye is moving from B towards A, and light is propagated in time, with a velocity that is to the velocity of the eye, as CA to BA; then light moving from C to A, whilst the eye moves from B to A, that particle of it by which the object will be discerned when the eye comes to A, is at C when the eye is at B. Joining the points BC, he supposed the line CB to be a tube, inclined to the line BD in the angle DBC, of such diameter as to admit but one particle of light. Then it was easy to conceive, that the particle of light at C, by which the object must be seen, when the eye, as it moves along, arrives at A, would pass through the tube BC, if it is inclined to BD in the angle DBC, and accompanies the eye in its motion from B to A; and that it could not come to the eye placed behind such a tube, if it had any other inclination to the line BD. If, instead of supposing CB so small a tube, we imagine it to be the axis of a larger; then, for the same reason, the particle of light at C would not pass through the axis, unless it is inclined to BD in the angle CBD. In like manner, if the eye moved the contrary way, from D towards A, with the same velocity, then the tube must be inclined in the angle BDC. Although, therefore, the true or real place of an object is perpendicular to the line in which the eye is moving, yet the visible place will not be so; since that, no doubt, must be in the direction of the tube; but the difference between the true and apparent place will be, _ceteris paribus_, greater or less, according to the different proportion between the velocity of light and that of the eye. So that, if we could suppose that light was propagated in an instant, then there would be no difference between the real and visible place of an object, although the eye was in motion; for in that case, AC being infinite with respect to AB, the angle ACB; the difference between the true and visible place, vanishes. But if light be propagated in time, it is evident, from the foregoing considerations, that there will be always a difference between the real and visible place of an object, unless the eye is moving either directly towards or from the object. And in all cases the fine of the difference between the real and visible place of the object will be to the fine of the visible inclination of the object to the line in which the eye is moving, as the velocity of the eye is to the velocity of light.

He then shews, that if the earth revolve round the sun annually, and the velocity of light be to the velocity of the earth's motion in its orbit, as 1000 to 1, that a star really placed in the very pole of the ecliptic would, to an eye carried along with the earth, seem to change its place continually; and, neglecting the small difference on the account of the earth's diurnal revolution on its axis, would seem to describe a circle round that pole every way distant from it 3½'; so that its longitude would be varied thro' all the points of the ecliptic every year, but its latitude would always remain the same. Its right ascension would also change, and its declination, according to the different situation of the sun with respect to the equinoctial points, and its apparent distance from the north pole of the equator, would be 7' less at the autumnal than at the vernal equinox.

By calculating exactly the quantity of aberration of velocity the fixed stars from their place, he found that light came light from the sun to us in 8°-13' ; so that its velocity is to the velocity of the earth in its orbit, as 10201 to 1.

It must here be taken notice of, however, that Mr Nevil errors in Mairlyne, in attempting to find the parallax of Sirius the observer with a ten-foot sector, observed, that by the friction of the plummet line on the pin which suspended it, an error of 10', 20', and sometimes 30', was committed. The pin was ¼" of an inch diameter; and though he reduced it to ⅛" of an inch, the error still amounted to 3'. All observations, therefore, that have hitherto been made in order to discover the parallax of the fixed stars, are to be disregarded; and even the quantity of aberration assigned by Dr Bradley to the fixed stars is to be doubted.

It is also objected, that the sun seems to change his place daily, so as to make a tour round the starry heavens in a year. But, whether the sun or earth moves, this appearance will be the same; for, when the earth is in any part of the heavens, the sun will appear in the opposite. And therefore, this appearance can be no objection against the motion of the earth.

It is well known to every person who has sailed on smooth water, or been carried by a stream in a calm, that, however fast the vessel goes, he does not feel its progressive motion. The motion of the earth is incomparably more smooth and uniform than that of a ship, or any machine made and moved by human art; and therefore it is not to be imagined that we can feel its motion.

We find that the sun, and those planets on which there are visible spots, turn round their axes; for the spots move regularly over their disks (a). From hence

---

(a) This, however, must be understood with some degree of limitation, as will evidently appear from what has been said concerning the spots of the sun. Nay, even in the planet Jupiter, whose rotation on his axis seems better ascertained than that of any other, the difference of time between the revolution of the spots about his equator, and those near his poles, is a phenomenon that hath puzzled the best astronomers. Mr M'Laurin (Phys. and Literary Essays, Vol. I. p. 366.) says, that it is "a phenomenon of that kind of which it is perhaps best not to attempt any explanation till confirmed by further experiments." Since his time it has been confirmed, but we have not heard of any satisfactory explanation. we may reasonably conclude, that the other planets on which we see no spots, and the earth, which is likewise a planet, have such rotations. But being incapable of leaving the earth, and viewing it at a distance, and its rotation being smooth and uniform, we can neither see it move on its axis as do the planets, nor feel ourselves affected by its motion. Yet there is one effect of such a motion, which will enable us to judge with certainty whether the earth revolves on its axis or not.

All globes which do not turn round their axes will be perfect spheres, on account of the equality of the weight of bodies on their surfaces; especially of the fluid parts. But all globes which turn on their axes will be oblate spheroids; that is, their surfaces will be higher, or farther from the centre, in the equatorial than in the polar regions: for, as the equatorial parts move quickest, they will recede farther from the axis of motion, and enlarge the equatorial diameter. That our earth is really of this figure, is demonstrable from the unequal vibrations of a pendulum, and the unequal lengths of degrees in different latitudes. Since then the earth is higher at the equator than at the poles, the sea, which naturally runs downward, or towards the places which are nearest the centre, would run towards the polar regions, and leave the equatorial parts dry, if the centrifugal force of these parts, by which the waters were carried thither, did not keep them from returning. The earth's equatorial diameter is 36 miles longer than its axis.

Bodies near the poles are heavier than those towards the equator, because they are nearer the earth's centre, where the whole force of the earth's attraction is accumulated. They are also heavier, because their centrifugal force is less, on account of their diurnal motion being slower. For both these reasons, bodies carried from the poles toward the equator gradually lose their weight. Experiments prove, that a pendulum, which vibrates seconds near the poles vibrates slower near the equator, which shows that it is lighter or less attracted there. To make it oscillate in the same time, it is found necessary to diminish its length. By comparing the different lengths of pendulums swinging seconds at the equator and at London, it is found that a pendulum must be 21580 lines shorter at the equator than at the poles. A line is a twelfth part of an inch.

If the earth turned round its axis in 84 minutes 43 seconds, the centrifugal force would be equal to the power of gravity at the equator; and all bodies there would entirely lose their weight. If the earth revolved quicker, they would all fly off, and leave it.

A person on the earth can no more be sensible of its undisturbed motion on its axis, than one in the cabin of a ship on smooth water can be sensible of the ship's motion when it turns gently and uniformly round. It is therefore no argument against the earth's diurnal motion, that we do not feel it: nor is the apparent revolutions of the celestial bodies every day a proof of the reality of these motions; for whether we or they revolve, the appearance is the very same. A person, looking through the cabin-windows of a ship, as strongly fancies the objects on land to go round when the ship turns, as if they were actually in motion.

If we could translate ourselves from planet to planet, we should still find that the stars would appear of the same magnitudes, and at the same distances from each other, as they do to us here; because the width of the remotest planet's orbit bears no sensible proportion to the distance of the stars. But then, the heavens would seem to revolve about very different axes; and consequently, those quiescent points, which are our poles in the heavens, would seem to revolve about other points, which, though apparently in motion as seen from the earth, would be at rest as seen from any other planet. Thus the axis of Venus, which lies at right angles to the axis of the earth, would have its motionless poles in two opposite points of the heavens lying almost in our equinoctial, where the motion appears quickest, because it is seemingly performed in the greatest circle: and the very poles, which are at rest to us, have the quickest motion of all as seen from Venus. To Mars and Jupiter the heavens appear to turn round with very different velocities on the same axis, whose poles are about 23½ degrees from ours. Were we on Jupiter, we should be at first amazed at the rapid motion of the heavens; the sun and stars going round in 9 hours 56 minutes. Could we go from thence to Venus, we should be as much surprised at the slowness of the heavenly motions; the sun going but once round in 34 hours, and the stars in 540. And could we go from Venus to the moon, we should see the heavens turn round with a yet slower motion; the sun in 708 hours, the stars in 655. As it is impossible these various circumvolutions in such different times, and on such different axes, can be real, so it is unreasonable to suppose the heavens to revolve about our earth more than it does about any other planet. When we reflect on the vast distance of the fixed stars, to which 190,000,000 of miles, the diameter of the earth's orbit, is but a point, we are filled with amazement at the immensity of their distance. But if we try to frame an idea of the extreme rapidity with which the stars must move, if they move round the earth in 24 hours, the thought becomes so much too big for our imagination, that we can no more conceive it than we do infinity or eternity. If the sun was to go round the earth in 24 hours, he must travel upwards of 300,000 miles in a minute: but the stars being at least 400,000 times as far from the sun, as the sun is from us, those about the equator must move 400,000 times as quick. And all this to serve no other purpose than what can be as fully and much more firmly obtained by the earth's turning round eastward, as on an axis, every 24 hours, causing thereby an apparent diurnal motion of the sun westward, and bringing about the alternate returns of day and night.

As to the common objections against the earth's motion on its axis, they are all easily answered and set aside. That it may turn without being seen or felt by us to do so, has been already shewn. But some are apt to imagine, that if the earth turns eastward (as it certainly does if it turns at all), a ball fired perpendicularly upward in the air must fall considerably westward of the place it was projected from. The objection, which at first seems to have some weight, will be found to have none at all, when we consider that the gun and ball partake of the earth's motion; and therefore the ball being carried forward with the air as quick as the earth and air turn, must fall down on the same place. A stone let fall from the top of a mainmast, mast, if it meets with no obstacle, falls on the deck as near the foot of the mast when the ship sails as when it does not. If an inverted bottle, full of liquor, be hung up to the ceiling of the cabin, and a small hole be made in the cork to let the liquor drop through on the floor, the drops will fall just as far forward on the floor when the ship sails as when it is at rest. And gnats or flies can as easily dance among one another in a moving cabin as in a fixed chamber. As for those scripture expressions which seem to contradict the earth's motion, this general answer may be made to them all, viz. It is plain from many instances, that the Scriptures were never intended to instruct us in philosophy or astronomy; and therefore on those subjects expressions are not always to be taken in the literal sense, but for the most part as accommodated to the common apprehensions of mankind. Men of sense in all ages, when not treating of the sciences purposely, have followed this method; and it would be in vain to follow any other in addressing ourselves to the vulgar, or bulk of any community.

The following experiment will give a plain idea of the diurnal and annual motions of the earth, together with the different lengths of days and nights, and all the beautiful variety of seasons, depending on those motions.

Take about seven feet of strong wire, and bend it into a circular form, as a b c d, which being viewed obliquely, appears elliptical, as in the figure. Place a lighted candle on a table; and having fixed one end of a silk thread K, to the north pole of a small terrestrial globe H, about three inches diameter, cause another person to hold the wire circle, so that it may be parallel to the table, and as high as the flame of the candle I, which should be in or near the centre. Then having twisted the thread as towards the left hand, that by untwisting it may turn the globe round eastward, or contrary to the way that the hands of a watch move, hang the globe by the thread within this circle, almost contiguous to it; and as the thread untwists, the globe (which is enlightened half round by the candle as the earth is by the sun) will turn round its axis, and the different places upon it will be carried through the light and dark hemispheres, and have the appearance of a regular succession of days and nights, as our earth has in reality by such a motion. As the globe turns, move your hand slowly, so as to carry the globe round the candle according to the order of the letters a b c d, keeping its centre even with the wire circle; and you will perceive, that the candle, being still perpendicular to the equator, will enlighten the globe from pole to pole in its whole motion round the circle; and that every place on the globe goes equally through the light and the dark, as it turns round by the untwisting of the thread, and therefore has a perpetual equinox. The globe thus turning round represents the earth turning round its axis; and the motion of the globe round the candle represents the earth's annual motion round the sun; and shows, that if the earth's orbit had no inclination to its axis, all the days and nights of the year would be equally long, and there would be no different seasons. Hence also it appears why the planets Mars and Jupiter have a perpetual equinox, namely, because their axis is perpendicular to the plane of their orbit, as the thread round which the

globe turns in this experiment is perpendicular to the plane of the area inclosed by the wire.—But now define the person who holds the wire to hold it obliquely in the position A B C D, raising the side E F just as much as he depresses the side G H, that the flame may be still in the plane of the circle; and twisting the thread as before, that the globe may turn round its axis the same way as you carry it round the candle; that is, from west to east; let the globe down into the lowermost part of the wire circle at G H; and if the circle be properly inclined, the candle will shine perpendicularly on the tropic of Cancer; and the frigid zone, lying within the arctic or north polar circle, will be all in the light, as in the figure; and will keep in the light let the globe turn round its axis ever so often. From the equator to the north polar circle, all the places have longer days and shorter nights; but from the equator to the south polar circle, just the reverse. The sun does not set to any part of the north frigid zone, as shown by the candle's shining on it, so that the motion of the globe can carry no place of that zone into the dark; and at the same time the south frigid zone is involved in darkness, and the turning of the globe brings none of its places into the light. If the earth were to continue in the like part of its orbit, the sun would never set to the inhabitants of the north frigid zone, nor rise to those of the south. At the equator, it would be always equal day and night; and as places are gradually more and more distant from the equator towards the arctic circle, they would have longer days and shorter nights; whilst those on the south side of the equator would have their nights longer than their days. In this case, there would be continual summer on the north side of the equator, and continual winter on the south side of it.

But as the globe turns round its axis, move your hand slowly forward, so as to carry the globe from H towards E, and the boundary of light and darkness will approach towards the north pole, and recede towards the south pole; the northern places will go through less and less of the light, and the southern places through more and more of it; showing how the northern days decrease in length, and the southern days increase, whilst the globe proceeds from H to E. When the globe is at E, it is at a mean state between the lowest and highest parts of its orbit; the candle is directly over the equator, the boundary of light and darkness just reaches to both the poles, and all places on the globe go equally through the light and dark hemispheres, showing that the days and nights are then equal at all places of the earth, the poles only excepted; for the sun is then setting to the north pole, and rising to the south pole.

Continue moving the globe forward, and as it goes through the quarter A, the north pole recedes still farther into the dark hemisphere, and the south pole advances more into the light, as the globe comes nearer to E F; and when it comes there at F, the candle is directly over the tropic of Capricorn; the days are at the shortest, and nights at the longest; in the northern hemisphere, all the way from the equator to the arctic circle; and the reverse in the southern hemisphere from the equator to the antarctic circle; within which circles it is dark to the north frigid zone, and light to the south. Continue both motions; and as the globe moves through the quarter B, the north pole advances towards the light, and the south pole recedes towards the dark; the days lengthen in the northern hemisphere, and shorten in the southern; and when the globe comes to G, the candle will be again over the equator (as when the globe was at E), and the days and nights will again be equal as formerly; and the north pole will be just coming into the light, the south pole going out of it.

Thus we see the reason why the days lengthen and shorten from the equator to the polar circles every year; why there is sometimes no day or night for many turnings of the earth, within the polar circles; why there is but one day and one night in the whole year at the poles; and why the days and nights are equally long all the year round at the equator, which is always equally cut by the circle bounding light and darkness.

The inclination of an axis or orbit is merely relative, because we compare it with some other axis or orbit which we consider as not inclined at all. Thus, our horizon being level to us, whatever place of the earth we are upon, we consider it as having no inclination; and yet, if we travel 90 degrees from that place, we shall then have an horizon perpendicular to the former; but it will still be level to us.

Let us now take a view of the earth in its annual course round the sun, considering its orbit as having no inclination; and its axis as inclining 23°4 minutes from a line perpendicular to the plane of its orbit, and keeping the same oblique direction in all parts of its annual course; or, as commonly termed, keeping always parallel to itself.

Plate XLV. Let \(a, b, c, d, e, f, g, h\) be the earth in eight different parts of its orbit, equidistant from one another; \(N\) its axis, \(N'\) its north pole, \(S\) its south pole, and \(S'\) the sun nearly in the centre of the earth's orbit. As the earth goes round the sun according to the order of the letters above, &c., its axis \(N'\) keeps the same obliquity, and is still parallel to the line \(MN\). When the earth is at \(a\), its north pole inclines towards the sun \(S\), and brings all the northern places more into the light than at any other time of the year. But when the earth is at \(e\) in the opposite time of the year, the north pole declines from the sun, which occasions the northern places to be more in the dark than in the light, and the reverse at the southern places; as is evident by the figure, which is taken from Dr Long's astronomy. When the earth is either at \(c\) or \(g\), its axis inclines not either to or from the sun, but lies sidewise to him, and then the poles are in the boundary of light and darkness; and the sun, being directly over the equator, makes equal day and night at all places. When the earth is at \(b\), it is half-way between the summer solstice and harvest equinox; when it is at \(d\), it is half-way from the harvest equinox to the winter solstice; at \(f\), half-way from the winter solstice to the spring equinox; and at \(h\), half-way from the spring equinox to the summer solstice.

From this oblique view of the earth's orbit, let us suppose ourselves to be raised far above it, and placed just over its centre \(S\), looking down upon it from its north pole; and as the earth's orbit differs but very little from a circle, we shall have its figure in such a view represented by the circle ABCDEFGH. Let us suppose this circle to be divided into 12 equal parts, fig. 1, called signs, having their names affixed to them; and each sign into 30 equal parts, called degrees, numbered 1, 2, 3, ..., 30, as in the outermost circle of the figure, which represents the great ecliptic in the heavens. The earth is shown in eight different positions in this circle; and in each position \(AE\) is the equator, \(T\) the tropic of Cancer, the dotted circle the parallel of London, \(U\) the arctic or north polar circle, and \(P\) the north pole, where all the meridians or hour-circles meet. As the earth goes round the sun, the north pole keeps constantly towards one part of the heavens, as it keeps in the figure towards the right-hand side of the plate.

When the earth is at the beginning of Libra, namely on the 20th of March, in this figure the sun \(S\) as seen from the earth appears at the beginning of Aries in the opposite part of the heavens, the north pole is just coming into the light, and the sun is vertical to the equator; which together with the tropic of Cancer, parallel of London, and arctic circle, are all equally cut by the circle bounding light and darkness, coinciding with the six-o'clock hour-circle, and therefore the days and nights are equally long at all places: for every part of the meridian \(AE\) comes into the light at six in the morning, and, revolving with the earth according to the order of the hour-letters, goes into the dark at six in the evening. There are 24 meridians or hour-circles drawn on the earth in this figure, to shew the time of sun-rising and setting at different seasons of the year.

As the earth moves in the ecliptic according to the order of the letters ABCD, &c. through the signs Libra, Scorpio, and Sagittarius, the north pole \(P\) comes more and more into the light; the days increase as the nights decrease in length, at all places north of the equator \(AE\); which is plain by viewing the earth at \(b\) on the 5th of May, when it is in the 15th degree of Scorpio, and the sun as seen from the earth appears in the 15th degree of Taurus. For then the tropic of Cancer \(T\) is in the light from a little after five in the morning till almost seven in the evening; the parallel of London, from half an hour past four till half an hour past seven; the polar circle \(U\), from three till nine; and a large track round the north pole \(P\) has day all the 24 hours, for many rotations of the earth on its axis.

When the earth comes to \(c\) (Pl. XLV. fig. 4.) at the beginning of Capricorn, and the sun as seen from the earth appears at the beginning of Cancer, on the 21st of June, as in this figure, it is in the position C in fig. 1.; Pl. XLVI., and its north pole inclines towards the sun, so as to bring all the north frigid zone into the light, and the northern parallels of latitude more into the light than the dark from the equator to the polar circle; and the more so as they are farther from the equator. The tropic of Cancer is in the light from five in the morning till seven at night, the parallel of London from a quarter before four till a quarter after eight; and the polar circle just touches the dark, so that the sun has only the lower half of his disk hid from the inhabitants on that circle for a few minutes about midnight, supposing no inequalities in the horizon, and no refractions.

A bare view of the figure is enough to shew, that as the earth advances from Capricorn towards Aries, and the sun appears to move from Cancer towards Libra; the north pole recedes from the light, which causes the days to decrease, and the nights to increase in length, till the earth comes to the beginning of Aries, and then they are equal as before; for the boundary of light and darkness cuts the equator and all its parallels equally, or in halves. The north pole then goes into the dark, and continues therein until the earth goes halfway round its orbit; or, from the 23rd of September till the 20th of March. In the middle between these times, viz. on the 22nd of December, the north pole is as far as it can be in the dark, which is 23° degrees, equal to the inclination of the earth's axis from a perpendicular to its orbit; and then the northern parallels are as much in the dark as they were in the light on the 21st of June; the winter nights being as long as the summer days, and the winter days as short as the summer nights. Here it must be noted, that of all that has been said of the northern hemisphere, the contrary must be understood of the southern; for on different sides of the equator the seasons are contrary, because, when the northern hemisphere inclines towards the sun, the southern declines from him.

The earth's orbit being elliptical, and the sun constantly keeping in its lower focus, which is 1,617,944 miles from the middle point of the longer axis, the earth comes twice so much, or 3,235,882 miles, nearer the sun at one time of the year than at another; for the sun appearing under a larger angle in our winter than summer, proves that the earth is nearer the sun in winter (1). But here this natural question will arise, Why have we not the hottest weather when the earth is nearest the sun? In answer it must be observed, that the eccentricity of the earth's orbit, or 1,617,944 miles, bears no greater proportion to the earth's mean distance from the sun than 17 does to 1000; and therefore, this small difference of distance cannot occasion any great difference of heat or cold. But the principal cause of this difference is, that in winter the sun's rays fall obliquely upon us, that any given number of them is spread over a much greater portion of the earth's surface where we live; and therefore each point must then have fewer rays than in summer. Moreover, there comes a greater degree of cold in the long winter nights, than there can return of heat in so short days; and on both these accounts the cold must increase. But in summer the sun's rays fall more perpendicularly upon us; and therefore come with greater force, and in greater numbers, on the same place; and by their long continuance, a much greater degree of heat is imparted by day than can fly off by night. Besides, those parts which are once heated, retain the heat for some time; which, with the additional heat daily imparted, makes it continue to increase, though the sun declines towards the south; and this is the reason why July is hotter than June, although the sun has withdrawn from the summer tropic; as we find it is generally hotter at three in the afternoon, when the sun has gone towards the west, than at noon when he is on the meridian. Likewise those places which are well cooled require time to be heated again; for the sun's rays do not heat even the surface of any body till they have been some time upon it. And therefore we find January for the most part colder than December, although the sun has withdrawn from the winter tropic, and begins to dart his beams more perpendicularly upon us. An iron bar is not heated immediately upon being in the fire, nor grows cold till some time after it has been taken out.

It has been already observed, that by the earth's motion on its axis, there is more matter accumulated all around the equatorial parts than anywhere else on the earth.

The sun and moon, by attracting this redundancy of matter, bring the equator sooner under them in every return towards it, than if there was no such accumulation. Therefore, if the sun sets out, as from any star, or other fixed point in the heavens, the moment when he is departing from the equinoctial or from either tropic, he will come to the same equinox or tropic again 20 min. 17½ sec. of time, or 50 seconds of a degree, before he completes his course, so as to arrive at the same fixed star or point from whence he set out. For the equinoctial points recede 50 seconds of a degree westward every year, contrary to the sun's annual progressive motion.

When the sun arrives at the same equinoctial or solstitial point, he finishes what we call the Tropical Year; which, by observation, is found to contain 365 days 5 hours 48 minutes 57 seconds; and when he arrives at the same fixed star again, as seen from the earth, he completes the sidereal year, which contains 365 days 6 hours 9 minutes 14½ seconds. The sidereal year is therefore 20 minutes 17½ seconds longer than the solar or tropical year, and 9 minutes 14½ seconds longer than the Julian or the civil year, which we state at 365 days 6 hours, so that the civil year is almost a mean between the sidereal and tropical.

As the sun describes the whole ecliptic, or 360 degrees, in a tropical year, he moves 59' 8" of a degree every day at a mean rate; and consequently 50' of a degree in 20 minutes 17½ seconds of time; therefore he will arrive at the same equinox or solstice when he is 50' of a degree short of the same star or fixed point in the heavens from which he set out the year before. So that, with respect to the fixed stars, the sun and equinoctial points fall back (as it were) 30 degrees in 2160 years, which will make the stars appear to have gone 30 deg. forward with respect to the signs of the ecliptic in that time: for the same signs always keep in the same points of the ecliptic, without regard to the constellations.

To explain this by a figure, let the sun be in conjunction with a fixed star at S, suppose in the 30th degree of Y, at any given time. Then, making 2160 revolutions through the ecliptic VWX, at the Pl. XLVII end of so many sidereal years, he will be found again at S; but at the end of so many Julian years, he will be found at M, short of S; and at the end of so many tropical years he will be found short of M, in the 30th deg. of Taurus at T, which has receded back from S to T in that time, by the precession of the equinoctial points Y Aries and Libra. The arc ST will be equal to the amount of the precession of the

(1) It is denied by some that the sun appears bigger in winter on account of his greater proximity to the earth than in summer; the reason they give is the increase of refraction by reason of the more oblique position of the sun, and greater quantity of vapours in the air, at that time. the equinox in 2160 years, at the rate of 50" of a degree, or 20 minutes 17½ seconds of time annually; this, in so many years, makes 30 days 10½ hours, which is the difference between 2160 sidereal and tropical years; and the arc MT will be equal to the space moved through by the sun in 2160 times 11 min. 8 sec. or 16 days 13 hours 48 minutes, which is the difference between 2160 Julian and tropical years.

The anticipation of the equinoxes, and consequently of the seasons, is by no means owing to the precession of the equinoctial and solstitial points in the heavens (which can only affect the apparent motions, places, and declinations, of the fixed stars), but to the difference between the civil and solar year, which is 11 minutes 3 seconds; the civil year containing 365 days 6 hours, and the solar year 365 days 5 hours 48 minutes 57 seconds.

The above 11 minutes 3 seconds, by which the civil or Julian year exceeds the solar, amounts to 11 days in 1433 years; and so much our seasons have fallen back with respect to the days of the months, since the time of the Nicene council in A.D. 325; and therefore, in order to bring back all the facts and festivals to the days then settled, it was requisite to suppress 11 nominal days: and, that the same seasons might be kept to the same times of the year for the future, to leave out the bissextile-day in February at the end of every century of years not divisible by 4; reckoning them only common years, as the 17th, 18th, and 19th centuries, viz. the years 1700, 1800, 1900, &c., because a day intercalated every fourth year was too much; and retaining the bissextile-day at the end of those centuries of years which are divisible by 4, as the 16th, 20th, and 24th centuries, viz. the years 1600, 2000, 2400, &c. Otherwise, in length of time, the seasons would be quite reversed with regard to the months of the year; though it would have required near 23,783 years to have brought about such a total change. If the earth had made exactly 365¼ diurnal rotations on its axis, whilst it revolved from any equinoctial or solstitial point to the same again, the civil and solar years would always have kept pace together, and the style would never have needed any alteration.

Having thus mentioned the cause of the precession of the equinoctial points in the heavens, which occasions a slow deviation of the earth's axis from its parallelism, and thereby a change of the declination of the stars from the equator, together with a slow apparent motion of the stars forward with respect to the signs of the ecliptic, we shall now explain the phenomena by a diagram.

Let NZSVL be the earth, SONA its axis produced to the starry heavens, and terminating in A, the present north pole of the heavens, which is vertical to N the north pole of the earth. Let EOQ be the equator, TZ the tropic of Cancer, and VT the tropic of Capricorn; VOZ the ecliptic, and BO its axis, both which are immovable among the stars. But as the equinoctial points recede in the ecliptic, the earth's axis SON is in motion upon the earth's centre O, in such a manner, as to describe the double cone NO and SO, round the axis of the ecliptic BO, in the time that the equinoctial points move quite round the ecliptic, which is 25,920 years; and in that length of time, the north pole of the earth's axis produced, describes the circle ABCDA in the starry heavens, round the pole of the ecliptic, which keeps immovable in the centre of that circle. The earth's axis being 23½ degrees inclined to the axis of the ecliptic, the circle ABCDA described by the north pole of the earth's axis produced to A, is 47 degrees in diameter, or double the inclination of the earth's axis. In consequence of this, the point A, which at present is the north pole of the heavens, and near to a star of the second magnitude in the tail of the constellation called the little bear, must be deflected by the earth's axis; which moving backwards a degree every 72 years, will be directed towards the star or point B in 6480 years hence; and in double of that time, or in 12,960 years, it will be directed towards the star or point C, which will then be the north pole of the heavens, although it is at present 8½ degrees south of the zenith of London L. The present position of the equator EOQ, will then be changed into eOg, the tropic of Cancer T Z into V Z, and the tropic of Capricorn VT into VTZ; as is evident by the figure. And the sun, in the same part of the heavens where he is now over the earthly tropic of Capricorn, and makes the shortest days and longest nights in the northern hemisphere, will then be over the earthly tropic of Cancer, and make the days longest and nights shortest. So that it will require 12,960 years yet more, or 25,920 from the then present time, to bring the north pole N quite round, so as to be directed toward that point of the heavens which is vertical to it at present. And then, and not till then, the same stars which at present describe the equator, tropics, and polar circles, &c. by the earth's diurnal motion, will describe them over again.

From the shifting of the equinoctial points, and with them all the signs of the ecliptic, it follows that those stars which in the infancy of astronomy were in Aries are now got into Taurus; those of Taurus into Gemini, &c. Hence likewise it is that the stars which rose or set at any particular season of the year, in the times of Hesiod, Eudoxus, Virgil, Pliny, &c. by no means answer at this time to their descriptions.

The moon is not a planet, but only a satellite or attendant of the earth, going round the earth from moon to moon, changing in 29 days 12 hours and 44 minutes, and round the sun with it every year. The moon's diameter is 2180 miles; and her distance from the earth's centre is 240,000. She goes round her orbit in 27 days 7 hours 43 minutes, moving about 2290 miles every hour; and turns round her axis exactly in the time that she goes round the earth, which is the reason of her keeping always the same side towards us, and that her day and night taken together is as long as our lunar month.

The moon is an opaque globe like the earth, and reflects the light of the sun; therefore, whilst that half of her which is towards the sun is enlightened, the other half must be dark and invisible. Hence she disappears when she comes between us and the sun; because her dark side is then towards us. When she is gone a little way forward, we see a little of her enlightened side; which still increases to our view, as she advances forward, until she comes to be opposite to the sun; and then her whole enlightened side is towards the earth, and she appears with a round illumined orb, which we call the full moon; her dark side being then turned... turned away from the earth. From the full she seems to decrease gradually as she goes through the other half of her course; shewing us less and less of her enlightened side every day, till her next change or conjunction with the sun, and then she disappears as before.

Her orbit is represented in the scheme by the little circle m, upon the earth's orbit O: but it is drawn more than fifty times too large in proportion to the earth's; and yet is almost too small to be seen in the diagram.

The moon has scarce any difference of seasons; her axis being almost perpendicular to the ecliptic. What is very singular, one half of her has no darkness at all; the earth constantly affording it a strong light in the sun's absence; while the other half has a fortnight's darkness and a fortnight's light by turns.

Our earth is thought to be a moon to the moon; waxing and waning regularly, but appearing 13 times as big, and affording her 13 times as much light as she does us. When she changes to us, the earth appears full to her; and when she is in her first quarter to us, the earth is in its third quarter to her; and vice versa.

But from one half of the moon, the earth is never seen at all: from the middle of the other half, it is always seen over head; turning round almost 30 times as quick as the moon does. From the circle which limits our view of the moon, only one half of the earth's side next her is seen; the other half being hid below the horizon of all places on that circle. To her the earth seems to be the biggest body in the universe; for it appears 13 times as big as she does to us.

As the earth turns round its axis, the several continents, seas, and islands, appear to the moon's inhabitants like so many spots of different forms and brightness, moving over its surface; but much fainter at some times than others, as our clouds cover them or leave them. By these spots the Lunarians can determine the time of the earth's diurnal motion, just as we do the motion of the sun; and perhaps they measure their time by the motion of the earth's spots; for they cannot have a truer dial.

The moon's axis is so nearly perpendicular to the ecliptic, that the sun never removes sensibly from her equator; and the obliquity of her orbit, which is next to nothing as seen from the sun, cannot cause the sun to decline sensibly from her equator. Yet her inhabitants are not destitute of means for ascertaining the length of their year, though their method and ours must differ. For we can know the length of our year by the return of our equinoxes; but the Lunarians, having always equal day and night, must have recourse to another method: and we may suppose, they measure their year by observing when either of the poles of our earth begins to be enlightened, and the other to disappear, which is always at our equinoxes; they being conveniently situated for observing great tracks of land about our earth's poles, which are entirely unknown to us. Hence we may conclude, that the year is of the same absolute length both to the earth and moon, though very different as to the number of days: we having 365 natural days, and the Lunarians only 12½, every day and night in the moon being as long as 29½ on the earth.

The moon's inhabitants on the side next the earth can find the latitude of ours. For the earth keeping constantly, or very nearly so, over one meridian of the moon, the east or west distances of places from that meridian are as easily found, as we can find our distance from the equator by the altitude of our celestial poles.

As the sun can only enlighten that half of the earth which is at any moment turned towards him, and, being withdrawn from the opposite half, leaves it in darkness; so he likewise doth to the moon: only with this difference, that as the earth is surrounded by an atmosphere, we have twilight after the sun sets; but if the moon has none of her own, nor is included in that of the earth, the lunar inhabitants have an immediate transition from the brightest sun-shine to the blackest darkness. For, let r k s w be the earth, and A, B, C, D, E, F, G, H, the moon in eight different parts of fig. 3. her orbit. As the earth turns round its axis from west to east, when any place comes to t the twilight begins there, and when it revolves from thence to r the sun rises; when the place comes to s the sun sets, and when it comes to w the twilight ends. But as the moon turns round her axis, which is only once a month, the moment that any point of her surface comes to r (see the moon at G), the sun rises there without any previous warning by twilight; and when the same point comes to s the sun sets, and that point goes into darkness as black as at midnight.

The moon being an opaque spherical body (for her hills take off no more from her roundness than the inequalities on the surface of an orange takes off from its roundness), we can only see that part of the enlightened half of her which is towards the earth. And therefore, when the moon is at A, in conjunction with the sun S, her dark half is towards the earth, and she disappears, as at a, there being no light on that half to render it visible. When she comes to her first octant at B, or has gone an eighth part of her orbit from her conjunction, a quarter of her enlightened side is towards the earth, and she appears horned, as at b. When she has gone a quarter of her orbit from between the earth and sun to C, she shews us one half of her enlightened side, as at c, and we say she is a quarter old. At D, she is in her second octant; and by shewing us more of her enlightened side she appears gibbous, as at d. At E, her whole enlightened side is towards the earth; and therefore she appears round, as at e; when we say it is full moon. In her third octant at F, part of her dark side being towards the earth, she again appears gibbous, and is on the decrease, as at f. At G, we see just one half of her enlightened side; and she appears half decreased, or in her third quarter, as at g. At H, we only see a quarter of her enlightened side, being in her fourth octant; where she appears horned, as at h. And at A, having completed her course from the sun to the sun again, she disappears; and we say it is new moon. Thus, in going from A to E, the moon seems continually to increase; and in going from E to A, to decrease in the same proportion; having like phases at equal distances from A to E, but as seen from the sun S she is always full.

The moon appears not perfectly round when she is full in the highest or lowest part of her orbit, because we have not a full view of her enlightened side at that time. When full in the highest part of her orbit, a small... small deficiency appears on her lower edge; and the contrary when full in the lowest part of her orbit.

It is plain by the figure, that when the moon changes to the earth, the earth appears full to the moon; and vice versa. For when the moon is at A, new to the earth, the whole enlightened side of the earth is towards the moon; and when the moon is at E, full to the earth, its dark side is towards her. Hence a new moon answers to a full earth, and a full moon to a new earth. The quarters are also reversed to each other.

Between the third quarter and change, the moon is frequently visible in the forenoon, even when the sun shines; and then she affords us an opportunity of seeing a very agreeable appearance, where-ever we find a globular stone above the level of the eye, as suppose on the top of a gate. For, if the sun shines on the stone, and we place ourselves so as the upper part of the stone may just seem to touch the point of the moon's lowermost horn, we shall then see the enlightened part of the stone exactly of the same shape with the moon; horned as she is, and inclined the same way to the horizon. The reason is plain; for the sun enlightens the stone the same way as he does the moon: and both being globes, when we put ourselves into the above situation, the moon and stone have the same position to our eyes; and therefore we must see as much of the illuminated part of the one as of the other.

The position of the moon's cusps, or a right line touching the points of her horns, is very differently inclined to the horizon at different hours of the same days of her age. Sometimes she stands, as it were, upright on her lower horn, and then such a line is perpendicular to the horizon: when this happens, she is in what the astronomers call the meridian degree; which is the highest point of the ecliptic above the horizon at that time, and is 90 degrees from both sides of the horizon where it is then cut by the ecliptic. But this never happens when the moon is on the meridian, except when she is at the very beginning of Cancer or Capricorn.

That the moon turns round her axis in the time that she goes round her orbit, is quite demonstrable; for, a spectator at rest, without the periphery of the moon's orbit, would see all her sides turned regularly towards him in that time. She turns round her axis from any star to the same star again in 27 days 8 hours; from the sun to the sun again in 29½ days: the former is the length of her sidereal day, and the latter the length of her solar day. A body moving round the sun would have a solar day in every revolution, without turning on its axis; the same as if it had kept all the while at rest; and the sun moved round it: but without turning round its axis it could never have one sidereal day, because it would always keep the same side towards any given star.

If the earth had no annual motion, the moon would go round it so as to complete a lunation, a sidereal, and a solar day, all in the same time. But, because the earth goes forward in its orbit while the moon goes round the earth in her orbit, the moon must go as much more than round her orbit from change to change in completing a solar day, as the earth has gone forward in its orbit during that time, i.e. almost a twelfth part of a circle.

If the earth had no annual motion, the moon's motion round the earth, and her track in open space, would be always the same (k). But as the earth and moon move round the sun, the moon's real path in the heavens is very different from her visible path round the earth; the latter being in a progressive circle, and the former in a curve of different degrees of concavity, which would always be the same in the same parts of the heavens, if the moon performed a complete number of lunations in a year without any fraction.

Let a nail in the end of the axle of a chariot-wheel represent the earth, and a pin in the nave the moon; if the body of the chariot be propped up so as to keep that wheel from touching the ground, and the wheel be then turned round by hand, the pin will describe a circle both round the nail and in the space it moves through. But if the props be taken away, the horses put to, and the chariot driven over a piece of ground which is circularly convex; the nail in the axle will describe a circular curve, and the pin in the nave will still describe a circle round the progressive nail in the axle, but not in the space through which it moves. In this case, the curve described by the nail will resemble in miniature as much of the earth's annual path round the sun, as it describes whilst the moon goes as often round the earth as the pin does round the nail: and the curve described by the pin will have some resemblance of the moon's path during so many lunations.

Let us now suppose that the radius of the circular curve described by the nail in the axle is to the radius fig. 4, of the circle which the pin in the nave describes round the axle, as 337:101; (l) which is the proportion of the radius or semidiameter of the earth's orbit to that of the moon's, or of the circular curve A 1 2 3 4 5 6 7 B, &c., to the little circle a; and then, whilst the progressive nail describes the said curve from A to E, the pin will go once round the nail with regard to the centre of its path, and in so doing will describe the curve a b c d e. The former will be a true representation of the earth's path for one lunation, and the latter of the moon's for that time. Here we may set aside the inequalities of the moon's motion, and also the earth's moving round its common centre of gravity and the moon's: all which, if they were truly copied in this experiment, would not sensibly alter the figure of the paths described by the nail and pin, even though they should rub against a plain upright surface all the way, and leave their traces visible upon it. And if the chariot was driven forward on such a convex piece of ground, so as to turn the wheel several times round, the track of the pin in the nave would still be concave toward the centre of the circular curve described by the nail in the axle; as the moon's path is always concave to the sun in the centre of the earth's annual orbit.

In this diagram, the thickest curve line ABCDE, with

(k) In this place, we may consider the orbits of all the satellites as circular, with respect to their primary planets; because the eccentricities of their orbits are too small to affect the phenomena here described.

(l) The figure by which this is illustrated is borrowed from Mr Ferguson, whom we principally follow in our explanations of the phenomena. Later observations have determined the proportions to be different; but we cannot find that any delineation of this kind hath been given by astronomers, according to the new proportions. with the numeral figures set to it, represents as much of the earth's annual orbit as it describes in 32 days from west to east; the little circles at A, B, C, D, E, show the moon's orbit in due proportion to the earth's; and the smallest curve A C f represents the line of the moon's path in the heavens for 32 days, accounted from any particular new moon at a. The sun is supposed to be in the centre of the curve A 1 2 3 4 5 6 7 B, &c., and the small dotted circles upon it represent the moon's orbit, of which the radius is in the same proportion to the earth's path in this scheme, that the radius of the moon's orbit in the heavens was supposed to bear to the radius of the earth's annual path round the sun; that is, as 240,000, to 81,000,000, or as 1 to 337½.

When the earth is at A, the new moon is at a; and in the seven days that the earth describes the curve 1 2 3 4 5 6 7, the moon in accompanying the earth describes the curve a b; and is in her first quarter at b when the earth is at B. As the earth describes the curve B 8 9 10 11 12 13 14, the moon describes the curve b c; and is at c, opposite to the sun, when the earth is at C. Whilst the earth describes the curve C 15 16 17 18 19 20 21 22, the moon describes the curve c d; and is in her third quarter at d when the earth is at D. And lastly, whilst the earth describes the curve D 23 24 25 26 27 28 29, the moon describes the curve d e; and is again in conjunction at e with the sun when the earth is at E, between the 29th and 30th day of the moon's age, accounted by the numeral figures from the new moon at A. In describing the curve a C e, the moon goes round the progressive earth as really as if she had kept in the dotted circle A, and the earth continued immovable in the centre of that circle.

And thus we see, that although the moon goes round the earth in a circle, with respect to the earth's centre, her real path in the heavens is not very different in appearance from the earth's path. To show that the moon's path is concave to the sun, even at the time of change, it is carried on a little farther into a second lunation, as to f.

The moon's absolute motion from her change to her first quarter, or from a to b, is so much slower than the earth's, that she falls 240,000 miles (equal to the semidiameter of her orbit) behind the earth at her first quarter in b, when the earth is in B; that is, she falls back a space equal to her distance from the earth. From that time her motion is gradually accelerated to her opposition or full at c; and then she is come up as far as the earth, having regained what she lost in her first quarter from a to b. From the full to the last quarter at d, her motion continues accelerated so as to be just as far before the earth at d, as she was behind it at her first quarter in b. But from d to e her motion is retarded so, that she loses as much with respect to the earth as is equal to her distance from it, or to the semidiameter of her orbit; and by that means she comes to e, and is then in conjunction with the sun as seen from the earth at E. Hence we find, that the moon's absolute motion is slower than the earth's from her third quarter to her first, and swifter than the earth's from her first quarter to her third; her path being less curved than the earth's in the former case, and more in the latter. Yet it is still bent the same way towards the sun; for if we imagine the concavity of the earth's orbit to be measured by the length of a perpendicular line C g, let down from the earth's place upon the straight line b g d at the full of the moon, and connecting the places of the earth at the end of the moon's first and third quarters, that length will be about 640,000 miles; and the moon when now only approaching nearer to the sun by 240,000 miles than the earth is, the length of the perpendicular let down from her place at that time upon the same straight line, and which shows the concavity of that part of her path, will be about 400,000 miles.

The moon's path being concave to the sun throughout, demonstrates that her gravity towards the sun, at her conjunction, exceeds her gravity towards the earth; and if we consider that the quantity of matter in the sun is vastly greater than the quantity of matter in the earth, and that the attraction of each body diminishes as the square of the distance from it increases, we shall soon find, that the point of equal attraction between the earth and the sun, is much nearer the earth than the moon is at her change. It may then appear surprising that the moon does not abandon the earth when she is between it and the sun, because she is considerably more attracted by the sun than by the earth at that time. But this difficulty vanishes when we consider, that a common impulse on any system of bodies affects not their relative motions; but that they will continue to attract, impel, or circulate round one another, in the same manner as if there was no such impulse. The moon is so near the earth, and both of them so far from the sun, that the attractive power of the sun may be considered as equal on both; and therefore the moon will continue to circulate round the earth in the same manner as if the sun did not attract them at all: like bodies in the cabin of a ship, which may move round or impel one another in the same manner when the ship is under sail, as when it is at rest; because they are all equally affected by the common motion of the ship. If by any other cause, such as the near approach of a comet, the moon's distance from the earth should happen to be so much increased, that the difference of their gravitating forces towards the sun should exceed that of the moon towards the earth; in that case, the moon, when in conjunction, would abandon the earth, and be either drawn into the sun, or comet, or circulate round about it.

The ruggedness of the moon's surface mentioned n°40, 41, is of great use to us, by reflecting the sun's light to all sides: for if the moon were smooth and polished like a looking-glass, or covered with water, she could never distribute the sun's light all round; only in some positions she would shew us his image, no bigger than a point, but with such a lustre as would be hurtful to our eyes.

The moon's surface being so uneven, many have wondered why her edge appears not jagged, as well as the edge appears curve bounding the light and dark places. But if we consider, that what we call the edge of the moon's disk is not a single line set round with mountains, in which case it would appear irregularly indented, but a large zone having many mountains lying behind one another from the observer's eye, we shall find that the mountains in some rows will be opposite to the vales in others; and so fill up the inequalities as to make her appear quite round: just as when one looks at an orange, although its roughness be very discernible on the The planet Mars comes next in order, being the first above the earth's orbit. His distance from the sun is computed to be 145,074,483 miles; and by travelling at the rate of 55,287 miles every hour, as in the circle of his equator, he goes round the sun in 686 of our days and 23 hours; which is the length of his year, and contains 667½ of his days; every day and night together being 40 minutes longer than with us. His diameter is 5400 miles, and by his diurnal rotation the inhabitants about his equator are carried 669 miles every hour. His quantity of light and heat is equal but to one half of ours; and the sun appears but half as big to him as to us.

This planet being but a fifth part so big as the earth, if any moon attends him, she must be very small, and has not yet been discovered by our best telescopes. To Mars, our earth and moon appear like two moons, a bigger and a less; changing places with one another, and appearing sometimes horned, sometimes half or three quarters illuminated, but never full; nor at most above one quarter of a degree from each other, although they are 240,000 miles asunder.

Our earth appears almost as big to Mars as Venus does to us, and at Mars it is never seen above 48 degrees from the sun; sometimes it appears to pass over the disk of the sun, and so do Mercury and Venus; but Mercury can never be seen from Mars by such eyes as ours, unassisted by proper instruments; and Venus will be as seldom seen as we see Mercury. Jupiter and Saturn will appear bigger to Mars than to us. His axis is perpendicular to the ecliptic, and his orbit is two degrees inclined to it.

Jupiter, the biggest of all the planets, is still higher in the system, being 494,990,976 miles from the sun; and going at the rate of 29,083 miles every hour in his orbit, as in the circle of his equator, finishes his annual period in eleven of our years 314 days and 12 hours. He is above 1300 times as big as the earth; for his diameter is 94,000 miles, which is more than eleven times the diameter of the earth.

Jupiter turns round his axis in 9 hours 56 minutes; so that his year contains 10,479 days; and the diurnal velocity of his equatorial parts is greater than the swiftness with which he moves in his annual orbit; a singular circumstance, as far as we know. By this prodigious quick rotation, his equatorial inhabitants are carried 29,542 miles every hour (which is 4000 miles in one hour more than an inhabitant of our earth's equator moves in 24 hours), besides the 29,083 above-mentioned, which is common to all parts of his surface, by his annual motion. By this prodigiously swift rotation on his axis the centrifugal force of his equatorial parts is able to shorten his polar diameter by ¼ of the whole, so that the difference is very perceptible through a good telescope.

The axis of Jupiter is so nearly perpendicular to his orbit, that he has no sensible change of seasons; which is a great advantage, and wisely ordered by the Author of Nature. For, if the axis of this planet were inclined any considerable number of degrees, just so many degrees round each pole would in their turn be almost six of our years together in darkness. And, as each degree of a great circle on Jupiter contains 820 of our miles at a mean rate, it is easy to judge what vast tracts of land would be rendered uninhabitable by any considerable inclination of his axis.

The sun appears but ¼ part so big to Jupiter as to us; and his light and heat are in the same small proportion, but compensated by the quick returns thereof, and by four moons (some bigger and some less than our earth) which revolve about him: so that there is scarce any part of this huge planet but what is during the whole night enlightened by one or more of these moons, except his poles, whence only the farthest moons can be seen, and where their light is not wanted, because the sun constantly circulates in or near the horizon, and is very probably kept in view of both poles by the refraction of Jupiter's atmosphere, which, if it be like ours, has certainly refractive power enough for that purpose.

The orbits of these moons are represented in the scheme of the solar system by four small circles marked 1, 2, 3, 4, on Jupiter's orbit; but they are drawn greatly too large in proportion to it. The first moon, or that nearest to Jupiter, goes round him in 1 day 18 hours and 36 minutes of our time; and is 266,332 miles distant from his centre: The second performs its revolution in 3 days 13 hours and 15 minutes, at 423,000 miles distance: The third in 7 days 3 hours and 59 minutes, at the distance of 676,078 miles: and the fourth, or outermost, in 16 days 18 hours and 30 minutes, at the distance of 1,189,148 miles from his centre.

The angles under which the orbits of Jupiter's moons are seen from the earth, at its mean distance from Jupiter, are as follow: The first, 3° 55′; the second, 6° 14′; the third, 9° 58′; and the fourth, 17° 30′. And their distances from Jupiter, measured by his semidiameters, are thus: The first 5½; the second, 9½; the third, 14½; and the fourth, 25½. This planet, seen from its nearest moon, appears 1000 times as large as our moon does to us; waxing and waning in all her monthly shapes, every 10 hours.

Jupiter's three nearest moons fall into his shadow, and Longitude are eclipsed in every revolution: but the orbit of the determined fourth moon is so much inclined, that it passeth by its opposition to Jupiter, without falling into his shadow, etc., two years in every five. By these eclipses, astronomers have not only discovered that the sun's light takes up eight minutes of time in coming to us; but they have also determined the longitudes of places on this earth with greater certainty and facility, than by any other method yet known.

Jupiter's orbit is 1 degree 20 minutes inclined to the ecliptic. His north node is in the 7th degree of Cancer, and his south node in the 7th degree of Capricorn.

The curves which Jupiter's satellites describe, are all of different sorts from the path described by our moon, although these satellites go round Jupiter, as the moon goes round the earth. Let ABCDE, &c. be as much of Jupiter's orbit as he describes in 18 days from A to T; and the curves α, β, γ, δ will be the paths of his four moons going round him in his progressive motion. Now let us suppose all these moons to set out from a conjunction with the sun, as seen from Jupiter at A; then, his first or nearest moon will be at α; his second at β; his third at γ; and his fourth at δ. At the end of of 24 terrestrial hours after this conjunction, Jupiter has moved to B, his first moon or satellite has described the curve a₁, his second the curve b₁, his third c₁, and his fourth d₁. The next day, when Jupiter is at C, his first satellite has described the curve a₂, from its conjunction, his second the curve b₂, his third the curve c₂, and his fourth the curve d₂, and so on. The numeral figures under the capital letters show Jupiter's place in his path every day for 18 days, accounted from A to T; and the like figures set to the paths of his satellites, show where they are at the like times. The first satellite, almost under C, is stationary at + as seen from the sun, and retrograde from + to 2; at 2 it appears stationary again, and thence it moves forward until it has past 3, and is twice stationary and once retrograde between 3 and 4. The path of this satellite intersects itself every 42½ hours, making such loops as in the diagram at 2, 3, 5, 7, 9, 10, 12, 14, 16, 18, a little after every conjunction. The second satellite b₁, moving slower, barely crosses its path every 3 days 13 hours; as at 4, 7, 11, 14, 18, making only five loops and as many conjunctions in the time that the first makes ten. The third satellite c₁ moving still slower, and having described the curve c₁, 2, 3, 4, 5, 6, 7, comes to an angle at 7 in conjunction with the sun at the end of 7 days four hours; and so goes on to describe such another curve 7, 8, 9, 10, 11, 12, 13, 14, and is at 14 in its next conjunction. The fourth satellite d₁ is always progressive, making neither loops nor angles in the heavens; but comes to its next conjunction at e between the numeral figures 16 and 17, or in 16 days 18 hours.

The method used by Mr Ferguson to delineate the paths of these satellites was the following. Having drawn their orbits on a card, in proportion to their relative distances from Jupiter, he measured the radius of the orbit of the fourth satellite, which was an inch and \( \frac{1}{12} \) parts of an inch; then multiplied this by 424 for the radius of Jupiter's orbit, because Jupiter is 424 times as far from the sun's centre as his fourth satellite is from his centre; and the product thence arising was 483\(\frac{1}{12}\) inches. Then taking a small cord of this length, and fixing one end of it to the floor of a long room by a nail, with a black-lead pencil at the other end he drew the curve ABCD &c. and set off a degree and half thereon, from A to T; because Jupiter moves only so much, whilst his outermost satellite goes once round him, and somewhat more; so that this small portion of so large a circle differs but very little from a straight line. This done, he divided the space AT into 18 equal parts, as AB, BC, &c. for the daily progress of Jupiter; and each part into 24 for his hourly progress. The orbit of each satellite was also divided into as many equal parts as the satellite is hours in finishing its synodical period round Jupiter. Then drawing a right line through the centre of the card, as a diameter to all the four orbits upon it, he put the card upon the line of Jupiter's motion, and transferred it to every horary division thereon, keeping always the said diameter-line on the line of Jupiter's path; and running a pin through each horary division in the orbit of each satellite as the card was gradually transferred along the line ABCD &c. of Jupiter's motion, he marked points for every hour through the card for the curves described by the satellites, as the primary planet in the centre of the card was carried forward on the line; and so finished the figure, by drawing the lines of each satellite's motion through those (almost innumerable) points: by which means, this is perhaps as true a figure of the paths of the satellites as can be desired. And in the same manner might those of Saturn's satellites be delineated.

It appears by the scheme, that the three first satellites come almost into the same line or position every seventh day; the first being only a little behind with the second, and the second behind with the third. But the period of the fourth satellite is so incommensurate to the periods of the other three, that it cannot be guessed at by the diagram when it would fall again into a line of conjunction with them, between Jupiter and the sun. And no wonder; for supposing them all to have been once in conjunction, it will require 3,087,043,493,260 years to bring them in conjunction again.

Saturn, the remotest of all the planets, is about Of Saturn 907,956,130 miles from the sun; and, travelling at PL XLIII. the rate of 22,101 miles every hour, as in the circle fig. marked B, performs its annual circuit in 29 years 167 days and 5 hours of our time; which makes only one year to that planet. Its diameter is 78,000 miles; and therefore it is near 1000 times as big as the earth.

To Saturn the sun appears only \( \frac{1}{70} \) part so big as to us; and the light and heat he receives from the sun are in the same proportion to ours. But, to compensate for the small quantity of sun-light, he has five moons, all going round him on the outside of his ring, and nearly in the same plane with it. The first or nearest moon to Saturn, goes round him in one day 21 hours 19 minutes; and is 140,000 miles from his centre: the second, in 2 days 17 hours 40 minutes, at the distance of 187,000 miles: the third, in 4 days 12 hours 25 minutes; at 263,000 miles distance: the fourth, in 15 days 22 hours 41 minutes; at the distance of 600,000 miles: and the fifth, or outermost, at 1,800,000 miles from Saturn's centre, goes round him in 79 days 7 hours 48 minutes. Their orbits in the scheme of the solar system are represented by the five small circles marked 1, 2, 3, 4, 5, on Saturn's orbit; but these, like the orbits of the other satellites, are drawn vastly too large in proportion to the orbits of their primary planets.

The sun shines almost 15 of our years together on one side of Saturn's ring without setting, and as long on the other in its turn. So that the ring is visible to the inhabitants of that planet for almost 15 of our years, and as long invisible, by turns, if its axis has no inclination to its ring; but if the axis of the planet be inclined to the ring, suppose about 30 degrees, the ring will appear and disappear once every natural day to all the inhabitants within 30 degrees of the equator, on both sides, frequently eclipsing the sun on a Saturnian day. Moreover, if Saturn's axis be so inclined to his ring, it is perpendicular to his orbit; and thereby the inconvenience of different seasons to that planet is avoided. For, considering the length of Saturn's year, which is almost equal to thirty of ours, what a dreadful condition must the inhabitants of his polar regions be in, if they be half that time deprived of the light and heat of the sun? which is not their case alone, if the axis of the planet be perpendicular to the ring; for then the ring... ring must hide the sun from vast tracks of land on each side of the equator for 13 or 14 of our years together, on the south side and north side by turns, as the axis inclines to or from the sun; the reverse of which inconvenience is another good presumptive proof of the inclination of Saturn's axis to his ring, and also of his axis being perpendicular to his orbit.

This ring, seen from Saturn, appears like a vast luminous arch in the heavens, as if it did not belong to the planet. When we see the ring most open, its shadow upon the planet is broadest; and from that time the shadow grows narrower, as the ring appears to do to us; until, by Saturn's annual motion, the sun comes to the plane of the ring, or even with its edge; which being then directed towards us, becomes invisible on account of its thinness; the ring disappears twice in every annual revolution of Saturn, namely, when he is in the 19th degree both of Pisces and of Virgo. And when Saturn is in the middle between these points, or in the 19th degree either of Gemini or of Sagittarius, his ring appears most open to us; and then its longest diameter is to its shortest, as 9 to 4.

As Saturn goes round the sun, his obliquely poised ring, like our earth's axis, keeps parallel to itself, and is therefore turned edgewise to the sun twice in a Saturnian year, which is almost as long as 30 of our years. But the ring, though considerably broad, is too thin to be seen by us when it is turned edgewise to the sun, at which time it is also edgewise to the earth; and therefore it disappears once in every 15 years to us. As the sun shines half a year together on the north pole of our earth, then disappears to it, and shines as long on the south pole; so, during one half of Saturn's year, the sun shines on the north side of his ring, then disappears to it, and shines as long on its south side. When the earth's axis inclines neither to nor from the sun, but sidewise to him, he instantly ceases to shine on one pole, and begins to enlighten the other; and when Saturn's ring inclines neither to nor from the sun, but sidewise to him, he ceases to shine on the one side of it, and begins to shine upon the other.

To such eyes as ours unassisted by instruments, Jupiter is the only planet that can be seen from Saturn, and Saturn the only planet that can be seen from Jupiter. So that the inhabitants of these two planets must either see much farther than we do, or have equally good instruments to carry their sight to remote objects, if they know that there is such a body as our earth in the universe: for the earth is no bigger seen from Jupiter, than his moons are seen from the earth; and if his large body had not first attracted our sight, and prompted our curiosity to view him with a telescope, we should never have known anything of his moons; unless by chance we had directed the telescope toward that small part of the heavens where they were at the time of observation. And the like is true of the moons of Saturn.

The orbit of Saturn is 2½ degrees inclined to the ecliptic, or orbit of our earth, and intersects it in the 21st degree of Cancer and of Capricorn; so that Saturn's nodes are only 14 degrees from Jupiter's.

The quantity of light afforded by the sun to Jupiter, being but ¼ part, and to Saturn only ⅛ part, of what we enjoy, may at first thought induce us to believe that these two planets are entirely unfit for rational beings to dwell upon. But that their light is not so weak as we imagine, is evident from their brightness in the night time; and also from this remarkable phenomenon, that when the sun is so much eclipsed to us, as to have only the 40th part of his disk left uncovered by the moon, the decrease of light is not very sensible; and just at the end of darkness in total eclipses when his western limb begins to be visible, and seems no bigger than a bit of fine silver wire, every one is surprized at the brightness therewith that small part of him shines. The moon, when full, affords travellers light enough to keep them from milking their way; and yet, according to Mr. Bower, the light of the sun is 300,000 times as strong as that of the moon. Consequently, the sun gives almost 12,360 times as much light to Saturn, as the full moon does to us; and above 30,000 times as much to Jupiter. So that these two planets, even without any moons, would be much more enlightened than we at first imagine; and, by having so many, they may be very comfortable places of residence. Their heat, so far as it depends on the force of the sun's rays, is certainly much less than ours; to which no doubt the bodies of their inhabitants are as well adapted as ours are to the feations we enjoy. And if we consider that Jupiter never has any winter, even at his poles, which probably is also the case with Saturn, the cold cannot be so intense on these two planets as is generally imagined. Besides, there may be something in the nature of their mould warmer than in that of our earth; and we find that all our heat depends not on the rays of the sun; for if it did, we should always have the same months equally hot or cold at their annual returns. But it is far otherwise, for February is sometimes warmer than May; which must be owing to vapours and exhalations from the earth.

In the Philosophical Transactions a method is given by Mr. Azot for knowing how much Jupiter or Saturn are illuminated experimentally. It is by admitting the sun's rays into a dark room through a convex lens; when being collected into a focus, they will afterwards diverge to any distance we please. This experiment, however, will be apt to raise doubts in the minds of those who try it, either with regard to the diminution of the sun's light at those distances, or with regard to the substances of the planets. For it is certain, that the brightness of the superior planets is far from being diminished in the proportion that it ought to be were all the wandering bodies in our system of a similar substance. Jupiter, for instance, as he hath not the 50th part of the light that Venus hath, seeing he is removed to such a distance as to appear as small as she does, ought to shine with only the 50th part of her lustre; but it is manifest he has much more; of consequence, either the sun's light must be stronger at Jupiter than is commonly supposed, or he must be formed of a substance more capable of reflecting the light than Venus.

In fig. 2, we have a view of the proportional breadth pl. XLIII. of the sun's face or disk, as seen from the different planets. The sun is represented No. 1, as seen from Mercury; No. 2, as seen from Venus; No. 3, as seen from the earth; No. 4, as seen from Mars; No. 5, as seen from Jupiter; and No. 6, as seen from Saturn.

Let the circle B (fig. 3.) be the sun as seen from any planet, at a given distance to another planet, at double that that distance, the sun will appear just half that breadth, as A; which contains only one fourth part of the area or surface of B. For all circles, as well as square surfaces, are to one another as the squares of their diameters. Thus, the square A (fig. 4.) is just half as broad as the square B; and yet it is plain to sight, that B contains four times as much surface as A. Hence, by comparing the diameters of the above circles (fig. 2.) together, it will be found, that, in round numbers, the sun appears 7 times larger to Mercury than to us, 90 times larger to us than to Saturn, and 630 times as large to Mercury as to Saturn.

Under fig. 3. are the names and characters of the twelve signs of the zodiac, which the reader should be perfectly well acquainted with, so as to know the characters without seeing the names. Each sign contains 30 degrees, as in the circle (fig. 1.) bounding the solar system; to which the characters of the signs are set in their proper places.

Part of the paths of three comets are delineated in the scheme of the solar system, and the years marked in which they made their appearance. It is believed, that there are at least 21 comets belonging to our system, moving in all sorts of directions. Of all these, the periods of the above-mentioned three only are known with any degree of certainty. The first of these comets appeared in the years 1531, 1607, 1682, and 1759; and is expected to appear every 75th year afterwards. The second of them appeared in 1532 and 1661; and may be expected to return in 1789, and every 129th year afterwards. The third, having last appeared in 1680, and its period being no less than 575 years, cannot return until the year 2225. This comet, at its greatest distance, is about 11,200,000,000 miles from the sun; and at its least distance from the sun's centre, which is 490,000 miles, is within less than a third part of the sun's semidiameter from his surface. In that part of its orbit which is nearest the sun, it flies with the amazing swiftness of 880,000 miles in an hour; and the sun, as seen from it, appears an hundred degrees in breadth; consequently, 40,000 times as large as he appears to us. The astonishing length that this comet runs out into empty space, suggests to our minds an idea of the vast distance between the sun and the nearest fixed stars; of whose attractions all the comets must keep clear, to return periodically, and go round the sun; and it shows us also, that the nearest stars, which are probably those that seem the largest, are as big as our sun, and of the same nature with him; otherwise, they could not appear so large and bright to us, as they do, at such an immense distance.

Sect. VI. Of the Ebbing and Flowing of the Sea, and the Phenomena of the Harvest and Horizontal Moon.

The cause of the tides was discovered by Kepler, who, in his Introduction to the Physics of the Heavens, thus explains it: "The orb of the attracting power, which is in the moon, is extended as far as the earth; and draws the waters under the torrid zone, acting upon places where it is vertical, insensibly on confined seas and bays, but sensibly on the ocean whose beds are large and where the waters have the liberty of reciprocation, that is, of rising and falling." And in the 70th page of his Lunar Astronomy—“But the cause of the tides of the sea appears to be the bodies of the sun and moon drawing the waters of the sea.” This hint being given, the immortal Sir Isaac Newton improved it, and wrote so amply on the subject, as to make the theory of the tides in a manner quite his own, by discovering the cause of their rising on the side of the earth opposite to the moon. For Kepler believed, that the presence of the moon occasioned an impulse which caused another in her absence.

It has been already observed, that the power of gravity diminishes as the square of the distance increases; fig. 4. and therefore the waters at Z on the side of the earth ABCDEFGH next the moon M are more attracted than the central parts of the earth O by the moon, and the central parts are more attracted by her than the waters on the opposite side of the earth at n; and therefore the distance between the earth's centre and the waters on its surface under and opposite to the moon will be increased. For, let there be three bodies at H, O, and D: if they are all equally attracted by the body M, they will all move equally fast toward it, their mutual distances from each other continuing the same. If the attraction of M is unequal, then that body which is most strongly attracted will move fastest, and this will increase its distance from the other body. Therefore, by the law of gravitation, M will attract H more strongly than it does O, by which the distance between H and O will be increased; and a spectator on O will perceive H rising higher toward Z. In like manner, O being more strongly attracted than D, it will move farther towards M than D does; consequently, the distance between O and D will be increased; and a spectator on O, not perceiving his own motion, will see D receding farther from him towards n: all effects and appearances being the same, whether D recedes from O, or O from D.

Suppose now there is a number of bodies, as A, B, C, D, E, F, G, H, placed round O, so as to form a flexible or fluid ring: then, as the whole is attracted towards M, the parts at H and D will have their distance from O increased; whilst the parts at B and F being nearly at the same distance from M as O is, these parts will not recede from one another; but rather, by the oblique attraction of M, they will approach nearer to O. Hence, the fluid ring will form itself into an ellipse ZIBL n KFNZ, whose longer axis nOZ produced will pass through M, and its shorter axis BOF will terminate in B and F. Let the ring be filled with fluid particles, so as to form a sphere round O; then, as the whole moves towards M, the fluid sphere being lengthened at Z and n, will assume an oblong or oval form. If M is the moon, O the earth's centre, ABCDEFGH the sea covering the earth's surface, it is evident, by the above reasoning, that whilst the earth by its gravity falls toward the moon, the water directly below her at B will swell and rise gradually towards her; also the water at D will recede from the centre [strictly speaking, the centre recedes from D], and rise on the opposite side of the earth; whilst the water at B and F is depressed, and falls below the former level. Hence as the earth turns round its axis from the moon to the moon again in 24½ hours, there will be two tides of flood and two of ebb in that time, as we find by experience. As this explanation of the ebbing and flowing of the sea is deduced from the earth's constantly falling towards the moon by the power of gravity, some may find a difficulty in conceiving how this is possible, when the moon is full, or in opposition to the sun; since the earth revolves about the sun, and must continually fall towards it, and therefore cannot fall contrary ways at the same time: or if the earth is constantly falling towards the moon, they must come together at last. To remove this difficulty, let it be considered, that it is not the centre of the earth that describes the annual orbit round the sun, but the (m) common centre of gravity of the earth and moon together: and that whilst the earth is moving round the sun, it also describes a circle round that centre of gravity; going as many times round it in one revolution about the sun as there are lunations or courses of the moon round the earth in a year: and therefore the earth is constantly falling towards the moon from a tangent to the circle it describes round the said common centre of gravity. Let M be the moon, T W part of the moon's orbit, and C the centre of gravity of the earth and moon; whilst the moon goes round her orbit, the centre of the earth describes the circle d g e round C, to which circle g a k is a tangent; and therefore when the moon has gone from M to a little past W, the earth has moved from g to e; and in that time has fallen towards the moon, from the tangent at a to e: and so on, round the whole circle.

The sun's influence in raising the tides is but small in comparison of the moon's; for though the earth's diameter bears a considerable proportion to its distance from the moon, it is next to nothing when compared to its distance from the sun. And therefore the difference of the sun's attraction on the sides of the earth under and opposite to him, is much less than the difference of the moon's attraction on the sides of the earth under and opposite to her; and therefore the moon must raise the tides much higher than they can be raised by the sun.

On this theory, the tides ought to be highest directly under and opposite to the moon; that is, when the moon is due north and south. But we find, that in open seas, where the water flows freely, the moon M is generally past the north and south meridian, as at p, when it is high water at Z and at n. The reason is obvious: for though the moon's attraction was to cease altogether when she was past the meridian, yet the motion of ascent communicated to the water before that time would make it continue to rise for some time after; much more must it do so when the attraction is only diminished; as a little impulse given to a moving ball will cause it still to move farther than otherwise it could have done; and as experience shows, that the day is hotter about three in the afternoon, than when the sun is on the meridian, because of the increase made to the heat already imparted.

The tides answer not always to the same distance of the moon from the meridian at the same places; but are variously affected by the action of the sun, which brings them on sooner when the moon is in her first and third quarters, and keeps them back later when she is in her second and fourth; because, in the former case, the tide raised by the sun alone would be earlier than the tide raised by the moon; and, in the latter case, later.

The moon goes round the earth in an elliptic orbit; and therefore, in every lunar month, she approaches nearer to the earth than her mean distance, and recedes farther from it. When she is nearest, she attracts strongest, and so raises the tides most; the contrary happens when she is farthest, because of her weaker attraction. When both luminaries are in the equator, and the moon in perigee, or at her least distance from the earth, she raises the tides highest of all, especially at her conjunction and opposition; both because the equatorial parts have the greatest centrifugal force from their describing the largest circle, and from the concurring actions of the sun and moon. At the change, the attractive forces of the sun and moon being united, they diminish the gravity of the waters under the moon, and their gravity on the opposite side is diminished by means of a greater centrifugal force. At the full, whilst the moon raises the tide under and opposite to her, the sun, acting in the same line, raises the tide under and opposite to him; whence their conjoint effect is the same as at the change; and, in both cases, occasion what we call the Spring tides. But at the quarters the sun's action on the waters at O and H diminishes the effect of the moon's action on the waters at Z and N; so that they rise a little under and opposite to the sun at O and H, and fall as much under and opposite to the moon at Z and N; making what we call the Neap tides, because the sun and moon then act cross-wise to each other. But these tides happen not till some time after; because in this, as in other cases, the actions do not produce the greatest effect when they are at the strongest, but some time afterward.

The sun, being nearer the earth in winter than in summer, is of course nearer to it in February and October than in March and September; and therefore the greatest tides happen not till some time after the autumnal equinox, and return a little before the vernal.

The sea, being thus put in motion, would continue to ebb and flow for several times, even though the sun and moon were annihilated, or their influence should cease; as, if a basin of water were agitated, the water would continue to move for some time after the basin was left to stand still; or like a pendulum, which, having been put in motion by the hand, continues to make several vibrations without any new impulse.

When the moon is in the equator, the tides are equally high in both parts of the lunar day, or time of the moon's revolving from the meridian to the meridian again, which is 24 hours 50 minutes. But as the moon declines from the equator towards either pole, the tides are alternately higher and lower at places having north or south latitude. For one of the highest elevations, which is that under the moon, follows her towards the pole to which she is nearest, and the other declines.

(m) This centre is as much nearer the earth's centre than the moon's as the earth is heavier, or contains a greater quantity of matter than the moon, namely, about 40 times. If both bodies were suspended on it, they would hang in equilibrium. So that dividing 240,000 miles, the moon's distance from the earth's centre, by 40, the excess of the earth's weight above the moon's, the quotient will be 6000 miles, which is the distance of the common centre of gravity of the earth and moon from the earth's centre. declines towards the opposite pole; each elevation describing parallels as far distant from the equator, on opposite sides, as the moon declines from it to either side; and consequently, the parallels described by these elevations of the water are twice as many degrees from one another, as the moon is from the equator; increasing their distance as the moon increases her declination, till it be at the greatest, when the said parallels are, at a mean state, 47 degrees from one another: and on that day, the tides are most unequal in their heights.

As the moon returns towards the equator, the parallels described by the opposite elevations approach towards each other, until the moon comes to the equator, and then they coincide. As the moon declines towards the opposite pole, at equal distances, each elevation describes the same parallel in the other part of the lunar day, which its opposite elevation described before.

Whilst the moon has north declination, the greatest tides in the northern hemisphere are when she is above the horizon; and the reverse whilst her declination is south.

Let NESQ be the earth, NCS its axis, EQ the equator, T the tropic of Cancer, T the tropic of Capricorn, a the arctic circle, c the antarctic, N the north pole, S the south pole, M the moon, F and G the two eminences of water, whose lowest parts are at a and d, at N and S, and at b and e, always 90 degrees from the highest. Now, when the moon is in her greatest north declination at M, the highest elevation G under her is on the tropic of Cancer T, and the opposite elevation F on the tropic of Capricorn T; and these two elevations describe the tropics by the earth's diurnal rotation. All places in the northern hemisphere ENQ have the highest tides when they come into the position b Q under the moon; and the lowest tides when the earth's diurnal rotation carries them into the position a TE, on the side opposite to the moon; the reverse happens at the same time in the southern hemisphere ESQ, as is evident to sight.

The axis of the tides a C d has now its poles a and d (being always 90 degrees from the highest elevations) in the arctic and antarctic circles; and therefore it is plain, that at these circles there is but one tide of flood, and one of ebb, in the lunar day. For, when the point a revolves half round to b, in 12 lunar hours, it has a tide of flood; but when it comes to the same point a again in 12 hours more, it has the lowest ebb.

In seven days afterward, the moon M comes to the equinoctial circle, and is over the equator EQ, when both elevations describe the equator; and in both hemispheres, at equal distances from the equator, the tides are equally high in both parts of the lunar day.

The whole phenomena being reversed, when the moon has south declination, to what they were when her declination was north, require no further description.

In the three last-mentioned figures, the earth is orthographically projected on the plane of the meridian; but, in order to describe a particular phenomenon, we now project it on the plane of the ecliptic. Let HZON (fig. 2.) be the earth and sea, FED the equator, T the tropic of Cancer, C the arctic circle, P the north pole, and the curves 1, 2, 3, &c., 24 meridians or hour-circles, intersecting each other in the poles; AGM is the moon's orbit, S the sun, M the moon, Z the water elevated under the moon, and N the opposite equal elevation. As the lowest parts of the water are always 90 degrees from the highest, when the moon is in either of the tropics (as at M), the elevation Z is on the tropic of Capricorn, and the opposite elevation N on the tropic of Cancer; the low-water circle HCO touches the polar circles at C; and the high-water circle ETP goes over the poles at P, and divides every parallel of latitude into two equal segments. In this case, the tides upon every parallel are alternately higher and lower; but they return in equal times: the point T, for example, on the tropic of Cancer (where the depth of the tide is represented by the breadth of the dark shade), has a shallower tide of flood at T, than when it revolves half round from thence to 6, according to the order of the numeral figures; but it revolves as soon from 6 to T as it did from T to 6. When the moon is in the equinoctial, the elevations Z and N are transferred to the equator at O and H, and the high and low water circles are got into each other's former places; in which case the tides return in unequal times, but are equally high in both parts of the lunar day: for a place at r (under D) revolving as formerly, goes sooner from r to 11 (under F) than from 11 to r, because the parallel it describes is cut into unequal segments by the high-water circle HCO: but the points r and 11 being equidistant from the pole of the tides at C, which is directly under the pole of the moon's orbit MGA, the elevations are equally high in both parts of the day.

And thus it appears, that as the tides are governed by the moon, they must turn on the axis of the moon's orbit, which is inclined 23° 28' degrees to the earth's axis at a mean state: and therefore the poles of the tides must be so many degrees from the poles of the earth, or in opposite points of the polar circles, going round these circles in every lunar day. It is true, that according to fig. 4., when the moon is vertical to the equator EQO, the poles of the tides seem to fall in with the poles of the world N and S; but when we consider that FGH is under the moon's orbit, it will appear, that when the moon is over H, in the tropic of Capricorn, the north pole of the tides (which can be no more than 90 degrees from under the moon) must be at C in the arctic circle, not at P the north pole of the earth; and as the moon ascends from H to G in her orbit, the north pole of the tides must shift from c to a in the arctic circle, and the south pole as much in the antarctic.

It is not to be doubted, but that the earth's quick rotation brings the poles of the tides nearer to the poles of the world, than they would be if the earth were at rest and the moon revolved about it only once a month; for otherwise the tides would be more unequal in their heights and times of their returns, than we find they are. But how near the earth's rotation may bring the poles of its axis and those of the tides together, or how far the preceding tides may affect those which follow, so as to make them keep up nearly to the same heights and times of ebbing and flowing, is a problem more fit to be solved by observation than by theory.

Those who have opportunity to make observations, and choose to satisfy themselves whether the tides are really affected in the above manner by the different positions of the moon, especially as to the unequal times of their returns, may take this general rule for knowing when they ought to be so affected. When the earth's axis inclines to the moon, the northern tides, if not not retarded in their passage through shoals and channels, nor affected by the winds, ought to be greatest when the moon is above the horizon, least when she is below it; and quite the reverse when the earth's axis declines from her: but in both cases, at equal intervals of time. When the earth's axis inclines sidewise to the moon, both tides are equally high, but they happen at unequal intervals of time. In every lunation the earth's axis inclines once to the moon, once from her, and twice sidewise to her, as it does to the sun every year; because the moon goes round the ecliptic every month, and the sun but once in a year. In summer, the earth's axis inclines towards the moon when new; and therefore the day-tides in the north ought to be highest, and night-tides lowest, about the change; at the full, the reverse. At the quarters, they ought to be equally high, but unequal in their returns; because the earth's axis then inclines sidewise to the moon. In winter, the phenomena are the same at full moon as in summer at new. In autumn, the earth's axis inclines sidewise to the moon when new and full; therefore the tides ought to be equally high and uneven in their returns at these times. At the first quarter, the tides of flood should be least when the moon is above the horizon, greatest when she is below it; and the reverse at her third quarter. In spring, the phenomena of the first quarter answer to those of the third quarter in autumn; and vice versa. The nearer any time is to either of these seasons, the more the tides partake of the phenomena of these seasons; and in the middle between any two of them the tides are at a mean state between those of both.

In open seas, the tides rise but to very small heights in proportion to what they do in wide-mouthed rivers, opening in the direction of the stream of tide. For, in channels growing narrower gradually, the water is accumulated by the opposition of the contracting bank: like a gentle wind, little felt on an open plain, but strong and brisk in a street; especially if the wider end of the street be next the plain, and in the way of the wind.

The tides are so retarded in their passage through different shoals and channels, and otherwise so variously affected by striking against capes and headlands, that to different places they happen at all distances of the moon from the meridian, consequently at all hours of the lunar day. The tide propagated by the moon in the German ocean, when she is three hours past the meridian, takes 12 hours to come from thence to London bridge; where it arrives by the time that a new tide is raised in the ocean. And therefore, when the moon has north declination, and we should expect the tide at London to be greatest when the moon is above the horizon, we find it least; and the contrary when she has south declination. At several places it is high-water three hours before the moon comes to the meridian; but that tide which the moon pushes as it were before her, is only the tide opposite to that which was raised by her when she was nine hours past the opposite meridian.

There are no tides in lakes, because they are generally so small, that when the moon is vertical she attracts every part of them alike, and therefore, by rendering all the water equally light, no part of it can be raised higher than another. The Mediterranean and Baltic seas have very small elevations, because the inlets by which they communicate with the Ocean are so narrow, that they cannot, in so short a time, receive or discharge enough to raise or sink their surfaces sensibly.

Air being lighter than water, and the surface of the atmosphere being nearer to the moon than the surface of the sea, it cannot be doubted that the moon raises much higher tides in the air than in the sea. And therefore many have wondered why the mercury does not sink in the barometer when the moon's action on the particles of air makes them lighter as she passes over the meridian.

But we must consider, that as these particles are rendered lighter, a greater number of them is accumulated, until the deficiency of gravity be made up by the height of the column; and then there is an equilibrium, and consequently an equal pressure upon the mercury as before; so that it cannot be affected by the aerial tides. It is very probable, however, that the stars which are seen through an aerial tide of this kind will have their light more refracted than those which are seen through the common depth of the atmosphere; and this may account for the supposed refractions by the lunar atmosphere that have been sometimes observed. See n° 43.

It is generally believed that the moon rises about 50 minutes later every day than on the preceding; but this is true only with regard to places on the equator. In places of considerable latitude there is a remarkable difference, especially in the harvest-time. Here the autumnal full moons rise very soon after sun-set for several evenings together. At the polar circles, where the mild season is of very short duration, the autumnal full moon rises at sun-set from the first to the third quarter. And at the poles, where the sun is for half a year absent, the winter full-moons shine constantly without setting from the first to the third quarter.

All these phenomena are owing to the different angles made by the horizon and different parts of the moon's orbit; and may be explained in the following manner.

The plane of the equinoctial is perpendicular to the earth's axis; and therefore, as the earth turns round its axis, all parts of the equinoctial make equal angles with the horizon both at rising and setting; so that equal portions of it always rise or set in equal times. Consequently, if the moon's motion were equable, and in the equinoctial, at the rate of 12 degrees 11 min. from the sun every day, as it is in her orbit, she would rise and set 50 minutes later every day than on the preceding; for 12 deg. 11 min. of the equinoctial rise or set in 50 minutes of time in all latitudes.

But the moon's motion is so nearly in the ecliptic, that we may consider her at present as moving in it. Now the different parts of the ecliptic, on account of its obliquity to the earth's axis, make very different angles with the horizon as they rise or set. Those parts or signs which rise with the smallest angles set with the greatest, and vice versa. In equal times, whenever this angle is least, a greater portion of the ecliptic rises than when the angle is larger; as may be seen by elevating the pole of a globe to any considerable latitude, and then turning it round its axis in the horizon. Consequently, when the moon is in those signs which rise or set with the smallest angles, she rises or sets with the least difference of time; and with Let FUP be the axis of a globe, TR the tropic of Cancer, L the tropic of Capricorn, EU the ecliptic touching both the tropics, which are 47 degrees from each other, and A B the horizon. The equator, being in the middle between the tropics, is cut by the ecliptic in two opposite points, which are the beginnings of Aries and Libra; K is the hour-circle with its index, F the north pole of the globe elevated to a considerable latitude, suppose 40 degrees above the horizon; and P the south pole depressed as much below it. Because of the oblique position of the sphere in this latitude, the ecliptic has the high elevation N above the horizon, making the angle NU of 73½ degrees with it when Cancer is on the meridian, at which time Libra rises in the east. But let the globe be turned half round its axis, till Capricorn comes to the meridian and Aries rises in the east; and then the ecliptic will have the low elevation NL above the horizon, making only an angle NUL of 26½ degrees with it; which is 47 degrees less than the former angle, equal to the distance between the tropics.

In northern latitudes, the smallest angle made by the ecliptic and horizon is when Aries rises, at which time Libra sets; the greatest when Libra rises, at which time Aries sets. From the rising of Aries to the rising of Libra (which is twelve (s) sidereal hours) the angle increases; and from the rising of Libra to the rising of Aries, it decreases in the same proportion.

By this article and the preceding, it appears that the ecliptic rises fastest about Aries, and slowest about Libra.

On the parallel of London, as much of the ecliptic rises about Pisces and Aries in two hours as the moon goes through in six days; and therefore, whilst the moon is in these signs, she differs but two hours in rising for five days together; that is, about 20 minutes later every day or night than on the preceding, at a mean rate. But in 14 days afterwards, the moon comes to Virgo and Libra, which are the opposite signs to Pisces and Aries; and then she differs almost four times as much in rising; namely, one hour and about fifteen minutes later every day or night than the former, whilst she is in these signs.

As the moon can never be full but when she is opposite to the sun, and the sun is never in Virgo and Libra but in our autumnal months, it is plain that the moon is never full in the opposite signs, Pisces and Aries, but in these two months. And therefore we can have only two full moons in the year, which rise to near the time of sun-set for a week together, as abovementioned. The former of these is called the harvest moon, and the latter the hunter's moon.

Here it will probably be asked, why we never observe this remarkable rising of the moon but in harvest, seeing she is in Pisces and Aries twelve times in the year besides; and must then rise with as little difference of time as in harvest? The answer is plain: for in winter these signs rise at noon; and being then only a quarter of a circle distant from the sun, the moon in them is in her first quarter: but when the sun is above the horizon, the moon's rising is neither regarded nor perceived. In spring, these signs rise with the sun, because he is then in them; and, as the moon changeth in them at that time of the year, she is quite invisible. In summer, they rise about midnight; and the sun being then three signs, or a quarter of a circle, before them, the moon is in them about her third quarter; when rising so late, and giving but very little light, her rising passes unobserved. And in autumn, these signs, being opposite to the sun, rise when he sets, with the moon in opposition, or at the full, which makes her rising very conspicuous.

At the equator, the north and south poles lie in the horizon; and therefore the ecliptic makes the same angle southward with the horizon when Aries rises, as it does northward when Libra rises. Consequently, as the moon rises and sets nearly at equal angles with the horizon all the year round, and about 50 minutes later every day or night than on the preceding, there can be no particular harvest-moon at the equator.

The farther that any place is from the equator, if it be not beyond the polar circle, the more the angle is diminished which the ecliptic and horizon make when Pisces and Aries rise: and therefore when the moon is in these signs, she rises with a nearly proportionable difference later every day than on the former; and is for that reason the more remarkable about the full, until we come to the polar circles, or 66 degrees from the equator; in which latitude the ecliptic and horizon become coincident every day for a moment, at the same sidereal hour (or 3 minutes 56 seconds sooner every day than the former), and the very next moment one half of the ecliptic containing Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini, rises, and the opposite half sets. Therefore, whilst the moon is going from the beginning of Capricorn to the beginning of Cancer, which is almost 14 days, she rises at the same sidereal hour; and in autumn just at sun-set, because all that half of the ecliptic, in which the sun is at that time, sets at the same sidereal hour, and the opposite half rises; that is, 3 minutes 56 seconds, of mean solar time, sooner every day than on the day before. So whilst the moon is going from Capricorn to Cancer, she rises earlier every day than on the preceding; contrary to what she does at all places between the polar circles. But during the above 14 days, the moon is 24 sidereal hours later in setting; for the six signs which rise all at once on the eastern side of the horizon are 24 hours in setting on the western side of it.

In northern latitudes, the autumnal full moons are in Pisces and Aries, and the vernal full moons in Virgo and Libra; in southern latitudes, just the reverse, because the seasons are contrary. But Virgo and Libra rise at small angles with the horizon in southern latitudes, as Pisces and Aries do in the northern; and therefore the harvest-moons are just as regular on one side of the equator as on the other.

As these signs, which rise with the least angles, set with the greatest, the vernal full moons differ as much in their times of rising every night as the autumnal full moons.

(n) The ecliptic, together with the fixed stars, make 366½ apparent diurnal revolutions about the earth in a year; the sun only 365½. Therefore the stars gain 5 minutes 56 seconds upon the sun every day; so that a sidereal day contains only 23 hours 56 minutes of mean solar time; and a natural or solar day 24 hours. Hence 12 sidereal hours are 11 minutes 58 seconds shorter than 12 solar. moons differ in their times of setting; and set with as little difference as the autumnal full moons rise: the one being in all cases the reverse of the other.

Hitherto, for the sake of plainness, we have supposed the moon to move in the ecliptic, from which the sun never deviates. But the orbit in which the moon really moves is different from the ecliptic; one half being elevated $5\frac{1}{2}$ degrees above it, and the other half as much depressed below it. The moon's orbit therefore intersects the ecliptic in two points diametrically opposite to each other; and these intersections are called the Moon's Nodes. So the moon can never be in the ecliptic but when she is in either of her nodes, which is at least twice in every course from change to change, and sometimes thrice. For, as the moon goes almost a whole sign more than round her orbit from change to change; if she passes by either node about the time of change, she will pass by the other in about 14 days after, and come round to the former node two days again before the next change. That node from which the moon begins to ascend northward, or above the ecliptic, in northern latitudes, is called the Ascending Node; and the other the Descending Node, because the moon, when she passes by it, descends below the ecliptic southward.

The moon's oblique motion with regard to the ecliptic causes some difference in the times of her rising and setting from what is already mentioned. For when she is northward of the ecliptic, she rises sooner and sets later than if she moved in the ecliptic: and when she is southward of the ecliptic, she rises later and sets sooner. This difference is variable, even in the same signs, because the nodes shift backward about $19\frac{1}{2}$ degrees in the ecliptic every year; and so go round it contrary to the order of signs in 18 years 225 days.

When the ascending node is in Aries, the southern half of the moon's orbit makes an angle of $5\frac{1}{2}$ degrees less with the horizon than the ecliptic does, when Aries rises in northern latitudes: for which reason the moon rises with less difference of time whilst she is in Pisces and Aries, than she would do if she kept in the ecliptic. But in 9 years and 112 days afterward, the defending node comes to Aries; and then the moon's orbit makes an angle $5\frac{1}{2}$ degrees greater with the horizon when Aries rises, than the ecliptic does at that time; which causes the moon to rise with greater difference of time in Pisces and Aries than if she moved in the ecliptic.

To be a little more particular; when the ascending node is in Aries, the angle is only $9\frac{1}{2}$ degrees on the parallel of London when Aries rises; but when the defending node comes to Aries, the angle is $20\frac{1}{2}$ degrees. This occasions as great a difference of the moon's rising in the same signs every nine years, as there would be on two parallels $10\frac{1}{2}$ degrees from one another, if the moon's course were in the ecliptic.

As there is a complete revolution of the nodes in 18$\frac{1}{2}$ years, there must be a regular period of all the varieties which can happen in the rising and setting of the moon during that time. But this shifting of the nodes never affects the moon's rising so much, even in her quickest descending latitude, as not to allow us still the benefit of her rising nearer the time of sun-set for a few days together about the full in harvest, than when she is full at any other time of the year.

At the polar circles, when the sun touches the summer tropic, he continues 24 hours above the horizon; and 24 hours below it, when he touches the winter tropic. For the same reason, the full moon neither rises in summer, nor sets in winter, considering her as moving in the ecliptic. For the winter full moon being as high in the ecliptic as the summer sun, must therefore continue as long above the horizon; and the summer full moon being as low in the ecliptic as the winter sun, can no more rise than he does. But these are only the two full moons which happen about the tropics, for all the others rise and set. In summer, the full moons are low, and their stay is short above the horizon, when the nights are short, and we have least occasion for moonlight: in winter they go high, and stay long above the horizon, when the nights are long, and we want the greatest quantity of moonlight.

At the poles, one half of the ecliptic never sets, and the other half never rises: and therefore, as the sun is always half a year in describing one half of the ecliptic, and as long in going through the other half, it is natural to imagine that the sun continues half a year together above the horizon of each pole in its turn, and as long below it; rising to one pole when he sets to the other. This would be exactly the case if there were no refraction; but by the atmosphere's refracting the sun's rays, he becomes visible some days sooner, and continues some days longer in sight, than he would otherwise do: so that he appears above the horizon of either pole before he has got below the horizon of the other. And, as he never goes more than $23\frac{1}{2}$ degrees below the horizon of the poles, they have very little dark night; it being twilight there, as well as at other places, till the sun be 18 degrees below the horizon. The full moon, being always opposite to the sun, can never be seen while the sun is above the horizon, except when she is in the northern half of her orbit; for whenever any point of the ecliptic rises, the opposite point sets. Therefore, as the sun is above the horizon of the north pole from the 20th of March till 23rd of September, it is plain that the moon, when full, being opposite to the sun, must be below the horizon during that half of the year. But when the sun is in the southern half of the ecliptic, he never rises to the north pole; during which half of the year, every full moon happens in some part of the northern half of the ecliptic which never sets. Consequently, as the polar inhabitants never see the full moon in summer, they have her always in the winter, before, at, and after the full, shining for 14 of our days and nights. And when the sun is at his greatest depression below the horizon, being then in Capricorn, the moon is at her first quarter in Aries, full in Cancer, and at her third quarter in Libra. And as the beginning of Aries is the rising point of the ecliptic, Cancer the highest, and Libra the setting point, the moon rises at her first quarter in Aries; is most elevated above the horizon, and full, in Cancer; and sets, at the beginning of Libra, in her third quarter, having continued visible for 14 diurnal rotations of the earth. Thus the poles are supplied one half of the winter-time with constant moon-light in the sun's absence; and only lose sight of the moon from her third to her first quarter, while she gives but very little light, and could be but of little and sometimes of no service to them. A bare view of the figure will make this plain. plain; in which let S be the sun; e, the earth in summer, when its north pole n inclines toward the sun; and E the earth in winter, when its north pole declines from him. SEN and NWS is the horizon of the north pole, which is coincident with the equator; and, in both these positions of the earth, VP ES is the moon's orbit, in which she goes round the earth, according to the order of the letters a b c d, A B C D. When the moon is at a, she is in her third quarter to the earth at e, and just rising to the north pole n; at b she changes, and is at the greatest height above the horizon, as the sun likewise is; at c she is in her first quarter, setting below the horizon; and is lowest of all under it at d, when opposite to the sun, and her enlightened side toward the earth. But then she is full in view to the south pole p; which is as much turned from the sun as the north pole inclines toward him. Thus, in our summer, the moon is above the horizon of the north pole whilst she describes the northern half of the ecliptic VP ES, or from her third quarter to her first; and below the horizon during her progress through the southern half ES VP; highest at the change, most depressed at the full. But in winter, when the earth is at E, and its north pole declines from the sun, the new moon at D is at her greatest depression below the horizon NWS, and the full moon at B at her greatest height above it; rising at her first quarter A, and keeping above the horizon till she comes to her third quarter C. At a mean state she is 23½ degrees above the horizon at B and b, and as much below it at D and d, equal to the inclination of the earth's axis F S ES, or S VP, are, as it were, a ray of light proceeding from the sun to the earth; and shews that when the earth is at e, the sun is above the horizon, vertical to the tropic of Cancer; and when the earth is at E, he is below the horizon, vertical to the tropic of Capricorn.

The sun and moon generally appear larger when near the horizon than when at a distance from it; for which there have been various reasons assigned. The following account is given by Mr Ferguson. "These luminaries, although at great distances from the earth, appear floating as it were on the surface of our atmosphere, HGF/EC, a little way beyond the clouds; of which, those about F, directly over our heads at E, are nearer us than those about H or c in the horizon HEc. Therefore, when the sun or moon appear in the horizon at e, they are not only seen in a part of the sky which is really farther from us than if they were at any considerable altitude, as about f; but they are also seen through a greater quantity of air and vapours at c than at f. Here we have two concurring appearances which deceive our imagination, and cause us to refer the sun and moon to a greater distance at their rising or setting about c, than when they are considerably high, as at f: first, their seeming to be on a part of the atmosphere at c, which is really farther than f from a spectator at E; and, secondly, their being seen through a groser medium when at c than when at f, which, by rendering them dimmer, causes us to imagine them to be at a yet greater distance. And as, in both cases, they are seen much under the same angle, we naturally judge them to be larger when they seem farther from us.

Any one may satisfy himself that the moon appears under no greater angle in the horizon than on the meridian, by taking a large sheet of paper, and rolling it up in the form of a tube, of such a width, that, observing the moon through it when she rises, she may as it were just fill the tube: then tie a thread round it to keep it of that size; and when the moon comes to the meridian, and appears much less to the eye, look at her again through the same tube, and she will fill it just as much, if not more, than she did at her rising.

"When the full moon is in her perigee, or at her least distance from the earth, she is seen under a larger angle, and must therefore appear bigger than when she is full at other times: And if that part of the atmosphere where she rises be more replete with vapours than usual, she appears so much the dimmer; and therefore we fancy her to be still the bigger, by referring her to an unusually great distance, knowing that no objects which are very far distant can appear big unless they really be so."

To others this solution has appeared unsatisfactory; and accordingly Mr Dunn has given the following dissertation on this phenomenon, Phil. Trans. Vol. LXIV.

1. "The sun and moon, when they are in or near the horizon, appear to the naked eye of the generality of persons, so very large in comparison with their apparent magnitudes when they are in the zenith, or somewhat elevated, that several learned men have been led to inquire into the cause of this phenomenon; and after endeavouring to find certain reasons, founded on the principles of physics, they have at last pronounced this phenomenon a mere optical illusion.

2. "The principal dissertations which I have seen conducing to give any information on this subject, or helping to throw any light on the same, have been those printed on the transactions of the Royal Society, the Academy of Sciences at Paris, the German Acts, and Dr Smith's Optics; but as all the accounts which I have met with in these writings any way relative to this subject, have not given me that satisfaction which I have desired, curiosity has induced me to inquire after the cause of this singular phenomenon in a manner somewhat different from that which others have done before me, and by such experiments and observations as have appeared to me pertinent; some of which have been as follows, viz.

3. "I have observed the rising and setting sun near the visible horizon, and near rising grounds elevated above the visible horizon about half a degree, and found him to appear largest when near to the visible horizon; and particularly a considerable alteration of his magnitude and light has always appeared to me from the time of his being in the horizon at rising, to the time of his being a degree or two above the horizon, and the contrary at his setting; which property I have endeavoured to receive as a prejudice, and an imposition on my sight and judgement, the usual reasons for this appearance.

4. "I have also observed that the sun near the horizon appears to put on the figure of a spheroid, having its vertical diameter appearing to the naked eye shorter than the horizontal diameter; and, by measuring those diameters in a telescope, have found the vertical one shorter than the other.

5. "I have made frequent observations and comparisons of the apparent magnitude of the sun's disk, with objects directly under him, when he has been near the horizon, and with such objects as I have found by measurement." measurement to be of equal breadth with the sun's diameter; but in the sudden transition of the eye from the sun to the object, and from the object to the sun, have always found the sun to appear least; and that when two right lines have been imaginarily produced by the sides of those equal magnitudes, they have not appeared to keep parallel, but to meet beyond the sun.

6. "From these and other like circumstances, I first began to suspect that a sudden dip of the sun into the horizontal vapours, might some how or other be the cause of a sudden apparent change of magnitude; although the horizontal vapours had been disallowed to be able to produce any other than a refraction in a vertical direction; and, reducing things to calculation, found, that from the time when the sun is within a diameter or two of the horizon, to the time when he is a semidiameter below the horizon, the sun's rays become passable through such a length of medium, reckoning in the direction of the rays, that the total quantity of medium (reckoning both depth and density) through which the rays pass, being compared with the like total depth and density through which they pass at several elevations, it was proportionable to the difference of apparent magnitude, as appearing to the naked eye.

7. "This circumstance of sudden increase and decrease of apparent magnitude, and as sudden decrease and increase of light (for they both go together), seemed to me no improbable cause of the phenomenon, although I could not then perceive how such vapours might contribute toward enlarging the diameter of the sun in a horizontal direction.

8. "I therefore examined the sun's disk again and again, by the naked eye and by telescopes, at different altitudes; and, among several circumstances, found the solar macula to appear larger and plainer to the naked eye; and through a telescope, the sun being near the horizon, than they had appeared the same days when the sun was on the meridian, and to appearance more strongly defined, yet obscured.

9. "A little before sun-setting, I have often seen the edge of the sun with such protuberances and indentures as have rendered him in appearance a very odd figure; the protuberances shooting out far beyond, and the indentures pressing into the disk of the sun; and always, through a telescope magnifying 55 times, the lower limb has appeared with a red glowing arch beneath it, and close to the edge of the sun, while the other parts have been clear.

10. "At sun-settings, these protuberances and indentures have appeared to slide along the vertical limbs, from the lower limb to the higher, and there vanishing, so as often to form a segment of the sun's upper limb, apparently separated from the disk for a small space of time.

11. "At sun-rising I have seen the like protuberances, indentures, and slices, above described; but with this difference of motion, that at sun-rising they first appear to rise in the sun's upper limb, and slide or move downward to the lower limb; or, which is the same thing, they always appear at the rising and setting of the sun, to keep in the same parallels of altitude by the telescope. This property has been many times so discernible, even by the naked eye, that I have observed the sun's upper limb to shoot out towards right and left, and move downwards, forming the upper part of the disk an apparent portion of a lesser spheroid than the lower part at rising, and the contrary at setting. Through the telescope this has appeared more plain in proportion to the power of magnifying.

12. "These protuberances and indentures so easily measurable by the micrometer, whilst the telescope wires appeared strait, enabled me to conclude, that certain strata of the atmosphere have different refractive powers; and, lying horizontally across the conical or cycloidal space traced out by the rays between the eye and that part of the atmosphere first touched by the rays, must have been the cause of such apparent protuberances and indentures in an horizontal direction across the sun's vertical limbs; and also that the bottoms of those protuberances and indentures must be considerably enlarged, and removed to appearance farther from the centre of the disk than they would have been had there been no such strata to refract.

13. "Before sun-rising, when the sun has been near the tropic; and the sky, at the utmost extent of the horizon, hath appeared very clear; and when certain fogs have appeared in strata placed alternately between the hills, and over intervening rivers, valleys, &c. so as to admit a sight of the rising sun over those fogs; I have observed with admiration, the most distant trees and bushments, which at other times have appeared small to the naked eye, but while the sun has been passing along a little beneath the horizon obliquely under them, just before sun-rising, when the sun has been thus approaching towards trees and bushments, they have grown apparently very large to the naked eye, and also through a telescope; and they have lost that apparent largeness as the sun has been passed by them. Thus a few trees standing together on the rising ground, at the distance of a few miles, have appeared to grow up into an apparent mountain. Such apparent mountains formed from trees put on all forms and shapes, as sloping, perpendicular, over-leaning, &c. but soon recover their natural appearance when the sun is past by them, or got above the horizon.

14. "Mountains themselves, at a distance, sometimes appear larger than at other times. Beasts and cattle in the midst of, and being surrounded with, water, appear nearer to us than when no water surrounds them. Cattle, houses, trees, all objects on the summit of a hill, when seen through a fog, and at a proper distance, appear enlarged. All bodies admit of larger apparent magnitudes when seen through some mediums than others.

"But more particularly,

15. "I took a cylindrical glass vessel about two feet high; and having graduated its sides to inches, I placed it upright on a table, with a piece of paper under the bottom of the glass, on which paper were drawn parallel right lines at proper distances from each other; and having placed a shilling at the bottom of the vessel, it was nearly as low as the paper. Pouring water into the vessel, and viewing the shilling through the medium of water with one eye, whilst I beheld with the other eye where the edges of the shilling were projected on the paper and its parallels, I found the shilling appear larger at every additional inch depth of the water; and this was the case if either eye was used; and the same when the eye was removed far from the surface or near to it, or in any proportion thereto." 16. "I took large vessels; and, filling them with water, placed different bodies at the bottoms of those vessels. It always followed, that the greater depth of water I looked through, in the direction from my eye to the objects in the water, the nearer those objects appeared to me. Thus light bodies appeared more mellow and faint, and dark bodies rather better defined, than out of the water, when they were not deeply immersed. And thus they appeared under whatever directions or positions I viewed the bodies.

17. "I placed different bodies in proper vessels of fair water, and immersed my face in the water; viewing the bodies in and through the water. They all appeared to me plain, when not too far from the eye; and although a little hazy at the edges, they appeared much enlarged, and always larger through a greater depth of water. Thus a shilling appeared nearly as large as half a crown, with a red glowing arch on that side opposite to the sun, when the sun shined on the water. From this experiment I concluded, that divers see light objects not only larger, but very distinctly, in the water."

From these experiments he draws a confirmation of his doctrine, that the appearances treated of arise from the different strata of the atmosphere; and then concludes, that the rays coming from the sun are by the horizontal vapours first obstructed, and many of them totally absorbed; the rest proceeding with a retarded motion, are thereby first reflected, and then less refracted through the humours of the eye; and lastly, that hereby the image on the retina becomes enlarged."

Sect. VII. Of drawing a Meridian Line. Of Solar and Sidereal Time, and of the Equation of Time.

The foundation of all astronomical observations is a knowledge of the exact time when the sun, or any other of the celestial bodies, comes to the meridian; and therefore astronomers have been very attentive to the most proper methods of drawing a meridian line, by which only this can be exactly known. The easiest method of doing this is the following, recommended by Mr Ferguson.

Make four or five concentric circles, about a quarter of an inch from one another, on a flat board about a foot in breadth; and let the outmost circle be but little less than the board will contain. Fix a pin perpendicularly in the centre, and of such a length that its whole shadow may fall within the innermost circle for at least four hours in the middle of the day. The pin ought to be about an eighth part of an inch thick, and to have a round blunt point. The board being set exactly level in a place where the sun shines, suppose from eight in the morning till four in the afternoon, about which hours the end of the shadow should fall without all the circles; watch the times in the forenoon when the extremity of the shortening shadow just touches the several circles, and there make marks. Then, in the afternoon of the same day, watch the lengthening shadow; and where its end touches the several circles in going over them, make marks also. Lastly, with a pair of compasses, find exactly the middle point between the two marks on any circle, and draw a straight line from the centre to that point; which line will be covered at noon by the shadow of a small upright wire, which should be put in the place of the pin. The reason for drawing several circles is, that in case one part of the day should prove clear, and the other part somewhat cloudy, if you miss the time when the point of the shadow should touch one circle, you may perhaps catch it in touching another. The best time for drawing a meridian line in this manner is about the summer solstice; because the sun changes his declination slowest, and his altitude fastest, in the longest days.

If the calefaction of a window on which the sun shines at noon be quite upright, you may draw a line along the edge of its shadow on the floor, when the shadow of the pin is exactly on the meridian line of the board; and as the motion of the shadow of the calefaction will be much more sensible on the floor than that of the shadow of the pin on the board, you may know to a few seconds when it touches the meridian line on the floor.

This method may suffice for ordinary purposes, but for astronomers the following is preferable. Take the common of an horizontal dial for the latitude of the place, and to the hypotenusa fix two sights, whose centres may be parallel to the same; let the eye-light be a small hole, but the other's diameter must be equal to the tangent of the double distance of the north-star from the pole; the distance of the sights being made radius, let the file be rivetted to the end of a straight ruler; then when you would make use of it, lay the ruler on an horizontal plane, so that the end to which the file is fixed may overhang; then look through the eye-light, moving the instrument till the north-star appears to touch the circumference of the hole in the other sight, on the same hand with the girdle of Cassiopeia, or on the opposite side to that whereon the star in the Great Bear's rump is at that time; then draw a line by the edge of the ruler, and it will be a true meridian line.

A meridian line being by either of these methods exactly drawn, the time when the sun or any other of the celestial bodies is exactly in the meridian may be found by a common quadrant, placing the edge of it along the line, and observing when the sun or other luminary can be seen exactly through its two sights, and noting exactly the time; which, supposing the luminary viewed to be the sun, will be exactly noon, or 12 o'clock; but as the apparent diameter of the sun is pretty large, it ought to be known exactly when his centre is in the meridian, which will be some short space after his western limb has arrived at it, and before his eastern limb comes thither. It will be proper, therefore, to observe exactly the time of the two limbs having been through the sights of the quadrant; and the half of the difference between these times added to the one, or subtracted from the other, will give the exact time when the sun's centre is in the meridian. What we say with regard to the sun, is also applicable to the moon; but not to the stars, which have no sensible diameter. To render this more intelligible, the following short description of the quadrant, and method of taking the altitudes of celestial bodies by it, is subjoined.

In Plate XLIX. let HOX be a horizontal line, supposed to be extended from the eye at A to X, where the sky and earth seem to meet at the end of a long and level plain; and let S be the sun. The arc XY will be the sun's height above the horizon at X, and is found by the instrument EDC, which is a quadrantal board, or plate of metal, divided into 90 equal parts or or degrees on its limb DPC; and has a couple of little brass plates, as \(a\) and \(b\), with a small hole in each of them, called sight-holes, for looking through, parallel to the edge of the quadrant whereon they stand. To the centre E is fixed one end of a thread F, called the plumb-line, which has a small weight or plummet P fixed to its other end. Now, if an observer holds the quadrant upright, without inclining it to either side, and so that the horizon at X is seen through the sight-holes \(a\) and \(b\), the plumb-line will cut or hang over the beginning of the degrees at o, in the edge EC; but if he elevates the quadrant so as to look through the sight-holes at any part of the heavens, suppose to the sun at S; just so many degrees as he elevates the sight-hole \(b\) above the horizontal line HOX, so many degrees will the plumb-line cut in the limb CP of the quadrant. For, let the observer's eye at A be in the centre of the celestial arc XYV (and he may be said to be in the centre of the sun's apparent and diurnal orbit, let him be on what part of the earth he will); in which arc the sun is at that time, suppose 25 degrees high, and let the observer hold the quadrant so that he may see the sun through the sight-holes; the plumb-line freely playing on the quadrant will cut the 25th degree in the limb CP, equal to the number of degrees of the sun's altitude at the time of observation. (N.B. Whoever looks at the sun, must have a smoked glass before his eyes, to save them from hurt. The better way is, not to look at the sun through the sight-holes, but to hold the quadrant facing the eye, at a little distance, and so that, the sun shining through one hole, the ray may be seen to fall on the other.)

By observations made in the manner above directed, it is found, that the stars appear to go round the earth in 23 hours 56 minutes 4 seconds, and the sun in 24 hours; so that the stars gain three minutes 56 seconds upon the sun every day, which amounts to one diurnal revolution in a year; and therefore, in 365 days as measured by the returns of the sun to the meridian, there are 366 days as measured by the stars returning to it; the former are called solar days, and the latter sidereal.

If the earth had only a diurnal motion, without an annual, any given meridian would revolve from the sun to the sun again in the same quantity of time as from any star to the same star again; because the sun would never change his place with respect to the stars. But, as the earth advances almost a degree eastward in its orbit in the time that it turns eastward round its axis, whatever star passes over the meridian on any day with the sun, will pass over the same meridian on the next day when the sun is almost a degree short of it; that is, 3 minutes 56 seconds sooner. If the year contained only 360 days, as the ecliptic does 360 degrees, the sun's apparent place, so far as his motion is equable, would change a degree every day; and then the sidereal days would be just 4 minutes shorter than the solar.

Let ABCDEFGHIKLM be the earth's orbit, in which it goes round the sun every year, according to the order of the letters; that is, from west to east; and turns round its axis in the same way from the sun to the sun again in every 24 hours. Let S be the sun, and R a fixed star at such an immense distance, that the diameter of the earth's orbit bears no sensible proportion to that distance. Let Nm be any particular meridian of the earth, and N a given point or place upon that meridian when the earth is at A, the sun S hides the star R, which would always be hid if the earth never removed from A; and consequently, as the earth turns round its axis, the point N would always come round to the sun and star at the same time. But when the earth has advanced, suppose a twelfth part of its orbit, from A to B, its motion round its axis will bring the point N a twelfth part of a natural day, or two hours, sooner to the star than to the sun; for the angle of NBm is equal to the angle ASB; and therefore any star, which comes to the meridian at noon with the sun when the earth is at A, will come to the meridian at 10 in the forenoon when the earth is at B. When the earth comes to C, the point N will have the star on its meridian at 8 in the morning, or four hours sooner than it comes round to the sun; for it must revolve from N to n, before it has the sun in its meridian. When the earth comes to D, the point N will have the star on its meridian at 6 in the morning; but that point must revolve six hours more from N to n, before it has mid-day by the sun; for now the angle ASD is a right angle, and so is NDn; that is, the earth has advanced 90 degrees in its orbit, and must turn 90 degrees on its axis to carry the point N from the star to the sun; for the star always comes to the meridian when Nm is parallel to RSA; because DS is but a point in respect of RS. When the earth is at E, the star comes to the meridian at 4 in the morning; at F, at two in the morning; and at G, the earth having gone half round its orbit, N points to the star R at midnight, it being then directly opposite to the sun; and therefore, by the earth's diurnal motion, the star comes to the meridian 12 hours before the sun. When the earth is at H, the star comes to the meridian at 10 in the evening; at I, it comes to the meridian at 8, that is, 16 hours before the sun; at K, 18 hours before him; at L, 20 hours; at M, 22; and at A, equally with the sun again.

Thus it is plain, that an absolute turn of the earth on its axis (which is always completed when any particular meridian comes to be parallel to its situation at any time of the day before) never brings the same meridian round from the sun to the sun again; but that the earth requires as much more than one turn on its axis to finish a natural day, as it has gone forward in that time; which, at a mean state, is a 365th part of a circle. Hence, in 365 days, the earth turns 366 times round its axis; and therefore, as a turn of the earth on its axis completes a sidereal day, there must be one sidereal day more in a year than the number of solar days, be the number what it will, on the earth or any other planet. One turn being lost with respect to the number of solar days in a year, by the planet's going round the sun; just as it would be lost to a traveller, who, in going round the earth, would lose one day by following the apparent diurnal motion of the sun; and consequently would reckon one day less at his return (let him take what time he would to go round the earth) than those who remained all the while at the place from which he set out. So, if there were two earths revolving equally on their axes, and if one remained at A until the other had gone round the sun from A to A again, that earth which kept its place at A would have its solar and sidereal days always of the same length; length; and so would have one solar day more than the other at its return. Hence, if the earth turned but once round its axis in a year, and if that turn was made the same way as the earth goes round the sun, there would be continual day on one side of the earth, and continual night on the other.

The earth's motion on its axis being perfectly uniform, and equal at all times of the year, the sidereal days are always precisely of an equal length; and so would the solar or natural days be, if the earth's orbit were a perfect circle, and its axis perpendicular to its orbit. But the earth's diurnal motion on an inclined axis, and its annual motion in an elliptic orbit, cause the sun's apparent motion in the heavens to be unequal: for sometimes he revolves from the meridian to the meridian again in somewhat less than 24 hours, shown by a well-regulated clock; and at other times in somewhat more: so that the time shown by an equal going clock and a true sun-dial is never the same but on the 15th of April, the 16th of June, the 31st of August, and the 24th of December. The clock, if he goes equally, and true all the year round, will be before the sun from the 24th of December till the 15th of April; from that time till the 16th of June, the sun will be before the clock; from the 16th of June till the 31st of August, the clock will be again before the sun; and from thence to the 24th of December, the sun will be faster than the clock.

As the equation of time, or difference between the time shown by a well-regulated clock and a true sundial, depends upon two causes, namely, the obliquity of the ecliptic, and the unequal motion of the earth in it, we shall first explain the effects of these causes separately considered, and then the united effects resulting from their combination.

The earth's motion on its axis being perfectly equable, or always at the same rate, and the plane of the equator being perpendicular to its axis, it is evident that in equal times equal portions of the equator pass over the meridian; and so would equal portions of the ecliptic, if it were parallel to, or coincident with, the equator. But, as the ecliptic is oblique to the equator, the equable motion of the earth carries unequal portions of the ecliptic over the meridian in equal times, the difference being proportionate to the obliquity; and as some parts of the ecliptic are much more oblique than others, those differences are unequal among themselves. Therefore, if two suns should start from the beginning either of Aries or Libra, and continue to move through equal arcs in equal times, one in the equator, and the other in the ecliptic, the equatorial sun would always return to the meridian in 24 hours time, as measured by a well regulated clock; but the sun in the ecliptic would return to the meridian sometimes sooner and sometimes later than the equatorial sun; and only at the same moments with him on four days of the year; namely, the 20th of March, when the sun enters Aries; the 21st of June, when he enters Cancer; the 23rd of September, when he enters Libra; and the 21st of December, when he enters Capricorn; and to this fictitious sun the motion of a well regulated clock always answers.

Let ZYPZ be the earth; ZFRz, its axis; abcd, &c. the equator; ABCDE, &c. the northern half of the ecliptic from Y to Z on the side of the globe next the eye; and MNOP, &c. the southern half on the opposite side from Z to Y. Let the points at A, B, C, D, E, F, &c. quite round from Y to Y again bound equal portions of the ecliptic, gone through in equal times by the real sun; and those at a, b, c, d, e, f, &c. equal portions of the equator described in equal times by the fictitious sun; and let ZYPZ be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun moves directly in the equator, with respect to the meridian; a degree, or any number of degrees, between YP and F on the ecliptic, must be nearer the meridian ZYPZ than a degree, or any corresponding number of degrees, on the equator from Y to F; and the more so, as they are the more oblique: and therefore the true sun comes sooner to the meridian every day whilst he is in the quadrant YF, than the fictitious sun does in the quadrant YF; for which reason, the solar noon precedes noon by the clock, until the real sun comes to F, and the fictitious to F; which two points, being equidistant from the meridian, both suns will come to it precisely at noon by the clock.

Whilst the real sun describes the second quadrant of the ecliptic FGHIKL from Cancer to Z, he comes later to the meridian every day, than the fictitious sun moving through the second quadrant of the equator from F to Z; for the points at G, H, I, K, and L, being farther from the meridian, their corresponding points at g, h, i, and l, must be later of coming to it: and as both suns come at the same moment to the point Z, they come to the meridian at the moment of noon by the clock.

In departing from Libra, through the third quadrant, the real sun going through MNOPQ towards YP at R, and the fictitious sun through MNOPQ towards YP; the former comes to the meridian every day sooner than the latter, until the real sun comes to YP, and the fictitious to YP; and then they both come to the meridian at the same time.

Lastly, as the real sun moves equably thro' STUVW, from YP towards Y; and the fictitious sun thro' STUVW, from YP towards Y; the former comes later every day to the meridian than the latter, until they both arrive at the point YP, and then they make it noon at the same time with the clock.

Having explained one cause of the difference of time shown by a well-regulated clock and a true sundial, and considered the sun, not the earth, as moving in the ecliptic; we now proceed to explain the other cause of this difference, namely, the inequality of the sun's apparent motion, which is slowest in summer, when the sun is farthest from the earth, and swiftest in winter when he is nearest to it. But the earth's motion on its axis is equable all the year round, and is performed from west to east; which is the way that the sun appears to change his place in the ecliptic.

If the sun's motion were equable in the ecliptic, the whole difference between the equal time as shown by the clock, and the unequal time as shown by the sun, would arise from the obliquity of the ecliptic. But the sun's motion sometimes exceeds a degree in 24 hours, though generally it is less; and when his motion is slowest, any particular meridian will revolve sooner to him than when his motion is quickest; for it will overtake him in less time when he advances a less space than Sect. VII.

when he moves through a larger.

Now, if there were two suns moving in the plane of the ecliptic, so as to go round it in a year; the one describing an equal arc every 24 hours, and the other describing sometimes a less arc in 24 hours, and at other times a larger, gaining at one time of the year what it lost at the opposite; it is evident, that either of these suns would come sooner or later to the meridian than the other; as it happened to be behind or before the other; and when they were both in conjunction, they would come to the meridian at the same moment.

As the real sun moves unequally in the ecliptic, let us suppose a fictitious sun to move equably in a circle coincident with the plane of the ecliptic. Let ABCD be the ecliptic or orbit in which the real sun moves, and the dotted circle about the imaginary orbit of the fictitious sun; each going round in a year according to the order of letters, or from west to east. Let HIKL be the earth turning round its axis the same way every 24 hours; and suppose both suns to start from A and a, in a right line with the plane of the meridian EH, at the same moment: the real sun at A, being then at his greatest distance from the earth, at which time his motion is slowest; and the fictitious sun at a, whose motion is always equable, because his distance from the earth is supposed to be always the same. In the time that the meridian revolves from H to H again, according to the order of the letters HIKL, the real sun has moved from A to F; and the fictitious with a quicker motion from a to f, thro' a large arc: therefore, the meridian EH will revolve sooner from H to E under the real sun at F, than from H E to k under the fictitious sun at f; and consequently it will then be noon by the sundial sooner than by the clock.

As the real sun moves from A towards C, the swiftness of his motion increases all the way to C, where it is at the quickest. But notwithstanding this, the fictitious sun gains so much upon the real, soon after his departing from A, that the increasing velocity of the real sun does not bring him up with the equally moving fictitious sun till the former comes to C, and the latter to c; when each has gone half round its respective orbit; and then being in conjunction, the meridian EH, revolving to EK, comes to both suns at the same time; and therefore it is noon by them both at the same moment.

But the increased velocity of the real sun, now being at the quickest, carries him before the fictitious one; and therefore, the same meridian will come to the fictitious sun sooner than to the real: for whilst the fictitious sun moves from c to g, the real sun moves thro' a greater arc from C to G; consequently the point K has its noon by the clock when it comes to k, but not its noon by the sun till it comes to l. And although the velocity of the real sun diminishes all the way from C to A, and the fictitious sun by an equable motion is still coming nearer to the real sun, yet they are not in conjunction till the one comes to A and the other to a, and then it is noon by them both at the same moment.

Thus it appears, that the solar noon is always later than noon by the clock whilst the sun goes from C to A; sooner, whilst he goes from A to C; and at these two points the sun and clock being equal, it is noon by them both at the same moment.

The point A is called the sun's apogee, because when he is there he is at his greatest distance from the earth; the point C his perigee, because when in it he is at his least distance from the earth: and a right line, as AEC, drawn through the earth's centre, from one of the points to the other, is called the line of the Apogees.

The distance that the sun has gone in any time from his apogee (not the distance he has to go to it, though ever so little) is called his mean anomaly, and is reckoned in signs and degrees, allowing 30 degrees to a sign. Thus, when the sun has gone suppose 174 degrees from his apogee at A, he is said to be 5 signs 14 degrees from it, which is his mean anomaly; and when he is gone suppose 355 degrees from his apogee, he is said to be 11 signs 25 degrees from it, although he be but 5 degrees short of A in coming round to it again.

From what was said above, it appears, that when the sun's anomaly is less than 6 signs, that is, when he is anywhere between A and C, in the half ABC of his orbit, the solar noon precedes the clock noon; but when his anomaly is more than 6 signs, that is, when he is anywhere between C and A, in the half CDA of his orbit, the clock noon precedes the solar. When his anomaly is 0 signs 0 degrees, that is, when he is in his apogee at A; or 6 signs 0 degrees, which is when he is in his perigee at C; he comes to the meridian at the moment that the fictitious sun does, and then it is noon by them both at the same instant.

Sect. VIII. Of calculating the Distances, Magnitudes, &c. of the Sun, Moon, and Planets.

This is accomplished by finding out the horizontal To find the parallax of the body whose distance you desire to know; that is, the angle under which the semidiameter of the earth would appear provided we could see it from that body: and this is to be found out in the following manner.

Let BAG be one half of the earth, AC its semidiameter, S the sun, m the moon, and EKOL a quarter of the circle described by the moon in revolving from the meridian to the meridian again. Let CRS be the rational horizon of an observer at A, extended to the sun in the heavens; and HAO, his sensible horizon extended to the moon's orbit. ALC is the angle under which the earth's semidiameter AC is seen from the moon at L; which is equal to the angle OAL, because the right lines AO and CL which include both these angles are parallel. ASC is the angle under which the earth's semidiameter AC is seen from the sun at S; and is equal to the angle OAF, because the lines AO and CRS are parallel. Now, it is found by observation, that the angle OAL is much greater than the angle OAF; but OAL is equal to ALC, and OAF is equal to ASC. Now as ASC is much less than ALC, it proves that the earth's semidiameter AC appears much greater as seen from the moon at L than from the sun at S; and therefore the earth is much farther from the sun than from the moon. The quantities of these angles may be determined by observation in the following manner.

Let a graduated instrument, as DAE (the larger the better), having a moveable index with sight-holes, be fixed in such a manner, that its plane surface may be parallel to the plane of the equator, and its edge AD. AD in the meridian: so that when the moon is in the equinoctial, and on the meridian ADE, she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at c, on the graduated limb DE; and when she is so seen, let the precise time be noted. Now as the moon revolves about the earth from the meridian to the meridian again in about 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that time, viz., in 6 hours 12 minutes, as seen from C, that is, from the earth's centre or pole. But as seen from A, the observer's place on the earth's surface, the moon will seem to have gone a quarter round the earth when she comes to the sensible horizon at O; for the index through the sights of which she is then viewed will be at d, 90 degrees from D, where it was when she was seen at E. Now let the exact moment when the moon is seen at O (which will be when she is in or near the sensible horizon) be carefully noted (o), that it may be known in what time she has gone from E to O; which time subtracted from six hours 12 minutes (the time of her going from E to L) leaves the time of her going from O to L, and affords an easy method for finding the angle OAL, (called the moon's horizontal parallax, which is equal to the angle ALC) by the following analogy: As the time of the moon's describing the arc EO is to 90 degrees, so is fix hours 12 minutes to the degrees of the arc DdE, which measures the angle EAL; from which subtract 90 degrees, and there remains the angle OAL, equal to the angle ALC, under which the earth's semidiameter AC is seen from the moon. Now, since all the angles of a right-lined triangle are equal to 180 degrees, or to two right angles, and the sides of a triangle are always proportional to the sines of the opposite angles, say, by the Rule of Three, As the sine of the angle ALC at the moon L, is to its opposite side AC the earth's semidiameter, which is known to be 3985 miles; so is radius, viz. the sine of 90 degrees, or of the right angle ACL, to its opposite side AL, which is the moon's distance at L from the observer's place at A on the earth's surface; or, so is the sine of the angle CAL to its opposite side CL, which is the moon's distance from the earth's centre, and comes out at a mean rate to be 240,000 miles. The angle CAL is equal to what OAL wants of 90 degrees.

Other methods have been fallen upon for determining the moon's parallax, of which the following is recommended as the best, by Mr Ferguson, tho' hitherto it has not been put in practice. "Let two observers be placed under the same meridian, one in the northern hemisphere, and the other in the southern, at such a distance from each other, that the arc of the celestial meridian included between their two zeniths may be at least 80 or 90 degrees. Let each observer take the distance of the moon's centre from his zenith, by means of an exceeding good instrument, at the moment of her passing the meridian: add these two zenith-distances of the moon together, and their excess above the distance between the two zeniths will be the distance between the two apparent places of the moon. Then, as the sum of the natural sines of the two zenith-distances of the moon is to radius, so is the distance between her two apparent places to her horizontal parallax: which being found, her distance from the earth's centre may be found by the analogy mentioned above.

Thus, in fig. 1. (2nd Plate L.) let VECQ be the earth, M the moon, and Zbax an arc of the celestial meridian. Let V be Vienna, whose latitude EV is 48° 20' north; and C the Cape of Good Hope, whose latitude EC is 34° 30' south: both which latitudes we suppose to be accurately determined before-hand by the observers. As these two places are on the same meridian πVEC, and in different hemispheres, the sum of their latitudes 82° 50' is their distance from each other. Z is the zenith of Vienna, and z the zenith of the Cape of Good Hope; which two zeniths are also 82° 50' distant from each other, in the common celestial meridian Zz. To the observer at Vienna, the moon's centre will appear at a in the celestial meridian; and at the same instant, to the observer at the Cape, it will appear at b. Now suppose the moon's distance Za from the zenith of Vienna to be 38° 1' 53", and her distance zb from the zenith of the Cape of Good Hope to be 46° 4' 41": the sum of these two zenith-distances (Za+zb) is 84° 6' 34"; from which subtract 82° 50', the distance of Zz between the zeniths of these two places, and there will remain 1° 16' 34" for the arc ba, or distance between the two apparent places of the moon's centre, as seen from V and from C. Then, supposing the tabular radius to be 10,000,000, the natural sine of 38° 1' 53" (the arc Za) is 6,160,816, and the natural sine of 46° 4' 41" (the arc zb) is 7,202,821: the sum of both these sines is 13,363,637. Say therefore, As 13,363,637 is to 10,000,000, so is 1° 16' 34" to 57° 18" which is the moon's horizontal parallax.

If the two places of observation be not exactly under the same meridian, their difference of longitude must be accurately taken, that proper allowance may be made for the moon's declination whilst she is passing from the meridian of the one to the meridian of the other.

The parallax, and consequently the distance and bulk, of any primary planet, might be found in the above manner, if the planet was near enough to the earth, so as to make the difference of its two apparent places sufficiently sensible: but the nearest planet is too remote for the accuracy required.

The sun's distance from the earth might be found the same way, though with more difficulty, if his horizontal parallax, or the angle OAS equal to the angle ASC, were not so small, as to be hardly perceptible, being found in this way to be scarce 10 seconds of a minute, or the 360th part of a degree. Hence all astronomers both ancient and modern have failed in taking the sun's parallax to a sufficient degree of exactness; but as some of the methods used are very ingenious, and show the great acuteness and sagacity of the ancient astronomers, we shall here give an account of them. The first method was invented by Hipparchus; and has been made use of by Ptolemy and his followers, and many other astronomers. It depends on an observation of an eclipse of the moon: And the principles on which it is founded are, In a lunar eclipse, the horizontal parallax of the sun is equal to the difference between the apparent semidiameter of the sun, sun, and half the angle of the conical shadow; which is easily made out in this manner. Let the circle AFG represent the sun, and DHE the earth; let DHM be the shadow, and DMC the half angle of the cone. Draw from the centre of the sun the right line SD touching the earth, and the angle DSC is the apparent semidiameter of the earth, seen from the sun, which is equal to the horizontal parallax of the sun; and the angle ADS is the apparent semidiameter of the sun, seen from the earth: The external angle ADS is equal to the two internals DSM and DSM, by the 32d Prop. Elem. I. And therefore the angle DSM, or DSC, is equal to the difference of the angles ADS and DSM. 2dly, Half the angle of the cone is equal to the difference of the horizontal parallax of the moon, and the apparent semidiameter of the shadow, seen from the earth at the distance of the moon. For let CDE be the earth, CME the shadow, which at the distance of the moon being cut by a plane, the section will be the circle FLH, whose semidiameter is FG, and is seen from the centre of the earth under the angle FTG. But by the 32d Prop. Elem. I. the angle CFT is equal to the two internals FMT and FTM. Wherefore the angle FMT is the difference of the two angles CFT and GTF: But the angle CFT is the angle under which the semidiameter of the earth is seen from the moon, and this is equal to the horizontal parallax of the moon; and the angle GTF is the apparent semidiameter of the shadow seen from the earth's centre. It is therefore evident that the half angle of the cone is equal to the difference of the horizontal parallax of the moon, and the apparent semidiameter of the shadow seen from the earth. Wherefore, if to the apparent semidiameter of the sun there be added the apparent semidiameter of the shadow, and from the sum you take away the horizontal parallax of the moon, there will remain the horizontal parallax of the sun; which therefore, if these were accurately known, would be likewise known accurately: But none of them can be so exactly and nicely obtained, as to be sufficient for determining the parallax of the sun; for very small errors, which cannot be easily avoided in measuring these angles, will produce very great errors in the parallax; and there will be a prodigious difference in the distances of the sun, when drawn from these parallaxes. For example: Suppose the horizontal parallax of the moon to be $6^\circ 15''$, the semidiameter of the sun $16'$, and the semidiameter of the shadow $44' 30''$; we shall conclude from thence, that the parallax of the sun was $15''$, and its distance from the earth about $13,700$ semidiameters of the earth. But if there be an error committed, in determining the semidiameter of the shadow, of $12''$ in defect (and certainly the semidiameter of the shadow cannot be had so precisely as not to be liable to such an error), that is, if instead of $44' 30''$ we put $44' 18''$ for the apparent diameter of the shadow, all the others remaining as before, we shall have the parallax of the sun $3''$, and its distance from the earth almost $70,000$ semidiameters of the earth, which is five times more than what it was by the first position. But if the fault were in excess, or the diameter of the shadow exceeded the true by $12''$, so that we should put in $44' 42''$ the parallax would arise to $27''$, and the distance of the sun only $7700$ of the earth's semidiameters; which is nine times less than what it comes to by a like error in defect. If an error in defect was committed of $15''$, which is still but a small mistake, the sun's parallax would be equal to nothing, and his distance infinite. Wherefore, since from so small mistakes the parallax and distance of the sun vary so much, it is plain that the distance of the sun cannot be obtained by this method.

Since, therefore, the angle that the earth's semidiameter subtends at the sun is so small, that it cannot be determined by any observation, Aristarchus Samius, an ancient and great philosopher and astronomer, contrived a very ingenious way for finding the angle which the semidiameter of the moon's orbit subtends when seen from the sun: This angle is about $60$ times bigger than the former, subtended only by the earth's semidiameter. To find this angle, he lays down the following principles.

From the phases of the moon, it hath been demonstrated, that if a plane passed through the moon's centre, to which the line joining the sun and moon's centre was perpendicular, this plane would divide the illuminated hemisphere of the moon from the dark one: And therefore, if this plane should likewise pass through the eye of a spectator on the earth, the moon would appear bisected, or like a half circle; and a right line, drawn from the earth to the centre of the moon, would be in the plane of illumination, and consequently would be perpendicular to the right line which joins the centres of the sun and moon. Let S be the sun, and T the earth, AL a quadrant of the moon's orbit; and let the line SL, drawn from the sun, touch the orbit of the moon in L; the angle TLS will be a right angle: And therefore, when the moon is seen in L, it will appear bisected, or just half a circle. At the same time take the angle LTS, the elongation of the moon from the sun, and then we shall have the angle LST, its complement to a right angle. But we have the side TL, by which we can find the side ST, the distance of the sun from the earth.

But the difficult point is to determine exactly the moment of time when the moon is bisected, or in its scientific true dichotomy; for there is a considerable space of time both before and after the dichotomy, nay even in the quadrature, when the moon will appear bisected, or half a circle; so that the exact moment of biflection cannot be known by observation, as experience tells us: And consequently, the true distance of the sun from the earth cannot be obtained by this method.

Since the moment in which the true dichotomy happens is uncertain, but it is certain that it happens before the quadrature; Ricciolus takes that point of Riccioli's time which is in the middle, between the time that the phasis begins to be doubtful whether it be bisected or not, and the time of quadrature: but he had done better, if he had taken the middle point between the time when it becomes doubtful whether the moon's side is concave or straight, and the time again when it is doubtful whether it is straight or convex; which point of time is after the quadrature: and if he had done this, he would have found the sun's distance a great deal more than he has made it.

There is no need to confine this method to the phases of a dichotomy or biflection, for it can be as well performed when the moon has any other phase bigger or less than a dichotomy: for observe by a very good telescope, telescope, with a micrometer, the phase of the moon, that is, the proportion of the illuminated part of the diameter to the whole; and at the same moment of time take her elongation from the sun: The illuminated part of the diameter, if it be less than the semidiameter, is to be subtracted from the semidiameter; but if it be greater, the semidiameter is to be subtracted from it, and mark the residue: then say, As the semidiameter of the moon is to the residue, so is the radius to the sine of an angle, which is therefore found: this angle added to, or subtracted from, a right angle, gives the exterior angle of the triangle at the moon: but we have the angle at the earth, which is the elongation observed; which therefore being subtracted from the exterior angle, leaves the angle at the sun. And in the triangle SLT, having all the angles and one side LT, we can find the other side ST, the distance of the sun from the earth. But it is almost impossible to determine accurately the quantity of the lunar phases, so that there may not be an error of a few seconds committed; and consequently, we cannot by this method find precisely enough the true distance of the sun.

However, from such observations, we are sure, that the sun is above 7000 semidiameters of the earth distant from us. Since therefore the true distance of the sun can neither be found by eclipses, nor by the phases of the moon, the astronomers are forced to recur to the parallaxes of the planets that are next to us, as Mars and Venus, which are sometimes much nearer to us than the sun is. Their parallaxes they endeavour to find by some of the methods above explained; and if these parallaxes were known, then the parallax and distance of the sun, which cannot directly by any observations be attained, would easily be deduced from them. For from the theory of the motions of the earth and planets, we know at any time the proportion of the distances of the sun and planets from us; and the horizontal parallaxes are in a reciprocal proportion to these distances. Wherefore, knowing the parallax of a planet, we may from thence find the parallax of the sun.

Mars, when he is in an achronycal position, that is, opposite to the sun, is twice as near to us as the sun is; and therefore his parallax will be twice as great. But Venus, when she is in her inferior conjunction with the sun, is four times nearer to us than he is; and her parallax is greater in the same proportion: Therefore, though the extreme smallness of the sun's parallax renders it unobservable by our senses, yet the parallaxes of Mars or Venus, which are twice or four times greater, may become sensible. The astronomers have bestowed much pains in finding out the parallax of Mars; but some time ago Mars was in his opposition to the sun, and also in his perihelion, and consequently in his nearest approach to the earth: And then he was most accurately observed by two of the most eminent astronomers of our age, who have determined his parallax to have been scarce 30 seconds; from whence it was inferred, that the parallax of the sun is scarce 1 second, and his distance about 19,000 semidiameters of the earth.

As the parallax of Venus is still greater than that of Mars, Dr Halley proposed a method by it of finding the distance of the sun to within a 500th part of the whole. The times of observation were at her transit over the sun in 1761 and 1769. At these times the greatest attention was given by astronomers, but it was found impossible to observe the exact times of immersion and emergence with such accuracy as had been expected; so that the matter is not yet determined so exactly as could be wished. The method of calculating the sun's distance by means of these transits, is as follows.

In fig 6. let DBA be the earth, V Venus, and TSR the eastern limb of the sun. To an observer at A, the point t of that limb will be on the meridian, its place referred to the heaven will be at E, and Venus will appear just within it at S. But at the same instant, to an observer at A, Venus is east of the sun, in the right line AVF; the point t of the sun's limb appears at e in the heavens; and if Venus were then visible, she would appear at F. The angle CVA is the horizontal parallax of Venus, which we seek; and is equal to the opposite angle FVE, whose measure is the arc FE. ASC is the sun's horizontal parallax, equal to the opposite angle e SE, whose measure is the arc e E; and FAe (the same as VAe) is Venus's horizontal parallax from the sun, which may be found by observing how much later in absolute time her ingress on the sun is, as seen from A than as seen from B, which is the time she takes to move from V to v in her orbit OVv.

It appears by the tables of Venus's motion and the sun's, that at the time of her transit in 1761 she moved 4' of a degree on the sun's disk in 60 minutes of time; and consequently 4" of a degree in one minute of time.

Now let us suppose, that A is 90° west of B, so that when it is noon at B it will be six in the morning at A; that the total ingress as seen from B is at one minute past 12, but that as seen from A it is at seven minutes 30 seconds past six; deduct six hours for the difference of meridians of A and B, and the remainder will be six minutes 30 seconds for the time by which the total ingress of Venus on the sun at S, is later as seen from A than as seen from B; which time being converted into parts of a degree is 26", or the arc Fe of Venus's horizontal parallax from the sun; for, as 1 minute of time is to 4 seconds of a degree, so is 6½ minutes of time to 26 seconds of a degree.

The times in which the planets perform their annual revolutions about the sun are already known by observation.—From these times, and the universal power of gravity by which the planets are retained in their orbits, it is demonstrable, that if the earth's mean distance from the sun be divided into 100,000 equal parts, Mercury's mean distance from the sun must be equal to 38,710 of these parts—Venus's mean distance from the sun, to 72,133—Mars's mean distance, 152,369—Jupiter's, 520,096—and Saturn's, 954,006. Therefore when the number of miles contained in the mean distance of any planet from the sun is known, we can by these proportions find the mean distance in miles of all the rest.

At the time of the above-mentioned transit, the earth's distance from the sun was 1015 (the mean distance being here considered as 1000), and Venus's distance from the sun 726 (the mean distance being considered as 723), which differences from the mean distances arise from the elliptical figure of the planets orbits— bits—Subtracting 726 parts from 1015, there remain 289 parts for Venus's distance from the earth at that time.

Now, since the horizontal parallaxes of the planets are inversely as their distances from the earth's centre, it is plain, that as Venus was between the earth and the sun on the day of her transit, and consequently her parallax at that time greater than the sun's, if her horizontal parallax was then ascertained by observation, the sun's horizontal parallax might be found, and consequently his distance from the earth.—Thus, suppose Venus's horizontal parallax was found to be 36°.34803; then, As the sun's distance 1015 is to Venus's distance 289, so is Venus's horizontal parallax 36°.34803 to the sun's horizontal parallax 10°.3493 on the day of her transit. And the difference of these two parallaxes, viz. 25°.9987 (which may be esteemed 26°), will be the quantity of Venus's horizontal parallax from the sun.

To find the sun's horizontal parallax at the time of his mean distance from the earth, say, As 1000 parts the sun's mean distance from the earth's centre, is to 1015, his distance therefrom on the day of the transit, so is 10°.3493, his horizontal parallax on that day, to 10°.5045, his horizontal parallax at the time of his mean distance from the earth's centre.

The sun's parallax being thus (or any other way supposed to be) found, at the time of his mean distance from the earth, we may find his true distance therefrom, in semidiameters of the earth, by the following analogy. As the fine (or tangent of so small an arc as that) of the sun's parallax 10°.5045 is to radius, so is unity or the earth's semidiameter to the number of semidiameters of the earth that the sun is distant from its centre; which number, being multiplied by 3985, the number of miles contained in the earth's semidiameter, will give the number of miles by which the sun is distant from the earth's centre.

Then, As 100,000, the earth's mean distance from the sun in parts, is to 38,710, Mercury's mean distance from the sun in parts, so is the earth's mean distance from the sun in miles to Mercury's mean distance from the sun in miles.—And,

As 100,000 is to 72,333, so is the earth's mean distance from the sun in miles to Venus's mean distance from the sun in miles.—Likewise,

As 100,000 is to 152,369, so is the earth's mean distance from the sun in miles to Mars's mean distance from the sun in miles.—Again,

As 100,000 is to 520,096, so is the earth's mean distance from the sun in miles to Jupiter's mean distance from the sun in miles.—Lastly,

As 100,000 is to 954,006, so is the earth's mean distance from the sun in miles to Saturn's mean distance from the sun in miles.

And thus, by having found the distance of any one of the planets from the sun, we have sufficient data for finding the distances of all the rest. And then from their apparent diameters at these known distances, their real diameters and bulk may be found. According to the calculations made from the transit in 1769, we have given the distance of each of the primary and secondary planets from one another, and from the sun. In Plate XLIII, their proportional bulks are shown, according to former calculations by Mr Ferguson; and in 2d Plate XLII, their relative magnitudes according to the latest calculations by Mr Dunn. In 3d Plate XLIII, fig. 3, 4, 5, are given three figures of Jupiter by Mr Wollaston; and in Plate XLVIII, fig. 1, the proportional distances of the satellites of Jupiter and Saturn, with the magnitudes of the sun, and orbit of our moon, by Mr Ferguson.

With regard to the fixed stars, no method of ascertaining their distance hath hitherto been found out. Those who have formed conjectures concerning them, have thought that they behoved to be at least 400,000 times farther from us than we are from the sun.

They are said to be fixed, because they have been generally observed to keep at the same distances from each other; their apparent diurnal revolutions being caused solely by the earth's turning on its axis. They appear of a sensible magnitude to the bare eye, because the retina is affected not only by the rays of light which are emitted directly from them, but by many thousands more, which falling upon our eye-lids, and upon the aerial particles about us, are reflected into our eyes so strongly as to excite vibrations not only in those points of the retina where the real images of the stars are formed, but also in other points at some distance round about. This makes us imagine the stars to be much bigger than they would appear if we saw them only by the few rays which come directly from them, so as to enter our eyes without being intermixed with others. Any one may be sensible of this, by looking at a star of the first magnitude through a long narrow tube; which, though it takes in as much of the sky as would hold 1000 such stars, yet scarce renders that one visible.

The more a telescope magnifies, the less is the aperture through which the star is seen; and consequently the fewer rays it admits into the eye. Now, since the stars appear less in a telescope which magnifies 200 times, than they do to the bare eye, inasmuch that they seem to be only indivisible points, it proves at once that the stars are at immense distances from us, and that they shine by their own proper light. If they shone by borrowed light, they would be as invisible without telescopes as the satellites of Jupiter are; for these satellites appear bigger when viewed with a good telescope than the largest fixed stars do.

The number of stars discoverable, in either hemisphere, by the naked eye, is not above a thousand. This at first may appear incredible, because they seem to be without number; but the deception arises from our looking confusedly upon them, without reducing them into order. For, look but steadfastly upon a pretty large portion of the sky, and count the number of stars in it, and you will be surprized to find them so few. Or, if one considers how seldom the moon meets with any stars in her way, although there are as many about her path as in other parts of the heavens, he will soon be convinced that the stars are much thinner down than he was aware of. The British catalogue, which, besides the stars visible to the bare eye, includes a great number which cannot be seen without the assistance of a telescope, contains no more than 3000, in both hemispheres.

As we have incomparably more light from the moon than from all the stars together, it were the greatest absurdity to imagine that the stars were made for no other purpose than to cast a faint light upon the earth; especially especially since many more require the assistance of a good telescope to find them out than are visible without that instrument. Our sun is surrounded by a system of planets and comets; all which would be invisible from the nearest fixed star. And from what we already know of the immense distance of the stars, the nearest may be computed at $32,000,000,000,000$ miles from us, which is farther than a cannon-bullet would fly in $7,000,000$ years. Hence it is easy to prove, that the sun, from such a distance, would appear no bigger than a star of the first magnitude. From all this it is highly probable, that each star is a sun to a system of worlds moving round it, tho' unseen by us; especially as the doctrine of a plurality of worlds is rational, and greatly manifests the power, wisdom, and goodness, of the great Creator. See § 66.

The stars, on account of their apparently various magnitudes, have been distributed into several classes, or orders. Those which appear largest are called stars of the first magnitude; the next to them in lustre, stars of the second magnitude; and so on to the sixth, which are the smallest that are visible to the bare eye. This distribution having been made long before the invention of telescopes, the stars which cannot be seen without the assistance of these instruments are distinguished by the name of telescopic stars.

The ancients divided the starry sphere into particular constellations, or systems of stars, according as they lay near one another, so as to occupy those spaces which the figures of different sorts of animals or things would take up, if they were there delineated. And those stars which could not be brought into any particular constellation were called unformed stars.

This division of the stars into different constellations, or asterisms, serves to distinguish them from one another, so that any particular star may be readily found in the heavens by means of a celestial globe; on which the constellations are so delineated, as to put the most remarkable stars into such parts of the figures as are most easily distinguished. The number of the ancient constellations is 48, and upon our present globes about 70. On Senex's globes are inserted Bayer's letters; the first in the Greek alphabet being put to the biggest star in each constellation, the second to the next, and so on: by which means, every star is as easily found as if a name were given to it. Thus, if the star γ in the constellation of the ram be mentioned, every astronomer knows as well what star is meant as if it were pointed out to him in the heavens.

There is also a division of the heavens into three Divisions of parts. 1. The zodiac (κατάλογος), from κατάλογος, the heavens, an animal, because most of the constellations in it, which are twelve in number, have the names of animals: As Aries the ram, Taurus the bull, Gemini the twins, Cancer the crab, Libra the balance, Scorpio the scorpion, Sagittarius the archer, Capricornus the goat, Aquarius the water-bearer, and Pisces the fishes. The zodiac goes quite round the heavens: it is about 16 degrees broad, so that it takes in the orbits of all the planets, and likewise the orbit of the moon. Along the middle of this zone or belt is the ecliptic, or circle which the earth describes annually as seen from the sun, and which the sun appears to describe as seen from the earth. 2. All that region of the heavens which is on the north side of the zodiac, containing 21 constellations. And, 3. That on the south side, containing 15.

The ancients divided the zodiac into the above 12 constellations or signs in the following manner. They took a vessel with a small hole in the bottom, and, having filled it with water, suffered the same to drip drop by drop into another vessel set beneath to receive it; beginning at the moment when some star rose, and continuing till it rose the next following night. The water falling down into the receiver they divided into twelve equal parts; and having two other small vessels in readiness, each of them fit to contain one part, they again poured all the water into the upper vessel; and, observing the rising of some star in the zodiac, they at the same time suffered the water to drop into one of the small vessels; and as soon as it was full, they shifted it, and set an empty one in its place. When each vessel was full, they took notice what star of the zodiac rose; and thus this could not be done in one night, yet in many they observed the rising of 12 stars or points, by which they divided the zodiac into 12 parts.

The names of the constellations, and the number of stars observed in each of them by different astronomers, are as follow.

| The ancient Constellations | Ptolemy | Tycho | Hevelius | Flamsteed | |---------------------------|---------|-------|----------|-----------| | Ursa minor | 8 | 7 | 12 | 24 | | Ursa major | 35 | 29 | 73 | 87 | | Draco | 31 | 32 | 40 | 80 | | Cepheus | 13 | 4 | 51 | 35 | | Bootes, Arctophilax | 23 | 18 | 52 | 54 | | Corona Borealis | 8 | 8 | 8 | 21 | | Hercules, Engonasin | 29 | 28 | 45 | 113 | | Lyra | 10 | 11 | 17 | 21 | | Cygnus, Gallina | 19 | 18 | 47 | 81 | | Cassiopea | 13 | 26 | 37 | 55 | | Perseus | 29 | 29 | 46 | 59 | | Auriga | 14 | 9 | 40 | 66 | | Serpentarius, Ophiuchus | 29 | 15 | 40 | 74 | | Serpens | 18 | 13 | 22 | 64 | | Sagitta | 5 | 5 | 5 | 18 | | Aquila, Vultur | 15 | 12 | 23 | 71 | | Antinous | 3 | 3 | 19 | 71 | | Delphinus | 10 | 10 | 14 | 18 | | Equuleus, Equi sellio | 4 | 4 | 6 | 10 |

The ### Sect. VIII. ASTRONOMY.

#### The ancient Constellations.

| Constellation | Ptolemy | Tycho | Hevelius | Flamsteed | |---------------------|---------|-------|----------|-----------| | Pegasis, Equus | | | | | | Andromeda | | | | | | Triangulum | | | | | | Aries | | | | | | Taurus | | | | | | Gemini | | | | | | Cancer | | | | | | Leo | | | | | | Coma Berenices | | | | | | Virgo | | | | | | Libra, Chelae | | | | | | Scorpius | | | | | | Sagittarius | | | | | | Capricornus | | | | | | Aquarius | | | | | | Pisces | | | | | | Cetus | | | | | | Orion | | | | | | Eridanus, Fluvius | | | | | | Lepus | | | | | | Canis major | | | | | | Canis minor | | | | | | Argo Navis | | | | | | Hydra | | | | | | Crater | | | | | | Corvus | | | | | | Centaurus | | | | | | Lupus | | | | | | Ara | | | | | | Corona Australis | | | | | | Piscis Australis | | | | |

#### The new Southern Constellations.

| Constellation | Ptolemy | Tycho | Hevelius | Flamsteed | |----------------------|---------|-------|----------|-----------| | Columba Noachi | | | | | | Robur Carolinum | | | | | | Grus | | | | | | Phoenix | | | | | | Indus | | | | | | Pavo | | | | | | Apus, Avis Indica | | | | | | Apis, Musca | | | | | | Chameleon | | | | | | Triangulum Australis | | | | | | Piscis volans, Passer| | | | | | Dorado, Xiphias | | | | | | Toucan | | | | | | Hydrus | | | | |

#### Hevelius's Constellations made out of the unformed Stars.

| Constellation | Hevel. | Flamst. | |----------------------|--------|---------| | Lynx | The Lynx | 19 44 | | Leo minor | The Little Lion | 53 | | Asterion & Chara | The Greyhounds | 23 25 | | Cerberus | Cerberus | 4 | | Vulpecula & Anser | The Fox and Goose | 27 35 | | Scutum Sobieski | Sobielki's Shield | 7 | | Lacerta | The Lizard | 10 16 | | Camelopardalus | The Camelopard | 32 58 | | Monocorns | The Unicorn | 19 31 | | Sextans | The Sextant | 11 41 (p) |

---

Some of the stars, particularly Arcturus, have been observed to change their places above a minute of a degree with respect to others. But whether this be owing to any real motion in the stars themselves, must require the observations of many ages to determine. If our solar system changeth its place with regard to absolute space, this must in process of time occasion an apparent change in the distances of the stars from each other; and in such a case, the places of the nearest stars to us being more affected than those which are very remote, their relative positions must seem to alter, tho' the stars themselves were really immoveable. On the other hand, if the stars are fixed in their places, and if the earth moves in its orbit, then the stars will appear to move with respect to each other.

---

(p) To the conjectures mentioned no. 65.—70., concerning the disappearance of some stars, we may add that of Mr. Maupertuis, who, in his Dissertation on the figures of the celestial bodies (p. 61.—63.), is of opinion, that some stars, by their prodigious quick rotations on their axes, may not only assume the figures of oblate spheroids, but that, by the great centrifugal force arising from such rotations, they may become of the figures of mill-stones; or be reduced to flat circular planes, so thin as to be quite invisible when their edges are turned towards us; as Saturn's ring is in such positions. But when very eccentric planets or comets go round any flat star, in orbits much inclined to its equator, the attraction of the planets or comets in their perihelions must alter the inclination of the axis of that star; on which account it will appear more or less large and luminous, as its broad side is more or less turned towards us. And thus he imagines we may account for the apparent changes of magnitude and lustre in those stars, and likewise for their appearing and disappearing. other hand, if our own system be at rest, and any of the stars in real motion, this must vary their positions; and the more so, the nearer they are to us, or swifter their motions are; or the more proper the direction of their motions is for our perception. See note on n° 89.

The obliquity of the ecliptic to the equinoctial is found at present to be above the third part of a degree less than Ptolemy found it. And most of the observers after him found it to decrease gradually down to Tycho's time. If it be objected, that we cannot depend on the observations of the ancients, because of the incorrect- ness of their instruments; we have to answer, that both Tycho and Flamsteed are allowed to have been very good observers; and yet we find that Flamsteed makes this obliquity 2½ minutes of a degree less than Tycho did about 100 years before him: and as Ptolemy was 1324 years before Tycho, so the gradual decrease an- swers nearly to the difference of time between these three astronomers. If we consider, that the earth is not a perfect sphere, but an oblate spheroid, having its axis shorter than its equatorial diameter; and that the sun and moon are constantly acting obliquely upon the greater quantity of matter about the equator, pulling it, as it were, towards a nearer and nearer coincidence with the ecliptic; it will not appear improbable that these actions should gradually diminish the angle be- tween those planes. Nor is it less probable that the mutual attractions of all the planets should have a ten- dency to bring their orbits to a coincidence: but this change is too small to become sensible in many ages.

Sect. IX. Of calculating the periodical Times, Pla- ces, &c. of the Sun, Moon, and Planets; Deli- neation of the Phases of the Moon for any parti- cular time; and the Construction of Astronomical Tables.

This title includes almost all of what may be called the Practical part of Astronomy; and as it is by far the most difficult and arbitrary, so the thorough investiga- tion of it would necessarily lead us into very deep geo- metrical demonstrations. The great labours of former astronomers have left little for succeeding ones to do in this respect: tables of the motions of all the celestial bodies have been made long ago, the periodical times, eccentricities, &c. of the planets determined; and as we suppose few will desire to repeat these laborious ope- rations, we shall here content ourselves with giving some general hints of the methods by which these things have been originally accomplished, that so the opera- tions of the young astronomer who makes use of tables already formed to his hand may not be merely mechanical.

It hath been already observed*, that the foundation of all astronomical operations was the drawing a meridian line. This being done, the next thing is to find out the latitude of the place where the observations are to be made, and for which the meridian line is drawn. From what hath been said n° 3. it will easily be un- derstood that the latitude of a place must always be equal to the elevation either of the north or south pole above the horizon; because when we are exactly on the equator, both poles appear in the horizon. There is, however, no star exactly in either of the celestial poles; therefore, to find the altitude of that invisible point call- ed the Pole of the Heavens, we must choose some star near it which does not set; and having by several ob- servations, according to the directions given n° 177, found its greatest and least altitudes, divide their dif- ference by 2; and half that difference added to the least, or subtracted from the greatest, altitude of the star, gives the exact altitude of the pole or latitude of the place. Thus, suppose the greatest altitude of the star observed is 60°, and its least 50°, we then know that the latitude of the place where the observation was made is exactly 55°.

The latitude being once found, the obliquity of the ecliptic, or the angle made by the sun's annual path with the earth's equator, is easily obtained by the fol- lowing method. Observe, about the summer solstice, the sun's meridian distance from the zenith, which is easily done by a quadrant with a moveable index furnished with sights; if this distance is subtracted from the latitude of the place, provided the sun is nearer the equator than the place of observation, the remainder will be the ob- liquity of the ecliptic. But if the place of observa- tion is nearer the equator than the sun at that time, the zenith distance must be added. By this method, the obli- quity of the ecliptic hath been determined to be 23° 29'.

By the same method the declination of the sun from the equator for any day may be found; and thus a ta- ble of his declination for every day in the year might be constructed: thus also the declination of the stars might be found.

Having the declination of the sun, his right ascen- sion and place in the ecliptic may be geometrically found by the solution of a case in spherical trigono- metry. For let EQ represent the celestial equator, y the sun, and y X the ecliptic; then, in the right-angled spherical triangle ECy, we have the side Ey, equal to the sun's declination: the angle ECy is always 23° 29', being the angle of the ecliptic with the equator; and the angle y EC is 90°, or a right angle. From these data we can find the side EC the right ascension; and Cy the sun's place in the ecliptic, or his distance from the equinoctial point; and thus a table of the sun's place for every day in the year, answerable to his de- clination, may be formed.

Having the sun's place in the ecliptic, the right as- cension of the stars may be found by the help of it and right as- cension of the stars.

To find the

Latitude of any place how found. to be found. This time converted into hours and minutes of the equator, will give the difference of right ascensions; from whence, by addition, we collect the right ascension of the star which was to be found out.

The right ascension and declination of a star being known, its longitude, and latitude, or distance from the first star of Aries, and north or south from the ecliptic, may thence be easily found, from the solution of a case in spherical trigonometry, similar to that already mentioned concerning the sun's place; and the places of the fixed stars being all marked in a catalogue according to their longitudes and latitudes, it may thence be conceived how the longitude and latitude of a planet or comet may be found for any particular time by comparing its distance from them, and its apparent path may thus be traced; and thus the paths of Mercury and Venus were traced by M. Cassini, though Mr Ferguson made use of an error for that purpose.

With regard to the planets, the first thing to be done is to find out their periodical times, which is done by observing when they have no latitude. At that time the planet is in the ecliptic, and consequently in one of its nodes; so that, by waiting till it returns to the same node again, and keeping an exact account of the time, the periodical time of its revolution round the sun may be known pretty exactly. By the same observations, from the theory of the earth's motion we can find the position of the line of the nodes; and when once the position of this line is found, the angle of inclination of that planet's orbit to the earth may also be known.

The eccentricity of the earth's orbit may be determined by observing the apparent diameters of the sun at different times; when the sun's diameter is least, the earth is at the greatest distance; and when this diameter is greatest, the earth is at its least distance from him. But as this method must necessarily be precarious, another is recommended by Dr Keil, by observing the velocity of the earth in its orbit, or the apparent velocity of the sun, which is demonstrated to be always reciprocally as the square of the distance.

The eccentricities of the orbits of the other planets may be likewise found by observing their velocities at different times; for all of them observe the same proportions with regard to the increase or decrease of their velocity that the earth does; only, in this case, care must be taken to observe the real, not the apparent, velocities of the planets, the last depending on the motion of the earth at the same time. Their aphelia, or points of their orbits where they are farthest from the sun, may be known by making several observations of their distances from him, and thus perceiving when these distances cease to increase.

The position of the aphelion being determined, the planet's distance from it at any time may also be found by observation, which is called its true or equated anomaly; but by supposing the motion of the planet to be regular and uniform, tables of that motion may easily be constructed. From thence the planet's mean place in its orbit may be found for any moment of time; and one of these moments being fixed upon as an epoch or beginning of the table, it is easy to understand, that from thence tables of the planet's place in its orbit for any number of years either preceding or consequent to that period may be constructed. These tables are to be constructed according to the meridian of equal time, and not true or apparent time, because of the inequalities of the earth's motion as well as of that of the planet, and equations must be made to be added to or subtracted from the mean motion of the planet, as occasion requires; which will be readily understood from what we have already mentioned concerning the unequal motion of the earth in its orbit. When all the necessary tables are constructed by this or similar methods, the calculating of the planetary places becomes a mere matter of mechanism, and consists only in the proper additions and subtractions according to the directions always given along with such tables.

It must be observed, however, that the accidental interference of the planets with one another by their mutual attractions render it impossible to construct any tables that shall remain equally perfect; and therefore frequent actual observations and corrections of the tables will be necessary. This disturbance, however, is inconsiderable, except in the planets Jupiter and Saturn, and they are in conjunction only once in 800 years.

What hath been already mentioned with regard to the planets, is also applicable to the moon; but with more difficulty, on account of the greater inequalities of her motions. She indeed moves in an ellipse as the rest do, and its eccentricity may be better computed from observing her diameter at different times than that of the earth's orbit; but that eccentricity is not always the same. The reason of this, and indeed of all the other lunar inequalities, is, that the sun has a sensible effect upon her by his attraction, as well as the earth. Consequently, when the earth is at its least distance from the sun, her orbit is dilated, and she moves more slowly; and, on the contrary, when the earth is in its aphelion, her orbit contracts, and she moves more swiftly. The eccentricity is always greatest when the line of the apsides coincides with that of the syzygies, and the earth at its least distance from the sun. When the moon is in her syzygies, i.e., in the line that joins the centres of the earth and sun, which is either in her conjunction or opposition, she moves swifter, ceteris paribus, than in the quadratures. According to the different distances of the moon from the syzygies, she changes her motion: from the conjunction to her first quadrature, she moves somewhat slower; but recovers her velocity in the second quarter. In the third quarter she again loses, and in the last again recovers it. The apogee of the moon is also irregular; being found to move forward when it coincides with the line of the syzygies, and backwards when it cuts that line at right angles. Nor is this motion in any degree equal: in the conjunction or opposition, it goes briskly forwards; and, in the quadratures, moves either slowly forwards, stands still, or goes backward. The motion of the nodes has been already taken notice of; but this motion is not uniform more than the rest; for when the line of the nodes coincides with that of the syzygies, they stand still; when their line cuts that at right angles, they go backwards, with the velocity, as Sir Isaac Newton hath shewn, of $16°, 19′, 24″$ an hour.

The only equable motion the moon has, is her revolution on her axis, which she always performs exactly in the space of time in which she moves round the earth. From hence arises what is called the Moon's libration; for as the motion round her axis is equable, and that in her orbit unequal, it follows, that when the moon is in her perigee, where she moves swiftest, that part of her surface, which on account of the motion in her orbit would be turned from the earth, is not so, by reason of the motion on her axis. Thus some parts in the limb or margin of the moon sometimes recede from, and sometimes approach towards, the centre of the disk. Yet this equable rotation produces an apparent irregularity; for the axis of the moon, not being perpendicular, but a little inclined to its orbit, and this axis maintaining its parallelism round the earth, it must necessarily change its situation with respect to an observer on the earth, to whom sometimes the one and sometimes the other pole of the moon becomes visible; whence it appears to have a kind of wavering or vacillatory motion.

From all these irregularities it may well be concluded, that the calculation of the moon's place in her orbit is a very difficult matter; and indeed, before Sir Isaac Newton, astronomers in vain laboured to subject the lunar irregularities to any rule. By his labours, however, and those of other astronomers, these difficulties are in a great measure overcome; and calculations with regard to this luminary may be made with as great certainty as concerning any other. Her periodical time may be determined from the observation of two lunar eclipses, at as great a distance from one another as possible; for in the middle of every lunar eclipse, the moon is exactly in opposition to the sun. Compute the time between these two eclipses or oppositions, and divide this by the number of lunations that have intervened, and the quotient will be the synodical month, or time the moon takes to pass from one conjunction to another, or from one opposition to another. Compute the sun's mean motion in the time of the synodical month, and add this to the entire circle described by the moon. Then, as that sum is to $360^\circ$, so is the quantity of the synodical month to the periodical, or time that the moon takes to move from one point of her orbit to the same point again. Thus, Copernicus in the year 1500, November 6th, at 2 hours 20 minutes, observed an eclipse of the moon at Rome; and August 1st, 1523, at 4 hours 25 minutes, another at Cracow: hence the quantity of the synodical month is thus determined

| Observ. 2d | 1573 | 237 | 4 | 25 | | Observ. 1st | 1500 | 210 | 2 | 20 |

Interval of time 22 292 2 5

Add the intercalary days for leap years 5

Exact interval 22 297 2 5,011991005'

This interval divided by 282, the number of months elapsed in that time, gives 29 days 12 hours 41 minutes for the length of the synodical month. But from the observations of two other eclipses, the same author more accurately determined the quantity of the synodical month to be 29 degrees 11 hours 45 minutes 3 seconds; from whence the mean periodical time of the moon comes to be 27 degrees 7 hours 43 minutes 5 seconds, which exactly agrees with the observations of later astronomers.

The quantity of the periodical month being given, by the Rule of Three we may find the moon's diurnal and horary motion; and thus may tables of the moon's mean motion be constructed; and if from the moon's mean diurnal motion that of the sun be subtracted, the remainder will be the moon's mean diurnal motion from the sun.

Having the moon's distance from the sun, her phasis for that time may be easily delineated by the following method laid down by Dr Keil. "Let the circle COBP represent the disk of the moon, which is turned towards the earth; and let OP be the line in which the semicircle OMP is projected, which suppose to be cut by the diameter BC, at right angles; and, making LP the radius, take LF equal to the cosine of the elongation of the moon from the sun: And then upon BC, as the great axis, and LF the lesser axis, describe the semi-ellipse BFC. This ellipse will cut off from the disk of the moon the portion BFCP of the illuminated face, which is visible to us from the earth."

Since in the middle of a total eclipse the moon is exactly in the node, if the sun's place be found for that time, and six figures added to it, if the eclipse is a lunar one the sun will give the place of the node, or if the eclipse observed is a solar one, the place of the node and of the sun are the same. From comparing two eclipses together, the mean motion of the nodes will thus be found out. The apogee of the moon may be known from her apparent diameter, as already observed; and by comparing her place when in the apogee at different times, the motion of the apogee itself may also be determined.

These short hints will be sufficient to give a general knowledge of the methods used for the solution of some of the most difficult problems in astronomy. As for the proper equations to be added or subtracted, in order to find out the true motion and place of the moon, together with the particular methods of constructing tables for calculating eclipses, they are given from Mr Ferguson, in the following section.

Sect. X. Of Eclipses: With Tables for their Calculation; the method of constructing them; rules for calculation, and directions for the delineation, of Solar and Lunar Eclipses.

Every planet and satellite is illuminated by the sun; and casts a shadow towards that point of the heavens which is opposite to the sun. This shadow is nothing but a privation of light in the space hid from the sun by the opaque body that intercepts his rays.

When the sun's light is so intercepted by the moon, that to any place of the earth the sun appears partly or wholly covered, he is said to undergo an eclipse; though, properly speaking, it is only an eclipse of that part of the earth where the moon's shadow or penumbra falls. When the earth comes between the sun and moon, the moon falls into the earth's shadow; and, having no light of her own, she suffers a real eclipse from the interception of the sun's rays. When the sun is eclipsed to us, the moon's inhabitants, on the side next the earth, see her shadow like a dark spot travelling over the earth, about twice as fast as its equatorial parts move, and the same way as they move. When the moon is in an eclipse, the sun appears eclipsed to her, total to all those parts on which the earth's shadow falls, and of as long continuance as they are in the shadow. That the earth is spherical (for the hills take off no more from the roundness of the earth, than grains of dust do from the roundness of a common globe) is evident from the figure of its shadow on the moon; which is always bounded by a circular line, although the earth is incessantly turning its different sides to the moon, and very seldom shows the same side to her in different eclipses, because they seldom happen at the same hours. Were the earth shaped like a round flat plate, its shadow would only be circular when either of its sides directly faced the moon, and more or less elliptical as the earth happened to be turned more or less obliquely towards the moon when she is eclipsed. The moon's different phases prove her to be round; for, as she keeps still the same side towards the earth, if that side were flat, as it appears to be, she would never be visible from the third quarter to the first; and from the first quarter to the third, she would appear as round as when we say she is full; because, at the end of her first quarter, the sun's light would come as suddenly on all her side next the earth, as it does on a flat wall, and go off as abruptly at the end of her third quarter.

If the earth and sun were equally large, the earth's shadow would be infinitely extended, and all of the same bulk; and the planet Mars, in either of its nodes and opposite to the sun, would be eclipsed in the earth's shadow. Were the earth larger than the sun, its shadow would increase in bulk the farther it extended, and would eclipse the great planets Jupiter and Saturn, with all their moons, when they were opposite to the sun. But as Mars, in opposition, never falls into the earth's shadow, although he is not then above 42,000,000 miles from the earth, it is plain that the earth is much less than the sun; for otherwise its shadow could not end in a point at so small a distance. If the sun and moon were equally large, the moon's shadow would go on to the earth with an equal breadth, and cover a portion of the earth's surface more than 2000 miles broad, even if it fell directly against the earth's centre, as seen from the moon; and much more if it fell obliquely on the earth: But the moon's shadow is seldom 150 miles broad at the earth, unless when it falls very obliquely on the earth, in total eclipses of the sun. In annular eclipses, the moon's real shadow ends in a point at some distance from the earth. The moon's small distance from the earth, and the shortness of her shadow, prove her to be less than the sun. And, as the earth's shadow is large enough to cover the moon, if her diameter were three times as large as it is (which is evident from her long continuance in the shadow when she goes through its centre), it is plain, that the earth is much bigger than the moon.

Though all opaque bodies, on which the sun shines, have their shadows, yet such is the bulk of the sun, and the distances of the planets, that the primary planets can never eclipse one another. A primary can eclipse only its secondary, or be eclipsed by it; and never but when in opposition or conjunction with the sun. The primary planets are very seldom in these positions, but the sun and moon are so every month: Whence one may imagine, that these two luminaries should be eclipsed every month. But there are few eclipses in respect of the number of new and full moons; the reason of which we shall now explain.

If the moon's orbit were coincident with the plane of the ecliptic, in which the earth always moves and the sun appears to move, the moon's shadow would fall upon the earth at every change, and eclipse the sun to some parts of the earth. In like manner, the moon would go through the middle of the earth's shadow, and be eclipsed at every full; but with this difference, that she would be totally darkened for above an hour and a half; whereas the sun never was above four minutes totally eclipsed by the interposition of the moon. But one half of the moon's orbit is elevated 5° degrees above the ecliptic, and the other half as much depressed below it; consequently, the moon's orbit intersects the ecliptic in two opposite points called the moon's nodes, as has been already taken notice of. When these points are in a right line with the centre of the sun at new or full moon, the sun, moon, and earth, are all in a right line; and if the moon be then new, her shadow falls upon the earth; if full, the earth's shadow falls upon her. When the sun and moon are more than 17 degrees from either of the nodes at the time of conjunction, the moon is then generally too high or too low in her orbit to cast any part of her shadow upon the earth; when the sun is more than 12 degrees from either of the nodes at the time of full moon, the moon is generally too high or too low in her orbit to go through any part of the earth's shadow; and in both these cases there will be no eclipse. But when the moon is less than 17 degrees from either node at the time of conjunction, her shadow or penumbra falls more or less upon the earth, as she is more or less within this limit. And when she is less than 12 degrees from either node at the time of opposition, she goes through a greater or less portion of the earth's shadow, as she is more or less within this limit. Her orbit contains 360 degrees; of which 17, the limit of solar eclipses on either side of the nodes, and 12, the limit of lunar eclipses, are but small portions: And as the sun commonly passes by the nodes but twice in a year, it is no wonder that we have so many new and full moons without eclipses.

To illustrate this, (Plate L. fig. 1.) let ABCD be the ecliptic, RSTU a circle lying in the same plane with the ecliptic, and VXYZ the moon's orbit, all thrown into an oblique view, which gives them an elliptical shape to the eye. One half of the moon's orbit, as VWX, is always below the ecliptic, and the other half XYV above it. The points V and X, where the moon's orbit intersects the circle RSTU, which lies even with the ecliptic, are the moon's nodes; and a right line, as XEV, drawn from one to the other, through the earth's centre, is the line of the nodes, which is carried almost parallel to itself round the sun in a year.

If the moon moved round the earth in the orbit RSTU, which is coincident with the plane of the ecliptic, her shadow would fall upon the earth every time she is in conjunction with the sun, and at every opposition she would go through the earth's shadow. Were this the case, the sun would be eclipsed at every change, and the moon at every full, as already mentioned.

But although the moon's shadow N must fall upon the earth at σ, when the earth is at E, and the moon in conjunction with the sun at i, because she is then very near one of her nodes; and at her opposition n she must go through the earth's shadow J, because she is then near the other node; yet, in the time that she goes round: round the earth to her next change, according to the order of the letters XYVVW, the earth advances from E to e, according to the order of the letters EFGH; and the line of the nodes VEX, being carried nearly parallel to itself, brings the point f of the moon's orbit in conjunction with the sun at that next change; and then the moon being at f, is too high above the ecliptic to cast her shadow on the earth; and as the earth is still moving forward, the moon at her next opposition will be at g, too far below the ecliptic to go through any part of the earth's shadow; for by that time the point g will be at a considerable distance from the earth as seen from the sun.

When the earth comes to F, the moon in conjunction with the sun Z is not at k in a plane coincident with the ecliptic, but above it at Y in the highest part of her orbit; and then the point b of her shadow O goes far above the earth (as in fig. 2, which is an edge view of fig. 1). The moon at her next opposition, is not at s (fig. 1), but at W, where the earth's shadow goes far above her (as in fig. 2). In both these cases the line of the nodes VFX (fig. 1) is about ninety degrees from the sun, and both luminaries are as far as possible from the limits of the eclipses.

When the earth has gone half round the ecliptic from E to G, the line of the nodes VGX is nearly, if not exactly, directed towards the sun at Z; and then the new-moon l casts her shadow P on the earth G; and the full moon p goes through the earth's shadow L; which brings on eclipses again, as when the earth was at E.

When the earth comes to H, the new moon falls not at m in a plane coincident with the ecliptic CD, but at W in her orbit below it; and then her shadow Q (see fig. 2.) goes far below the earth. At the next full she is not at q (fig. 1) but at Y in her orbit 5½ degrees above q, and at her greatest height above the ecliptic CD; being then as far as possible, at any opposition, from the earth's shadow M, as in fig. 2.

So, when the earth is at E and G, the moon is about her nodes at new and full, and in her greatest north and south declination (or latitude as it is generally called) from the ecliptic at her quarters; but when the earth is at F or H, the moon is in her greatest north and south declination from the ecliptic at new and full, and in the nodes about her quarters.

The point X, where the moon's orbit crosses the ecliptic, is called the ascending node, because the moon ascends from it above the ecliptic; and the opposite point of intersection V is called the descending node, because the moon descends from it below the ecliptic. When the moon is at Y in the highest point of her orbit, she is in her greatest north latitude; and when she is at W in the lowest point of her orbit, she is in her greatest south latitude.

If the line of the nodes, like the earth's axis, was carried parallel to itself round the sun, there would be just half a year between the conjunctions of the sun and nodes. But the nodes shift backwards, or contrary to the earth's annual motion, 19½ deg. every year; and therefore the same node comes round the sun 19 days sooner every year than on the year before. Consequently, from the time that the ascending node X (when the earth is at E) passes by the sun as seen from the earth, it is only 173 days (not half a year) till the descending node V passes by him. Therefore in whatever time of the year we have eclipses of the luminaries about either node, we may be sure that in 173 days afterward we shall have eclipses about the other node. And when at any time of the year the line of the nodes is in the situation VGX, at the same time next year it will be in the situation rGx; the ascending node having gone backward, that is, contrary to the order of signs, from X to z, and the descending node from V to r; each 19½ deg. At this rate, the nodes shift through all the signs and degrees of the ecliptic in 18 years and 225 days; in which time there would always be a regular period of eclipses, if any complete number of lunations were finished without a fraction. But this never happens: for if both the sun and moon should start from a line of conjunction with either of the nodes in any point of the ecliptic, the sun would perform 18 annual revolutions and 222 degrees over and above, and the moon 230 lunations and 85 degrees of the 231st, by the time the node came round to the same point of the ecliptic again; so that the sun would then be 138 degrees from the node, and the moon 85 degrees from the sun.

But, in 233 mean lunations, after the sun, moon, and nodes, have been once in a line of conjunction, they return so nearly to the same state again, as that the same node, which was in conjunction with the sun and moon at the beginning of the first of these lunations, will be within 28' 12" of a degree of a line of conjunction with the sun and moon again, when the last of these lunations is completed. And therefore in that time there will be a regular period of eclipses, or return of the same eclipse, for many ages.—In this period (which was first discovered by the Chaldeans) there are 18 Julian years 11 days 7 hours 43 minutes 20 seconds, when the last day of February in leap-years is four times included; but when it is five times included, the period consists of only 18 years 10 days 7 hours 43 minutes 20 seconds. Consequently, if to the mean time of any eclipse, either of the sun or moon, you add 18 Julian years 11 days 7 hours 43 minutes 20 seconds, when the last day of February in leap-years comes in four times, or a day less when it comes in five times, you will have the mean time of the return of the same eclipse.

But the falling back of the line of conjunctions or oppositions of the sun and moon 28' 12" with respect to the line of the nodes in every period, will wear it out in process of time; and after that, it will not return again in less than 12,492 years.—These eclipses of the sun, which happen about the ascending node, and begin to come in at the north pole of the earth, will go a little southerly at each return, till they go quite off the earth at the fourth pole; and those which happen about the descending node, and begin to come in at the south pole of the earth, will go a little northerly at each return, till at last they quite leave the earth at the north pole.

To exemplify this matter, we shall first consider the sun's eclipse, (March 21st old style, April 1st new style), A.D. 1764, according to its mean revolutions, without equating the times, or the sun's distance from the node; and then according to its true equated times.

This eclipse fell in open space at each return, quite clear of the earth, ever since the creation, till A.D. ### TABLE I. The mean Time of New Moon in March, Old Style; with the mean Anomalies of the Sun and Moon, and the Sun's mean Distance from the Moon's ascending Node, from A.D. 1700 to A.D. 1800 inclusive.

| Year | Mean New Moon in March | Sun's mean Anomaly | Moon's mean Anomaly | Sun's mean Diff. from the Node | |------|------------------------|--------------------|--------------------|-------------------------------| | 1700 | 8 16 11 25 | 8 19 58 48 | 1 22 30 37 | 6 14 31 7 | | 1701 | 27 13 44 | 9 8 20 59 | 0 28 7 4 | 7 23 14 8 | | 1702 | 16 22 32 | 8 27 36 51 | 1 17 55 47 | 8 11 16 55 | | 1703 | 6 7 21 18 | 8 16 52 43 | 9 17 43 52 | 8 9 19 42 | | 1704 | 24 4 53 57 | 9 5 14 54 | 8 23 20 57 | 9 18 2 43 | | 1705 | 13 13 42 34 | 8 24 30 47 | 7 3 9 2 | 9 26 5 30 | | 1706 | 2 22 31 11 | 8 13 46 39 | 1 52 57 10 | 10 4 8 17 | | 1707 | 21 20 3 5 | 9 2 8 50 | 4 18 34 13 | 11 12 51 18 | | 1708 | 10 4 52 27 | 8 21 24 43 | 2 28 22 18 | 11 20 54 5 | | 1709 | 29 2 25 | 7 9 46 54 | 2 3 59 24 | 0 29 37 6 | | 1710 | 18 11 13 43 | 8 29 2 47 | 1 43 37 3 | 1 7 39 54 | | 1711 | 7 20 2 20 | 8 18 18 39 | 10 23 35 36 | 1 15 42 41 | | 1712 | 25 17 34 59 | 9 6 40 51 | 9 29 12 42 | 2 14 25 43 | | 1713 | 15 2 23 36 | 8 25 56 43 | 8 9 0 47 | 3 2 28 30 | | 1714 | 4 11 12 13 | 8 15 12 35 | 6 18 48 52 | 3 10 31 17 | | 1715 | 23 8 44 52 | 9 3 34 47 | 5 24 25 57 | 4 19 14 18 | | 1716 | 11 17 33 29 | 8 22 50 39 | 4 4 4 14 | 4 27 17 5 | | 1717 | 1 2 22 | 8 12 6 32 | 2 14 2 8 | 5 5 19 52 | | 1718 | 19 23 54 45 | 9 0 28 44 | 1 19 39 13 | 6 14 2 54 | | 1719 | 9 8 43 22 | 8 19 44 37 | 11 29 27 18 | 6 22 5 41 | | 1720 | 27 6 16 19 | 9 8 6 49 | 11 5 24 8 | 8 0 48 43 | | 1721 | 16 15 4 38 | 8 27 22 41 | 9 14 52 29 | 8 8 51 29 | | 1722 | 5 23 53 14 | 8 16 38 33 | 7 24 40 34 | 8 16 54 16 | | 1723 | 24 21 25 54 | 9 5 0 45 | 7 17 40 9 | 9 25 37 18 | | 1724 | 13 6 14 31 | 8 24 16 37 | 5 10 5 45 | 10 3 40 5 | | 1725 | 2 15 3 7 | 8 13 32 29 | 3 19 53 50 | 10 11 42 52 | | 1726 | 21 12 35 47 | 9 1 54 41 | 2 25 30 56 | 11 20 25 54 | | 1727 | 10 21 24 23 | 8 21 10 34 | 1 5 19 11 | 11 28 28 41 | | 1728 | 28 18 57 39 | 9 5 2 40 | 0 10 56 7 | 1 7 11 42 | | 1729 | 18 3 45 49 | 8 28 48 39 | 10 20 44 12 | 1 15 14 29 | | 1730 | 7 12 34 16 8 | 8 18 4 31 | 9 0 32 17 | 1 23 17 16 | | 1731 | 26 10 6 56 9 | 6 26 42 | 8 6 9 23 | 3 2 0 17 | | 1732 | 14 18 55 33 | 8 25 42 34 | 6 15 57 28 | 3 10 3 4 | | 1733 | 4 3 44 9 | 8 14 58 26 | 4 25 45 33 | 3 18 5 51 | | 1734 | 23 1 16 49 9 | 3 20 38 | 1 2 22 39 | 4 26 48 53 | | 1735 | 12 10 5 25 | 8 22 36 30 | 2 11 10 44 | 5 4 51 40 | | 1736 | 0 18 54 2 | 8 11 52 22 | 0 20 58 49 | 5 12 54 27 | | 1737 | 19 16 26 42 9 | 0 14 11 26 | 35 55 6 21 | 37 29 | | 1738 | 9 1 15 18 8 | 19 30 26 | 10 6 24 0 | 6 29 40 16 | | 1739 | 27 22 47 58 9 | 7 52 38 | 9 12 1 6 | 8 8 23 18 | | 1740 | 16 7 30 4 8 | 8 27 8 30 | 7 21 49 1 | 8 16 26 5 | | 1741 | 5 16 25 11 8 | 8 16 24 2 | 6 1 37 16 | 8 24 28 52 | | 1742 | 24 13 57 52 | 9 4 46 34 | 5 7 14 22 | 10 3 11 54 | | 1743 | 13 22 46 27 | 8 24 2 27 | 3 17 2 27 | 10 1 14 41 | | 1744 | 2 7 35 4 | 8 13 18 20 | 1 26 50 19 | 10 19 27 18 | | 1745 | 21 5 7 44 9 | 1 40 32 | 1 2 27 38 | 11 28 0 30 | | 1746 | 10 13 56 20 | 8 20 56 24 | 11 12 15 43 | 0 6 3 17 | | 1747 | 29 11 29 9 | 9 18 36 | 10 17 52 49 | 1 14 46 19 | | 1748 | 17 20 17 36 8 | 8 28 34 28 | 8 27 40 54 | 1 22 49 5 | | 1749 | 7 5 6 13 8 | 17 50 20 | 7 7 28 59 | 2 0 51 52 | | 1750 | 26 2 38 53 9 | 6 12 32 | 6 13 6 5 | 3 9 34 53 | | 1751 | 15 11 27 29 8 | 25 28 24 | 4 22 54 10 | 3 17 37 40 | | 1752 | 3 20 16 6 14 44 16 | 3 2 42 15 | 3 25 40 27 | |

---

**TABLE II.** ### TABLE II. Mean New Moon, &c. in March, New Style, from A.D. 1752 to A.D. 1800.

| Year | Mean New Moon in March | Sun's Mean Anomaly | Moon's mean Anomaly | Sun's Mean Dist. from the Node | |------|------------------------|--------------------|---------------------|-------------------------------| | 1752 | 14 20 16 | 68 14 44 | 16 | 3 2 42 15 | | 1753 | 5 4 42 | 8 4 0 | 8 | 1 12 30 20 | | 1754 | 3 2 37 | 22 22 20 | 18 | 7 26 | | 1755 | 12 11 25 | 59 11 38 | 10 27 55 | 31 | | 1756 | 30 8 58 | 38 9 0 | 24 10 3 | 37 | | 1757 | 19 17 47 | 15 8 19 | 16 16 | 8 13 20 42 | | 1758 | 9 2 35 | 51 8 32 | 8 6 23 | 8 47 | | 1759 | 8 0 31 | 26 54 50 | 5 28 45 | 54 | | 1760 | 6 8 57 | 88 16 10 | 12 4 34 | 9 2 3 26 | | 1761 | 5 17 45 | 44 8 5 26 | 4 2 18 22 | 5 9 10 6 13 | | 1762 | 4 15 18 | 24 8 23 | 48 16 | 1 23 59 11 | | 1763 | 4 0 7 | 13 4 8 | 3 0 47 | 16 10 26 52 | | 1764 | 2 8 55 | 36 8 20 | 10 13 35 | 21 11 4 54 48 | | 1765 | 21 6 28 | 17 8 20 42 | 13 9 19 12 | 26 10 13 37 49 | | 1766 | 10 15 16 | 53 8 9 58 | 5 7 29 | 31 0 21 40 37 | | 1767 | 9 12 49 | 33 8 28 | 20 17 | 7 4 37 37 | | 1768 | 7 21 38 | 9 8 17 36 | 9 5 14 25 | 42 3 | | 1769 | 7 6 26 | 46 8 6 52 | 1 3 24 13 | 47 2 16 29 13 | | 1770 | 6 3 59 | 26 8 25 14 | 13 2 29 50 | 53 3 25 12 14 | | 1771 | 5 12 48 | 2 8 14 30 | 5 1 9 38 | 58 4 3 15 | | 1772 | 3 21 36 | 39 8 3 45 | 57 11 19 27 | 3 4 11 17 48 | | 1773 | 2 22 9 | 19 9 8 22 | 9 10 25 4 | 9 5 20 0 50 | | 1774 | 12 3 57 | 55 8 11 24 | 1 9 4 52 | 14 5 28 3 37 | | 1775 | 11 12 46 | 31 8 0 39 | 39 7 14 40 | 19 6 6 6 24 | | 1776 | 10 19 10 | 12 19 8 2 | 5 6 20 17 | 25 7 14 49 25 | | 1777 | 9 19 7 | 48 8 8 17 | 57 5 0 30 | 7 22 52 12 | | 1778 | 7 16 40 | 28 8 26 40 | 9 4 5 42 | 36 9 1 35 13 | | 1779 | 7 1 29 | 4 8 15 56 | 1 2 15 30 | 41 9 3 8 30 | | 1780 | 5 10 17 | 40 8 5 11 | 53 0 25 18 | 46 17 14 40 47 | | 1781 | 4 24 7 | 50 21 8 23 | 34 5 0 55 | 52 10 26 23 48 | | 1782 | 3 16 38 | 57 8 12 49 | 58 10 10 43 | 57 11 4 26 35 | | 1783 | 3 1 27 | 33 8 2 5 | 50 8 20 32 | 9 9 26 8 0 | | 1784 | 2 20 23 | 0 13 8 20 | 28 9 6 5 | 8 0 21 12 23 | | 1785 | 2 7 48 | 50 8 9 4 | 35 6 5 57 | 13 0 29 15 10 | | 1786 | 1 5 21 | 30 8 28 6 | 7 5 11 34 | 19 2 7 5 8 12 | | 1787 | 18 14 10 | 6 8 17 21 | 59 3 21 22 | 24 2 16 0 59 | | 1788 | 6 22 58 | 42 8 6 37 | 51 2 1 10 | 29 2 24 3 46 | | 1789 | 5 20 31 | 23 8 25 0 | 3 1 6 47 | 35 4 2 46 48 | | 1790 | 5 5 19 | 59 3 14 15 | 55 11 16 35 | 40 4 10 49 35 | | 1791 | 4 14 8 | 35 3 31 47 | 9 26 23 45 | 4 18 52 22 | | 1792 | 2 11 41 | 15 8 21 53 | 59 9 2 0 52 | 5 27 35 24 | | 1793 | 1 20 29 | 51 8 11 9 | 51 7 11 48 | 57 6 5 38 11 | | 1794 | 0 18 2 | 32 8 29 32 | 3 6 17 26 | 4 7 14 21 13 | | 1795 | 0 2 51 | 8 8 18 47 | 55 4 27 14 | 9 7 22 24 0 | | 1796 | 8 11 39 | 44 8 8 3 47 | 3 7 2 14 | 8 0 26 47 | | 1797 | 7 9 12 | 24 8 26 25 | 59 2 12 39 | 19 9 9 9 48 | | 1798 | 6 18 1 | 18 15 41 | 51 0 22 27 | 25 9 17 12 35 | | 1799 | 6 2 49 | 37 8 4 57 | 43 11 2 15 | 30 9 25 15 22 | | 1800 | 5 0 22 | 17 8 23 19 | 55 10 7 52 | 36 11 3 58 24 |

### TABLE III. Mean Anomalies, and Sun's mean Distance from the Nodes, for 13½ mean Lunations.

| N.D.H.M.S. | Sun's mean Anomaly | Moon's mean Anomaly | Sun's mean Dist. from the Node | |------------|--------------------|---------------------|-------------------------------| | 1 29 12 44 | 3 0 29 6 19 | 0 25 49 | 0 1 0 40 14 | | 2 59 1 28 6 | 1 28 12 39 | 1 21 38 | 1 2 1 20 28 | | 3 88 14 12 9 | 2 27 18 58 | 2 17 27 | 1 3 2 0 42 | | 4 118 2 56 12 | 3 26 25 17 | 3 13 16 2 | 4 2 4 20 56 | | 5 147 15 40 15 | 4 25 31 37 | 4 9 5 2 | 5 3 2 1 10 | | 6 177 4 24 18 | 5 24 37 56 | 5 4 54 3 | 6 4 1 24 | | 7 206 17 8 21 | 6 23 44 15 | 6 0 43 3 | 7 4 41 38 | | 8 236 5 52 4 | 7 22 50 35 | 6 26 3 8 | 8 5 21 52 | | 9 265 18 36 27 | 8 21 56 54 | 7 22 21 4 | 9 6 2 6 6 | | 10 295 7 20 30 | 9 21 3 14 | 8 18 10 4 | 10 6 42 20 | | 11 324 20 4 33 | 10 20 9 33 | 9 13 59 1 | 11 7 22 34 | | 12 354 8 48 36 | 11 19 15 52 | 10 9 48 5 | 12 8 2 47 | | 13 383 21 32 40 | 0 18 22 12 | 5 37 6 1 | 13 8 43 1 |

### TABLE IV. The Days of the Year, reckoned from the beginning of March.

| Month | Days | |-------|------| | March | 1 | | April | 13 | | May | 25 | | June | 37 | | July | 49 | | Aug. | 61 | | Sep. | 73 | | Oct. | 85 | | Nov. | 97 | | Dec. | 109 | | Jan. | 121 | | Feb. | 133 |

### Table V. ### TABLE V. Mean Lunations from 1 to 100,000.

| Lunations | Days, Decimal Parts | Days, Hrs., M. S. Th. Fo. | |-----------|---------------------|--------------------------| | 1 | 29.530590851080 | 29 12 44 3 2 38 | | 2 | 59.061181702160 | 59 1 28 6 5 57 | | 3 | 88.591772533240 | 88 14 12 9 8 55 | | 4 | 118.22263404320 | 118 2 56 12 11 53 | | 5 | 147.652954553401 | 147 15 40 15 14 52 | | 6 | 177.183545106481 | 177 4 24 18 17 50 | | 7 | 206.714135957561 | 206 17 8 21 20 48 | | 8 | 236.244726808641 | 236 5 52 24 23 47 | | 9 | 265.77317659722 | 265 18 36 27 26 45 | | 10 | 295.30596551080 | 295 7 20 30 29 43 | | 11 | 324.83674622160 | 324 12 44 33 22 40 | | 12 | 354.36753683240 | 354 5 52 24 23 47 |

### TABLE VI. The first mean New Moon, with the mean Anomalies of the Sun and Moon, and the Sun's mean Distance from the Ascending Node, next after complete Centuries of Julian years.

| Lunations | First New Moon | Sun's mean Anomaly | M.'s mean Anomaly | Sun from Node | |-----------|----------------|--------------------|-------------------|--------------| | 1237 | 100 4 8 10 52 | 3 21 8 15 22 | 4 19 27 | | 2474 | 200 8 16 21 44 | 6 42 5 0 44 | 9 8 55 | | 3711 | 300 13 0 32 37 | 10 3 1 16 6 | 1 28 22 | | 4948 | 400 17 8 43 29 | 13 24 10 1 28 | 6 17 49 |

### TABLE VII. ### TABLE VII. The annual, or first Equation of the mean to the true Syzygy.

**Argument:** Sun's mean Anomaly.

**Subtract.**

| Degrees | 0 Signs | 1 Sign | 2 Signs | 3 Signs | 4 Signs | 5 Signs | |---------|---------|--------|---------|---------|---------|---------| | H.M.S. | H.M.S. | H.M.S. | H.M.S. | H.M.S. | Degrees | | 0 | 0 | 0 | 0 | 0 | 0 | | | | | | | | | | | | | | |

### TABLE VIII. Equation of the Moon's mean Anomaly.

**Argument:** Sun's mean Anomaly.

**Subtract.**

| Degrees | 0 Signs | 1 Sign | 2 Signs | 3 Signs | 4 Signs | 5 Signs | |---------|---------|--------|---------|---------|---------|---------| | H.M.S. | H.M.S. | H.M.S. | H.M.S. | H.M.S. | Degrees | | 0 | 0 | 0 | 0 | 0 | 0 | | | | | | | |

### TABLE IX. The second Equation of the mean to the true Syzygy.

**Argument:** Moon's equated Anomaly.

**Add.**

| Degrees | 0 Signs | 1 Sign | 2 Signs | 3 Signs | 4 Signs | 5 Signs | |---------|---------|--------|---------|---------|---------|---------| | H.M.S. | H.M.S. | H.M.S. | H.M.S. | H.M.S. | Degrees | | 0 | 0 | 0 | 0 | 0 | 0 | | | | | | | | ### TABLE IX. Concluded.

| Degrees | Signs | H.M.S. | H.M.S. | H.M.S. | H.M.S. | |---------|-------|--------|--------|--------|--------| | | | | | | | | | | | | | |

### TABLE X. The third equation of the mean to the true Syzygy.

| Degrees | Signs | Signs | Signs | Degrees | |---------|-------|-------|-------|---------| | | | | | | | | | | | |

### TAB. XI. The fourth equation of the mean to the true Syzygy.

| Degrees | Add | Degrees | |---------|-----|---------| | | | | | | | |

### TABLE XII. The Sun's mean Longitude, Motion, and Anomaly:

| Years | Sun's mean Longitude | Sun's mean Anomaly | |---------|----------------------|--------------------| | | | | | | | |

### TABLE XIII.

In Leap-years, after February, add one day, and one day's motion. ### TABLE XIII. Equation of the Sun's centre, or the difference between his mean and true place.

| Argument | Sun's mean Anomaly | |----------|--------------------| | Degrees | Signs | | | 1 Sign | | | 2 Signs | | | 3 Signs | | | 4 Signs | | | 5 Signs |

### TABLE XIV. The Sun's Declination.

| Argument | Sun's true Place | |----------|------------------| | Degrees | Signs | | | 1 N | | | 2 N | | | 3 N | | | 4 N | | | 5 N |

### TABLE XV. Equation of the Sun's mean Distance from the Node.

| Argument | Sun's mean Anomaly | |----------|--------------------| | Degrees | Signs | | | 1 Sign | | | 2 Signs | | | 3 Signs | | | 4 Signs | | | 5 Signs |

### TABLE XVI. The Moon's Latitude in Eclipses.

| Arg. Moon's equated Distance from the Node. | |---------------------------------------------| | Degrees | | Signs | | North Ascend | | South Descend |

### TABLE XVII. The Moon's horizontal Parallax, with the Semidiameters and true Horary Motions of the Sun and Moon, to every sixth degree of their mean Anomalies, the quantities for the intermediate degrees being easily proportioned by sight.

| Moon's Horizontal Parallax. | |----------------------------| | Degrees | | Signs | | North Ascend | | South Descend |

This Table shows the Moon's Latitude a little beyond the utmost Limits of Eclipses. 1295, June 13th old style, at 12 h. 52 m. 59 sec. p.m. meridian, when the moon's shadow first touched the earth at the north pole; the sun being then 17° 48' 27" from the ascending node. In each period since that time, the sun has come 28° 12' nearer and nearer the same node, and the moon's shadow has therefore gone more and more southerly.—In the year 1692, July 18th old style, at 10 h. 36 m. 21 sec. p.m., when the same eclipse will have returned 48 times, the sun will be only 24° 45" from the ascending node, and the centre of the moon's shadow will fall a little northward of the earth's centre.—At the end of the next following period, A.D. 1980, July 28th old style, at 18 h. 19 m. 41 sec. p.m., the sun will have receded back 3° 27" from the ascending node, and the moon will have a very small degree of southern latitude, which will cause the centre of her shadow to pass a very small matter south of the earth's centre.—After which, in every following period, the sun will be 28° 12' farther back from the ascending node than in the period last before; and the moon's shadow will go still farther and farther southward, until September 14th old style, at 23 h. 46 m. 22 sec. p.m. A.D. 2665; when the eclipse will have completed its 77th periodical return, and will go quite off the earth at the south pole (the sun being then 17° 55' 22" back from the node), and cannot come in at the north pole, so as to begin the same course over again, in less than 12,492 years afterwards.—And such will be the case of every other eclipse of the sun: For, as there is about 18 degrees on each side of the node within which there is a possibility of eclipses, their whole revolution goes through 36 degrees about that node, which, taken from 360 degrees, leaves remaining 324 degrees for the eclipses to travel in expansion. And as this 36 degrees is not gone through in less than 77 periods, which takes up 1388 years, the remaining 324 degrees cannot be so gone through in less than 12,492 years. For, as 36 is to 1388, so is 324 to 12,492.

To illustrate this a little farther, we shall examine some of the most remarkable circumstances of the returns of the eclipse which happened July 14th 1748, about noon. This eclipse, after traversing the voids of space from the creation, at last began to enter the Terra Australis Incognita about 88 years after the conquest, which was the last of king Stephen's reign: every Chaldean period it has crept more northerly, but was still invisible in Britain before the year 1622; when, on the 20th of April, it began to touch the fourth parts of England about 2 in the afternoon; its central appearance rising in the American south seas, and traversing Peru and the Amazon's country, through the Atlantic ocean into Africa, and setting in the Ethiopian continent, not far from the beginning of the Red sea.

Its next visible period was, after three Chaldean revolutions, in 1676, on the first of June, rising central in the Atlantic ocean, passing us about 9 in the morning, with four digits eclipsed on the under limb, and setting in the gulf of Cochinchina in the East Indies.

It being now near the solstice, this eclipse was visible the very next return in 1694, in the evening; and in two periods more, which was in 1730, on the 4th of July, was seen about half eclipsed just after sun-rise, and observed both at Wurtemberg in Germany, and Pekin in China, soon after which it went off.

Eighteen years more afforded us the eclipse which fell on the 14th of July 1748.

The next visible return happened on July 25th 1766 in the evening, about four digits eclipsed; and, after two periods more, will happen on August 16th 1802, early in the morning, about five digits, the centre coming from the north frozen continent, by the capes of Norway, through Tartary, China, and Japan, to the Ladrones islands, where it goes off.

Again, in 1820, August 26th, between one and two, there will be another great eclipse at London, about 10 digits; but, happening so near the equinox, the centre will leave every part of Britain to the west, and enter Germany at Emden, passing by Venice, Naples, Grand Cairo, and set in the gulf of Baffora near that city.

It will be no more visible till 1874, when five digits will be obscured (the centre being now about to leave the earth) on September 28th. In 1892, the sun will go down eclipsed in London; and again, in 1928, the passage of the centre will be in the expansion, tho' there will be two digits eclipsed at London, October the 31st of that year, and about the year 2090 the whole penumbra will be wore off; whence no more returns of this eclipse can happen till after a revolution of 10,000 years.

From these remarks on the entire revolution of this eclipse, we may gather, that a thousand years more or less, (for there are some irregularities that may protract or lengthen this period 100 years), complete the whole terrestrial phenomena of any single eclipse: and since 20 periods of 54 years each, and about 33 days, comprehend the entire extent of their revolution, it is evident, that the times of the returns will pass thro' a circuit of one year and ten months, every Chaldean period being 10 or 11 days later, and of the equable appearances, about 32 or 33 days. Thus, though this eclipse happens about the middle of July, no other subsequent eclipse of this period will return till the middle of the same month again; but wear constantly each period 10 or 11 days forward, and at last appear in winter, but then it begins to cease from affecting us.

Another conclusion from this revolution may be drawn, that there will seldom be any more than two great eclipses of the sun in the interval of this period, and these follow sometimes next return, and often at greater distances. That of 1715 returned again in 1733 very great; but this present eclipse will not be great till the arrival of 1820, which is a revolution of four Chaldean periods; so that the irregularities of their circuits must undergo new computations to assign them exactly.

Nor do all eclipses come in at the south pole: that depends altogether on the position of the lunar nodes, which will bring in as many from the expansion one way as the other; and such eclipses will wear more southerly by degrees, contrary to what happens in the present case.

The eclipse, for example, of 1736 in September, had its centre in the expansion, and set about the middle of its obscurity in Britain; it will wear in at the north pole, and in the year 2600, or thereabouts, go off into the expansion on the south side of the earth.

The eclipses therefore which happened about the crea- Astronomy.

It is particularly to be noted, that eclipses which have happened many centuries ago will not be found by our present tables to agree exactly with ancient observations, by reason of the great anomalies in the lunar motions; which appears an incontrovertible demonstration of the non-eternity of the universe. For it seems confirmed by undeniable proofs, that the moon now finishes her period in less time than formerly, and will continue, by the centripetal law, to approach nearer and nearer the earth, and to go sooner and sooner round it: nor will the centrifugal power be sufficient to compensate the different gravitations of such an assemblage of bodies as constitute the solar system, which would come to ruin of itself, without some regulation and adjustment of their original motions. See No. 89.

We are credibly informed from the testimony of the ancients, that there was a total eclipse of the sun predicted by Thales to happen in the fourth year of the 48th Olympiad, either at Sardis or Miletus in Asia, where Thales then resided. That year corresponds to the 587th year before Christ; when accordingly there happened a very signal eclipse of the sun, on the 28th of May, answering to the present 16th of that month, central through North America, the south parts of France, Italy, &c., as far as Athens, or the isles in the Aegean sea; which is the farther that even the Caroline tables carry it; and consequently make it invisible to any part of Asia, in the total character; tho' there are good reasons to believe that it extended to Babylon, and went down central over that city. We are not however to imagine, that it was felt before it past Sardis and the Asiatic towns, where the predictor lived; because an invisible eclipse could have been of no service to demonstrate his ability in astronomical sciences to his countrymen, as it could give no proof of its reality.

For a farther illustration, Thucydides relates, That a solar eclipse happened on a summer's day, in the afternoon, in the first year of the Peloponnesian war, so great, that the stars appeared. Rhodius was victor in the Olympic games the fourth year of the said war, being also the fourth year of the 87th Olympiad, on the 428th year before Christ. So that the eclipse must have happened in the 431st year before Christ; and by computation it appears, that on the third of August there was a signal eclipse which would have past over Athens, central about 6 in the evening, but which our present tables bring no farther than the ancient Syrtes on the African coast, above 400 miles from Athens; which, suffering in that case but 9 digits, could by no means exhibit the remarkable darkness recited by this historian: the centre therefore seems to have past Athens about 6 in the evening, and probably might go down about Jerusalem, or near it, contrary to the construction of the present tables. These things are only mentioned by way of caution to the present astronomers, in re-computing ancient eclipses; and they may examine the eclipse of Nicias, so fatal to the Athenian fleet; that which overthrew the Macedonian army, &c.

In any year, the number of eclipses of both luminaries cannot be less than two, nor more than seven; the most usual number is four, and it is very rare to have more than six. For the sun passes by both the nodes but once a-year, unless he passes by one of them in the beginning of the year; and if he does, he will pass by the same node again a little before the year be finished; because, as these points move 19° degrees backwards every year, the sun will come to either of them 17 days after the other. And when either node is within 17 degrees of the sun at the time of new moon, the sun will be eclipsed. At the subsequent opposition, the moon will be eclipsed in the other node, and come round to the next conjunction again ere the former node be 17 degrees past the sun, and will therefore eclipse him again. When three eclipses fall about either node, the like number generally falls about the opposite; as the sun comes to it in 17 days afterward; and six lunations contain but four days more. Thus, there may be two eclipses of the sun and one of the moon about each of her nodes. But when the moon changes in either of the nodes, she cannot be near enough the other node at the next full to be eclipsed; and in six lunar months afterward she will change near the other node: in these cases, there can be but two eclipses in a year, and they are both of the sun.

A longer period than the abovementioned, for comparing and examining eclipses which happen at long intervals of time, is 557 years, 21 days, 18 hours, 30 minutes, 11 seconds; in which time there are 6890 mean lunations; and the sun and node meet again so nearly as to be but 11 seconds distant; but then it is not the same eclipse that returns, as in the shorter period abovementioned.

Eclipses of the sun are more frequent than of the moon, because the sun's ecliptic limits are greater than the moon's; yet we have more visible eclipses of the moon than of the sun, because eclipses of the moon are seen from all parts of that hemisphere of the earth which is next her, and are equally great to each of those parts; but the sun's eclipses are visible only to that small portion of the hemisphere next him whereon the moon's shadow falls.

The moon's orbit being elliptical, and the earth in one of its focuses, she is once at her least distance from the earth, and once at her greatest, in every lunation. When the moon changes at her least distance from the earth, and so near the node that her dark shadow falls upon the earth, she appears big enough to cover the whole disk of the sun from that part on which her shadow falls; and the sun appears totally eclipsed there for some minutes: but when the moon changes at her greatest distance from the earth, and so near the node that her dark shadow is directed towards the earth, her diameter subtends a less angle than the sun's; and therefore she cannot hide his whole disk from any part of the earth, nor does her shadow reach it at that time; and to the place over which the point of her shadow hangs, the eclipse is annular, the sun's edge appearing like a luminous ring all around the body of the moon. When the change happens within 17 degrees of the node, and the moon at her mean distance from the earth, the point of her shadow just touches the earth, and she eclipses the sun totally to that small spot whereon her shadow falls; but the darkness is not of a moment's continuance.

The moon's apparent diameter, when largest, exceeds ceeds the sun's, when least, only 1 minute 38 seconds of a degree; and in the greatest eclipse of the sun that can happen at any time and place, the total darkness continues no longer than whilst the moon is going 1 minute 38 seconds from the sun in her orbit, which is about 3 minutes and 13 seconds of an hour.

The moon's dark shadow covers only a spot on the earth's surface about 180 English miles broad, when the moon's diameter appears largest, and the sun's least; and the total darkness can extend no farther than the dark shadow covers. Yet the moon's partial shadow or penumbra may then cover a circular space 4900 miles in diameter, within all which the sun is more or less eclipsed, as the places are less or more distant from the centre of the penumbra. When the moon changes exactly in the node, the penumbra is circular on the earth at the middle of the general eclipse; because at that time it falls perpendicularly on the earth's surface; but at every other moment it falls obliquely, and will therefore be elliptical; and the more so, as the time is longer before or after the middle of the general eclipse; and then much greater portions of the earth's surface are involved in the penumbra.

When the penumbra first touches the earth, the general eclipse begins; when it leaves the earth, the general eclipse ends: from the beginning to the end the sun appears eclipsed in some part of the earth or other. When the penumbra touches any place, the eclipse begins at that place, and ends when the penumbra leaves it. When the moon changes in the node, the penumbra goes over the centre of the earth's disk as seen from the moon; and consequently, by describing the longest line possible on the earth, continues the longest upon it; namely, at a mean rate, 5 hours 50 minutes; more, if the moon be at her greatest distance from the earth, because she then moves slower; less, if she be at her least distance, because of her quicker motion.

To make several of the above and other phenomena plainer, (Plate L fig. 3.), let S be the sun, E the earth, M the moon, and AMP the moon's orbit. Draw the right line Wc from the western side of the sun at W, touching the western side of the moon at c, and the earth at e; draw also the right line Vd from the eastern side of the sun at V, touching the eastern side of the moon at d', and the earth at e': the dark space ced included between those lines is the moon's shadow, ending in a point at e, where it touches the earth; because in this case the moon is supposed to change at M in the middle between A the apogee, or farthest point of her orbit from the earth, and P the perigee, or nearest point to it. For, had the point P been at M, the moon had been nearer the earth; and her dark shadow at e would have covered a space upon it about 180 miles broad, and the sun would have been totally darkened, with some continuance; but had the point A been at M, the moon would have been farther from the earth, and her shadow would have ended in a point a little above e, and therefore the sun would have appeared like a luminous ring all around the moon. Draw the right lines WXab and VXeg, touching the contrary sides of the sun and moon, and ending on the earth at a and b; draw also the right line SXM, from the centre of the sun's disk, thro' the moon's centre, to the earth; and suppose the two former lines WXab and VXeg to revolve on the line SXM as an axis, and their points a and b will describe the limits of the penumbra TT on the earth's surface, including the large space aba; within which the sun appears more or less eclipsed, as the places are more or less distant from the verge of the penumbra ab.

Draw the right line yz across the sun's disk, perpendicular to SXM the axis of the penumbra; then divide the line yz into twelve equal parts, as in the figure, for the twelve digits or equal parts of the sun's diameter; and, at equal distances from the centre of the penumbra at e (on the earth's surface YY) to its edge ab, draw twelve concentric circles, marked with the numeral figures 1 2 3 4 &c. and remember that the moon's motion in her orbit AMP is from west to east, as from s to t. Then,

To an observer on the earth at A, the eastern limb of the moon at d seems to touch the western limb of the sun at W, when the moon is at M; and the sun's eclipse begins at b, appearing as at A in Plate LI. fig. 1., at the left hand; but, at the same moment of absolute time, to an observer at a in Plate L fig. 3., the western edge of the moon at e leaves the eastern edge of the sun at V, and the eclipse ends, as at the right hand C, Plate LI. fig. 1. At the very same instant, to all those who live on the circle marked 1 on the earth E, in Plate L fig. 3., the moon M cuts off or darkens a twelfth part of the sun S, and eclipses him one digit, as at 1 in Plate LI. fig. 1.: to those who live on the circle marked 2 in Plate L fig. 3., the moon cuts off two twelfth parts of the sun, as at 2 in Plate LI. fig. 1.; to those on the circle 3, three parts; and so on to the centre at 12 in Plate L fig. 3., where the sun is centrally eclipsed, as at B in the middle of fig. 1. Plate LI.; under which figure there is a scale of hours and minutes, to show at a mean rate how long it is from the beginning to the end of a central eclipse of the sun on the parallel of London; and how many digits are eclipsed at any particular time from the beginning at A to the middle at B, or the end at C. Thus, in 16 minutes from the beginning, the sun is two digits eclipsed; in an hour and five minutes, eight digits; and in an hour and 37 minutes, 12 digits.

By Plate L fig. 3., it is plain, that the sun is totally or centrally eclipsed but to a small part of the earth at any time, because the dark conical shadow e of the moon M falls but on a small part of the earth; and that the partial eclipse is confined at that time to the space included by the circle ab, of which only one half can be projected in the figure, the other half being supposed to be hid by the convexity of the earth E: and likewise, that no part of the sun is eclipsed to the large space YY of the earth, because the moon is not between the sun and any of that part of the earth; and therefore to all that part the eclipse is invisible. The earth turns eastward on its axis, as from g to h, which is the same way that the moon's shadow moves; but the moon's motion is much swifter in her orbit from s to t; and therefore, although eclipses of the sun are of longer duration on account of the earth's motion on its axis than they would be if that motion was stopt, yet, in four minutes of time at most, the moon's swifter motion carries her dark shadow quite over any place that its centre touches at the time of greatest obscuration. The motion of the shadow on the earth's disk is equal to the moon's motion from the sun, which is about $30\frac{1}{2}$ minutes of a degree every hour at a mean rate; but so much of the moon's orbit is equal to $30\frac{1}{2}$ degrees of a great circle on the earth; and therefore the moon's shadow goes $30\frac{1}{2}$ degrees, or 1830 geographical miles, on the earth in an hour, or $30\frac{1}{2}$ miles in a minute, which is almost four times as swift as the motion of a cannon-ball.

As seen from the sun or moon, the earth's axis appears differently inclined every day of the year, on account of keeping its parallelism throughout its annual course. In Plate LI. fig. 2, let EDON be the earth at the two equinoxes and the two solstices, NS its axis, N the north pole, S the south pole, AEQ the equator, T the tropic of Cancer, and ABC the circumference of the earth's enlightened disk as seen from the sun or new moon at these times. The earth's axis has the position NES at the vernal equinox, lying towards the right hand, as seen from the sun or new moon; its poles N and S being then in the circumference of the disk; and the equator and all its parallels seem to be straight lines, because their planes pass through the observer's eye looking down upon the earth from the sun or moon directly over E, where the ecliptic FG intersects the equator AE. At the summer solstice, the earth's axis has the position NDS; and that part of the ecliptic FG, in which the moon is then new, touches the tropic of Cancer T' at D. The north pole N at that time inclining $23\frac{1}{2}$ degrees towards the sun, falls so many degrees within the earth's enlightened disk, because the sun is then vertical to D $23\frac{1}{2}$ degrees north of the equator AEQ; and the equator with all its parallels seem elliptic curves bending downward, or towards the south pole, as seen from the sun; which pole, together with $23\frac{1}{2}$ degrees all round it, is hid behind the disk in the dark hemisphere of the earth. At the autumnal equinox, the earth's axis has the position NOS, lying to the left hand as seen from the sun or new moon, which are then vertical to O, where the ecliptic cuts the equator AEQ. Both poles now lie in the circumference of the disk, the north pole just going to disappear behind it; and the south pole just entering into it; and the equator, with all its parallels, seem to be straight lines, because their planes pass through the observer's eye, as seen from the sun, and very nearly so as seen from the moon. At the winter solstice, the earth's axis has the position NNS, when its south pole S inclining $23\frac{1}{2}$ degrees towards the sun, falls $23\frac{1}{2}$ degrees within the enlightened disk, as seen from the sun or new moon, which are then vertical to the tropic of Capricorn T', $23\frac{1}{2}$ degrees south of the equator AEQ; and the equator, with all its parallels, seem elliptic curves bending upward; the north pole being as far hid behind the disk in the dark hemisphere, as the south pole is come into the light. The nearer that any time of the year is to the equinoxes or solstices, the more it partakes of the phenomena relating to them.

Thus it appears, that from the vernal equinox to the autumnal, the north pole is enlightened; and the equator and all its parallels appear elliptical as seen from the sun, more or less curved as the time is nearer to, or farther from, the summer solstice; and bending downwards, or towards the south pole; the reverse of which happens from the autumnal equinox to the vernal. A little consideration will be sufficient to convince the reader, that the earth's axis inclines towards the sun at the summer solstice; from the sun at the winter solstice; and sideways to the sun at the equinoxes; but towards the right hand, as seen from the sun at the vernal equinox; and towards the left hand at the autumnal. From the winter to the summer solstice, the earth's axis inclines more or less to the right hand, as seen from the sun; and the contrary from the summer to the winter solstice.

The different positions of the earth's axis, as seen from the sun at different times of the year, affect solar eclipses greatly with regard to particular places; yea, the position so far as would make central eclipses which fall at one time of the year invisible if they fell at another, even though the moon should always change in the nodes, and at the same hour of the day; of which indefinitely various affections, we shall only give examples for the times of the equinoxes and solstices.

In the same diagram, (Plate LI. fig. 2.), let FG be part of the ecliptic, and IK, ik, ik, ik, part of the moon's orbit; both seen edgeways, and therefore projected into right lines; and let the intersections NODE be one and the same node at the above times, when the earth has the aforementioned different positions; and let the spaces included by the circles Pppp be the penumbra at these times, as its centre is passing over the centre of the earth's disk. At the winter solstice, when the earth's axis has the position NNS, the centre of the penumbra P touches the tropic of Capricorn T in N at the middle of the general eclipse; but no part of the penumbra touches the tropic of Cancer T'. At the summer solstice, when the earth's axis has the position NDS (iD being then part of the moon's orbit whose node is at D), the penumbra p has its centre at D, on the tropic of Cancer T', at the middle of the general eclipse, and then no part of it touches the tropic of Capricorn T. At the autumnal equinox, the earth's axis has the position NOS, (iO being then part of the moon's orbit), and the penumbra equally includes part of both tropics T' and T at the middle of the general eclipse: at the vernal equinox it does the same, because the earth's axis has the position NES; but, in the former of these two last cases, the penumbra enters the earth at A, north of the tropic of Cancer T', and leaves it at m, south of the tropic of Capricorn T'; having gone over the earth obliquely southward, as its centre described the line AOM; whereas, in the latter case, the penumbra touches the earth at n, south of the equator AEQ, and describing the line nEq (similar to the former line AOM in open space), goes obliquely northward over the earth, and leaves it at g, north of the equator.

In all these circumstances, the moon has been supposed to change at noon in her descending node: Had she changed in her ascending node, the phenomena would have been as various the contrary way, with respect to the penumbras going northward or southward over the earth. But because the moon changes at all hours, as often in one node as in the other, and at all distances from them both at different times as it happens, the variety of the phases of eclipses are almost innumerable, even at the same places; considering also how variously the same places are situated on the en- enlightened disk of the earth, with respect to the penumbra's motion, at the different hours when eclipses happen.

When the moon changes 17 degrees short of her descending node, the penumbra just touches the northern part of the earth's disk near the north pole N; and as seen from that place, the moon appears to touch the sun, but hides no part of him from sight. Had the change been as far short of the ascending node, the penumbra would have touched the southern part of the disk near the south pole S. When the moon changes 12 degrees short of the descending node, more than a third part of the penumbra falls on the northern parts of the earth at the middle of the general eclipse: Had she changed as far past the same node, as much of the other side of the penumbra about P would have fallen on the southern part of the earth; all the rest in the expanse, or open space. When the moon changes 6 degrees from the node, almost the whole penumbra falls on the earth at the middle of the general eclipse. And lastly, when the moon changes in the node at N, the penumbra PN takes the longest course possible on the earth's disk; its centre falling on the middle thereof, at the middle of the general eclipse. The farther the moon changes from either node, within 17 degrees of it, the shorter is the penumbra's continuance on the earth, because it goes over a less portion of the disk, as is evident by the figure.

The nearer that the penumbra's centre is to the equator at the middle of the general eclipse, the longer is the duration of the eclipse at all those places where it is central; because, the nearer that any place is to the equator, the greater is the circle it describes by the earth's motion on its axis; and so, the place moving quicker, keeps longer in the penumbra, whose motion is the same way with that of the place, though faster, as has been already mentioned. Thus (see the earth at D and the penumbra at 12) whilst the point b in the polar circle abcd is carried from b to c by the earth's diurnal motion, the point d' on the tropic of Cancer T is carried a much greater length from d' to D'; and therefore, if the penumbra's centre goes one time over c and another time over D', the penumbra will be longer in passing over the moving place d' than it was in passing over the moving place b. Consequently, central eclipses about the poles are of the shortest duration; and about the equator, of the longest.

In the middle of summer, the whole frigid zone, included by the polar circle abcd, is enlightened; and if it then happens that the penumbra's centre goes over the north pole, the sun will be eclipsed much the same number of digits at a as at e; but whilst the penumbra moves eastward over e, it moves westward over a; because, with respect to the penumbra, the motions of a and c are contrary: for c moves the same way with the penumbra towards d', but a moves the contrary way towards b; and therefore the eclipse will be of longer duration at c than at a. At a the eclipse begins on the sun's eastern limb, but at e on his western: at all places lying without the polar circles, the sun's eclipses begin on his western limb, or near it, and end on or near his eastern. At those places where the penumbra touches the earth, the eclipse begins with the rising sun, on the top of his western or uppermost edge; and at those places where the penumbra leaves the earth, the eclipse ends with the setting sun, on the top of his eastern edge, which is then the uppermost, just at its disappearing in the horizon.

If the moon were surrounded by an atmosphere of any considerable density, it would seem to touch the sun a little before the moon made her appulse to his edge, and we should see a little faintness on that edge before it were eclipsed by the moon; but as no such faintness has been observed, it seems plain, that the moon has no such atmosphere as that of the earth. The faint ring of light surrounding the sun in total eclipses, called by Cassini la chevelure du soleil, is said to be the atmosphere of the sun; because it has been observed to move equally with the sun, not with the moon.

Having been so prolix concerning eclipses of the sun, we shall drop that subject at present, and proceed to the doctrine of lunar eclipses; which, being more simple, may be explained in less time.

That the moon can never be eclipsed but at the time of her being full, and the reason why she is not eclipsed at every full, has been shewn already. In Plate L. fig. 3, let S be the sun, E the earth, RR the earth's shadow, and B the moon in opposition to the sun: In this situation the earth intercepts the sun's light in its way to the moon; and when the moon touches the earth's shadow at v, she begins to be eclipsed on her eastern limb x, and continues eclipsed until her western limb y leaves the shadow at w: at B she is in the middle of the shadow, and consequently in the middle of the eclipse.

The moon, when totally eclipsed, is not invisible if she be above the horizon and the sky be clear; but appears generally of a dusky colour, like tarnished copper, which some have thought to be the moon's native light*. But the true cause of her being visible is the scattered beams of the sun, bent into the earth's shadow by going through the atmosphere; which, being more or less dense near the earth than at considerable heights above it, refracts or bends the sun's rays more inward, the nearer they are passing by the earth's surface, than those rays which go through higher parts of the atmosphere, where it is less dense according to its height; until it be so thin or rare as to lose its refractive power. Let the circle fghk, concentric to the earth, include the atmosphere whose refractive power vanishes at the heights f and g; so that the rays Wv and Vw go on straight without suffering the least refraction: but all those rays which enter the atmosphere between f and g, and between i and k, on opposite sides of the earth, are gradually more bent inwards as they go thro' a greater portion of the atmosphere, until the rays Wv and Vw touching the earth at v and w, are bent so much as to meet at q, a little short of the moon; and therefore the dark shadow of the earth is contained in the space mognp, where none of the sun's rays can enter: all the rest RR, being mixed by the scattered rays which are refracted as above, is in some measure enlightened by them; and some of these rays falling on the moon, give her the colour of tarnished copper, or of iron almost red hot. So that if the earth had no atmosphere, the moon would be as invisible in total eclipses as she is when new. If the moon were so near the earth as to go into its dark shadow, suppose about 109, she would be invisible during her. her stay in it; but visible before and after in the fainter shadow RR.

When the moon goes thro' the centre of the earth's shadow, she is directly opposite to the sun: yet the moon has been often seen totally eclipsed in the horizon when the sun was also visible in the opposite part of it: for, the horizontal refraction being almost 34 minutes of a degree, and the diameter of the sun and moon being each at a mean flate but 32 minutes, the refraction causes both luminaries to appear above the horizon when they are really below it.

When the moon is full at 12 degrees from either of her nodes, she just touches the earth's shadow, but enters not into it. In Plate LI. fig. 2. let GH be the ecliptic, ef the moon's orbit where she is 12 degrees from the node at her full; cd her orbit where she is 6 degrees from the node, ab her orbit where she is full in the node, A B the earth's shadow, and M the moon. When the moon describes the line ef, she just touches the shadow, but does not enter into it; when she describes the line cd, she is totally, tho' not centrally, immersed in the shadow; and when she describes the line ab, she passes by the node at M in the centre of the shadow, and takes the longest line possible, which is a diameter, through it: and such an eclipse being both total and central is of the longest duration, namely, 5 hours 57 minutes 6 seconds from the beginning to the end, if the moon be at her greatest distance from the earth; and 3 hours 37 minutes 26 seconds, if she be at her least distance. The reason of this difference is, that when the moon is farther from the earth, she moves slower; and when nearest to it, quicker.

The moon's diameter, as well as the sun's, is supposed to be divided into 12 equal parts, called digits; and so many of these parts as are darkened by the earth's shadow, so many digits is the moon eclipsed. All that the moon is eclipsed above 12 digits, shows how far the shadow of the earth is over the body of the moon, on that edge to which she is nearest at the middle of the eclipse.

It is difficult to observe exactly either the beginning or ending of a lunar eclipse, even with a good telescope, because the earth's shadow is so faint and ill defined about the edges, that when the moon is either just touching or leaving it, the obscuration of her limb is scarce sensible; and therefore the nicest observers can hardly be certain to four or five seconds of time. But both the beginning and ending of solar eclipses are visibly instantaneous; for the moment that the edge of the moon's disk touches the sun's, his roundness seems a little broken on that part; and the moment the leaves it, he appears perfectly round again.

In astronomy, eclipses of the moon are of great use for ascertaining the periods of her motions; especially such eclipses as are observed to be alike in all her circumstances, and have long intervals of time between them. In geography, the longitudes of places are found by eclipses: but for this purpose eclipses of the moon are more useful than those of the sun, because they are more frequently visible, and the same lunar eclipse is of equal largeness and duration at all places where it is seen. In chronology, both solar and lunar eclipses serve to determine exactly the time of any past event: for there are so many particulars observable in every eclipse, with respect to its quantity, the places where it is visible (if of the sun), and the time of the day or night, that it is impossible there can be two solar eclipses in the course of many ages which are alike in all circumstances.

From the above explanation of the doctrine of eclipses, it is evident, that the darkness at our Saviour's crucifixion was supernatural. For he suffered on the day on which the passover was eaten by the Jews, on which day it was impossible that the moon's shadow could fall on the earth; for the Jews kept the passover at the time of full moon: nor does the darkness in total eclipses of the sun last above four minutes in any place; whereas the darkness at the crucifixion lasted three hours, Matth. xxviii. 15. and overspread at least all the land of Judca.

The theory of eclipses being now, we hope, pretty plainly laid down, the construction of tables for their calculation will be understood from the following considerations.

The motions of the sun and moon are observed to be continually accelerated from the apogee to the perigee, and as gradually retarded from the perigee to the apogee; being slowest of all when the mean anomaly is nothing, and swiftest of all when it is six signs.

When the luminary is in its apogee or perigee, its place is the same as it would be if its motion were equable in all parts of its orbit. The supposed equable motions are called mean; the unequable are justly called the true.

The mean place of the sun or moon is always forwarder than the true place, whilst the luminary is moving from its apogee to its perigee; and the true place is always forwarder than the mean, whilst the luminary is moving from its perigee to its apogee. In the former case, the anomaly is always less than six signs; and in the latter case, more.

It has been found, by a long series of observations, that the sun goes through the ecliptic, from the vernal equinox to the same equinox again, in 365 days 5 hours 48 minutes 55 seconds; from the first star of Aries to the same star again, in 365 days 6 hours nine minutes 24 seconds; and from his apogee to the same again, in 365 days 6 hours 14 minutes 0 seconds.—The first of these is called the solar year; the second, the sidereal year; and the third, the anomalistic year. So that the solar year is 20 minutes 29 seconds shorter than the sidereal; and the sidereal year is 4 minutes 36 seconds shorter than the anomalistic. Hence it appears, that the equinoctial point, or intersection of the ecliptic and equator at the beginning of Aries, goes backward with respect to the fixed stars, and that the sun's apogee goes forward.

It is also observed, that the moon goes through her orbit, from any given fixed star to the same star again, in 27 days 7 hours 42 minutes 4 seconds at a mean rate; from her apogee to her apogee again, in 27 days 13 hours 18 minutes 43 seconds; and from the fun to the fun again, in 29 days 12 hours 44 minutes 3½ seconds. This shows, that the moon's apogee moves forward in the ecliptic, and that at a much quicker rate than the sun's apogee does; since the moon is five hours 55 minutes 39 seconds longer in revolving from her apogee to her apogee again, than from any star to the same star again.

The moon's orbit crosses the ecliptic in two oppo- site points, which are called her Nodes; and it is observed, that she revolves sooner from any node to the node again, than from any star to the star again, by 2 hours 38 minutes 27 seconds; which shews, that her nodes move backward, or contrary to the order of signs, in the ecliptic.

The time in which the moon revolves from the sun to the sun again (or from change to change) is called a Lunation; which, according to Dr. Pound's mean measures, would always consist of 29 days 12 hours 44 minutes 3 seconds 2 thirds 58 fourths, if the motions of the sun and moon were always equable. Hence, 12 mean lunations contain 354 days 8 hours 48 minutes 36 seconds 35 thirds 40 fourths, which is 10 days 21 hours 11 minutes 23 seconds 24 thirds 20 fourths less than the length of a common Julian year, consisting of 365 days 6 hours; and 13 mean lunations contain 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, which exceeds the length of a common Julian year, by 18 days 15 hours 32 minutes 39 seconds 38 thirds 38 fourths.

The mean time of new moon being found for any given year and month, as suppose for March 1700, old style, if this mean new moon falls later than the 11th day of March, then 12 mean lunations added to the time of this mean new moon will give the time of the mean new moon in March 1701, after having thrown off 365 days. But when the mean new moon happens to be before the 11th of March, we must add 13 mean lunations, in order to have the time of mean new moon in March the year following; always taking care to subtract 365 days in common years, and 366 days in leap-years, from the sum of this addition.

Thus, A.D. 1700, old style, the time of mean new moon in March was the 8th day, at 16 hours 11 minutes 25 seconds after the noon of that day (viz. at 11 minutes 25 seconds past four in the morning of the 9th day, according to common reckoning). To this we must add 13 mean lunations, or 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, and the sum will be 392 days 13 hours 44 minutes 4 seconds 38 thirds 38 fourths: from which subtract 365 days, because the year 1701 is a common year, and there will remain 27 days 13 hours 44 minutes 4 seconds 38 thirds 38 fourths for the time of mean new moon in March, A.D. 1701.

Carrying on this addition and subtraction till A.D. 1703, we find the time of mean new moon in March that year to be on the 6th day, at 7 hours 21 minutes 17 seconds 49 thirds 46 fourths past noon; to which add 13 mean lunations, and the sum will be 390 days 4 hours 53 minutes 57 seconds 28 thirds 20 fourths; from which subtract 366 days, because the year 1704 is a leap-year, and there will remain 24 days 4 hours 53 minutes 57 seconds 28 thirds 20 fourths, for the time of mean new moon in March, A.D. 1704.

In this manner was the first of the following tables constructed to seconds, thirds, and fourths; and then wrote out to the nearest seconds.—The reason why we chose to begin the year with March, was to avoid the inconvenience of adding a day to the tabular time in leap-years after February, or subtracting a day therefrom in January and February in those years; to which all tables of this kind are subject, which begin the year with January, in calculating the times of new or full moons.

The mean anomalies of the sun and moon, and the sun's mean motion from the ascending node of the moon's orbit, are set down in Table III. from 1 to 13 mean lunations.—These numbers, for 13 lunations, being added to the radical anomalies of the sun and moon, and to the sun's mean distance from the ascending node, at the time of mean new moon in March 1700 (Table I.), will give their mean anomalies, and the sun's mean distance from the node, at the time of mean new moon in March 1701; and being added for 12 lunations to those for 1701, give them for the time of mean new moon in March 1702. And so on, as far as you please to continue the table (which is here carried on to the year 1800), always throwing off 12 signs when their sum exceeds 12, and setting down the remainder as the proper quantity.

If the numbers belonging to A.D. 1700 (in Table I.) be subtracted from those belonging to 1800, we shall have their whole differences in 100 complete Julian years; which accordingly we find to be 4 days 8 hours 10 minutes 52 seconds 15 thirds 40 fourths, with respect to the time of mean new moon.—These being added together 60 times (always taking care to throw off a whole lunation when the days exceed 29½), make up 60 centuries, or 6000 years, as in Table VI. which was carried on to seconds, thirds, and fourths; and then wrote out to the nearest seconds. In the same manner were the respective anomalies and the sun's distance from the node found, for these centurial years; and then (for want of room) wrote out only to the nearest minutes, which is sufficient in whole centuries. By means of these two tables, we may find the time of any mean new moon in March, together with the anomalies of the sun and moon, and the sun's distance from the node, at these times, within the limits of 6000 years, either before or after any given year in the 18th century; and the mean time of any new or full moon in any given month after March, by means of the third and fourth tables, within the same limits, as shown in the precepts for calculation.

Thus it would be a very easy matter to calculate the time of any new or full moon, if the sun and moon moved equably in all parts of their orbits.—But we have already shewn, that their places are never the same as they would be by equable motions, except when they are in apogee or perigee; which is, when their mean anomalies are either nothing, or six signs; and that their mean places are always forwarder than their true places, whilst the anomaly is less than six signs; and their true places are forwarder than the mean, whilst the anomaly is more.

Hence it is evident, that whilst the sun's anomaly is less than six signs, the moon will overtake him, or be opposite to him, sooner than she could if his motion were equable; and later whilst his anomaly is more than six signs.—The greatest difference that can possibly happen between the mean and true time of new or full moon, on account of the inequality of the sun's motion, is 3 hours 48 minutes 28 seconds; and that is, when the sun's anomaly is either 3 signs 1 degree, or 8 signs 29 degrees; sooner in the first case, and later in the last.—In all other signs and degrees of anomaly, the difference is gradually less, and vanishes when the anomaly is either nothing or six signs. The sun is in his apogee on the 30th of June, and in his perigee on the 30th of December, in the present age: so that he is nearer the earth in our winter than in our summer.—The proportional difference of distance, deduced from the difference of the sun's apparent diameter at these times, is as 983 to 1017.

The moon's orbit is dilated in winter, and contracted in summer; therefore the lunations are longer in winter than in summer. The greatest difference is found to be 22 minutes 29 seconds; the lunations increasing gradually in length whilst the sun is moving from his apogee to his perigee, and decreasing in length whilst he is moving from his perigee to his apogee.—On this account, the moon will be later every time coming to her conjunction with the sun, or being in opposition to him, from December till June, and sooner from June till December, than if her orbit had continued of the same size all the year round.

As both these differences depend on the sun's anomaly, they may be fitly put together into one table, and called The annual or first equation of the mean to the true syzygy (see Table VII.) This equational difference is to be subtracted from the time of the mean syzygy when the sun's anomaly is less than six signs, and added when the anomaly is more.—At the greatest it is 4 hours 10 minutes 57 seconds, viz. 3 hours 48 minutes 28 seconds, on account of the sun's unequal motion, and 22 minutes 29 seconds, on account of the dilatation of the moon's orbit.

This compound equation would be sufficient for reducing the mean time of new or full moon to the true time thereof, if the moon's orbit were of a circular form, and her motion quite equable in it. But the moon's orbit is more elliptical than the sun's, and her motion in it so much the more unequal. The difference is so great, that she is sometimes in conjunction with the sun, or in opposition to him, sooner by 9 hours 47 minutes 54 seconds, than she would be if her motion were equable; and at other times as much later. The former happens when her mean anomaly is 9 signs 4 degrees, and the latter when it is 2 signs 26 degrees. See Table IX.

At different distances of the sun from the moon's apogee, the figure of the moon's orbit becomes different. It is longest of all, or most eccentric, when the sun is in the same sign and degree either with the moon's apogee or perigee; shortest of all, or least eccentric, when the sun's distance from the moon's apogee is either three signs or nine signs; and at a mean state when the distance is either 1 sign 15 degrees, 4 signs 15 degrees, 7 signs 15 degrees, or 10 signs 15 degrees. When the moon's orbit is at its greatest eccentricity, her apogal distance from the earth's centre is to her perigal distance therefrom, as 1067 is to 933; when least eccentric, as 1043 is to 957; and when at the mean state, as 1055 is to 945.

But the sun's distance from the moon's apogee is equal to the quantity of the moon's mean anomaly at the time of new moon, and by the addition of five signs it becomes equal in quantity to the moon's mean anomaly at the time of full moon. Therefore, a table may be constructed to answer to all the various inequalities depending on the different eccentricities of the moon's orbit, in the syzygies; and called The second equation of the mean to the true syzygy (See Table IX.) and the moon's anomaly, when equated by Table VIII., may be made the proper argument for taking out this second equation of time; which must be added to the former equated time, when the moon's anomaly is less than six signs, and subtracted when the anomaly is more.

There are several other inequalities in the moon's motion, which sometimes bring on the true syzygy a little sooner, and at other times keep it back a little later, than it would otherwise be; but they are so small, that they may be all omitted except two; the former of which (see Table X.) depends on the difference between the anomalies of the sun and moon in the syzygies, and the latter (see Table XI.) depends on the sun's distance from the moon's nodes at these times.—The greatest difference arising from the former is 4 minutes 58 seconds; and from the latter, 1 minute 34 seconds.

The tables here inserted being calculated by Mr Fer-Direction for the use of those tables.

To calculate the true time of New or full Moon.

Precept I. If the required time be within the limits of the 18th century, write out the mean time of new moon in March, for the proposed year, from Table I. in the old file, or from Table II. in the new; together with the mean anomalies of the sun and moon, and the sun's mean distance from the moon's ascending node. If you want the time of full moon in March, add the half lunation at the foot of Table III. with its anomalies, &c. to the former numbers, if the new moon falls before the 15th of March; but if it falls after, subtract the half lunation, with the anomalies, &c. belonging to it, from the former numbers, and write down the respective sums or remainders.

II. In these additions or subtractions, observe, that 60 seconds make a minute, 60 minutes make a degree, 30 degrees make a sign, and 12 signs make a circle. When you exceed 12 signs in addition, reject 12, and set down the remainder. When the number of signs to be subtracted is greater than the number you subtract from, add 12 signs to the lesser number, and then you will have a remainder to set down. In the table signs are marked thus *, degrees thus °, minutes thus ', and seconds thus ".

III. When the required new or full moon is in any given month after March, write out as many lunations with their anomalies, and the sun's distance from the node from Table III. as the given month is after March, setting them in order below the numbers taken out for March.

VI. Add all these together, and they will give the mean time of the required new or full moon, with the mean anomalies and sun's mean distance from the ascending node, which are the arguments for finding the proper equations.

V. With the number of days added together, enter Table IV. under the given month; and against that number you have the day of mean new or full moon in the left-hand column, which set before the hours, minutes, and seconds, already found.

But (as it will sometimes happen) if the said number of days falls short of any in the column under the given month, add one lunation and its anomalies, &c. (from Table III.) to the foreaid sums, and then you will will have a new sum of days wherewith to enter Table IV. under the given month, where you are sure to find it the second time, if the first falls short.

VI. With the signs and degrees of the sun's anomaly, enter Table VIII. and therewith take out the annual or first equation for reducing the mean syzygy to the true; taking care to make proportions in the table for the odd minutes and seconds of anomaly, as the table gives the equation only to whole degrees.

Observe, in this and every other case of finding equations, that if the signs are at the head of the table, their degrees are at the left hand, and are reckoned downwards; but if the signs are at the foot of the table, their degrees are at the right hand, and are counted upward; the equation being in the body of the table, under or over the signs, in a collateral line with the degrees. The titles Add or Subtract at the head or foot of the tables where the signs are found, shew whether the equation is to be added to the mean time of new or full moon, or to be subtracted from it. In this table, the equation is to be subtracted, if the signs of the sun's anomaly are found at the head of the table; but it is to be added, if the signs are at the foot.

VII. With the signs and degrees of the sun's mean anomaly, enter Table VIII. and take out the equation of the moon's mean anomaly; subtract this equation from her mean anomaly, if the signs of the sun's anomaly be at the head of the table, but add it if they are at the foot; the result will be the moon's equated anomaly, with which enter Table IX. and take out the second equation for reducing the mean to the true time of new or full moon; adding this equation, if the signs of the moon's anomaly are at the head of the table, but subtracting it if they are at the foot; and the result will give you the mean time of the required new or full moon twice equated, which will be sufficiently near for common almanacs.—But when you want to calculate an eclipse, the following equations must be used: thus,

VIII. Subtract the moon's equated anomaly from the sun's mean anomaly, and with the remainder in signs and degrees, enter Table X. and take out the third equation, applying it to the former equated time, as the titles, Add or Subtract, do direct.

IX. With the sun's mean distance from the ascending node enter Table XI. and take out the equation answering to that argument, adding it to, or subtracting it from, the former equated time, as the titles direct, and the result will give the time of new or full moon, agreeing with well regulated clocks or watches, very near the truth. But, to make it agree with the solar, or apparent time, you must apply the equation of natural days, taken from an equation table, as it is leap-year, or the first, second, or third after. This, however, unless in very nice calculations, needs not be regarded, as the difference between true and apparent time is never very considerable.

The method of calculating the time of any new or full moon without the limits of the 18th century, will be shewn further on. And a few examples compared with the precepts, will make the whole work plain.

N.B. The tables begin the day at noon, and reckon forward from thence to the noon following.—Thus, March the 31st, at 22 h. 30 min. 25 sec. of tabular time, is April 1st (in common reckoning) at 30 min. 25 sec. after 10 o'clock in the morning.

**EXAMPLE I.**

Required the true time of New Moon in April 1764, New Style?

| By the Precepts. | New Moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |-----------------|-----------|----------------|-----------------|----------------| | | D. H. M. S. | s o ' " | s o ' " | s o ' " | | March 1764, | 2 8 55 36 | 8 2 20 0 | 10 13 35 21 | 11 4 54 48 | | Add 1 Lunation, | 29 12 44 3 | 0 29 6 19 | 0 25 49 0 | 1 0 40 14 | | Mean New Moon, | 31 21 39 39 | 9 1 26 19 | 11 9 24 21 | 0 5 35 2 | | First Equation, | + 4 10 40 11 | 10 59 18 | + 1 34 57 | | | Time once equated, | 32 1 50 19 | 9 20 27 11 | 10 59 18 | Sun from Node, Arg. 3rd equation. Arg. 2nd equation. | | Second Equation, | - 3 24 49 | | | | | Time twice equated, | 31 22 25 30 | | | | | Third Equation, | + 4 37 | | | | | Time thrice equated, | 31 22 30 7 | | | | | Fourth Equation, | + 18 | | | | | True New Moon, | 31 22 30 25 | | | | | Equation of days, | - 3 48 | | | | | Apparent time, | 31 22 26 37 | | | |

So the true time is 22 h. 30 min. 25 sec. after the noon of the 31st March; that is, April 1st, at 30 min. 25 sec. after ten in the morning. But the apparent time is 26 min. 37 sec. after ten in the morning. ### Example II

**Qu.** The true time of Full Moon in May 1762, New Style?

| By the Precepts | New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |-----------------|----------|---------------|----------------|--------------| | March 1762, | | | | | | Add 2 Lunations | | | | | | New Moon, May, | | | | | | Subt. ½ Lunation| | | | | | Full Moon, May, | | | | | | First Equation, | | | | | | Time once equated, | | | | | | Second Equation, | | | | | | Time twice equated, | | | | | | Third Equation, | | | | | | Time thrice equated, | | | | | | Fourth Equation, | | | | | | The Full Moon, | | | | |

Anf. May 7th at 15 h. 50 min. 50 sec. past noon, viz. May 8th at 3 h. 50 min. 50 sec. in the morning.

---

**To calculate the time of New and Full Moon in a given year and month of any particular century, between the Christian era and the 18th century.**

**Precept I.** Find a year of the same number in the 18th century with that of the year in the century proposed, and take out the mean time of new moon in March, old style, for that year, with the mean anomalies and sun's mean distance from the node at that time, as already taught.

II. Take as many complete centuries of years from Table VI. as, when subtracted from the above said year in the 18th century, will answer to the given year; and take out the first mean new moon and its anomalies, &c.

---

**Example III.**

Required the true time of Full Moon in April, Old Style, A.D. 30.

From 1730 subtract 1700 (or 17 centuries) and there remains 30.

| By the Precepts | New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |-----------------|----------|---------------|----------------|--------------| | March 1730, | | | | | | Add ½ Lunation, | | | | | | Full Moon, | | | | | | 1700 years subtr.| | | | | | Full D March A.D. 30. | | | | | | Add 1 Lunation, | | | | | | Full Moon, April, | | | | | | First Equation, | | | | | | Time once equated, | | | | | | Second Equation, | | | | | | Time twice equated, | | | | | | Third Equation, | | | | | | Time thrice equated, | | | | | | Fourth Equation, | | | | | | True Full Moon, April, | | | | |

Hence it appears, that the true time of Full Moon in April, A.D. 30, old style, was on the 6th day at 25 m. 45 f. past eight in the evening. To calculate the true time of New or Full Moon in any given year and month before the Christian era.

Precept I. Find a year in the 18th century, which being added to the given number of years before Christ diminished by one, shall make a number of complete centuries.

II. Find this number of centuries in Table VI. and

**Example IV.**

Required the true time of New Moon in May, Old Style, the year before Christ 585?

The years 584 added to 1716, make 2300, or 23 centuries.

| By the Precepts. | New moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |-----------------|-----------|---------------|-----------------|---------------| | March 1716, | | | | | | 2300 years subtract. | 11 17 33 29 | 8 22 50 39 | 4 4 14 24 | 4 27 17 5 | | March before Christ 585, | 11 5 57 53 | 11 19 47 0 | 1 5 59 0 | 7 25 27 0 | | Add 3 Lunations, | 0 11 35 36 | 9 3 3 39 | 2 28 15 2 | 9 1 50 5 | | May before Christ 585, | 88 14 12 9 | 2 27 18 58 | 2 17 27 1 | 3 2 0 42 | | First Equation, | 28 1 47 45 | 0 0 22 37 | 5 15 42 3 | 0 3 50 47 | | Time once equated, | 28 1 46 8 | 6 14 41 20 | 5 15 41 17 | Sun from Node, Arg. 3rd equation | | Second Equation, | + 2 15 1 | Arg. 2nd equation | Arg. 2nd equation | equation. | | Time twice equated, | 28 4 1 9 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Third Equation, | + 1 9 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Time thrice equated, | 28 4 2 18 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Fourth Equation, | + 12 | Arg. 3rd equation | Arg. 2nd equation | equation. | | True new moon, | 28 4 2 30 | Arg. 3rd equation | Arg. 2nd equation | equation. |

So the true time was May 28th, at 2 minutes 30 seconds past four in the afternoon.

These Tables are calculated for the meridian of London; but they will serve for any other place, by subtracting four minutes from the tabular time, for every degree that the meridian of the given place is westward of London, or adding four minutes for every degree that the meridian of the given place is eastward: as in

**Example V.**

Required the true time of Full Moon at Alexandria in Egypt in September, Old Style, the year before Christ 201?

The years 200 added to 1800, make 2000, or 20 centuries.

| By the Precepts. | New Moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |-----------------|-----------|---------------|-----------------|---------------| | March 1800, | | | | | | Add 1 lunation, | 13 0 22 17 | 8 23 19 55 | 10 7 52 36 | 11 3 58 24 | | From the sum, | 29 12 44 3 | 0 29 6 19 | 0 25 49 0 | 1 0 40 14 | | Subtract 2000 years, | 42 13 6 20 | 9 22 26 14 | 11 3 41 36 | 0 4 38 38 | | N. M. bef. Chr. 201, | 27 18 9 19 | 0 8 50 0 | 0 15 42 0 | 6 27 45 7 | | Add {6 lunations, | 14 18 57 1 | 9 13 36 14 | 10 17 59 36 | 5 6 53 38 | | half lunations, | 177 4 24 18 | 5 24 37 56 | 5 4 54 3 | 6 4 1 24 | | Full Moon, September, | 14 18 22 2 | 0 14 33 10 | 0 12 54 30 | 0 15 20 7 | | First equation, | 22 17 43 21 | 3 22 47 20 | 10 5 48 9 | 11 26 15 9 | | Time once equated, | 22 13 51 15 | 5 18 27 25 | 10 4 19 55 | Sun from Node, Arg. 3rd equation | | Second equation, | - 8 25 4 | Arg. 2nd equation | Arg. 2nd equation | fourth equation. | | Time twice equated, | 22 5 26 11 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Third equation, | - 58 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Time thrice equated, | 22 5 25 13 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Fourth equation, | - 12 | Arg. 3rd equation | Arg. 2nd equation | equation. | | True time at London, | 22 5 25 1 | Arg. 3rd equation | Arg. 2nd equation | equation. | | Add for Alexandria, | 2 1 27 | Arg. 3rd equation | Arg. 2nd equation | equation. | | True time there, | 22 7 26 28 | Arg. 3rd equation | Arg. 2nd equation | equation. |

Thus it appears, that the true time of Full Moon at Alexandria, in September, old style, the year before Christ 201, was the 22nd day, at 26 minutes 28 seconds after seven in the evening. **Example VI.**

Required the true time of Full Moon at Babylon in October, Old Style, the 4008 year before the first year of Christ, or 4007 before the year of his birth?

The years 4007 added to 1793, make 5800, or 58 centuries.

| By the Precepts. | New Moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |------------------|-----------|----------------|-----------------|---------------| | March 1793, | D. H. M. S. | 8 0 0 | 8 0 0 | 8 0 0 | | Subtract 5800 years, | 30 9 13 55 | 9 10 16 11 | 8 7 37 58 | 7 6 18 26 | | N. M. bef. Chr. 4007, | 15 12 38 7 | 10 21 35 0 | 6 24 43 0 | 9 13 1 0 | | Add {7 lunations, | 14 20 35 48 | 10 18 41 11 | 1 12 54 58 | 9 23 17 26 | | half lunations,} | 206 17 8 21 | 6 23 44 15 | 6 0 43 3 | 7 4 41 38 | | Full Moon, October, | 14 18 22 2 | 0 14 33 10 | 6 12 54 30 | 0 15 20 7 | | First Equation, | 22 8 6 11 | 5 26 58 36 | 1 26 32 31 | 5 13 19 11 | | Time once equated, | — 13 26 1 | 26 27 26 | — 5 5 | Sun from Node, | | Second Equation, | 22 7 52 45 | 4 0 31 10 | 1 20 27 26 | and Argument of | | + 8 29 21 Arg. 3rd equation. | Arg. 2nd equation. | | | fourth equation. |

So that, on the meridian of London, the true time was October 23rd, at 17 minutes 5 seconds past four in the morning; but at Babylon, the true time was October 23rd, at 42 minutes 46 seconds past six in the morning.—This is supposed by some to have been the year of the creation.

To calculate the true time of new or full moon in any given year and month after the 18th century.

**Precept I.** Find a year of the same number in the 18th century with that of the year proposed, and take out the mean time and anomalies, &c. of new moon in March, old style, for that year, in Table I.

II. Take so many years from Table VI. as when added to the abovementioned year in the 18th century will answer to the given year in which the new or full moon is required; and take out the first new moon, with its anomalies for these complete centuries.

III. Add all these together, and then work in all respects as above shewn, only remember to subtract a lunation and its anomalies, when the above said addition carries the new moon beyond the 31st of March; as in the following example.

**Example VII.**

Required the true time of New Moon in July, Old Style, A.D. 2180?

Four centuries (or 4000 years) added to A.D. 1780, make 2180.

| By the Precepts. | New Moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |------------------|-----------|----------------|-----------------|---------------| | March 1780, | D. H. M. S. | 8 0 0 | 8 0 0 | 8 0 0 | | Add 400 years, | 23 23 1 44 | 9 4 18 13 | 1 21 7 47 | 10 18 21 1 | | From the sum | 17 8 43 29 | 0 13 24 0 | 10 1 28 0 | 6 17 49 0 | | Subtract 1 lunation, | 41 7 45 13 | 9 17 42 13 | 11 22 35 47 | 6 10 1 | | New Moon March 2180, | 29 12 44 3 | 0 29 6 19 | 0 25 49 0 | 0 40 14 | | Add 4 lunations, | 11 19 1 10 | 8 18 35 54 | 10 26 46 47 | 4 5 29 47 | | New Moon July 2180, | 118 2 56 12 | 3 26 25 17 | 3 13 16 2 | 4 2 40 56 | | First equation, | 7 21 57 22 | 0 15 1 11 | 2 10 2 49 | 8 8 10 43 | | Time once equated, | — 1 3 39 3 | 9 38 37 | — 24 12 | Sun from Node, | | Second equation, | 7 20 53 43 | 10 5 22 34 | 2 9 38 37 | and argument of | | + 9 24 8 Arg. 3rd equation. | Arg. 2nd equation. | | | fourth equation. |

True time, July 8th, at 22 minutes 55 seconds past six in the evening. In keeping by the old style, we are always sure to be right, by adding or subtracting whole hundreds of years to or from any given year in the 18th century. But in the new style we may be very apt to make mistakes, on account of the leap-year's not coming in regularly every fourth year; and therefore, when we go without the limits of the 18th century, we had best keep to the old style, and at the end of the calculation reduce the time to the new. Thus, in the 22nd century, there will be fourteen days difference between the styles; and therefore, the true time of new moon in this last example being reduced to the new style, will be the 22nd of July, at 22 minutes 55 seconds past six in the evening.

To calculate the true place of the Sun for any given moment of time.

Precept I. In Table XII find the next lesser year in number to that in which the Sun's place is sought, and write out his mean longitude and anomaly answering thereto: to which add his mean motion and anomaly for the complete residue of years, months, days, hours, minutes, and seconds, down to the given time, and this will be the Sun's mean place and anomaly at that time, in the old style, provided the said time be in any year after the Christian era. See the first following Example.

II. Enter Table XIII, with the Sun's mean anomaly, and making proportions for the odd minutes and seconds thereof, take out the equation of the Sun's centre: which, being applied to his mean place as the title Add or Subtract directs, will give his true place or longitude from the vernal equinox, at the time for which it was required.

III. To calculate the Sun's place for any time in a given year before the Christian era, take out his mean longitude and anomaly for the first year thereof, and from these numbers subtract the mean motions and anomalies for the complete hundreds or thousands next above the given year; and, to the remainders, add those for the residue of years, months, &c. and then work in all respects as above. See the second Example following.

Example I.

Required the Sun's true place, March 20th Old Style, 1764, at 22 hours 30 minutes 25 seconds past noon?

In common reckoning, March 21st, at 10 hours 30 minutes 25 seconds in the afternoon.

| Sun's Longitude | Sun's Anomaly | |-----------------|--------------| | 8° 0' | 8° 0' | | 9° 20' 43" | 6° 13' 1" | | 0° 0' 27" | 11° 29' 26" | | 3° 11' 29" | 11° 39' 14" | | 1° 28' 9" | 1° 28' 9" | | 20° 41' 55" | 20° 41' 55" | | 54° 13' | 54° 13' | | 1° 14' | 1° 14' | | | |

Sun's mean place at the given time

Equation of the Sun's centre, add

Sun's true place at the same time

Example II.

Required the Sun's true place, October 23rd Old Style, at 16 hours 57 minutes past noon, in the 4006th year before the year of Christ 1; which was the 4007th before the year of his birth, and the year of the Julian period 706.

By the Precepts.

From the radical numbers after Christ:

Subtract those for 5000 complete years

Remains for a new radix

Complete years

To which add,

to bring it to the given time.

Days

Hours

Minutes

Sun's mean place at the given time

Equation of the Sun's centre subtract

Sun's true place at the same time So that in the meridian of London, the sun was then just entering the sign of Libra, and consequently was upon the point of the autumnal equinox.

If to the above time of the autumnal equinox at London, we add 2 hours, 25 minutes 41 seconds for the longitude of Babylon, we shall have for the time of the same equinox, at that place, October 23rd, at 19 hours 22 minutes 41 seconds; which, in the common way of reckoning, is October 24th, at 22 minutes 41 seconds past seven in the morning.

And it appears by Example VI. that in the same year, the true time of full moon at Babylon was October 23rd, at 42 minutes 46 seconds after six in the morning; so that the autumnal equinox was on the day next after the day of full moon.—The dominical letter for that year was G, and consequently the 24th of October was on a Wednesday.

To find the sun's distance from the moon's ascending node, at the time of any given new or full moon; and consequently, to know whether there is an eclipse at that time, or not.

The sun's distance from the moon's ascending node is the argument for finding the moon's fourth equation in the syzygies, and therefore it is taken into all the foregoing examples in finding the times thereof. Thus, at the time of mean new moon in April 1764, the sun's mean distance from the ascending node, is 0° 5° 35' 2".

See Example I. p. 829.

The descending node is opposite to the ascending one, and they are just six signs distant from each other.

When the sun is within 17 degrees of either of the nodes at the time of new moon, he will be eclipsed at that time; and when he is within 12 degrees of either of the nodes at the time of full moon, the moon will be then eclipsed. Thus we find, that there will be an eclipse of the sun at the time of new moon in April 1764.

But the true time of that new moon comes out by the equations to be 50 minutes 46 seconds later than the mean time thereof, by comparing these times in the above example; and therefore, we must add the sun's motion from the node during that interval, to the above mean distance 0° 5° 35' 2", which motion is found in Table XII. for 50 minutes 46 seconds, to be 2° 12'.

And to this we must apply the equation of the sun's mean distance from the node, in Table XV. found by the sun's anomaly, which, at the mean time of new moon in Example I. is 9° 1° 26' 15"; and then we shall have the sun's true distance from the node, at the true time of new moon, as follows:

| Sun from Node. | |----------------| | At the mean time of new moon in April 1764. | | Sun's motion from the node for 50 minutes 46 seconds | | Sun's mean distance from node at true new moon | | Equation of mean distance from node, add |

Sun's true distance from the ascending node

Which being far within the above limit of 17 degrees, shows that the sun must then be eclipsed.

And now we shall shew how to project this, or any other eclipse, either of the sun or moon.

To project an Eclipse of the Sun.

In order to this, we must find the ten following elements, by means of the tables.

1. The true time of conjunction of the sun and moon; and at that time. 2. The semidiameter of the earth's disk, as seen from the moon, which is equal to the moon's horizontal parallax. 3. The sun's distance from the fulcrum colure to which he is then nearest. 4. The sun's declination. 5. The angle of the moon's visible path with the ecliptic. 6. The moon's latitude. 7. The moon's true horary motion from the sun. 8. The sun's semidiameter. 9. The moon's. 10. The semidiameter of the penumbra.

We shall now proceed to find these elements for the sun's eclipse in April 1764.

To find the true time of new moon. This, by Example I. p. 829, is found to be on the first day of the said month, at 30 minutes 25 seconds after ten in the morning.

2. To find the moon's horizontal parallax, or semidiameter of the earth's disk, as seen from the moon. Enter Table XVII. with the signs and degrees of the moon's anomaly (making proportions because the anomaly is in the table only to every 6° degree), and thereby take out the moon's horizontal parallax; which for the above time, answering to the anomaly 11° 9' 24" 21", is 54° 53".

3. To find the sun's distance from the nearest solstice, viz. the beginning of Cancer, which is 3° or 90° from the beginning of Aries. It appears by Example I. on p. 833, (where the sun's place is calculated to the above time of new moon) that the sun's longitude from the beginning of Aries is then 0° 12° 10' 12"; that is, the sun's place at that time is 9° Aries, 12° 10' 12".

Therefore from

Subtract the sun's longitude or place 0° 12° 10' 12"

Remains the sun's distance from the solstice

Or 77° 49' 48"; each sign containing 30 degrees.

4. To find the sun's declination. Enter Table XIV. with the signs and degrees of the sun's true place, viz. 0° 12°, and making proportions for the 10° 12", take out the sun's declination answering to his true place, and it will be found to be 4° 49' north.

5. To find the moon's latitude. This depends on her distance from her ascending node, which is the same as the sun's distance from it at the time of new moon; and is thereby found in Table XVI.

But we have already found, that the sun's equated distance from the ascending node, at the time of new moon in April 1764, is 0° 5° 42' 14". See above.

Therefore, enter Table XVI. with o signs at the top, and 7 and 8 degrees at the left hand, and take out 3° 6' and 39°, the latitude for 7°; and 4° 51', the latitude for 8°; and by making proportions between these latitudes for the 4° 14', by which the moon's distance from the node exceeds 7 degrees; her true latitude will be found to be 4° 18' north ascending.

6. To find the moon's true horary motion from the sun. With With the moon's anomaly, viz. $11^\circ 9' 24''$, enter Table XVII, and take out the moon's horary motion; which, by making proportions in that table, will be found to be $30' 22''$. Then, with the sun's anomaly, $9^\circ 16' 19''$, take out his horary motion $2' 28''$ from the same table; and subtracting the latter from the former, there will remain $27' 54''$ for the moon's true horary motion from the sun.

7. To find the angle of the moon's visible path with the ecliptic. This, in the projection of eclipses, may be always rated at $5^\circ 35'$, without any sensible error.

8. To find the semidiameters of the sun and moon. These are found in the same table, and by the same arguments, as their horary motions. In the present case, the sun's anomaly gives her semidiameter $16' 6''$, and the moon's anomaly gives her semidiameter $14' 57''$.

9. To find the semidiameter of the penumbra. Add the moon's semidiameter to the sun's, and their sum will be the semidiameter of the penumbra, viz. $31' 3''$.

Now collect these elements, that they may be found the more readily when they are wanted in the construction of this eclipse.

1. True time of new moon in April, 1764

| Time | 1 | 10 | 30 | 25 | |------|---|----|----|----| | | | | | |

2. Semidiameter of the earth's disk $54' 53''$ 3. Sun's distance from the nearest solst $77' 49' 48''$ 4. Sun's declination, north $4' 49'$ 5. Moon's latitude, north ascending $40' 18''$ 6. Moon's horary motion from the sun $27' 54''$ 7. Angle of the moon's visible path with the ecliptic $5' 35'$ 8. Sun's semidiameter $16' 6''$ 9. Moon's semidiameter $14' 57''$ 10. Semidiameter of the penumbra $31' 3''$

To project an eclipse of the sun geometrically.

Make a scale of any convenient length, as AC, and divide it into as many equal parts as the earth's semi-disk contains minutes of a degree; which, at the time of the eclipse in April 1764, is $54' 53''$. Then, with the whole length of the scale as a radius, describe the semicircle AMB upon the centre C; which semicircle shall represent the northern half of the earth's enlightened disk, as seen from the sun.

Upon the centre C raise the straight line CH, perpendicular to the diameter ACB; so ACB shall be a part of the ecliptic, and CH its axis.

Being provided with a good sector, open it to the radius CA in the line of chords; and taking from thence the chord of $23\frac{1}{2}$ degrees in your compasses, let it off both ways from H, to g and to h, in the periphery of the semidisk; and draw the straight line gVh, in which the north pole of the disk will always be found.

When the sun is in Aries, Taurus, Gemini, Cancer, Leo, and Virgo, the north pole of the earth is enlightened by the sun; but whilst the sun is in the other six signs, the south pole is enlightened, and the north pole is in the dark.

And when the sun is in Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini, the northern half of the earth's axis C XII P lies to the right hand of the axis of the ecliptic, as seen from the sun; and to the left hand, whilst the sun is in the other six signs.

Open the sector till the radius (or distance of the two 90's) of the lines be equal to the length of Vh, and take the line of the sun's distance from the solstice ($77' 49' 48''$) as nearly as you can guess, in your compasses, from the line of lines, and set off that distance from V to P in the line gVh, because the earth's axis lies to the right hand of the axis of the ecliptic in this case, the sun being in Aries; and draw the straight line C XII P for the earth's axis, of which P is the north pole. If the earth's axis had lain to the left hand from the axis of the ecliptic, the distance VP would have been set off from V towards g.

To draw the parallel of latitude of any given place, as suppose London, or the path of that place on the earth's enlightened disk as seen from the sun, from sunrise till sun-set, take the following method.

Subtract the latitude of London, $51\frac{1}{2}$ from $90^\circ$, and the remainder $38\frac{1}{2}$ will be the co-latitude, which take in your compasses from the line of chords, making CA or CB the radius, and set it from b (where the earth's axis meets the periphery of the disk) to VI and VI, and draw the occult or dotted line VI K VI.

Then, from the points where this line meets the earth's disk, set off the chord of the sun's declination $4' 49'$ to D and F, and to E and G, and connect these points by the two occult lines F XII G and DLE.

Bisect LK XII in K, and through the point K draw the black line VI K VI. Then making CB the radius of a line of lines on the sector, take the colatitude of London $38\frac{1}{2}$ from the lines in your compasses, and set it both ways from K, to VI and VI.

These hours will be just in the edge of the disk at the equinoxes, but at no other time in the whole year.

With the extent K VI taken into your compasses, set one foot in K (in the black line below the occult one) as a centre, and with the other foot describe the semicircle VI 7 8 9 10, &c. and divide it into 12 equal parts. Then, from these points of division, draw the occult lines 7p, 8q, 9r, &c. parallel to the earth's axis C XII P.

With the small extent K XII as a radius, describe the quadrantial arc XIIj; and divide it into six equal parts, as XII a, ab, bc, cd, de, and ef; and through the division-points a, b, c, d, e draw the occult lines VII e V, VIII d IV, IX c III, X b II, and XI a I, all parallel to VI K VI, and meeting the former occult lines 7p, 8q, &c. in the points VII VIII IX X XI, V IV III II and I; which points shall mark the several situations of London on the earth's disk, at these hours respectively, as seen from the sun; and the elliptic curve VI VII VIII, &c. being drawn through these points, shall represent the parallel of latitude, or path of London on the disk, as seen from the sun, from its rising to its setting.

N.B. If the sun's declination had been south, the diurnal path of London would have been on the upper side of the line VI K VI, and would have touched the line DLE in L. It is requisite to divide the horary spaces into quarters (as some are in the figure), and, if possible, into minutes also.

Make CB the radius of a line of chords on the sector, and taking therefrom the chord of $5' 35'$, the angle of the moon's visible path with the ecliptic, set it off from H to M on the left hand of CH, the axis of of the ecliptic, because the moon's latitude is north ascending. Then draw CM for the axis of the moon's orbit, and bisect the angle MCH by the right line Cz. If the moon's latitude had been north descending, the axis of her orbit would have been on the right hand from the axis of the ecliptic.—N. B. The axis of the moon's orbit lies the same way when her latitude is south ascending, as when it is north ascending; and the same way when south descending, as when north descending.

Take the moon's latitude $40^\circ 18''$ from the scale CA in your compasses, and set it from i to x in the bisecting line Cz, making ix parallel to Cy; and thro' x, at right angles to the axis of the moon's orbit CM, draw the straight line N wyS for the path of the penumbra's centre over the earth's disk.—The point w, in the axis of the moon's orbit, is that where the penumbra's centre approaches nearest to the centre of the earth's disk, and consequently is the middle of the general eclipse: the point x is that where the conjunction of the sun and moon falls, according to equal time by the tables; and the point y is the ecliptical conjunction of the sun and moon.

Take the moon's true horary motion from the sun, $27^\circ 54''$, in your compasses, from the scale CA (every division of which is a minute of a degree), and with that extent make marks along the path of the penumbra's centre; and divide each space from mark to mark into sixty equal parts or horary minutes, by dots; and set the hours to every 60th minute in such a manner, that the dot signifying the instant of new moon by the tables, may fall into the point x, half way between the axis of the moon's orbit and the axis of the ecliptic; and then, the rest of the dots will shew the points of the earth's disk, where the penumbra's centre is at the instants denoted by them, in its transit over the earth.

Apply one side of a square to the line of the penumbra's path, and move the square backwards and forwards until the other side of it cuts the same hour and minute (as at m and n) both in the path of London, and in the path of the penumbra's centre; and the particular minute or instant which the square cuts at the same time in both paths, shall be the instant of the visible conjunction of the sun and moon, or greatest obscuration of the sun, at the place for which the construction is made, namely London, in the present example; and this instant is at $47\frac{1}{2}$ minutes past ten o'clock in the morning; which is 17 minutes five seconds later than the tabular time of true conjunction.

Take the sun's semidiameter, $16^\circ 6''$, in your compasses, from the scale CA, and setting one foot in the path of London, at m, namely at $47\frac{1}{2}$ minutes past ten, with the other foot describe the circle UX, which shall represent the sun's disk as seen from London at the greatest obscuration.—Then take the moon's semidiameter, $14^\circ 57''$, in your compasses from the same scale; and setting one foot in the path of the penumbra's centre at m, in the $47\frac{1}{2}$ minute after ten, with the other foot describe the circle TY for the moon's disk, as seen from London, at the time when the eclipse is at the greatest, and the portion of the sun's disk which is hid or cut off by the moon's, will shew the quantity of the eclipse at that time; which quantity may be measured on a line equal to the sun's diameter, and divided into 12 equal parts for digits.

Lastly, take the semidiameter of the penumbra, $31^\circ 3''$, from the scale CA in your compasses; and setting one foot in the line of the penumbra's central path, on the left hand from the axis of the ecliptic, direct the other foot toward the path of London; and carry that extent backwards and forwards, till both the points of the compasses fall into the same instants in both the paths: and these instants will denote the time when the eclipse begins at London.—Then, do the like on the right hand of the axis of the ecliptic; and where the points of the compasses fall into the same instants in both the paths, they will shew at what time the eclipse ends at London.

These trials give 20 minutes after nine in the morning for the beginning of the eclipse at London, at the points N and O; $47\frac{1}{2}$ minutes after ten, at the points m and n, for the time of greatest obscuration; and 18 minutes after twelve, at R and S, for the time when the eclipse ends; according to mean or equal time.

From these times we must subtract the equation of natural days, viz. three minutes 48 seconds, in leap-year April 1, and we shall have the apparent times; namely, nine hours 16 minutes 12 seconds for the beginning of the eclipse, ten hours 43 minutes 42 seconds for the time of greatest obscuration, and 12 hours 14 minutes 12 seconds for the time when the eclipse ends.—But the best way is to apply this equation to the true equal time of new moon, before the projection be begun; as is done in example I. For the motion or position of places on the earth's disk answer to apparent or solar time.

In this construction it is supposed, that the angle under which the moon's disk is seen, during the whole time of the eclipse, continues invariably the same; and that the moon's motion is uniform and rectilineal during that time. But these suppositions do not exactly agree with the truth; and therefore, supposing the elements given by the tables to be accurate, yet the times and phases of the eclipse, deduced from its construction, will not answer exactly to what passeth in the heavens; but may be at least two or three minutes wrong, though done with the greatest care. Moreover, the paths of all places of considerable latitudes, are nearer the centre of the earth's disk, as seen from the sun, than those constructions make them; because the disk is projected as if the earth were a perfect sphere although it is known to be a spheroid. Consequently, the moon's shadow will go farther northward in all places of northern latitude, and farther southward in all places of southern latitude, than it is shewn to do in these projections.—According to Meyer's Tables, this eclipse was about a quarter of an hour sooner than either these tables, or Mr Flamsteed's, or Dr Halley's make it; and was not annular at London. But M. De la Caille's make it almost central.

The projection of lunar eclipses.

When the moon is within 12 degrees of either of her nodes at the time when she is full, she will be eclipsed, otherwise not.

We find by example second, page 830, that at the time of mean full moon in May 1762, the sun's distance from the ascending node was only $4^\circ 49' 35''$. THE SUN

with the Great Spot in 1769.

Diagonal Scale of Miles.

Distance of the fourth Satellite from the third.

Fig. 2.

Fig. 3. The appearance of Saturn as emerging from behind the dark Limb of the Moon 18 June 1762 at 14:22 apparent time at Chelsea. and the moon being then opposite to the sun, must have been just as near her descending node, and was therefore eclipsed.

The elements for constructing an eclipse of the moon, are eight in number, as follow:

1. The true time of full moon; and at that time, 2. The moon's horizontal parallax. 3. The sun's semidiameter. 4. The moon's semidiameter of the earth's shadow at the moon. 5. The moon's latitude. 6. The angle of the moon's visible path with the ecliptic. 7. The moon's true horary motion from the sun.

Therefore,

1. To find the true time of new or full moon. Work as already taught in the precepts.—Thus we have the true time of full moon in May 1762 (see example II. page 830) on the 8th day, at 50 minutes 50 seconds past three o'clock in the morning.

2. To find the moon's horizontal parallax. Enter Table XVII. with the moon's mean anomaly (at the above full) \(9^\circ 29' 42''\), and thereby take out her horizontal parallax; which by making the requisite proportions, will be found to be \(57^\circ 23''\).

3. To find the semidiameters of the sun and moon. Enter Table XVII. with their respective anomalies, the sun's being \(10^\circ 70' 27''\) (by the above example) and the moon's \(9^\circ 29' 42''\); and thereby take out their respective semidiameters; the sun's \(15^\circ 56''\), and the moon's \(15^\circ 38''\).

4. To find the semidiameter of the earth's shadow at the moon. Add the sun's horizontal parallax, which is always \(10''\), to the moon's which in the present case is \(57^\circ 23''\), the sum will be \(57^\circ 33''\), from which subtract the sun's semidiameter \(15^\circ 56''\), and there will remain \(41^\circ 37''\) for the semidiameter of that part of the earth's shadow which the moon then passes through.

5. To find the moon's latitude. Find the sun's true distance from the ascending node (as already taught in page 834) at the true time of full moon; and this distance increased by six signs, will be the moon's true distance from the same node; and consequently the argument for finding her true latitude, as shown in page 834.

Thus, in example II. the sun's mean distance from the ascending node was \(0^\circ 4^\circ 49' 35''\), at the time of mean full moon; but it appears by the example, that the true time thereof was six hours 33 minutes 38 seconds sooner than the mean time; and therefore we must subtract the sun's motion from the node (found in Table XII.) during this interval, from the above mean distance \(0^\circ 4^\circ 49' 35''\), in order to have his mean distance from it at the true time of full moon. Then to this apply the equation of his mean distance from the node, found in Table XV. by his mean anomaly \(10^\circ 70' 27''\); and lastly add six signs: so shall the moon's true distance from the ascending node be found as follows:

| Sun from node at mean full moon | 0° 4° 49' 35'' | |---------------------------------|----------------| | His motion from it in 6 hours | 15 35 | | 33 minutes | 1 26 | | 38 seconds | 2 | | Sum, subtract from the uppermost line | 17 3 | | Remains his mean distance at true full moon | 0° 4° 32' 32'' |

Equation of his mean distance, add

Sun's true distance from the node

To which add

And the sum will be

Which is the moon's true distance from her ascending node at the true time of her being full; and consequently the argument for finding her true latitude at that time.—Therefore, with this argument, enter Table XVI. making proportions between the latitudes belonging to the 6th and 7th degree of the argument at the left hand (the signs being at top) for the \(10^\circ 32''\) and it will give \(32' 21''\) for the moon's true latitude, which appears by the table to be south descending.

7. To find the angle of the moon's visible path with the ecliptic. This may be stated at \(5^\circ 35''\), without any error of consequence in the projection of the eclipse.

8. To find the moon's true horary motion from the sun. With their respective anomalies take out their horary motions from Table XVII. and the sun's horary motion subtracted from the moon's, leaves remaining the moon's true horary motion from the sun: in the present case \(30' 52''\).

Now collect these elements together for use.

| D. H. M. S. | |-------------| | True time of full moon in May, 1762 | 8 3 50 50 | | Moon's horizontal parallax | 0 57 23 | | Sun's semidiameter | 0 15 56 | | Moon's semidiameter | 0 15 38 | | Semidiameter of the earth's shadow at the moon | 0 41 37 | | Moon's true latitude, south descending | 0 32 21 | | Angle of her visible path with the ecliptic | 5 33 0 | | Her true horary motion from the sun | 0 30 52 |

These elements being found for the construction of the moon's eclipse in May, 1762, proceed as follows:

Make a scale of any convenient length, as W X, and Plate L, and divide it into 60 equal parts, each part standing for a minute of a degree.

Draw the right line ACB (fig. 4.) for part of the ecliptic, and CD perpendicular thereto for the southern part of its axis; the moon having south latitude.

Add the semidiameters of the moon and earth's shadow together, which, in this eclipse, will make \(57' 15''\); and take this from the scale in your compasses, and setting one foot in the point C as a centre, with the other foot describe the semicircle ADB; in one point of which the moon's centre will be at the beginning of the eclipse, and in another at the end thereof.

Take the semidiameter of the earth's shadow, \(41' 37''\), in your compasses from the scale, and setting one foot in the centre C, with the other foot describe the semicircle KLM for the southern half of the earth's shadow, because the moon's latitude is south in this eclipse. Make CD equal to the radius of a line of chords on the sector, and set off the angle of the moon's visible path with the ecliptic, $5^\circ 35'$, from D to E, and draw the right line CFE for the southern half of the axis of the moon's orbit, lying to the right hand from the axis of the ecliptic CD, because the moon's latitude is south descending.—It would have been the same way (on the other side of the ecliptic) if her latitude had been north descending; but contrary in both cases, if her latitude had been either north ascending or south ascending.

Bisect the angle DCE by the right line CG; in which line the true equal time of opposition of the sun and moon falls, as given by the tables.

Take the moon's latitude, $32^\circ 21''$, from the scale with your compasses, and set it from C to G, in the line CG; and through the point G, at right angles to CFE, draw the right line PHGFN for the path of the moon's centre. Then, F shall be the point in the earth's shadow, where the moon's centre is at the middle of the eclipse; G, the point where her centre is at the tabular time of her being full; and H, the point where her centre is at the instant of her elliptical opposition.

Take the moon's horary motion from the sun, $50^\circ 52''$, in your compasses from the scale; and with that extent make marks along the line of the moon's path PGN: then divide each space from mark to mark, into 60 equal parts, or horary minutes, and set the hours to the proper dots in such a manner, that the dot signifying the instant of full moon (viz. 50 minutes 50 seconds after III in the morning) may be in the point G, where the line of the moon's path cuts the line that bisects the angle DCE.

Take the moon's semidiameter, $15^\circ 38''$, in your compasses from the scale, and with that extent, as a radius, upon the points N, F, and P, as centres, describe the circle Q for the moon at the beginning of the eclipse, when she touches the earth's shadow at V; the circle R for the moon at the middle of the eclipse; and the circle S for the moon at the end of the eclipse, just leaving the earth's shadow at W.

The point N denotes the instant when the eclipse began, namely, at 15 minutes 10 seconds after II in the morning; the point F the middle of the eclipse at 47 minutes 44 seconds past III; and the point P the end of the eclipse, at 18 minutes after V.—At the greatest obscuration the moon was 10 digits eclipsed.

Sect. XI. The method of finding the Longitude by the Eclipses of Jupiter's Satellites; The amazing Velocity of Light demonstrated by these Eclipses; and of Cometary Eclipses.

In the former section, having explained at great length, how eclipses of the sun and moon happen at certain times, it must be evident, that similar eclipses will be observed by the inhabitants of Jupiter and Saturn, which are attended by so many moons. These eclipses indeed very frequently happen to the satellites of Jupiter; and as they are of the greatest service in determining the longitudes of places on this earth, astronomers have been at great pains to calculate tables for the eclipses of these satellites by their primary, for the satellites themselves have never been observed to eclipse one another. The construction of such tables is indeed much easier for these satellites than of any other celestial bodies, as their motions are much more regular.

The English tables are calculated for the meridian of Greenwich, and by these it is very easy to find how many degrees of longitude any place is distant either east or west from Greenwich; for, let an observer, who has these tables, with a good telescope and a well-regulated clock at any other place of the earth, observe the beginning or ending of an eclipse of one of Jupiter's satellites, and note the precise moment of time that he saw the satellite either immerse into, or emerge out of the shadow, and compare that time with the time shown by the tables for Greenwich; then 15 degrees difference of longitude being allowed for every hour's difference of time, will give the longitude of that place from Greenwich; and if there be any odd minutes of time, for every minute a quarter of a degree, east or west, must be allowed, as the time of observation is later or earlier than the time shown by the tables. Such eclipses are very convenient for this purpose at land, because they happen almost every day; but are of no use at sea, because the rolling of the ship hinders all nice telecopic observations.

To explain this by a figure, in Plate XLVI. fig. 1, let J be Jupiter, K, L, M, N his four satellites in their respective orbits, 1, 2, 3, 4; and let the earth be at F (suppose in November, although that month is no otherwise material than to find the earth readily in this scheme, where it is shown in eight different parts of the orbit). Let Q be a place on the meridian of Greenwich, and R a place on some other meridian eastward from Greenwich. Let a person at R observe the instantaneous vanishing of the first satellite K into Jupiter's shadow, suppose at three o'clock in the morning; but by the tables he finds the immersion of that satellite to be at midnight at Greenwich; he then can immediately determine, that as there are three hours difference of time between Q and R, and that R is three hours forwarder in reckoning than Q, it must be 45 degrees of east longitude from the meridian of Q. Were this method as practicable at sea as at land, any sailor might almost as easily, and with equal certainty, find the longitude as the latitude.

Whilst the earth is going from C to F in its orbit, when the only the immersions of Jupiter's satellites into his shadow are generally seen; and their emergences out of it while the earth goes from G to B. Indeed, both served, these appearances may be seen of the second, third, and fourth satellite when eclipsed, whilst the earth is between D and E, or between G and A; but never of the first satellite, on account of the smallness of its orbit and the bulk of Jupiter, except only when Jupiter is directly opposite to the sun, that is, when the earth is at G; and even then, strictly speaking, we cannot see either the immersions or emergences of any of his satellites, because his body being directly between us and his conical shadow, his satellites are hid by his body a few moments before they touch his shadow; and are quite emerged from thence before we can see them, as it were just dropping from him. And when the earth is at C, the sun, being between it and Jupiter, hides both him and his moons from us.

In this diagram, the orbits of Jupiter's moons are drawn. drawn in true proportion to his diameter; but in proportion to the earth's orbit, they are drawn vastly too large.

In whatever month of the year Jupiter is in conjunction with the sun, or in opposition to him, in the next year it will be a month later at least. For whilst the earth goes once round the sun, Jupiter describes a twelfth part of his orbit. And, therefore, when the earth has finished its annual period, from being in a line with the sun and Jupiter, it must go as much forward as Jupiter has moved in that time, to overtake him again; just like the minute hand of a watch, which, from any conjunction with the hour-hand, goes once round the dial-plate and somewhat above a twelfth part more, to overtake the hour-hand again.

It is found by observation, that when the earth is between the sun and Jupiter, as at G, his satellites are eclipsed about 8 minutes sooner than they should be according to the tables; and when the earth is at B or C, these eclipses happen about 8 minutes later than the tables predict them. Hence it is undeniably certain, that the motion of light is not instantaneous, since it takes about $16\frac{1}{2}$ minutes of time to go through a space equal to the diameter of the earth's orbit, which is $180,000,000$ miles in length; and consequently the particles of light fly almost $200,000$ miles every second of time, which is above a million of times swifter than the motion of a cannon bullet. And as light is $16\frac{1}{2}$ minutes in travelling across the earth's orbit, it must be $8\frac{1}{2}$ minutes in coming from the sun to us: therefore if the sun were annihilated, we should see him for $8\frac{1}{2}$ minutes after; and if he were again created, he would be $8\frac{1}{2}$ minutes old before we could see him.

To illustrate this progressive motion of light, (Plate XLVI. fig. 2.), let A and B be the earth in two different parts of its orbit, whose distance from each other is $95,000,000$ of miles, equal to the earth's distance from the sun S. It is plain, that if the motion of light were instantaneous, the satellite 1 would appear to enter into Jupiter's shadow FF at the same moment of time to a spectator in A, as to another in B. But by many years observations it has been found, that the immersion of the satellite into the shadow is seen $8\frac{1}{2}$ minutes sooner when the earth is at B, than when it is at A. And so, as Mr Romeur first discovered, the motion of light is thereby proved to be progressive, and not instantaneous, as was formerly believed. It is easy to compute in what time the earth moves from A to B; for the chord of 60 degrees of any circle is equal to the semidiameter of that circle; and as the earth goes through all the 360 degrees of its orbit in a year, it goes through 60 of those degrees in about 61 days. Therefore, if on any given day, suppose the first of June, the earth is at A, on the first of August it will be at B; the chord, or straight line AB, being equal to DS the radius of the earth's orbit, the same with AS its distance from the sun.

As the earth moves from D to C, through the side AB of its orbit, it is constantly meeting the light of Jupiter's satellites sooner, which occasions an apparent acceleration of their eclipses; and as it moves through the other half H of its orbit, from C to D, it is receding from their light, which occasions an apparent retardation of their eclipses, because their light is then longer before it overtakes the earth.

That these accelerations of the immersions of Jupiter's satellites into his shadow, as the earth approaches towards Jupiter, and the retardations of their emergences out of his shadow, as the earth is going from him, are not occasioned by any inequality arising from the motions of the satellites in eccentric orbits, is plain, because it affects them all alike, in whatever parts of their orbits they are eclipsed. Besides, they go often round their orbits every year, and their motions are no way commensurate to the earth's. Therefore, a phenomenon not to be accounted for from the real motions of the satellites, but so easily deducible from the earth's motion, and so answerable thereto, must be allowed to result from it. This affords one very good proof of the earth's annual motion.

From what we have said in general concerning eclipses, it is plain that secondary planets are not the only bodies that may occasion them. The primary planets would eclipse one another, were it not for their great distances; but as the comets are not subject to the same laws with the planets, it is possible they may sometimes approach so near to the primary planets, as to cause an eclipse of the sun to those planets; and as the body of a comet bears a much larger proportion to the bulk of a primary planet than any secondary, it is plain that a cometary eclipse would both be of much longer continuance, and attended with much greater darkness, than that occasioned by a secondary planet. This behoved to be the case at any rate: but if we suppose the primary planet and comet to be moving both the same way, the duration of such an eclipse would be prodigiously lengthened; and thus, instead of four minutes, the sun might be totally darkened to the inhabitants of certain places for as many hours. Hence we may account for that prodigious darkness which we sometimes read of in history at times when no eclipse of the sun by the moon could possibly happen (see No. 12).

It is remarkable, however, that no comet hath ever been observed passing over the disk of the sun like a spot, as Venus and Mercury are; yet this must certainly happen, when the comet is in its perihelion, and the earth on the same side of its annual orbit. Such a phenomenon well deserves the watchful attention of astronomers, as it would be a greater confirmation of the planetary nature of comets, than any thing hitherto observed.

Sect. XII. Of the Division of Time. A perpetual Table of New Moons. The Times of the Birth and Death of CHRIST. A Table of remarkable Ages or Events.

The parts of time are Seconds, Minutes, Hours, Days, Years, Cycles, Ages, and Periods.

The original standard, or integral measure of time, is a year; which is determined by the revolution of some celestial body in its orbit, viz. the sun or moon.

The time measured by the sun's revolution in the ecliptic, from any equinox or solstice to the same again, is called the Solar or Tropical Year, which contains 365 days, 5 hours, 48 minutes, 47 seconds; and is the only proper or natural year, because it always keeps the same seasons to the same months.

The quantity of time measured by the sun's revolution, tion, as from any fixed star to the same star again, is called the sidereal year; which contains 365 days 6 hours 9 minutes 14\(\frac{1}{2}\) seconds; and is 20 minutes 17\(\frac{1}{2}\) seconds longer than the true solar year.

The time measured by 12 revolutions of the moon, from the sun to the sun again, is called the lunar year; it contains 354 days 8 hours 48 minutes 36 seconds; and is therefore 10 days 21 hours 6 minutes 21 seconds shorter than the solar year. This is the foundation of the epact.

The civil year is that which is in common use among the different nations of the world; of which, some reckon by the lunar, but most by the solar. The civil solar year contains 365 days, for three years running, which are called common years; and then comes in what is called the bissextile or leap-year, which contains 366 days. This is also called the Julian year, on account of Julius Caesar, who appointed the intercalary day every fourth year, thinking thereby to make the civil and solar year keep pace together. And this day, being added to the 23rd of February, which in the Roman calendar was the sixth of the kalends of March, that sixth day was twice reckoned, or the 23rd and 24th were reckoned as one day, and was called bis sextus dies; and thence came the name bissextile for that year. But in our common almanacs this day is added at the end of February.

The civil lunar year is also common or intercalary. The common year consists of 12 lunations, which contain 354 days; at the end of which, the year begins again. The intercalary, or embolismic year is that wherein a month was added, to adjust the lunar year to the solar. This method was used by the Jews, who kept their account by the lunar motions. But by intercalating no more than a month of 30 days, which they called Ve-Adar, every third year, they fell 3\(\frac{1}{2}\) days short of the solar year in that time.

The Romans also used the lunar embolismic year at first, as it was settled by Romulus their first king, who made it to consist only of 10 months or lunations, which fell 61 days short of the solar year, and so their year became quite vague and unfixed; for which reason, they were forced to have a table published by the high-priest, to inform them when the spring and other seasons began. But Julius Caesar, as already mentioned, taking this troublesome affair into consideration, reformed the calendar, by making the year to consist of 365 days 6 hours.

The year thus settled, is what we still make use of in Britain; but as it is somewhat more than 11 minutes longer than the solar tropical year, the times of the equinoxes go backward, and fall earlier by one day in about 130 years. In the time of the Nicene council, (A.D. 325), which was 1444 years ago, the vernal equinox fell on the 21st of March; and if we divide 1444 by 130 it will quote 11, which is the number of days which the equinox has fallen back since the council of Nice. This causing great disturbances, by unfixing the times of the celebration of Easter, and consequently of all the other moveable feasts;

Pope Gregory XIII, in the year 1582, ordered ten days to be at once struck out of that year; and the next day after the 4th of October was called the 15th. By this means the vernal equinox was restored to the 21st of March; and it was endeavoured, by the omission of three intercalary days in 400 years, to make the civil or political year keep pace with the solar for time to come. This new form of the year is called the Gregorian account, or new style; which is received in all countries where the pope's authority is acknowledged, and ought to be in all places where truth is regarded.

The principal division of the year is into months, which are of two sorts, namely, astronomical and civil. The astronomical month is the time in which the moon and civil runs through the zodiac, and is either periodical or synodical. The periodical month is the time spent by the moon in making one complete revolution from any point of the zodiac to the same again; which is 27\(\frac{1}{3}\) days 7 hours 43 minutes. The synodical month, called a lunation, is the time contained between the moon's parting with the sun at a conjunction, and returning to him again, which is 29\(\frac{1}{3}\) days 12 hours 44 minutes. The civil months are those which are framed for the uses of civil life; and are different as to their names, number of days, and times of beginning, in several different countries. The first month of the Jewish year fell according to the moon in our August and September, old style; the second in September and October; and so on. The first month of the Egyptian year began on the 29th of our August. The first month of the Arabic and Turkish year began the 16th of July. The first month of the Grecian year fell according to the moon in June and July, the second in July and August, and so on, as in the following table.

A month is divided into four parts called weeks, and weeks a week into seven parts called days; so that in a Julian year there are 13 such months, or 52 weeks, and one day over. The Gentiles gave the names of the sun, moon, and planets, to the days of the week. To the first, the name of the Sun; to the second, of the Moon; to the third, of Mars; to the fourth, of Mercury; to the fifth, of Jupiter; to the sixth, of Venus; and to the seventh, of Saturn. ### Sect. XII.

#### ASTRONOMY.

| No | The Jewish year | Days | |----|-----------------|------| | 1 | Tifri | Aug.—Sept. 30 | | 2 | Marcheshvan | Sept.—Oct. 29 | | 3 | Caslen | Oct.—Nov. 30 | | 4 | Tebeth | Nov.—Dec. 29 | | 5 | Shebat | Dec.—Jan. 30 | | 6 | Adar | Jan.—Feb. 29 | | 7 | Nisan or Abib | Feb.—Mar. 30 | | 8 | Ijar | Mar.—Apr. 29 | | 9 | Sivan | Apr.—May 30 | | 10 | Tamuz | May—June 29 | | 11 | Ab | June—July 30 | | 12 | Elul | July—Aug. 29 |

Days in the year = 354

In the embolismic year after Adar they added month called Ve-Adar of 30 days.

| No | The Egyptian year | Days | |----|-------------------|------| | 1 | Thoth | August 29 | | 2 | Paophi | September 28 | | 3 | Athir | October 28 | | 4 | Chojac | November 27 | | 5 | Tybi | December 27 | | 6 | Mechir | January 26 | | 7 | Phamenoth | February 25 | | 8 | Parmuthi | March 27 | | 9 | Pachon | April 26 | | 10 | Payni | May 26 | | 11 | Epiphi | June 25 | | 12 | Mcfori | July 25 |

Epagomenae or days added = 5

Days in the year = 365

---

| No | The Arabic and Turkish year | Days | |----|----------------------------|------| | 1 | Muharram | July 16 | | 2 | Safar | August 15 | | 3 | Rabia I | September 13 | | 4 | Rabia II | October 13 | | 5 | Jomada I | November 11 | | 6 | Jomada II | December 11 | | 7 | Rajab | January 9 | | 8 | Shaaban | February 8 | | 9 | Ramadam | March 9 | | 10 | Shawal | April 8 | | 11 | Dulhaadah | May 7 | | 12 | Dulheggia | June 5 |

Days in the year = 354

The Arabians add 11 days at the end of every year, which keep the same months to the same feasts.

| No | The ancient Grecian year | Days | |----|--------------------------|------| | 1 | Hecatombaeon | June—July 30 | | 2 | Metagittion | July—Aug. 29 | | 3 | Boedromion | Aug.—Sept. 30 | | 4 | Pyanepstion | Sept.—Oct. 29 | | 5 | Mainacterion | Oct.—Nov. 30 | | 6 | Posideon | Nov.—Dec. 29 | | 7 | Gamelion | Dec.—Jan. 30 | | 8 | Anthesterion | Jan.—Feb. 29 | | 9 | Elaphebolion | Feb.—Mar. 30 | | 10 | Munichion | Mar.—Apr. 29 | | 11 | Thargelion | Apr.—May 30 | | 12 | Schirrophorion | May—June 29 |

Days in the year = 354

---

A day is either natural or artificial. The natural day contains 24 hours; the artificial the time from sunrise to sun-set. The natural day is either astronomical or civil. The astronomical day begins at noon, because the increase and decrease of days terminated by the horizon are very unequal among themselves; which inequality is likewise augmented by the inconstancy of the horizontal refractions, and therefore the astronomer takes the meridian for the limit of diurnal revolutions, reckoning noon, that is, the instant when the sun's centre is on the meridian, for the beginning of the day. The British, French, Dutch, Germans, Spaniards, Portuguese, and Egyptians, begin the civil day at midnight; the ancient Greeks, Jews, Bohemian's, Sileans, with the modern Italians, and Chinese, begin it at sun-setting; and the ancient Babylonians, Persians, Syrians, with the modern Greeks, at sun-rising.

An hour is a certain determinate part of the day, and is either equal or unequal. An equal hour is the 24th part of a mean natural day, as shewn by well regulated clocks and watches: but these hours are not quite equal as measured by the returns of the sun to the meridian, because of the obliquity of the ecliptic and sun's unequal motion in it. Unequal hours are those by which the artificial day is divided into twelve parts, and the night into as many.

An hour is divided into 60 equal parts called minutes, a minute into 60 equal parts called seconds, and so on, &c., these again into 60 equal parts called thirds. The Jews, Chaldeans, and Arabians, divide the hour into 1080 equal parts called scruples; which number contains 18 times 60, so that one minute contains 18 scruples.

A cycle is a perpetual round, or circulation of the same parts of time of any sort. The cycle of the sun is a revolution of 28 years, in which time the days of the months return again to the same days of the week; the sun's place to the same signs and degrees of the ecliptic on the same months and days, so as not to differ one degree in 100 years; and the leap-years begin the same course over again with respect to the days of the week on which the days of the months fall. The cycle of the moon, commonly called the golden number, is a revolution of 19 years; in which time, the conjunctions, oppositions, and other aspects of the moon, are within an hour and half of being the same as they were on the same days of the months 19 years before. dition is a revolution of 15 years, used only by the Romans for indicating the times of certain payments made by the subjects to the republic: It was established by Constantine, A.D. 312.

The year of our Saviour's birth, according to the vulgar era, was the 9th year of the solar cycle, the first year of the lunar cycle, and the 312th year after his birth was the first year of the Roman indiction. Therefore, to find the year of the solar cycle, add 9 to any given year of Christ, and divide the sum by 28; the quotient is the number of cycles elapsed since his birth, and the remainder is the cycle for the given year: If nothing remains, the cycle is 28. To find the lunar cycle, add 1 to the given year of Christ, and divide the sum by 19; the quotient is the number of cycles elapsed in the interval, and the remainder is the cycle for the given year: If nothing remains, the cycle is 19. Lastly, subtract 312 from the given year of Christ, and divide the remainder by 15; and what remains after this division is the indiction for the given year: If nothing remains, the indiction is 15.

Although the above deficiency in the lunar circle of an hour and an half every 19 years be but small, yet in time it becomes so sensible as to make a whole natural day in 310 years. So that, although this cycle be of use, when the golden numbers are rightly placed against the days of the months in the calendar, as in the Common Prayer Books, for finding the days of the mean conjunctions or oppositions of the sun and moon, and consequently the time of Easter; it will only serve for 310 years, old style. For as the new and full moons anticipate a day in that time, the golden numbers ought to be placed one day earlier in the calendar for the next 310 years to come. These numbers were rightly placed against the days of new moon in the calendar, by the council of Nice, A.D. 325; but the anticipation, which has been neglected ever since, is now grown almost into 5 days: And therefore, all the golden numbers ought now to be placed 5 days higher in the calendar for the old style than they were at the time of the said council; or 6 days lower for the new style, because at present it differs 11 days from the old.

In the following table the golden numbers under the months stand against the days of new moon in the left-hand column, for the new style; adapted chiefly to the second year after leap-year, as being the nearest mean for all the four; and will serve till the year 1900. Therefore, to find the day of new moon in any month of a given year till that time, look for the golden number of that year under the desired month, and against it you have the day of new moon in the left-hand column. Thus, suppose it were required to find the day of new moon in September 1769; the golden number for that year is 3, which I look for under September, and right against it in the left-hand column you will find 30, which is the day of new moon in that month.

N.B. If all the golden numbers, except 17 and 6, were set one day lower in the table, it would serve from the beginning of the year 1900 till the end of the year 2199. The table at the end of this section shows the golden number for 4000 years after the birth of Christ, by looking for the even hundreds of any given year at the left hand, and for the rest to make up that year at the head of the table; and where the columns meet, you have the golden number (which is the same both in old and new style) for the given year. Thus, suppose the golden number was wanted for the year 1769; look for 1700 at the left hand of the table, and for 69 at the top of it; then guiding your eye downward from 69 to over-against 1700, you will find 3, which is the golden number for that year.

But because the lunar cycle of 19 years sometimes includes five leap-years, and at other times only four, this table will sometimes vary a day from the truth in leap-years after February. And it is impossible to have one more correct, unless we extend it to four times 19 or 76 years; in which there are 19 leap-years without a remainder. But even then to have it of perpetual use, it must be adapted to the old style; because, in every centurial year not divisible by 4, the regular course of leap-years is interrupted in the new; as will be the case in the year 1800.

The cycle of Easter, also called the Dionysian period, Dionysia is a revolution of 532 years, found by multiplying the period, solar cycle 28 by the lunar cycle 19. If the new cycle of moons did not anticipate upon this cycle, Easter-day would always be the Sunday next after the first full moon, which follows the 21st of March. But, on account of the above anticipation, to which no proper regard was had before the late alteration of the style, the ecclesiastic Easter has several times been a week different from the true Easter within this last century: which inconvenience is now remedied by making the table, which used to find Easter for ever, in the Common Prayer Book, of no longer use than the lunar difference from the new style will admit of.

The earliest Easter possible is the 22nd of March, the latest the 25th of April. Within these limits are 35 days, and the number belonging to each of them is called the number of direction; because thereby the time of Easter is found for any given year.

The first seven letters of the alphabet are commonly placed in the annual almanacs, to show on what days of the week the days of the months fall throughout the year. And because one of those seven letters must necessarily stand against Sunday, it is printed in a capital form, and called the dominical letter; the other six being inserted in small characters, to denote the other six days of the week. Now, since a common Julian year contains 365 days, if this number be divided by 7 (the number of days in a week) there will remain one day. If there had been no remainder, it is plain the year would constantly begin on the same day of the week: but since one remains, it is plain, that the year must begin and end on the same day of the week; and therefore the next year will begin on the day following. Hence, when January begins on Sunday, A is the dominical or Sunday letter for that year: Then, because the next year begins on Monday, the Sunday will fall on the seventh day, to which is annexed the seventh letter G, which therefore will be the dominical letter for all that year: and as the third year will begin on Tuesday, the Sunday will fall on the sixth day; therefore F will be the Sunday letter for that year. Whence it is evident, that the Sunday letters will go annually in a retrograde order thus, G, F, E, D, C, B, A. And, in the course of seven years, if they were all common ones, the same days of the week and dominical letters would return to the same days of the months. But because there are 366 days in a leap-year, if this number... ber be divided by 7, there will remain two days over and above the 52 weeks of which the year consists. And therefore, if the leap-year begins on Sunday, it will end on Monday; and the next year will begin on Tuesday, the first Sunday whereof must fall on the sixth of January, to which is annexed the letter F, and not G, as in common years. By this means, the leap-year returning every fourth year, the order of the dominical letters is interrupted; and the series cannot return to its first state till after four times seven, or 28 years; and then the same days of the months return in order to the same days of the week as before.

From the multiplication of the solar cycle of 28 years into the lunar cycle of 19 years, and the Roman indiction of 15 years, arises the great Julian period, consisting of 7980 years, which had its beginning 764 years before Strouhius's supposed year of the creation (for no later could all the three cycles begin together), and it is not yet completed: And therefore it includes all other cycles, periods, and eras. There is but one year in the whole period that has the same numbers for the three cycles of which it is made up: And therefore, if historians had remarked in their writings the cycles of each year, there had been no dispute about the time of any action recorded by them.

The Dionysian or vulgar era of Christ's birth was about the end of the year of the Julian period 4713; and consequently the first year of his age, according to that account, was the 4714th year of the said period. Therefore, if to the current year of Christ we add 4713, the sum will be the year of the Julian period. So the year 1769 will be found to be the 6482nd year of that period. Or, to find the year of the Julian period answering to any given year before the first year of Christ, subtract the number of that given year from 4714, and the remainder will be the year of the Julian period. Thus, the year 585 before the first year of Christ (which was the 584th before his birth) was the 4129th year of the said period. Lastly, to find the cycles of the sun, moon, and indiction for any given year of this period, divide the given year by 28, 19, and 15; the three remainders will be the cycles sought, and the quotients the numbers of cycles run since the beginning of the period. So in the above 4714th year of the Julian period, the cycle of the sun was 10, the cycle of the moon 2, and the cycle of indiction 4; the solar cycle having run through 168 courses, the lunar 248, and the indiction 314.

The vulgar era of Christ's birth was never settled till the year 527, when Dionysius Exiguus, a Roman abbot, fixed it to the end of the 4713th year of the Julian period, which was four years too late. For our Saviour was born before the death of Herod, who sought to kill him as soon as he heard of his birth. And, according to the testimony of Josephus (B. xvii. ch. 8.) there was an eclipse of the moon in the time of Herod's last illness; which eclipse appears by our astronomical tables to have been in the year of the Julian period 4710, March 13th, at 3 hours past midnight, at Jerusalem. Now, as our Saviour must have been born some months before Herod's death, since in the interval he was carried into Egypt, the latest time in which we can fix the true era of his birth is about the end of the 4709th year of the Julian period.

As there are certain fixed points in the heavens from which astronomers begin their computations, so there are certain points of time from which historians begin to reckon; and these points or roots of time are called eras or epochs. The most remarkable eras are, those of the Creation, the Greek Olympiads, the building of Rome, the era of Nabonassar, the death of Alexander, the birth of Christ, the Arabian Hegira, and the Persian Jafdegird: All which, together with several others of less note, have their beginnings in the following table fixed to the years of the Julian period, to the age of the world at those times, and to the years before and after the year of Christ's birth. ### A Table of remarkable Eras and Events

| Event | Julian Period | Y.of the World | Before Christ | |----------------------------------------------------------------------|---------------|----------------|---------------| | 1. The creation of the world | 706 | 0 | 4007 | | 2. The deluge, or Noah's flood | 2302 | 1656 | 2351 | | 3. The Assyrian monarchy founded by Nimrod | 2537 | 1831 | 2176 | | 4. The birth of Abraham | 2714 | 2008 | 1999 | | 5. The destruction of Sodom and Gomorrah | 2816 | 2110 | 1897 | | 6. The beginning of the kingdom of Athens by Cecrops | 3157 | 2451 | 1556 | | 7. Moses receives the ten commandments from God | 3222 | 2516 | 1491 | | 8. The entrance of the Israelites into Canaan | 3262 | 2556 | 1451 | | 9. The destruction of Troy | 3529 | 2823 | 1184 | | 10. The beginning of king David's reign | 3650 | 2944 | 1063 | | 11. The foundation of Solomon's temple | 3701 | 2995 | 1012 | | 12. The Argonautic expedition | 3776 | 3070 | 937 | | 13. Lycurgus forms his excellent laws | 3829 | 3103 | 884 | | 14. Arbaces, the first king of the Medes | 3838 | 3132 | 875 | | 15. Mandaecus, the second | 3865 | 3159 | 848 | | 16. Sofarmus, the third | 3915 | 3209 | 798 | | 17. The beginning of the Olympiads | 3938 | 3232 | 775 | | 18. Artica, the fourth king of the Medes | 3945 | 3239 | 768 | | 19. The Catonian epocha of the building of Rome | 3961 | 3255 | 752 | | 20. The era of Nabonassar | 3967 | 3261 | 746 | | 21. The destruction of Samaria by Salmanefer | 3992 | 3286 | 721 | | 22. The first eclipse of the moon on record | 3993 | 3287 | 720 | | 23. Cardicea, the fifth king of the Medes | 3996 | 3290 | 717 | | 24. Phraortes, the sixth | 4058 | 3352 | 655 | | 25. Cyaxares, the seventh | 4080 | 3374 | 633 | | 26. The first Babylonish captivity by Nebuchadnezzar | 4107 | 3401 | 606 | | 27. The long war ended between the Medes and Lydians | 4111 | 3405 | 602 | | 28. The second Babylonish captivity, and birth of Cyrus | 4114 | 3408 | 599 | | 29. The destruction of Solomon's temple | 4125 | 3419 | 588 | | 30. Nebuchadnezzar struck with madness | 4144 | 3438 | 569 | | 31. Daniel's vision of the four monarchies | 4158 | 3452 | 555 | | 32. Cyrus begins to reign in the Persian empire | 4177 | 3471 | 536 | | 33. The battle of Marathon | 4223 | 3517 | 490 | | 34. Artaxerxes Longimanus begins to reign | 4249 | 3543 | 464 | | 35. The beginning of Daniel's seventy weeks of years | 4256 | 3550 | 457 | | 36. The beginning of the Peloponnesian war | 4282 | 3576 | 431 | | 37. Alexander's victory at Arbela | 4383 | 3677 | 330 | | 38. The death of Alexander | 4390 | 3684 | 323 | | 39. The captivity of 100,000 Jews by king Ptolemy | 4393 | 3687 | 320 | | 40. The colossus of Rhodes thrown down by an earthquake | 4491 | 3875 | 222 | | 41. Antiochus defeated by Ptolemy Philopater | 4496 | 3790 | 217 | | 42. The famous Archimedes murdered at Syracuse | 4506 | 3800 | 207 | | 43. Jason butchers the inhabitants of Jerusalem | 4543 | 3837 | 170 | | 44. Corinth plundered and burnt by consul Mummius | 4567 | 3861 | 146 | | 45. Julius Caesar invades Britain | 4659 | 3953 | 54 | | 46. He corrects the calendar | 4677 | 3961 | 46 | | 47. Is killed in the Senate-house | 4671 | 3965 | 42 | | 48. Herod made king of Judea | 4673 | 3967 | 40 | | 49. Anthony defeated at the battle of Actium | 4683 | 3977 | 30 | | 50. Agrippa builds the Pantheon at Rome | 4688 | 3982 | 25 | | 51. The true era of Christ's birth | 4709 | 4003 | 4 | | 52. The death of Herod | 4710 | 4004 | 3 | | 53. The Dionysian, or vulgar era of Christ's birth | 4713 | 4007 | 0 | | 54. The true year of his crucifixion | 4746 | 4040 | 33 | | 55. The destruction of Jerusalem | 4783 | 4077 | 70 | | 56. Adrian builds the long wall in Britain | 4833 | 4127 | 120 | | 57. Constantius defeats the Picts in Britain | 5019 | 4313 | 306 | | 58. The council of Nice | 5038 | 4332 | 325 | | 59. The death of Constantine the great | 5050 | 4344 | 337 | | 60. The Saxons invited into Britain | 5158 | 4452 | 445 | | 61. The Arabian Hegira | 5335 | 4629 | 622 |

After Christ.

---

**Note:** This table lists various historical events along with their Julian periods and years of the world before Christ. 62. The death of Mohammed the pretended prophet 63. The Persian Yeddegird 64. The sun, moon, and all the planets, in Libra, Sep. 14, as seen from the earth 65. The art of printing discovered 66. The reformation begun by Martin Luther

In fixing the year of the creation to the 706th year of the Julian period, which was the 4007th year before the year of Christ's birth, we have followed Mr Bedford in his scripture chronology, printed A.D. 1730, and Mr Kennedy in a work of the same kind, printed A.D. 1762.—Mr Bedford takes it only for granted that the world was created at the time of the autumnal equinox: But Mr Kennedy affirms, that the said equinox was at the noon of the fourth day of the creation-week, and that the moon was then 24 hours past her opposition to the sun.—If Moses had told us the same things, we should have had sufficient data for fixing the era of the creation: But as he has been silent on these points, we must consider the best accounts of chronologers as entirely hypothetical and uncertain.

Table, showing the Golden Number, (which is the same both in the Old and New Style) from the Christian Era, to A.D. 4000.

| Hundreds of Years | 0 | 1900 | 3800 | 5700 | 7600 | 9500 | |------------------|-----|------|------|------|------|------| | | 1 | 2000 | 3900 | 5800 | 7700 | 9600 | | | 2 | 2100 | 4000 | 5900 | 7800 | 9700 | | | 3 | 2200 | 4100 | 6000 | 7900 | 9800 | | | 4 | 2300 | 4200 | 6100 | 8000 | 9900 | | | 5 | 2400 | 4300 | 6200 | 8100 | 10000 | | | 6 | 2500 | 4400 | 6300 | 8200 | 10100 | | | 7 | 2600 | 4500 | 6400 | 8300 | 10200 | | | 8 | 2700 | 4600 | 6500 | 8400 | 10300 | | | 9 | 2800 | 4700 | 6600 | 8500 | 10400 | | | 10 | 2900 | 4800 | 6700 | 8600 | 10500 | | | 11 | 3000 | 4900 | 6800 | 8700 | 10600 | | | 12 | 3100 | 5000 | 6900 | 8800 | 10700 | | | 13 | 3200 | 5100 | 7000 | 8900 | 10800 | | | 14 | 3300 | 5200 | 7100 | 9000 | 10900 | | | 15 | 3400 | 5300 | 7200 | 9100 | 11000 | | | 16 | 3500 | 5400 | 7300 | 9200 | 11100 | | | 17 | 3600 | 5500 | 7400 | 9300 | 11200 | | | 18 | 3700 | 5600 | 7500 | 9400 | 11300 |

Sect. XIII. A Description of the Astronomical Machinery serving to explain and illustrate the foregoing part of this Treatise.

The Orrery, (Plate LI. fig. 3.) This machine shews the motions of the sun, Mercury, Venus, earth, and moon; and occasionally the superior planets, Mars, Jupiter, and Saturn, may be put on; Jupiter's four satellites are moved round him in their proper times by a small wheel; and Saturn has his five satellites, and his ring which keeps its parallelism round the sun; and by a lamp put in the sun's place, the ring shews all its various phases already described.

Vol. II.

In the centre, No. 1, represents the sun, supported by its axis, inclining almost 8 degrees from the axis of the ecliptic, and turning round in 25½ days on its axis, of which the north pole inclines toward the eighth degree of Pisces in the great ecliptic, (No. 11.), whereas the months and days are engraven over the signs and degrees in which the sun appears, as seen from the earth, on the different days of the year.

The nearest planet (No. 2.) to the sun is Mercury, which goes round him in 87 days 23 hours, or 87½ diurnal rotations of the earth; but has no motion round its axis in the machine, because the time of its diurnal motion in the heavens is not known to us. The next planet in order is Venus, (No 3.), which performs her annual course in 224 days 17 hours, and turns round her axis in 24 days 8 hours, or in 24½ diurnal rotations of the earth. Her axis inclines 75 degrees from the axis of the ecliptic, and her north pole inclines towards the 26th degree of Aquarius, according to the observations of Bianchini. She shews all the phenomena described in Sect. ii.

Next, without the orbit of Venus, is the earth, (No 4.), which turns round its axis, to any fixed point at a great distance, in 23 hours 56 minutes four seconds of mean solar time; but from the sun to the sun again, in 24 hours of the same time. No 6. is a sidereal dial-plate under the earth, and No 7. a solar dial-plate on the cover of the machine. The index of the former shews sidereal, and of the latter, solar time; and hence the former index gains one entire revolution on the latter every year, as 365 solar or natural days contain 366 sidereal days, or apparent revolutions of the stars. In the time that the earth makes 365½ diurnal rotations on its axis, it goes once round the sun in the plane of the ecliptic; and always keeps opposite to a moving index (No 10.) which shews the sun’s daily change of place, and also the days of the months.

The earth is half covered with a black cap, for dividing the apparently enlightened half next the sun from the other half, which, when turned away from him, is in the dark. The edge of the cap represents the circle bounding light and darkness, and shews at what time the sun rises and sets to all places throughout the year. The earth’s axis inclines 23½ degrees from the axis of the ecliptic; the north pole inclines toward the beginning of Cancer, and keeps its parallelism throughout its annual course; so that in summer the northern parts of the earth incline towards the sun, and in winter from him; by which means, the different lengths of days and nights, and the cause of the various seasons, are demonstrated to sight.

There is a broad horizon, to the upper side of which is fixed a meridian semicircle in the north and south points, graduated on both sides from the horizon to 90° in the zenith or vertical point. The edge of the horizon is graduated from the east and west to the south and north points, and within these divisions are the points of the compass. From the lower side of this thin horizontal plate stand out four small wires, to which is fixed a twilight-circle 18 degrees from the graduated side of the horizon all round. This horizon may be put upon the earth, (when the cap is taken away), and rectified to the latitude of any place; and then by a small wire called the solar ray, which may be put so as to proceed directly from the sun’s centre towards the earth’s, but to come no farther than almost to touch the horizon. The beginning of twilight, time of sun-rising, with his amplitude, meridian altitude, time of setting, amplitude then, and end of twilight, are shewn for every day of the year, at that place to which the horizon is rectified.

The moon (No 5.) goes round the earth, from between it and any fixed point at a great distance, in 27 days 7 hours 43 minutes, or through all the signs and degrees of her orbit, which is called her periodical revolution; but she goes round from the sun to the sun again, or from change to change, in 29 days 12 hours 45 minutes, which is her synodical revolution; and in that time she exhibits all the phases already described.

When the abovementioned horizon is rectified to the latitude of any given place, the times of the moon’s rising and setting, together with her amplitude, are shewn to that place as well as the sun’s; and all the various phenomena of the harvest-moon are made obvious to sight.

The moon’s orbit (No 9.) is inclined to the ecliptic (No 11.), one half being above, and the other below it. The nodes, or points at o and o, lie in the plane of the ecliptic, as before described, and shift back and forth through all its signs and degrees in 18½ years. The degrees of the moon’s latitude to the highest at NL (north latitude) and lowest at SL (south latitude), are engraven both ways from her nodes at o and o, and as the moon rises and falls in her orbit according to its inclination, her latitude and distance from her nodes are shewn for every day, having first rectified her orbit so as to set the nodes to their proper places in the ecliptic; and then as they come about at different and almost opposite times of the year, and then point towards the sun, all the eclipses may be shewn for hundreds of years, (without any new rectification), by turning the machinery backward for time past, or forward for time to come. At 17 degrees distance from each node, on both sides is ingraved a small sun; and at 12 degrees distance, a small moon, which shew the limits of solar and lunar eclipses; and when, at any change the moon falls between either of these suns and the node, the sun will be eclipsed on the day pointed to by the annual index, (No 10.); and as the moon has then north or south latitude, one may easily judge whether that eclipse will be visible in the northern or southern hemisphere; especially as the earth’s axis inclines toward the sun or from him at that time. And when at any full, the moon falls between either of the little moon’s and node, she will be eclipsed, and the annual index shews the day of that eclipse. There is a circle of 29½ equal parts (No 8.) on the cover of the machine, on which an index shews the days of the moon’s age.

There are two semicircles (Plate LII. fig. 1.) fixed to an elliptical ring, which being put like a cap upon the earth, and the forked part F upon the moon, shews the tides as the earth turns round within them, and they are led round it by the moon. When the different places come to the semicircle AaEbB, they have tides of flood; and when they come to the semicircle CED, they have tides of ebb; the index on the hour circle (No 7. Plate LII.) shewing the times of these phenomena.

There is a jointed wire, of which one end being put into a hole in the upright stem that holds the earth’s cap, and the wire laid into a small forked piece which may be occasionally put upon Venus or Mercury, shews the direct and retrograde motions of these two planets, with their stationary times and places, as seen from the earth.

The whole machinery is turned by a winch or handle (No 12.), and is so easily moved, that a clock might turn it without any danger of stopping.

To give a plate of the wheel-work of this machine, would answer no purpose, because many of the wheels lie so behind others as to hide them from sight in any view whatever. The Cometarium, (Plate LII. fig. 2.) This curious machine shews the motion of a comet or eccentric body moving round the sun, describing equal areas in equal times, and may be so contrived as to shew such a motion for any degree of eccentricity. It was invented by the late Dr Deaguliers.

The dark elliptical groove round the letters abcdefghijklm is the orbit of the comet Y; this comet is carried round in the groove according to the order of letters, by the wire W fixed in the sun S, and slides on the wire as it approaches nearer to or recedes farther from the sun, being nearest of all in the perihelion a, and farthest in the aphelion g. The areas, aSb, bSc, cSd, &c. or contents of these several triangles, are all equal; and in every turn of the winch N, the comet Y is carried over one of these areas; consequently, in as much time as it moves from f to g, or from g to h, it moves from m to a, or from a to b; and so of the rest, being quickest of all at a, and slowest at g. Thus the comet's velocity in its orbit continually decreases from the perihelion a to the aphelion g; and increases in the same proportion from g to a.

The elliptic orbit is divided into 12 equal parts or signs, with their respective degrees, and so is the circle nopersstu, which represents a great circle in the heavens, and to which the comet's motion is referred by a small knob on the point of the wire W. Whilst the comet moves from f to g in its orbit, it appears to move only about five degrees in this circle, as is shown by the small knob on the end of the wire W; but in as short time as the comet moves from m to a, or from a to b, it appears to describe the large space tn or no in the heavens, either of which spaces contains 120 degrees, or four signs. Were the eccentricity of its orbit greater, the greater still would be the difference of its motion, and vice versa.

ABCDEFGHIKLMA is a circular orbit for shewing the equable motion of a body round the sun S, describing equal areas ASB, BSC, &c. in equal times with those of the body Y in its elliptical orbit above mentioned; but with this difference, that the circular motion describes the equal arcs AB, BC, &c. in the same equal times that the elliptical motion describes the unequal arcs ab, bc, &c.

Now, suppose the two bodies Y and I to start from the points a and A at the same moment of time, and each having gone round its respective orbit, to arrive at these points again at the same instant, the body Y will be forwarder in its orbit than the body I all the way from a to g, and from A to G; but I will be forwarder than Y through all the other half of the orbit; and the difference is equal to the equation of the body Y in its orbit. At the points aA, and gG, that is, that in the perihelion and aphelion, they will be equal; and then the equation vanishes. This shews why the equation of a body moving in an elliptic orbit, is added to the mean or supposed circular motion from the perihelion to the aphelion, and subtracted from the aphelion to the perihelion, in bodies moving round the sun, or from the perigee to the apogee, and from the apogee to the perigee in the moon's motion round the earth.

This motion is performed in the following manner by the machine, (Plate LII. fig. 3.). ABC is a wooden bar, (in the box containing the wheel-work), above which are the wheels D and E, and below it the elliptic plates FF and GG; each plate being fixed on an axis in one of its focuses, at E and K; and the wheel E is fixed on the same axis with the plate FF. These plates have grooves round their edges precisely of equal diameters to one another, and in these grooves is the cat-gut string gg, gg crossing between the plates at h. On H, the axis of the handle or winch N in fig. 2. is an endless screw in fig. 4. working in the wheels D and E, whose numbers of teeth being equal, and should be equal to the number of lines aS, bS, cS, &c. in fig. 2., they turn round their axes in equal times to one another, and to the motion of the elliptic plates. For, the wheels D and E having equal numbers of teeth, the plate FF being fixed on the same axis with the wheel E, and the plate II turning the equally big plate GG by a cat-gut string round them both, they must all go round their axes in as many turns of the handle N as either of the wheels has teeth.

It is easy to see, that the end of b of the elliptical plate FF being farther from its axis E than the opposite end I is, must describe a circle so much the larger in proportion, and therefore move through so much more space in the same time; and for that reason the end b moves so much faster than the end I, although it goes no sooner round the centre E. But then the quick-moving end b of the plate FF leads about the short end bK of the plate GG with the same velocity; and the slow-moving end I of the plate FF coming half round as to B, must then lead the long end k of the plate GG as slowly about; so that the elliptical plate FF and its axis E move uniformly and equally quick in every part of its revolution; but the elliptical plate GG, together with its axis K, must move very unequally in different parts of its revolution; the difference being always inversely as the distance of any point of the circumference of GG from its axis at K: or in other words, to influence in two points, if the distance Kk be four, five, or six times as great as the distance Ka, the point b will move in that position four, five, or six times as fast as the point k does, when the plate GG has gone half round; and so on for any other eccentricity or difference of the distances Kk and Ka. The tooth I on the plate FF falls in between the two teeth at k on the plate GG, by which means the revolution of the latter is so adjusted to that of the former, that they can never vary from one another.

On the top of the axis of the equally-moving wheel D in fig. 3. is the sun S in fig. 2.; which sun, by the wire fixed to it, carries the ball I round the circle ABCD, &c. with an equable motion, according to the order of the letters: and on the top of the axis K of the unequally-moving ellipse GG, in fig. 3. is the sun S in fig. 2., carrying the ball Y unequally round in the elliptical groove abcde, &c. N.B. This elliptical groove must be precisely equal and similar to the verge of the plate GG, which is also equal to that of FF.

In this manner machines may be made to shew the true motion of the moon about the earth, or of any planet about the sun, by making the elliptical plates of the same eccentricities, in proportion to the radius, as the orbits of the planets are, whose motions they represent; and so their different equations in different parts of their orbits may be made plain to sight, and clearer ideas of these motions and equations acquired in half an hour, than could be gained from reading half The Improved Celestial Globe, (Pl. XLVIII. fig. 2.) On the north pole of the axis, above the hour-circle, is fixed an arch MKH of 23° degrees; and at the end H is fixed an upright pin HG, which stands directly over the north pole of the ecliptic, and perpendicular to that part of the surface of the globe. On this pin are two moveable collets at D and H, to which are fixed the quadrantile wires N and O, having two little balls on their ends for the sun and moon, as in the figure. The collet D is fixed to the circular plate F, whereon the 29½ days of the moon's age are engraven, beginning just under the sun's wire N; and as this wire is moved round the globe, the plate F turns round with it. These wires are easily turned, if the screw G be slackened; and when they are set to their proper places, the screw serves to fix them there so as in turning the ball of the globe, the wires with the sun and moon go round with it; and these two little balls rise and set at the same times, and on the same points of the horizon, for the day to which they are rectified, as the sun and moon do in the heavens.

Because the moon keeps not her course in the ecliptic, (as the sun appears to do,) but has a declination of 5½ degrees on each side from it in every lunation, her ball may be screwed as many degrees to either side of the ecliptic as her latitude or declination from the ecliptic amounts to at any given time; and for this purpose S, Plate LV. fig. 1. is a small piece of pasteboard, of which the curved edge at S, is to be let upon the globe at right angles to the ecliptic, and the dark line over S to stand upright upon it. From this line, on the convex edge, are drawn the 5½ degrees of the moon's latitude on both sides of the ecliptic; and when this piece is let upright on the globe, its graduated edge reaches to the moon on the wire O, by which means she is easily adjusted to her latitude found by an ephemeris.

The horizon is supported by two semicircular arches, because pillars would stop the progress of the balls when they go below the horizon in an oblique sphere.

To rectify this globe. Elevate the pole to the latitude of the place; then bring the sun's place in the ecliptic for the given day to the brazen meridian, and let the hour index to 12 at noon, that is to the upper 12 on the hour circle; keeping the globe in that situation, slacken the screw G, and let the sun directly over his place on the meridian; which done set the moon's wire under the number that expresses her age for that day on the plate F, and she will then stand over her place in the ecliptic, and shew what constellation she is in. Lastly, fasten the screw G, and laying the curved edge of the pasteboard S over the ecliptic below the moon, adjust the moon to her latitude over the graduated edge of the pasteboard; and the globe will be rectified.

Having thus rectified the globe, turn it round, and observe on what points of the horizon the sun and moon balls rise and set, for these agree with the points of the compass on which the sun and moon rise and set in the heavens on the given day; and the hour index shews the times of their rising and setting; and likewise the time of the moon's passing over the meridian.

This simple apparatus shews all the varieties that can happen in the rising and setting of the sun and moon; and makes the forementioned phenomena of the harvest moon plain to the eye. It is also very useful in reading lectures on the globes, because a large company can see this sun and moon go round, rising above and setting below the horizon at different times according to the seasons of the year; and making their appurtenances to different fixed stars. But in the usual way, where there is only the places of the sun and moon in the ecliptic to keep the eye upon, they are easily lost sight of, unless they be covered with patches.

The Planetary Globe, (Plate LIII. fig. 1.) In this machine, a terrestrial globe is fixed on its axis standing upright on the pedestal CDE, on which is an hour-circle, having its index fixed on the axis, which turns somewhat tightly in the pedestal, so that the globe may not be liable to shake; to prevent which, the pedestal is about two inches thick, and the axis goes quite through it, bearing on a shoulder. The globe is hung in a graduated brazen meridian, much in the usual way; and the thin plate N, NE, E is a moveable horizon graduated round the outer edge, for shewing the bearings and amplitudes of the sun, moon, and planets. The brazen meridian is grooved round the outer edge; and in this groove is a slender semicircle of brass, the ends of which are fixed to the horizon in its north and south points; this semicircle slides in the groove as the horizon is moved in rectifying it for different latitudes. To the middle of this semicircle is fixed a pin, which always keeps in the zenith of the horizon, and on this pin the quadrant of altitude q turns; the lower end of which, in all positions touches the horizon as it is moved round the same.

This quadrant is divided into 90 degrees from the horizon to the zenithal pin, on which it is turned, at 90°. The great flat circle or plate AB is the ecliptic, on the outer edge of which the signs and degrees are laid down; and every fifth degree is drawn through the rest of the surface of this plate towards its centre. On this plate are seven grooves, to which seven little balls are adjusted by sliding wires, so that they are easily moved in the grooves, without danger of starting them. The ball next the terrestrial globe is the moon, the next without it is Mercury, the next Venus, the next the sun, then Mars, then Jupiter, and lastly Saturn. This plate or ecliptic, is supported by four strong wires, having their lower ends fixed into the pedestal, at C, D, E, the fourth being hid by the globe. The ecliptic is inclined 23½ degrees to the pedestal, and is therefore properly inclined to the axis of the globe which stands upright on the pedestal.

To rectify this machine. Set the sun, and all the planetary balls, to their geocentric places in the ecliptic for any given time, by an ephemeris; then set the north point of the horizon to the latitude of your place on the brazen meridian, and the quadrant of altitude to the south point of the horizon; which done, turn the globe with its furniture till the quadrant of altitude comes right against the sun, viz. to his place in the ecliptic; and keeping it there, set the hour index to the XII next the letter C; and the machine will be rectified, not only for the following problems, but for several others which the artist may easily find out.

Problem. I. To find the amplitudes, meridian altitudes, and times of rising, culminating, and setting of the sun, moon, and planets. Turn the globe round eastward, or according to the order of signs; and as the eastern edge of the horizon comes right against the sun, moon, or any planet, the hour index will shew the time of its rising; and the inner edge of the ecliptic will cut its rising amplitude in the horizon. Turn on, and as the quadrant of altitude comes right against the sun, moon, or planets, the ecliptic cuts their meridian altitudes in the quadrant, and the hour index shews the times of their coming to the meridian. Continue turning, and as the western edge of the horizon comes right against the sun, moon, or planets, their setting amplitudes are cut in the horizon by the ecliptic; and the times of their setting are shewn by the index on the hour-circle.

**Prob. II. To find the altitude and azimuth of the sun, moon, and planets, at any time of their being above the horizon.**

Turn the globe till the index comes to the given time in the hour-circle, then keep the globe steady, and moving the quadrant of altitude to each planet respectively, the edge of the ecliptic will cut the planet's mean altitude on the quadrant, and the quadrant will cut the planet's azimuth, or point of bearing on the horizon.

**Prob. III. The sun's altitude being given at any time either before or after noon, to find the hour of the day, and variation of the compass, in any known latitude.**

With one hand hold the edge of the quadrant right against the sun; and, with the other hand, turn the globe westward, if it be in the forenoon, or eastward if it be in the afternoon, until the sun's place at the inner edge of the ecliptic cuts the quadrant in the sun's observed altitude; and then the hour-index will point out the time of the day, and the quadrant will cut the true azimuth, or bearing of the sun for that time: the difference between which, and the bearing shewn by the azimuth-compass, shews the variation of the compass in that place of the earth.

The **Trajectorium Lunare**, Plate LIII. fig. 2. This machine is for delineating the paths of the earth and moon, shewing what sort of curves they make in the ethereal regions. S is the sun, and E the earth, whose centres are 95 inches distant from each other; every inch answering to 1,000,000 of miles. M is the moon, whose centre is $\frac{3}{4}$ parts of an inch from the earth's in this machine, this being in just proportion to the moon's distance from the earth. A A is a bar of wood, to be moved by hand round the axis g which is fixed in the wheel Y. The circumference of this wheel is to the circumference of the small wheel L (below the other end of the bar) as $365\frac{1}{4}$ days is to $29\frac{1}{2}$, or as a year is to a lunation. The wheels are grooved round their edges, and in the grooves is the cat-gut string GG crossing between the wheels at X. On the axis of the wheel L is the index F, in which is fixed the moon's axis M for carrying her round the earth E (fixed on the axis of the wheel L) in the time that the index goes round a circle of $29\frac{1}{2}$ equal parts, which are the days of the moon's age. The wheel Y has the months and days of the year all round its limb; and in the bar A A is fixed the index I, which points out the days of the months answering to the days of the moon's age, shewn by the index F, in the circle of $29\frac{1}{2}$ equal parts at the other end of the bar. On the axis of the wheel L is put the piece D, below the cock C, in which this axis turns round; and in D are put the pencils e and m, directly under the earth E and moon M; so that m is carried round e, as M is round E.

Lay the machine on an even floor, pressing gently on the wheel Y, to cause its spiked feet (of which two appear at P and P, the third being supposed to be hid from sight by the wheel) enter a little into the floor to secure the wheel from turning. Then lay a paper about four feet long under the pencils e and m, crosswise to the bar; which done, move the bar slowly round the axis g of the wheel Y; and as the earth E goes round the sun S, the moon M will go round the earth with a duly proportioned velocity; and the friction wheel W running on the floor, will keep the bar from bearing too heavily on the pencils e and m, which will delineate the paths of the earth and moon. As the index I points out the days of the months, the index F shews the moon's age on these days, in the circle of $29\frac{1}{2}$ equal parts. And as this last index points to the different days in its circle, the like numeral figures may be set to those parts of the curves of the earth's path and moon's, where the pencils e and m are at those times respectively, to shew the places of the earth and moon. If the pencil e be pulled a very little off, as if from the pencil m, to about $\frac{1}{27}$ part of their distance, and the pencil m pulled as much towards e, to bring them to the same distances again, though not to the same points of space; then, as m goes round e, e will go as it were round the centre of gravity between the earth e and moon m; but this motion will not sensibly alter the figure of the earth's path or the moon's.

If a pin, as p, be put through the pencil m, with its head towards that of the pin q in the pencil e, its head will always keep thereto as m goes round e, or as the same side of the moon is still obverted to the earth. But the pin p, which may be considered as an equatorial diameter of the moon, will turn quite round the point m, making all possible angles with the line of its progress, or line of the moon's path. This is an ocular proof of the moon's turning round her axis.

The **Tide-dial**, Plate LIV. fig. 1. The outside parts of this machine consist of, 1. An eight-sided box, on the top of which at the corners is shewn the phases of the moon at the octants, quarters, and full. Within there is a circle of $29\frac{1}{2}$ equal parts, which are the days of the moon's age accounted from the sun at new moon, round to the sun again. Within this circle is one of $24$ hours divided into their respective halves and quarters. 2. A moving elliptical plate, painted blue, to represent the rising of the tides under and opposite to the moon; and has the words, high water, tide falling, low water, tide rising, marked upon it. To one end of this plate is fixed the moon M by the wire W, and goes along with it. 3. Above this elliptical plate is a round one, with the points of the compass upon it, and also the names of above 200 places in the large machine (but only 32 in the figure, to avoid confusion) set over those points in which the moon bears when she raises the tides to the greatest heights at these places twice in every lunar day: And to the north and south points of this plate are fixed two indexes.

Indexes I and K, which shew the times of high water, in the hour circle, at all these places. Below the elliptical plate are four small plates, two of which project out from below its ends at new and full moon; and so, by lengthening the ellipse, shew the spring-tides, which are then raised to the greatest heights by the united attractions of the sun and moon. The other two of these small plates appear at low water when the moon is in her quadratures, or at the sides of the elliptic plate, to shew the neap-tides; the sun and moon then acting cross-wise to each other. When any two of these small plates appear, the other two are hid; and when the moon is in her octants, they all disappear, their being neither spring nor neap tides at those times. Within the box are a few wheels for performing these motions by the handle or winch H.

Turn the handle until the moon M comes to any given day of her age in the circle of 29½ equal parts, and the moon's wire W will cut the time of her coming to the meridian on that day, in the hour circle; the XII under the sun being mid-day, and the opposite XII mid-night: Then looking for the name of any given place on the round plate (which makes 29½ rotations whilst the moon M makes only one revolution from the sun to the sun again), turn the handle till that place comes to the word high water under the moon, and the index which falls among the forenoon hours will shew the time of high water at that place in the forenoon of the given day: then turn the plate half round, till the same place comes to the opposite high-water mark, and the index will shew the time of high water in the afternoon at that place. And thus, as all the different places come successively under and opposite to the moon, the indexes shew the times of high water at them in both parts of the day: and, when the same places come to the low-water marks, the indexes shew the times of low water. For about three days before and after the times of new and full moon, the two small plates come out a little way from below the high-water marks on the elliptical plate, to shew that the tides rise still higher about these times: and about the quarters, the other two plates come out a little from under the low-water mark, towards the sun, and on the opposite side, shewing that the tides of flood rise not then so high, nor do the tides of ebb fall so low, as at other times.

By pulling the handle a little way outward, it is disengaged from the wheel-work, and then the upper plate may be turned round quickly by hand, so as the moon may be brought to any given day of her age in about a quarter of a minute; and by pushing in the handle, it takes hold of the wheel-work again.

On AB, (fig. 2.) the axis of the handle H, is an endless screw C, which turns the wheel FED of 24 teeth round in 24 revolutions of the handle: this wheel turns another ONG of 48 teeth, and on its axis is the pinion PQ of four leaves, which turns the wheel LKI of 59 teeth round in 29½ turnings or rotations of the wheel FED, or in 708 revolutions of the handle, which is the number of hours in a synodical revolution of the moon. The round plate, with the names of places upon it, is fixed on the axis of the wheel FED; and the elliptical or tide-plate with the moon fixed to it, is upon the axis of the wheel LKI; consequently, the former makes 29½ revolutions in the time that the latter makes one. The whole wheel FED, with the endless screw C, and dotted part of the axis of the handle AB, together with the dotted part of the wheel ONG, lie hid below the large wheel LKI.