ASTRONOMY is the science which treats of the nature and properties of the heavenly bodies.

CHAP. I. Of Astronomy in general.

By astronomy we discover that the earth is at so great a distance from the sun, that if seen from thence it would appear no bigger than a point, although its circumference is known to be 25,020 miles. Yet that distance is so small compared with the earth's distance from the fixed stars, that if the orbit in which the earth moves round the sun were solid, and seen from the nearest star, it would likewise appear no bigger than a point, although it is at least 162 millions of miles in diameter. For the earth, in going round the sun, is 162 millions of miles nearer to some of the stars at one time of the year than at another; and yet their apparent magnitudes, situations, and distances from one another still remain the same; and a telescope which magnifies above 200 times does not sensibly magnify them; which proves them to be at least 400 thousands times farther from us than we are from the sun.

It is not to be imagined that all the stars are placed in one concave surface, so as to be equally distant from us; but that they are scattered at immense distances from one another through unlimited space. So that there may be as great a distance between any two neighbouring stars, as between our sun and those which are nearest to him. Therefore an observer, who is nearest any fixed star, will look upon it alone as a real sun; and consider the rest as so many shining points, placed at equal distances from him in the firmament.

By the help of telescopes, we discover thousands of stars which are invisible to the naked eye; and the better our glasses are, still the more become visible; so that no limits can be set either to their number or their distances.

The sun appears very bright and large in comparison of the fixed stars, because we keep constantly near the sun, in comparison of our immense distance from the stars. For a spectator, placed as near to any star as we are to the sun, would see that star a body as large and bright as the sun appears to us: and a spectator, as far distant from the sun as we are from the stars, would see the sun as small as we see a star, divested of all its circumvolving planets; and would reckon it one of the stars in numbering them.

The stars, being at such immense distances from the sun, cannot possibly receive from him so strong a light as they seem to have; nor any brightness sufficient to make them visible to us. For the sun's rays must be so scattered and dissipated before they reach such remote objects, that they can never be transmitted back to our eyes, so as to render these objects visible by reflexion. The stars therefore shine with their own native and unborrowed lustre, as the sun does; and since each particular star, as well as the sun, is confined to a particular portion of space, it is plain that the stars are of the same nature with the sun.

It is nowadays probable that the Almighty, who always acts with infinite wisdom, and does nothing in vain, should create so many glorious suns, fit for so many important purposes, and place them at such distances from one another, without proper objects near enough to be benefited by their influences. Whoever imagines they were created only to give a faint glimmering light to the inhabitants of this globe, must have a very superficial knowledge of astronomy, and a mean opinion of the Divine Wisdom; since, by an infinitely less exertion of creating power, the Deity could have given our earth much more light by one single additional moon.

Instead then of one sun and one world only in the universe, astronomy discovers to us such an inconceivable number of suns, systems, and worlds, dispersed through boundless space, that if our sun, with all the planets, moons, and comets belonging to it, were annihilated, they would be no more missed, by an eye that could take in the whole creation, than a grain of sand from the sea-shore. The space they possess being comparatively so small, that it would scarce be a sensible blank in the universe, although Saturn, the outermost of our planets, revolves about the sun in an orbit of 483.4 millions of miles in circumference, and some of our comets make excursions upwards of ten thousand millions of miles beyond Saturn's orbit; and yet, at that amazing distance, they are incomparably nearer to the sun than to any of the stars; as is evident from their keeping clear of the attractive power of all the stars, and returning periodically by virtue of the sun's attraction.

From what we know of our own system, it may be reasonably concluded, that all the rest are with equal wisdom contrived, situated, and provided with accommodations for rational inhabitants. Let us therefore take a survey of the system to which we belong; the only one accessible to us; and from thence we shall be the better enabled to judge of the nature and end of the other systems of the universe. For although there is almost an infinite variety in the parts of the creation which we have opportunities of examining, yet there is a general analogy running through, and connecting all the parts into one great and universal system.

To an attentive considerer, it will appear highly probable, that the planets of our system, together with their attendants called satellites or moons, are much of the same nature with our earth, and destined for the like purposes. For they are solid opaque globes, capable of supporting animals and vegetable. Some of them are larger, some less, and some much about the size of our earth. They all circulate round the sun, as the earth does, in a shorter or longer time, according to their respective distances from him; and have, where it would not be inconvenient, regular returns of summer and winter, spring and autumn. They have warmer and colder climates, as the various productions of our earth require: And, in such as afford a possibility of discovering it, we observe a regular motion round their axes like that of our earth, causing an alternate return of day and night; which is necessary for labour, rest, and vegetation, and that all parts of their surfaces may be exposed to the rays of the sun.

Such of the planets as are farthest from the sun, and therefore enjoy least of his light, have that deficiency made up by several moons, which constantly accompany and revolve about them, as our moon revolves about the earth. The remotest planet has, over and above, a broad ring encompassing it; which like a lucid zone in the heavens reflects the sun's light very copiously on that planet; so that if the remoter planets have the sun's light fainter by day than we, they have an addition made to it morning and evening by one or more of their moons, and a greater quantity of light in the night-time.

On the surface of the moon, because it is nearer us than any other of the celestial bodies are, we discover a nearer resemblance of our earth. For, by the assistance of telescopes, we observe the moon to be full of high mountains, large valleys, and deep cavities. These similarities leave us no room to doubt, but that all the planets and moons in the system are designed as commodious habitations for creatures endowed with capacities of knowing and adoring their beneficent Creator.

Since the fixed stars are prodigious spheres of fire like our sun, and at inconceivable distances from one another as well as from us, it is reasonable to conclude they are made for the same purposes that the sun is; each to below light, heat, and vegetation, on a certain number of inhabited planets, kept by gravitation within the sphere of its activity.

**Chap. II. Of the Solar System.**

The planets and comets which move round the sun 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. Their names, distances, bulks, and periodical revolutions, are as follow.

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; and turns round his axis in 25 days 6 hours, as is evident by the motion of spots seen on his surface. His diameter is computed to be 763,000 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 & Ξ, &c. Plate XL. fig. 2, which represents the great ecliptic in the heavens; But, as seen from any one planet, the rest 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. 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 through its centre, about which it revolves as on a 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 30th 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 different parts of the ecliptic; and therefore, if the planetary tracks 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 Ξ, Jupiter's in 7 deg. 29 min. of the same sign, Mars's in 17 deg. 17 min. of Taurus Ψ, Venus's in 13 deg. 59 min. of Gemini Π, 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 up- on them at first; between which power and force there is so exact an adjustment, that they continue in the same tracks without any solid orbits to confine them.

Mercury, the nearest planet to the sun, goes round him (as in a circle marked $C$, Plate XXXIX. fig. 1.) 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 32 millions of miles, and his diameter 2600. In his course round the sun, he moves at the rate of 95 thousand 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 fit the bodies and constitutions of its inhabitants to the heat of their dwelling, as he has done ours 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.

This planet appears to us with all the various phases of the moon, when viewed at different times by a good telescope; excepting only that he never appears quite full, because his enlightened side is never turned directly towards us but when he is so near the sun as to be lost to our sight in its beams. And, as his enlightened side is always toward the sun, it is plain that he shines not by any light of his own; for if he did, he would constantly appear round. That he moves about the sun in an orbit within the earth’s orbit is also plain, (as will be shewn afterwards), because he is never seen opposite to the sun, nor above 56 times the sun’s breadth from his centre.

His orbit 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 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 min. in the afternoon; and 1799, May 7th, at 2 h. 34 m. in the afternoon. There will be several intermediate transits, but none of them visible to us.

Venus, the next planet in order, is computed to be 59 millions of miles from the sun; and by moving at the rate of 69 thousand miles every hour in her orbit, (as in the circle marked $C$), she goes round the sun in 224 days 17 hours of our time nearly. But though this be the full length of her year, as she performs only $9\frac{1}{2}$ revolutions on her own axis in that time, her year consists only of $9\frac{1}{2}$ days; so that in her, every day and night together is as long as $2\frac{1}{2}$ 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 7906 miles; and by her diurnal motion the inhabitants about her equator are carried 43 miles every hour, besides the 69,000 above mentioned.

Her orbit includes that of Mercury within it; for at her greatest elongation, or apparent distance from the sun, she is 96 times his breadth from his centre; which is almost double of Mercury’s. Her orbit is included by the earth’s; for if it were not, the 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. To us she appears, through a telescope, in all the various shapes of the moon.

The axis of Venus is inclined 75 degrees to the axis of her orbit; which is $51\frac{1}{2}$ 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 $112\frac{1}{2}$ 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 $9\frac{1}{2}$ 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 $18\frac{1}{2}$ of our days and nights.

Because her day is so great a part of her year, the sun changes his declination in one day so 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 from it; and whatever place he passes vertically over when in the equator, one day's revolution will remove him $36\frac{1}{4}$ 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 by that means 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 things.

The sun will rise $22\frac{1}{2}$ degrees north of the east; and going on $112\frac{1}{2}$ degrees, as measured on the plane of the horizon, he will cross the meridian at an altitude of $12\frac{1}{2}$ degrees; then making an entire revolution without setting, he will cross it again at an altitude of $48\frac{1}{2}$ 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 $48\frac{1}{2}$ degrees; next at the altitude of $12\frac{1}{2}$ degrees; and going on thence $112\frac{1}{2}$ degrees, he will let $22\frac{1}{2}$ degrees north of the west; so that, after having been $48\frac{1}{2}$ 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 midnight, on Venus, is much greater than on the earth; because in 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 $23\frac{1}{2}$ 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 betwixt 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 fifteen 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 $9\frac{1}{2}$ 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 $18\frac{1}{2}$ 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 $3\frac{1}{2}$ revolutions above the horizon without setting. Therefore, no place has the forenoon and afternoon of the same day equally long, unless it be in the equator, or at the poles.

The sun's altitude at noon, or at any other time of the 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 enjoy, because the daily change of the sun's declination is by 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. 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 fort 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.

Venus's orbit is inclined $3\frac{1}{2}$ degrees to the earth's; and crosses it in the 14th degree of Gemini and of Sagittarius; and therefore, when the earth is about these points of the ecliptic at the time that Venus is in her inferior conjunction, she will appear like a spot on the sun, and afford a more certain method of finding the distances of all the planets from the sun, than any other yet known. But these appearances happen very seldom. The first was in the year 1639. The second in the year 1761, June 6. In the morning of that day, when the sun rose at London, Venus had passed both the external and internal contacts. At 38 minutes 21 seconds past 7 o'clock, (apparent time) at Greenwich, the Rev. Dr Bliss, astronomer royal, first saw Venus on the sun; at which instant, the centre of Venus preceded the sun's centre, by $6'18''$ of right ascension, and was south of the sun's centre by $18'42''$ of declination.—From that time to the beginning of egresses, the Doctor made several observations, both of the difference of right ascension and declination of the centres of the sun and Venus; and at last found the beginning of egresses, or instant of the internal contact of Venus with the sun's limb, to be at 8 hours 19 minutes 0 seconds apparent time.—From the Doctor's own observations, and those which were made at Shirburn by another gentleman, he has computed, that the mean time at Greenwich of the ecliptical conjunction of the sun and Venus was at 51 minutes 20 seconds after 5 o'clock in the morning; that the place of the sun and Venus was Gemini $15'36''33''$; that the geocentric latitude of Venus was $9'44''9$ south,—her horary motion from the sun $3'57''13$ retrograde, and the angle then formed by the axis of the equator and the axis of the ecliptic was $6'9'34''$, decreasing hourly 1 minute of a degree.—By the mean of three good observations, the diameter of Venus on the sun was $58''$.

Mr Short made his observations at Savile-house, in London, 30 seconds in time west from Greenwich, in presence of his royal highnesses the duke of York, accompanied by their royal highnesses prince William, prince Henry, and prince Frederick.—He first saw Venus on the sun, through flying clouds, at 46 minutes 37 seconds after 5 o'clock; and at 6 hours 15 minutes 12 seconds he measured the diameter of Venus $59''8$.—He afterward found it to be $58''9$, when the sky was more favourable.—And, through a reflecting telescope of two feet focus, magnifying 140 times, he found the internal contact of Venus with the sun's limb to be at 8 hours 18 minutes 21 seconds, apparent time; which being reduced to the apparent time at Greenwich, was 8 hours 18 minutes 51 seconds; so that his time of seeing the contact was 8 seconds sooner (in absolute time) than the instant of its being seen at Greenwich.

Messrs Ellicott and Dollond observed the internal contact at Hackney; and their time of seeing it, reduced to the time at Greenwich, was at 8 hours 18 minutes 56 seconds, which was 4 seconds sooner in absolute time than the contact was seen at Greenwich.

Mr Canton in Spittle-Square, London, $4'11''$ west of Greenwich, (equal to 16 seconds 44 thirds of time), measured the sun's diameter $31'33''24''$, and the diameter of Venus on the sun $58''$; and, by observation, found the apparent time of the internal contact of Venus with the sun's limb to be at 8 hours 18 minutes 41 seconds; which, by reduction, was only $2\frac{1}{2}$ seconds short of the time at the Royal Observatory at Greenwich.

The Reverend Mr Richard Haydon, at Liskeard in Cornwall, (16 minutes 10 seconds in time west from London, as stated by Dr Bevis), observed the internal contact to be at 8 hours 0 minutes 20 seconds, which, by reduction, was 8 hours 16 minutes 30 seconds at Greenwich; so that he must have seen it 2 minutes 30 seconds sooner in absolute time than it was seen at Greenwich;—a difference by much too great to be occasioned by the difference of parallaxes. But by a memorandum of Mr Hayden's some years before, it appears that he then supposed his west longitude to be near two minutes more; which brings his time to agree within half a minute of the time at Greenwich; to which the parallaxes will very nearly answer.

At Stockholm Observatory, latitude $59°20'3$ north, and longitude 1 hour 12 minutes east from Greenwich, the whole of the transit was visible: the total ingress was observed by Mr Wargentin to be at 3 hours 39 minutes 23 seconds in the morning, and the beginning of egress at 9 hours 30 minutes 8 seconds; so that the whole duration between the two internal contacts, as seen at that place, was 5 hours 50 minutes 45 seconds.

At Torneo in Lapland, (1 hour 27 minutes 28 seconds east of Paris), Mr Hellant, who is esteemed a very good observer, found the total ingress to be at 4 hours 3 minutes 59 seconds, and the beginning of egress to be 9 hours 54 minutes 8 seconds.—So that the whole duration between the two internal contacts was 5 hours 50 minutes 9 seconds.

At Hernsund in Sweden, (latitude $6°38'$ north, and longitude 1 hour 2 minutes 12 seconds east of Paris), Mr Gilter observed the total ingress to be at 3 hours 38 minutes 26 seconds; and the beginning of egress to be at 9 hours 29 minutes 21 seconds;—the duration between these two internal contacts 5 hours 50 minutes 56 seconds.

Mr De La Lande, at Paris, observed the beginning of egresses to be at 8 hours 28 minutes 26 seconds apparent time.—But Mr Ferner (who was then at Conflans, $14''$ west of the Royal Observatory at Paris) observed the beginning of egresses to be at 8 hours 28 minutes 29 seconds true time. The equation, or difference between the true and apparent time, was 1 minute 54 seconds.—The total ingress, being before the sun rose, could not be seen.

At Tobolsk in Siberia, Mr Chappe observed the total ingress to be at 7 hours 0 minutes 28 seconds in the morning, and the beginning of egresses to be at 49 minutes 20 seconds after 12 at noon.—So that the whole duration of the transit between the internal contacts was 5 hours... hours 48 minutes 52 seconds, as seen at that place; which was 2 minutes 3 seconds less than as seen at Herneford in Sweden.

At Madras, the Reverend Mr Hirst observed the total ingress to be at 7 hours 47 minutes 55 seconds apparent time in the morning, and the beginning of egress at 1 hour 39 minutes 38 seconds past noon.—The duration between these two internal contacts was 5 hours 51 minutes 43 seconds.

Professor Mathenici at Bologna observed the beginning of egress to be at 9 hours 4 minutes 58 seconds.

At Calcutta, (latitude 22° 30' north, nearly 92° east longitude from London), Mr William Magee observed the total ingress to be at 8 hours 20 minutes 58 seconds in the morning, and the beginning of egress to be at 2 hours 11 minutes 34 seconds in the afternoon; the duration between the two internal contacts 5 hours 50 minutes 36 seconds.

At the Cape of Good Hope, (1 hour 13 minutes 35 seconds east from Greenwich), Mr Mafon observed the beginning of egress to be at 9 hours 39 minutes 50 seconds in the morning.

All these times are collected from the observers accounts, printed in the Philosophical Transactions for the years 1762 and 1763, in which there are several other other accounts that are not transcribed.—The instants of Venus's total exit from the sun are likewise mentioned, but they are here left out, as not of any use for finding the sun's parallax.

Whoever compares these times of the internal contacts, as given in by different observers, will find such differences among them, even those which were taken upon the same spot, as will shew, that the instant of either contact could not be so accurately perceived by the observers as Dr Halley thought it could; which probably arises from the difference of peoples eyes, and the different magnifying powers of those telescopes through which the contacts were seen.—If all the observers had made use of equal magnifying powers, there can be no doubt but that the times would have more nearly coincided; since it is plain, that supposing all their eyes to be equally quick and good, they whose telescopes magnified most would perceive the point of internal contact loohest, and of the total exit lastest.

Mr Short, in a paper published in the Philosophical Transactions, Vol. LII. Part II., has taken an incredible deal of pains in deducing the quantity of the sun's parallax, from the best of those observations which were made both in Britain and abroad; and finds it to have been 8".52 on the day of the transit when the sun was very nearly at his greatest distance from the earth; and consequently 8".65 when the sun is at his mean distance from the earth.

The log. sine (or tangent) of 8".65 is 5.6219140, which being subtracted from the radius 10,0000000, leaves remaining the logarithm 4.3780860, whose number is 23882.84; which is the number of semidiameters of the earth that the sun is distant from it.—And this last number, 23882.84, being multiplied by 3985, the number of English miles contained in the earth's semidiameter, gives 95,173,117 miles from the earth's mean distance from the sun.—But because it is impossible, from the nicest observations of the sun's parallax, to be sure of his true distance from the earth within 100 miles, we shall at present, for the sake of round numbers, state the earth's mean distance from the sun at 95,173,000 English miles.

And then, from the numbers and analogies in § 11. & 14. of Mr Short's dissertation, we find the mean distances of all the rest of the planets from the sun, in miles, to be as follows.—Mercury's distance, 36,841,468; Venus's distance, 68,891,486; Mars's distance, 145,014,148; Jupiter's distance, 494,990,976; and Saturn's distance, 907,956,130.

The semidiameter of the earth's annual orbit being equal to the earth's mean distance from the sun, viz. 95,173,000 miles; the whole diameter thereof is 190,346,000 miles. And since the circumference of a circle is to its diameter as 355 is to 113, the circumference of the earth's orbit is 597,989,646 miles.

And, as the earth describes this orbit in 365 days 6 hours (or in 8766 hours) it is plain that it travels at the rate of 68,216.9 miles every hour; and consequently 1136.9 miles every minute; so that its velocity in its orbit is at least 142 times as great as the velocity of a cannon-ball, supposing the ball to move through 8 miles in a minute, which it is found to do very nearly: And at this rate it would take 22 years 228 days for a cannon-ball to go from the earth to the sun.

On the 3d of June, in the year 1769, Venus again passed over the sun's disk, in such a manner, as to afford a much easier and better method of investigating the sun's parallax than her transit in the year 1761. But as few of the observations upon this transit have as yet been made public, we can only give the following, made by different observers at London.

| External contact | Regular circumferences in contact | Thread of light completed, or the internal contact | Telescopes made use of | Magnifying power | |------------------|---------------------------------|---------------------------------|------------------------|----------------| | N. Maskelyne | 7 10 58 | 7 28 31 | 7 29 23 | 2 feet reflector, 140 | | M. Hitchins | 7 10 54 | 7 28 47 | 7 28 57 | 6 f. reflector, 90 | | W. Hirst | 7 11 11 | 7 29 18 | 7 29 28 | 2 f. reflector, 55 | | J. Horley | 7 10 44 | 7 28 15 | 7 29 48 | 10 f. achromatic, 50 | | S. Dunn | 7 10 37 | 7 29 28 | 7 29 20 | 3½ f. achromatic, 140| | P. Dollond | 7 11 19 | 7 29 20 | 7 29 20 | 3½ f. achromatic, 150| | E. Nairne | 7 11 30 | 7 29 20 | 7 29 20 | 2 f. reflector, 120 |

When When Venus was little more than half emerged into the sun's disk, Mr Malklyne saw her whole circumference completed, by means of a vivid, but narrow and ill-defined border of light, which illuminated that part of her circumference which was off the sun, and otherwise not visible. They all observed the black protuberance in the internal contact. They likewise, after the internal contact, saw a luminous ring round the body of Venus, about the thickness of half her semi-diameter; it was brightest towards Venus's body, and gradually diminished in splendor at greater distance, but the whole was excessive white and faint.

Venus may have a satellite or moon, although it be undiscovered by us: which will not appear very surprising, if we consider how inconveniently we are placed from seeing it. For its enlightened side can never be fully turned towards us, but when Venus is beyond the sun; and then, as Venus appears little bigger than an ordinary star, her moon may be too small to be perceived at such a distance. When she is between us and the sun, her full moon has its dark side towards us; and then we cannot see it any more than we can our own moon at the time of change. When Venus is at her greatest elongation, we have but one half of the enlightened side of her full moon towards us; and even then it may be too far distant to be seen by us.

The Earth is the next planet above Venus in the system. It is 82 millions of 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 and 9 minutes; the former being the length of the tropical year, and the latter the length of the sidereal. It travels the rate of 58 thousand miles every hour; which motion, though 120 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 58 thousand 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 towards the same fixed stars throughout its annual course, which causes the returns of spring, summer, autumn, and winter; as will be explained afterwards.

The earth is round like a globe; as appears, 1. By its shadow in eclipses of the moon, which shadow is always bounded by a circular ring. 2. By our seeing the masts of a ship whilst the hull is hid by the convexity of the water. 3. By its having been sailed round by many navigators. The hills take off no more from the roundness of the earth in comparison, than grains of dust do from the roundness of a common globe.

The seas and unknown parts of the earth (by a measurement of the best maps) contain 160 million 522 thousand and 26 square miles; the inhabited parts 38 million 990 thousand 569; Europe 4 million 456 thousand and 65; Asia, 10 million 768 thousand 823; Africa, 9 million 654 thousand 807; America, 14 million 110 thousand 874. In all, 199 million 512 thousand 595; which is the number of square miles on the whole surface of our globe.

The Moon is not a planet, but only a satellite or attendant of the earth; going round the earth from change to change 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 240 thousand. She goes round her orbit in 27 days 7 hours 43 minutes, moving about 2260 miles every hour; and turns round her axis exactly in the same 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 shines only by reflecting the light of the sun: Therefore whilst that half of her which is toward 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 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 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.

This continual change of the moon's phases demonstrates that she shines not by any light of her own; for if she did, being globular, we should always see her with a round full orb like the sun. Her orbit is represented in the scheme by the little circle m, upon the earth's orbit ⊕, Plate XXXIX, fig. 1; but it is drawn 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 a moon to the moon, waxing and waning regularly, but appearing thirteen times as big; and affording her thirteen times as much light as she does to 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 thirty 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 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 largest body in the universe, for it appears thirteen times as large as she does to us.

The moon has no atmosphere of any visible density surrounding her; for if she had, we could never see her edge so well defined as it appears; but there would be a sort of a mist or haziness around her, which would make the stars look fainter, when they are seen through it. But observation proves, that the stars which disappear behind the moon retain their full lustre until they seem to touch her very edge, and then they vanish in a moment. The faint light which has been seen all around the moon in total eclipses of the sun, has been observed, during the time of darkness, to have its centre coincident with the centre of the sun; and was therefore much more likely to arise from the atmosphere of the sun than from that of the moon; for if it had been owing to the latter, its centre would have gone along with the moon's.

If there were seas in the moon, she could have no clouds, rains, nor storms, as we have; because she has no such atmosphere to support the vapours which occasion them. And every one knows, that when the moon is above our horizon in the night-time, she is visible, unless the clouds of our atmosphere hide her from our view, and all parts of her appear constantly with the same clear, serene, and calm aspect. But those dark parts of the moon, which were formerly thought to be seas, are now found to be only vast deep cavities, and places which reflect not the sun's light so strongly as others, having many caverns and pits, whose shadows fall within them, and are always dark on the sides next the sun, which demonstrates their being hollow; and most of these pits have little knobs like hillocks standing within them, and casting shadows also; which cause these places to appear darker than others which have fewer or less remarkable caverns. All these appearances shew, that there are no seas in the moon; for if there were any, their surfaces would appear smooth and even, like those on the earth.

There being no atmosphere about the moon, the heavens in the day-time have the appearance of night to a lunarian who turns his back toward the sun; and when he does, the stars appear as bright to him as they do in the night to us. For it is entirely owing to our atmosphere that the heavens are bright about us in the day.

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, being only $5^\circ$ degrees, 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 $365\frac{1}{4}$ natural days, and the lunarians only $12\frac{1}{2}$ every day and night in the moon—being as long as $29\frac{1}{2}$ on the earth.

The moon's inhabitants on the side next the earth may as easily find the longitude of their places as we 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.

The planet Mars is next in order, being the first above the earth's orbit. His distance from the sun is computed to be $125$ millions of miles; and by travelling at the rate of $47$ thousand miles every hour, as in the circle $\mathcal{C}$, he goes round the sun in $686$ of our days and $23$ hours; which is the length of his year, and contains $667\frac{1}{4}$ of his days, every day and night together being $40$ minutes longer than with us. His diameter is $4444$ miles, and by his diurnal rotation the inhabitants about his equator are carried $556$ 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, the must be very small, and has not yet been discovered by our best telescopes. He is of a fiery red colour, and by his appulses to some of the fixed stars seems to be encompassed by a very gross atmosphere. He appears sometimes gibbous, but never horned; which both shews that his orbit includes the earth's within it, and that he shines not by his own light.

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 $240$ thousand miles afar.

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 are as visible to Mars as to us. His axis is perpendicular to the ecliptic, and his orbit is $2$ degrees inclined to it.

Jupiter, the largest of all the planets, is still higher in the system, being about $426$ millions of miles from the sun; and going at the rate of $25$ thousand miles every hour in his orbit, as in the circle $\mathcal{H}$, finishes his annual period. Period in eleven of our years 314 days and 12 hours. He is about 1000 times as big as the earth, for his diameter is 81,000 miles; which is more than ten times the diameter of the earth.

Jupiter turns round his axis in 9 hours 56 minutes; so that his year contains 10 thousand 470 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 25 thousand 920 miles every hour, (which is 920 miles an hour more than an inhabitant of our earth's equator moves in twenty-four hours), besides the 25 thousand above mentioned, which is common to all parts of his surface, by his annual motion.

Jupiter is surrounded by faint substances, called belts, in which so many changes appear, that they are generally thought to be clouds; for some of them have been first interrupted and broken, and then have vanished entirely. They have sometimes been observed of different breadths, and afterwards have all become nearly of the same breadth. Large spots have been seen in these belts; and when a belt vanishes, the contiguous spots disappear with it. The broken ends of some belts have been generally observed to revolve in the same time with the spots; only those nearer the equator in somewhat less time than those near the poles, perhaps on account of the sun's greater heat near the equator, which is parallel to the belts and course of the spots. Several large spots, which appear round at one time, grow oblong by degrees, and then divide into two or three round spots.

The periodical time of the spots near the equator is 9 hours 50 minutes, but of those near the poles 9 hours 56 minutes.

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 706 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 1/8 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 larger 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 24; but they are drawn fifty times 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 229 thousand miles distant from his centre; the second performs its revolution in three days 13 hours and 15 minutes, at 364 thousand miles distance; the third in seven days three hours and 59 minutes, at the distance of 580 thousand miles; and the fourth, or outermost, in 16 days 18 hours and 30 minutes, at the distance of one million of miles from his centre. The periods of these moons are so incommensurate to one another, that if ever they were all in a right line between Jupiter and the sun, it will require more than 3,000,000,000,000 years from that time to bring them all into the same right line again, as any one will find who reduces all their periods into seconds, then multiplies them into one another, and divides the product by 432; which is the highest number that will divide the product of all their periodical times, namely, 42,085,303,376,931,994,955,104 seconds, without a remainder.

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 semi-diameters, are thus: The first, 5°; the second, 9°; the third, 14° 3′; and the fourth, 25° 8′. 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 42½ hours.

Jupiter's three nearest moons fall into his shadow, and are eclipsed in every revolution; but the orbit of the fourth moon is so much inclined, that it pasleth by its opposition to Jupiter, without falling into his shadow, two years in every six. 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.

The difference between the equatorial and polar diameters of Jupiter is 6230 miles; for his equatorial diameter is to his polar, as 13 to 12. So that his poles are 3115 miles nearer his centre than his equator is.

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.

Saturn, the remotest of all the planets, is about 780 millions of miles from the sun; and, travelling at the rate of 18 thousand miles every hour, as in the circle 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 67,000 miles; and therefore it is near 600 times as big as the earth.

This planet is surrounded by a thin broad ring, as an artificial globe is by a horizon, fig. 5. The ring appears double when seen through a good telescope, and is represented by the figure in such an oblique view as it is generally seen. It is inclined 30 degrees to the ecliptic, and is about 21 thousand miles in breadth; which is equal to its distance from Saturn on all sides. There is reason reason to believe that the ring turns round its axis, because, when it is almost edge-wise to us, it appears somewhat thicker on one side of the planet than on the other; and the thickest edge has been seen on different sides at different times. But Saturn having no visible spots on his body, whereby to determine the time of his turning round his axis, the length of his days and nights, and the position of his axis, are unknown to us.

To Saturn, the sun appears only $\frac{1}{5}$th 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 on the same plane with it. The first, or nearest moon to Saturn, goes round him in 1 day 21 hours 19 minutes; and is 140 thousand miles from his centre: The second, in 2 days 17 hours 40 minutes; at the distance of 187 thousand miles: The third, in 4 days 12 hours 25 minutes, at 263 thousand miles distance: The fourth, in 15 days 22 hours 41 minutes, at the distance of 600 thousand miles: And the fifth or outermost, at one million 800 thousand 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 small five circles, marked 1, 2, 3, 4, 5, on Saturn's orbit; but these, like the orbits of the other satellites, are drawn fifty times too large in proportion to the orbits of their primary planets.

The sun shines almost fifteen 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 fifteen 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 in 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 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 its 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; as shall be explained afterwards. The ring disappears twice in every annual revolution of Saturn, namely, when he is in the 15th degree both of Pisces and of Virgo. And when Saturn is in the middle between these points, or in the 15th 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.

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, fewer 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 the 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 $2\frac{1}{2}$ 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 $\frac{1}{8}$th part, and to Saturn only $\frac{1}{5}$th 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 surprised at the brightness wherewith that small part of him shines. The moon, when full, affords travellers light enough to keep them from mistaking their way; and yet, according to Dr Smith, it is equal to no more than a 90 thousandth part of the light of the sun: That is, the sun's light is 90 thousand times as strong as the light of the moon when full. Consequently, the sun gives a thousand times as much light to Saturn as the full moon does to us; and above three thousand 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 seasons 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 their nature or soil much warmer than 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.

Every person who looks upon, and compares the systems of moons together, which belong to Jupiter and Saturn, must be amazed at the vast magnitude of these two planets, and the noble attendance they have in respect of our little earth; and can never bring himself to think, that an infinitely wise Creator should dispose of all his animals and vegetables here, leaving the other planets bare and destitute of rational creatures. To suppose that he had any view to our benefit, in creating these moons, and giving them their motions round Jupiter and Saturn; to imagine that he intended these vast bodies for any advantage to us, when he well knew that they could never be seen but by a few astronomers peeping through telescopes; and that he gave to the planets regular returns of days and nights, and different seasons to all where they would be convenient; but of no manner of service to us, except only what immediately regards our own planet the earth; to imagine that he did all this on our account, would be charging him impiously with having done much in vain; and as absurd, as to imagine that he has created a little sun and a planetary system within the shell of our earth, and intended them for our use. These considerations amount to little less than a positive proof, that all the planets are inhabited: For if they are not, why all this care in furnishing them with so many moons, to supply those with light which are at the greater distances from the sun? Do we not see, that the farther a planet is from the sun, the greater apparatus it has for that purpose? save only Mars, which being but a small planet, may have moons too small to be seen by us. We know that the earth goes round the sun, and turns round its own axis, to produce the vicissitudes of summer and winter by the former, and of day and night by the latter motion, for the benefit of its inhabitants. May we not then fairly conclude, by parity of reason, that the end and design of all the other planets is the same? and is not this agreeable to the beautiful harmony which exists throughout the universe?

In fig. 2, we have a view of the proportional breadth 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 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, (fig. 4.) the square A 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.

In fig. 5, we have a view of the bulks of the planets in proportion to each other, and to a supposed globe of two feet diameter for the sun. The earth is 27 times as big as Mercury, very little bigger than Venus, five times as big as Mars; but Jupiter is 1049 times as big as the earth; Saturn 586 times as big, exclusive of his ring; and the sun is 877 thousand 650 times as big as the earth. If the planets in this figure were set at their due distances from a sun of two feet diameter, according to their proportional bulks, as in our system, Mercury would be 28 yards from the sun's centre; Venus 51 yards 1 foot; the earth 70 yards 2 feet; Mars 107 yards 2 feet; Jupiter 370 yards 2 feet; and Saturn 760 yards two feet; the comet of the year 1680, at its greatest distance, 10 thousand 760 yards. In this proportion, the moon's distance from the centre of the earth would be only 7½ inches.

To assist the imagination in forming an idea of the vast distances of the sun, planets, and stars, let us suppose, that a body projected from the sun should continue to fly with the swiftness of a cannon-ball, i.e., 480 miles every hour; this body would reach the orbit of Mercury, in 7 years 221 days; of Venus, in 14 years 8 days; of the earth, in 19 years 91 days; of Mars, in 29 years 85 days; of Jupiter, in 100 years 280 days; of Saturn, in 184 years 240 days; to the comet of 1680, at its greatest distance from the sun, in 2660 years; and to the nearest fixed stars, in about 7 million 600 thousand years.

As the earth is not the centre of the orbits in which the planets move, they come nearer to it and go farther from it, and at different times; on which account they appear bigger and less by turns. Hence, the apparent magnitudes of the planets are not always a certain rule to know them by.

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

The Comets are solid opaque bodies, with long transparent trains or tails, issuing from that side which is turned away from the sun. They move about the sun in very eccentric ellipses; and are of a much greater density than the earth; for some of them are heated in every period to such a degree, as would vitrify or dissipate any substance known to us. Sir Isaac Newton computed the heat of the comet which appeared in the year 1680, when nearest the sun, to be 2000 times hotter than red-hot iron; and that, being thus heated, it must retain its heat until it comes round again, although its period should be more than twenty thousand years; and it is computed to be only 575.

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 that there are at least 21 comets belonging to our system, moving in all sorts of directions; and all those which have been observed, have moved through the etherial regions and the orbits of the planets without suffering the least sensible resistance in their motions; which plainly proves that the planets do not move in solid orbs. Of all the comets, 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, and 1682; was expected to appear again in the year 1758, and 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 thousand two hundred millions of 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 semi-diameter 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 thousand 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 thaws 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.

The extreme heat, the dense atmosphere, the gross vapours, the chaotic state of the comets, seem at first sight to indicate them altogether unfit for the purposes of animal life, and a most miserable habitation for rational beings; and therefore some are of opinion that they are so many hells for tormenting the damned with perpetual vicissitudes of heat and cold. But when we consider, on the other hand, the infinite power and goodness of the Deity, the latter inclining, and the former enabling him to make creatures suited to all states and circumstances; that matter exists only for the sake of intelligent beings; and that where-ever we find it, we always find it pregnant with life, or necessarily subservient thereto; the numberless species, the afflicting diversity of animals in earth, air, water, and even on other animals; every blade of grass, every leaf, every fluid swarming with life; and every one of these enjoying such gratifications as the nature and state of each requires: When we reflect moreover, that some centuries ago, till experience undeceived us, a great part of the earth was judged uninhabitable, the torrid zone by reason of excessive heat, and the frigid zones because of their intolerable cold; it seems highly probable, that such numerous and large masses of durable matter as the comets are, however unlike they be to our earth, are not destitute of beings capable of contemplating with wonder, and acknowledging with gratitude, the wisdom, symmetry, and beauty of the creation; which is more plainly to be observed in their extensive tour through the heavens, than in our more confined circuit. If farther conjecture is permitted, may we not suppose them instrumental in recruiting the expanded fuel of the sun, and supplying the exhausted moisture of the planets? However difficult it may be, circumstanced as we are, to find out their particular delineation, this is an undoubted truth, that where-ever the Deity exerts his power, there he also manifests his wisdom and goodness.

The solar system here described is not a late invention, for it was known and taught by the wise Samian philosopher Pythagoras, and others among the ancients; but in latter times was lost, till the 15th century, when it was again restored by the famous Polish philosopher, Nicholas Copernicus, who was born at Thorn in the year 1473. In this he was followed by the greatest mathematicians and philosophers that have since lived; as Kepler, Galileo, Descartes, Gassendus, and Sir Isaac Newton; the last of whom has established this system on such a foundation of mathematical and physical demonstration, as can never be shaken.

In the Ptolemaic system, the earth was supposed to be fixed in the centre of the universe; and that the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn, moved round the earth: Above the planets this hypothesis placed the firmament of stars, and then the two crystalline spheres; all which were included in and received motion from the primum mobile, which constantly revolved about the earth in 24 hours from east to west. But as this rude scheme was found incapable to stand the test of art and observation, it was soon rejected by all true philosophers.

The Tychonic system succeeded the Ptolemaic, but was never so generally received. In this the earth was supposed to stand still in the centre of the universe or firmament of stars, and the sun to revolve about it every 24 hours; the planets, Mercury, Venus, Mars, Jupiter, and Saturn, going round the sun in the times already mentioned. But some of Tycho's disciples supposed the earth to have a diurnal motion round its axis, and the sun, with all the above planets, to go round the earth in a year; the planets moving round the sun in the foresaid times. This hypothesis, being partly true, and partly false, was embraced by few; and soon gave way to the only true and rational scheme, restored by Copernicus, and demonstrated by Sir Isaac Newton.

CHAP. III. The Phenomena of the Heavens as seen from different Parts of the Earth.

We are kept to the earth's surface on all sides by the power of its central attraction; which, laying hold of all bodies according to their densities or quantities of matter, without regard to their bulks, constitutes what we call their weight. And having the sky over our heads, go where we will, and our feet towards the centre of the earth, we call it up over our heads, and down under our feet: Although the same right line which is down to us, if continued through, and beyond the opposite side of the earth, would be up to the inhabitants on the opposite side. For, the inhabitants n, i, e, m, s, o, q, l (Plate XXXIX. fig. 6.) stand with their feet towards the earth's centre C; and have the same figure of sky, N, I, E, M, S, O, Q, L over their heads. Therefore the point S is as directly upward to the inhabitant (s) on the south pole, as N is to the inhabitant n on the north pole; so is E to the inhabitant e, supposed to be on the north end of Peru; and Q to the opposite inhabitant q on the middle of the island of Sumatra. Each of these observers is surprised that his opposite or antipode can stand with his head hanging downwards. But let either go to the other, and he will tell him that he stood as upright and firm upon the place where he was, as he now stands where he is. To all these observers, the sun, moon, and stars, seem to turn round the points N and S, as the poles of the fixed axis NCS; because the earth does really turn round the mathematical line NCs as round an axis, of which n is the north pole, and s the south pole. The inhabitant U (Plate XL. fig. 1.) affirms that he is on the uppermost side of the earth, and wonders how another at L can stand on the undermost side with his head hanging downwards. But U, in the meantime, forgets that in twelve hours time he will be carried half round with the earth, and then be in the very situation that L now is, although as far from him as before. And yet, when U comes there, he will find no difference as to his manner of standing; only he will see the opposite half of the heavens, and imagine the heavens to have gone half round the earth.

When we see a globe hung up in a room, we cannot help imagining it to have an upper and an under side, and immediately form a like idea of the earth; from whence we conclude, that it is as impossible for people to stand on the under side of the earth, as for pebbles to lie on the under side of a common globe, which instantly fall down from it to the ground; and well they may, because the attraction of the earth, being greater than the attraction of the globe, pulls them away. Just so would be the case with our earth, if it were placed near a globe much bigger than itself, such as Jupiter; for then it would really have an upper and an under side, with respect to that large globe; which, by its attraction, would pull away every thing from the side of the earth next to it; and only those on the top of the opposite or upper side could remain upon it. But there is no larger globe near enough our earth to overcome its central attraction; and therefore it has no such thing as an upper and an under side; for all bodies, on or near its surface, even to the moon, gravitate towards its centre.

The earth's bulk is but a point, as that at C, compared to the heavens; and therefore every inhabitant upon it, let him be where he will, as at n, e, m, s, &c., sees half of the heavens. The inhabitant n, on the north pole of the earth, constantly sees the hemisphere ENQ; and having the north pole N of the heavens just over his head, his horizon coincides with the celestial equator ECQ. Therefore, all the stars in the northern hemisphere ENQ, between the equator and north pole, appear to turn round the line NC, moving parallel to the horizon. The equatorial stars keep in the horizon, and all those in the southern hemisphere ESQ are invisible. The like phenomena are seen by the observer (s) on the south pole, with respect to the hemisphere ESQ; and to him the opposite hemisphere is always invisible. Hence, under either pole, only half of the heavens is seen; for those parts which are once visible never set, and those which are once invisible never rise. But the ecliptic YCX, or orbit which the sun appears to describe once a year by the earth's annual motion, has the half YC constantly above the horizon ECQ of the north pole n; and the other half CX always below it. Therefore, whilst the sun describes the northern half YC of the ecliptic, he neither sets to the north pole, nor rises to the south; and whilst he describes the southern half CX, he neither sets to the south pole nor rises to the north. The same things are true with respect to the moon; only with this difference, that as the sun describes the ecliptic but once a year, he is for half that time visible to each pole in its turn, and as long invisible; but as the moon goes round the ecliptic in 27 days 8 hours, she is only visible for 13 days 16 hours, and as long invisible to each pole by turns. All the planets likewise rise and set to the poles, because their orbits are cut obliquely in halves by the horizon of the poles. When the sun (in his apparent way from X) arrives at C, which is on the 20th of March, he is just rising to an observer n on the north pole, and setting to another at s on the south pole. From C he rises higher and higher in every apparent diurnal revolution, till he comes to the highest point of the ecliptic y, on the 21st of June, and then he is at his greatest altitude, which is $23\frac{1}{2}$ degrees, or the arc Ey, equal to his greatest north declination; and from thence he seems to descend gradually in every apparent circumvolution, till he sets at C on the 23rd of September; and then he goes to exhibit the like appearances at the south pole for the other half of the year. Hence, the sun's apparent motion round the earth is not in parallel circles, but in spirals; such as might be represented by a thread wound round a globe from tropic to tropic; the spirals being at some distance from one another about the equator, and gradually nearer to each other as they approach toward the tropics.

If the observer be anywhere on the terrestrial equator ECQ, as suppose at e, he is in the plane of the celestial equator; or under the equinoctial ECQ; and the axis of the earth nCs is coincident with the plane of his horizon, extended out to N and S, the north and south poles of the heavens. As the earth turns round the line NCS, the whole heavens MOLL seem to turn round the same line, but the contrary way. It is plain that this observer has the celestial poles constantly in his horizon; and that his horizon cuts the diurnal paths of all the celestial bodies perpendicularly and in halves. Therefore the sun, planets and stars, rise every day, and ascend perpendicularly above the horizon for six hours; and, passing over the meridian, descend in the same manner for the six following hours; then set in the horizon, and continue twelve hours below it. Consequently at the equator the days and nights are equally long throughout the year. When the observer is in the situation e, he sees the hemisphere SEN; but in twelve hours after, he he is carried half round the earth's axis to q, and then the hemisphere SQN becomes visible to him; and SEN disappears. Thus we find, that to an observer at either of the poles, one half of the sky is always visible, and the other half never seen; but to an observer on the equator, the whole sky is seen every 24 hours.

The figure here referred to, represents a celestial globe of glass, having a terrestrial globe within it; after the manner of the glass-sphere invented by Dr Long, Lowndes's professor of astronomy in Cambridge.

If a globe be held sidewise to the eye, at some distance, and so that neither of its poles can be seen, the equator ECQ, and all circles parallel to it, as DL, yzx, abX, MO, &c. will appear to be straight lines, as projected in this figure; which is requisite to be mentioned here, because we shall have occasion to call them circles in the following articles of this chapter.

Let us now suppose that the observer has gone from the equator towards the north pole n, and that he stops at i, from which place he then sees the hemisphere MEINL; his horizon MCL having shifted as many degrees from the celestial poles N and S, as he has travelled from under the equinoctial E. And as the heavens seem constantly to turn round the line NC'S as an axis, all those stars which are not so many degrees from the north pole N as the observer is from the equinoctial, namely, the stars north of the dotted parallel DL, never set below the horizon; and those which are south of the dotted parallel MO never rise above it. Hence the former of these two parallel circles is called the circle of perpetual apparition, and the latter the circle of perpetual occultation; but all the stars between these two circles rise and set every day. Let us imagine many circles to be drawn between these two, and parallel to them; those which are on the north side of the equinoctial will be unequally cut by the horizon MCL, having larger portions above the horizon than below it; and the more so, as they are nearer to the circle of perpetual apparition; but the reverse happens to those on the south side of the equinoctial, whilst the equinoctial is divided into two equal parts by the horizon. Hence, by the apparent turning of the heavens, the northern stars describe greater arcs or portions of circles above the horizon than below it; and the greater, as they are farther from the equinoctial towards the circle of perpetual apparition; whilst the contrary happens to all stars south of the equinoctial; but those upon it describe equal arcs both above and below the horizon, and therefore they are just as long above as below it.

An observer on the equator has no circle of perpetual apparition or occultation, because all the stars, together with the sun and moon, rise and set to him every day. But, as a bare view of the figure is sufficient to show that these two circles DL and MO are just as far from the poles N and S as the observer at i (or one opposite to him at o) is from the equator ECQ, it is plain, that if an observer begins to travel from the equator towards either pole, his circle of perpetual apparition rises from that pole as from a point, and his circle of perpetual occultation from the other. As the observer advances toward the nearer pole, these two circles enlarge their diameters, and come nearer one another, until he comes to the pole; and then they meet and coincide in the equinoctial. On different sides of the equator, to observers at equal distances from it, the circle of perpetual apparition to one is the circle of perpetual occultation to the other.

Because the stars never vary their distances from the equinoctial, so as to be sensible in an age, the lengths of their diurnal and nocturnal arcs are always the same to the same places on the earth. But as the earth goes round the sun every year in the ecliptic, one half of which is on the north side of the equinoctial, and the other half on its south side, the sun appears to change his place every day, so as to go once round the circle YCX every year. Therefore whilst the sun appears to advance northward, from having described the parallel abX touching the ecliptic in X, the days continually lengthen and the nights shorten, until he comes to y and describes the parallel yzx, when the days are at the longest and the nights at the shortest; for then, as the sun goes no farther northward, the greatest portion that is possible of the diurnal arc yz is above the horizon of the inhabitant i, and the smallest portion zx below it. As the sun declines southward from y, he describes smaller diurnal and greater nocturnal arcs, or portions of circles every day; which cause the days to shorten and nights to lengthen, until he arrives again at the parallel abX; which having only the small part ab above the horizon MCL, and the great part bX below it, the days are at the shortest and the nights at the longest; because the sun recedes no farther south, but returns northward as before. It is easy to see that the sun must be in the equinoctial ECQ twice every year, and then the days and nights are equally long; that is, 12 hours each. These hints serve at present to give an idea of some of the appearances resulting from the motions of the earth; which will be more particularly described in the tenth chapter.

To an observer at either pole, the horizon and equinoctial are coincident; and the sun and stars seem to move parallel to the horizon; therefore, such an observer is said to have a parallel position of the sphere. To an observer anywhere between either pole and equator, the parallels described by the sun and stars are cut obliquely by the horizon, and therefore he is said to have an oblique position of the sphere. To an observer anywhere on the equator, the parallels of motion, described by the sun and stars, are cut perpendicularly, or at right angles, by the horizon; and therefore he is said to have a right position of the sphere. And these three are all the different ways that the sphere can be pointed to all people on the earth.

**Chap. IV. The Phenomena of the Heavens as seen from different parts of the Solar System.**

So vastly great is the distance of the starry heavens, that if viewed from any part of the solar system, or even many millions of miles beyond it, its appearance would be the very same to us. The sun and stars would all seem to be fixed on one concave surface, of which the spectator's eye would be the centre. But the planets being much nearer than the stars, their appearances will vary considerably with the place from which they are viewed.

If the spectator is at rest without their orbits, the planets will seem to be at the same distance as the stars, but continually changing their places with respect to the stars and to one another, assuming various phases of increase and decrease like the moon; and, notwithstanding their regular motions about the sun, will sometimes appear to move quicker, sometimes slower, be as often to the west as to the east of the sun, and at their greatest distances seem quite stationary. The duration, extent, and distance of those points in the heavens where these digressions begin and end, would be more or less, according to the respective distances of the several planets from the sun; but in the same planet they would continue invariably the same at all times; like pendulums of unequal lengths oscillating together, the shorter move quick and go over a small place, the longer move slow and go over a large space. If the observer is at rest within the orbits of the planets, but not near the common centre, their apparent motions will be irregular, but less so than in the former case. Each of the several planets will appear larger and less by turns, as they approach nearer or recede farther from the observer, the nearest varying most in their size. They will also move quicker or slower with regard to their fixed stars, but will never be retrograde or stationary.

If an observer in motion views the heavens, the same apparent irregularities will be observed, but with some variation resulting from its own motion. If he is on a planet which has a rotation on its axis, not being sensible of his own motion, he will imagine the whole heavens, sun, planets, and stars, to revolve about him in the same time that his planet turns round, but the contrary way, and will not be easily convinced of the deception. If his planet moves round the sun, the same irregularities and aspects as above mentioned will appear in the motions of the other planets; and the sun will seem to move among the fixed stars or signs, directly opposite to those in which his planet moves, changing its place every day as he does. In a word, whether our observer be in motion or at rest, whether within or without the orbits of the planets, their motions will seem irregular, intricate, and perplexed, unless he is in the centre of the system; and from thence the most beautiful order and harmony will be seen by him.

The sun being the centre of all the planets' motions, the only place from which their motions could be truly seen is the sun's centre; where the observer, being supposed not to turn round with the sun, (which, in this case, we must imagine to be a transparent body), would see all the stars at rest, and seemingly equidistant from him. To such an observer, the planets would appear to move among the fixed stars, in a simple, regular, and uniform manner; only, that as in equal times they describe equal areas, they would describe spaces somewhat unequal, because they move in elliptic orbits. Their motions would also appear to be what they are in fact, the same way round the heavens, in paths which cross at small angles in different parts of the heavens, and then separate a little from one another: so that if the solar astronomer should make the path or orbit of any one planet a standard, and consider it as having no obliquity, he would judge the paths of all the rest to be inclined to it, each planet having one half of its path on one side, and the other half on the opposite side of the standard path or orbit. And if he should ever see all the planets start from a conjunction with each other, Mercury would move so much faster than Venus, as to overtake her again (though not in the same point of the heavens) in a quantity of time almost equal to 145 of our days and nights, or, as we commonly call them, natural days, which include both the days and nights; Venus would move so much faster than the earth, as to overtake it again in 585 natural days; the earth so much faster than Mars, as to overtake him again in 773 such days; Mars so much faster than Jupiter, as to overtake him again in 817 such days; and Jupiter so much faster than Saturn, as to overtake him again in 7236 days, all of our time.

But as our solar astronomer could have no idea of measuring the courses of the planets by our days, he would probably take the period of Mercury, which is the quickest moving planet, for a measure to compare the periods of the others by. As all the stars would appear quiescent to him, he would never think that they had any dependence upon the sun; but would naturally imagine that the planets have, because they move round the sun. And it is by no means improbable, that he would conclude those planets whose periods are quickest, to move in orbits proportionably less than those do which make slower circuits. But being destitute of a method for finding their parallaxes, or, more properly speaking, as they could have no parallax to him, he could never know anything of their real distances or magnitudes. Their relative distances he might perhaps guess at by their periods, and from thence infer something of truth concerning their relative bulk, by comparing their apparent bulks with one another. For example, Jupiter appearing bigger to him than Mars, he would conclude it to be much bigger in fact; because it appears so, and must be farther from him on account of its longer period. Mercury and the earth would seem much of the same bulk; but, by comparing its period with the earth's, he would conclude that the earth is much farther from him than Mercury, and consequently that it must be really larger, though apparently of the same bulk; and so of the rest. And as each planet would appear somewhat larger in one part of its orbit than in the opposite, and to move quicker when it seems biggest, the observer would be at no loss to determine that all the planets move in orbits, of which the sun is not precisely in the centre.

The apparent magnitudes of the planets continually change as seen from the earth; which demonstrates that they approach nearer to it, and recede farther from it by turns. From these phenomena, and their apparent motions among the stars, they seem to describe looped curves which never return into themselves, Venus's path excepted. excepted. And if we were to trace out all their apparent paths, and put the figures of them together in one diagram, they would appear so anomalous and confused, that no man in his senses could believe them to be representations of their real paths; but would immediately conclude, that such apparent irregularities must be owing to some optic illusions: And after a good deal of inquiry, he might perhaps be at a loss to find out the true cause of these inequalities; especially if he were one of those who would rather, with the greatest justice, charge frail man with ignorance, than the Almighty with being the author of such confusion.

Dr Long, in his first volume of Astronomy, has given us figures of the apparent paths of all the planets separately from Cassini; from them Mr Ferguson first thought of attempting to trace some of them by an orrery, that shews the motions of the sun, Mercury, Venus, the earth, and moon, according to the Copernican system. Having taken off the sun, Mercury, and Venus, he put black lead pencils in their places, with the points turned upward, and fixed a circular sheet of pasteboard so that the earth kept constantly under its centre in going round the sun, and the pasteboard kept its parallelism. Then, pressing gently with one hand upon the pasteboard to make it touch the three pencils, with the other hand he turned the winch that moves the whole machinery: and as the earth together with the pencils in the places of Mercury and Venus had their proper motions round the sun's pencils, which kept at rest in the centre of the machine, all the three pencils described a diagram, from which fig. 2. of Plate XL. is truly copied in a smaller size. As the earth moved round the sun, the sun's pencil described the dotted circle of months, whilst Mercury's pencil drew the curve with the greatest number of loops, and Venus's that with the fewest. In their inferior conjunctions they come as much nearer the earth, or within the circle of the sun's apparent motion round the heavens, as they go beyond it in their superior conjunctions. On each side of the loops they appear stationary; in that part of each loop next the earth retrograde; and in all the rest of their paths direct.

If Cassini's figures of the paths of the sun, Mercury, and Venus, were put together, the figure as above traced out would be exactly like them. It represents the sun's apparent motion round the ecliptic, which is the same every year; Mercury's motion for seven years, and Venus's for eight; in which time Mercury's path makes 23 loops, crizzling itself so many times, and Venus's only five. In eight years, Venus falls so nearly into the same apparent path again, as to deviate very little from it in some ages; but in what number of years Mercury and the rest of the planets would describe the same visible paths over again, it is hard to determine. Having finished the above figure of the paths of Mercury and Venus, he put the ecliptic round them as in the Doctor's book, and added the dotted lines from the earth to the ecliptic 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.

In Plate XL. fig. 2. 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 Taurus at F, as shewn 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 shew, 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 through 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.

**CHAP. V. The physical Causes of the Motions of the Planets. The Eccentricities of their Orbits. The Times in which the Action of Gravity alone would bring them to the Sun.**

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. In Plate XL. fig. 3, if the body A be projected along the right line ABX, 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 ABX, and forced to revolve about S in the circle BYTU. When the body A comes to U, or any other part of its orbit, if the small body u, 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 u 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 y in the time that the projectile force would have carried it from B to X; it will describe the curve BY 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 one another, the planet obeying them both, will move in the circle BYTU.

But if, whilst the projectile force carries the planet from B to b, the sun's attraction (which constitutes the planet's gravitation) should bring it down from B to I, the gravitating power would then be too strong for the projectile force, and would cause the planet to describe the curve BC. 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, BC, CD, DE, EF, &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 Kk, 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 Klmn, &c. it goes off, and ascends in the curve KLMN, &c. its motion decreasing as gradually from K to B, as it increased from B to K, because the sun's attraction acts now 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 removes them quite out of the sphere of his attraction, and causes them to go on in straight lines for ever afterward. But by considering the effects of these powers, 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 I; by these two forces it will describe the curve BC. 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 I 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 to go round him in the circle Klmn, &c. which would require its falling from K to w, through 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 cause already assigned.

The orbits of all the planets are ellipses, very little different from circles; but the orbits of the comets are very long ellipses, and the lower focus of them all is in the sun. If we suppose the mean distance (or middle between the greatest and least) of every planet and comet from the sun to be divided into 1000 equal parts, the eccentricities of their orbits, both in such parts and in English miles, will be as follow. Mercury's 210 parts, or 6,720,000 miles; Venus's, 7 parts, or 413,000 miles; the earth's, 17 parts, or 1,377,000 miles; Mars's, 93 parts, or 11,439,000 miles; Jupiter's, 48 parts, or 20,352,000 miles; Saturn's, 55 parts, or 42,735,000 miles. Of the nearest of the three aforementioned comets, 1,458,000 miles; of the middlemost, 2,025,000,000 miles; and of the outermost, 6,600,000,000.

By the laws of gravity and the projectile force, 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 so contrary to the motions of those bodies, causes them to move slower and slower, until their their projectile forces are diminished almost to nothing; and then they are brought back again by the sun's attraction, as before.

If the projectile forces of all the planets and comets were destroyed at their mean distances from the sun, their gravities would bring them down so, that Mercury would fall to the sun in 15 days 13 hours; Venus, in 39 days 17 hours; the earth or moon, in 64 days 10 hours; Mars, in 121 days; Jupiter, in 290; and Saturn, in 767. The nearest comet, in 13 thousand days; the middlemost, in 23 thousand days; and the outermost, in 66 thousand days. The moon would fall to the earth in 4 days 20 hours: Jupiter's first moon would fall to him in 7 hours; his second, in 15; his third, in 30; and his fourth, in 71 hours: Saturn's first moon would fall to him in 8 hours; his second, in 12; his third, in 19; his fourth, in 68; and the fifth, in 336. A stone would fall to the earth's centre, if there were an hollow passage, in 21 minutes 9 seconds. Mr Whiston gives the following rule for such computations. "It is demonstrable, that half the period of any planet, when it is diminished in the sesquialteral proportion of the number 1 to the number 2, or nearly in the proportion of 1000 to 2828, is the time that it would fall to the centre of its orbit." This proportion is, when a quantity or number contains another once and a half as much more.

The quick motions of the moons of Jupiter and Saturn round their primaries, demonstrate that these two planets have stronger attractive powers than the earth has: for the stronger that one body attracts another, the greater must be the projectile force, and consequently the quicker must be the motion of that other body to keep it from falling to its primary or central planet. Jupiter's second moon is 124 thousand miles farther from Jupiter than our moon is from us; and yet this second moon goes almost eight times round Jupiter whilst our moon goes only once round the earth. What a prodigious attractive power must the sun then have, to draw all the planets and satellites of the system towards him; and what an amazing power must it have required to put all these planets and moons into such rapid motions at first!

**Chap. VI. Reasons why the Sun, Moon, and Stars, when rising or setting, appear larger than when they rise higher in the Heavens.**

The sun and moon appear larger in the horizon than at any considerable height above it. These luminaries, although at great distances from the earth, appear floating, as it were, on the surface of our atmosphere, (Plate XLI, fig. 1.) HGEfeG, 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 e 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 e 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 e, than when they are considerably high, as at f: first, their seeming to be on a part of the atmosphere at e, which is really farther than f from a spectator at E; and, secondly, their being seen through a grosser medium when at e 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 largest when they seem farthest 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 that 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 be really so.

**Chap. VII. Use of the common Quadrant, and the Method of finding the Distances of the Sun, Moon, and Planets.**

To enable the young astronomer to understand the method of finding the distances of the heavenly bodies, we shall here give a short description of the quadrant. This instrument (Plate XLV, fig. 6.) is chiefly used in taking altitudes.

The altitude of any celestial phenomenon is an arc of the sky intercepted between the horizon and the phenomenon. In fig. 6. of Plate XLV, 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 degrees on its limb DPG; 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... 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 \( AV \) (and he may be said to be in the centre of the sun's apparent 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 \( GP \), 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, so that the sun shining through one hole, the ray may be seen to fall on the other.]

In fig. 2. Plate XLI. let \( BAG \) be one half of the earth, \( AG \) 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 are 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 \( AD \) in the meridian: so that when the moon is in the equinoctial, and on the meridian at \( E \), she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at \( O \), 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 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, that it may be known in what time she has gone from \( E \) to \( O \); which time subtracted from 6 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 \( ALG \)) by the following analogy. As the time of the moon's describing the arc \( EO \) is to 90 degrees, so is 6 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-angled 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 the radius, viz. the sine of 90 degrees, or of the right angle \( AGL \), 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.

The sun's distance from the earth is found the same way, but with much greater difficulty; because his horizontal parallax, or the angle \( OAS \) equal to the angle \( ASC \), is so small as to be hardly perceptible, being only 10 seconds of a minute, or the 360th part of a degree. But the moon's horizontal parallax, or angle \( OAL \), equal to the angle \( ALC \), is very discernible, being 57° 49" or 3459" at its mean state; which is more than 340 times as great as the sun's: And therefore the distances of the heavenly bodies being inversely as the tangents of their horizontal parallaxes, the sun's distance from the earth is at least 340 times as great as the moon's; and is rather understated at 81 millions of miles, when the moon's distance is certainly known to be 240 thousand. But because, according to some astronomers, the sun's horizontal parallax is 11 seconds, and according to others only 10, the former parallax making the sun's distance to be about 75,000,000 of miles, and the latter 82,000,000; we may take it for granted, that the sun's distance is not less than as deduced from the former, nor more than as shown by the latter: And every one who is accustomed to make such observations, knows how hard it is, if not impossible, to avoid an error of a second, especially on account of the inconstancy of horizontal refractions: And here, the error of one second, in so small an angle, will make an error of seven millions of miles in so great a distance as that of the sun's.

The sun and moon appear much about the same bulk; and every one who understands geometry, knows how their true bulks may be deduced from the apparent, when their real distances are known. Spheres are to one another another as the cubes of their diameters; whence, if the sun be 81 millions of miles from the earth, to appear as big as the moon, whose distance does not exceed 240 thousand miles, he must, in solid bulk, be 42 millions 875 thousand times as big as the moon.

The horizontal parallaxes are best observed at the equator. 1. Because the heat is so nearly equal every day, that the refractions are almost constantly the same. 2. Because the parallactic angle is greater there, as at A (the distance from thence to the earth's axis being greater) than upon any parallel of latitude, as a or b.

The earth's distance from the sun being determined, the distances of all the other planets from him are easily found by the following analogy, their periods round him being ascertained by observation. As the square of the earth's period round the sun is to the cube of its distance from him, so is the square of the period of any other planet to the cube of its distance, in such parts or measures as the earth's distance was taken. This proportion gives us the relative mean distances of the planets from the sun to the greatest degree of exactness; and they are as follow, having been deduced from their periodical times, according to the law just mentioned, which was discovered by Kepler, and demonstrated by Sir Isaac Newton.

Periodical Revolution to the same fixed Star in Days, and decimal Parts of a Day.

| Of Mercury | Venus | The Earth | Mars | Jupiter | Saturn | |------------|-------|-----------|------|---------|--------| | 87.9692 | 224.6176 | 365.2564 | 686.9785 | 4332.514 | 10759.275 |

Relative mean distances from the sun.

| 38710 | 72333 | 100000 | 152369 | 520096 | 954006 |

From these numbers we deduce, that if the sun's horizontal parallax be 10", the real mean distances of the planets from the sun in English miles are,

| 31,742,200 | 59,313,060 | 82,000,000 | 124,942,580 | 426,478,720 | 782,284,920 |

But if the sun's parallax be 11", their distances are no more than

| 29,032,500 | 54,238,570 | 75,000,000 | 114,276,750 | 390,034,500 | 715,504,500 |

Errors in distance, arising from the mistake of 1" in the sun's parallax.

| 2,709,700 | 5,074,490 | 7,000,000 | 10,665,830 | 36,444,220 | 66,780,420 |

But, from the transit of Venus, A.D. 1761, the sun's parallax appears to be only 8" 4' 5"; and according to that, their real distance in miles are

| 36,668,373 | 68,518,044 | 94,725,840 | 144,588,575 | 492,665,307 | 903,690,197 |

These numbers shew, that although we have the relative distances of the planets from the sun to the greatest nicety, yet the best observers could not ascertain their true distances until the above transit appeared, which we must confess was embarrassed with several difficulties. But the late transit of Venus over the sun, on the third of June, was much better suited to this great problem.

The earth's axis produced to the stars, being carried parallel to itself during the earth's annual revolution, describes a circle in the sphere of the fixed stars equal to the orbit of the earth. But this orbit, though very large, would seem no bigger than a point if it were viewed from the stars; and consequently, the circle described in the sphere of the stars, by the axis of the earth produced, if viewed from the earth, must appear but as a point; that is, its diameter appears too little to be measured by observation: For Dr Bradley has assured us, that if it had amounted to a single second, or two at most, he should have perceived it in the great number of observations he has made, especially upon γ draconis; and that it seemed to him very probable that the annual parallax of this star is not so great as a single second; and consequently, that it is above 400 thousand times farther from us than the sun. Hence, the celestial poles seem to continue in the same points of the heavens throughout the year; which, by no means, disproves the earth's annual motion, but plainly proves the distance of the stars to be exceeding great.

The small apparent motion of the stars, discovered by that great astronomer, he, found to be no ways owing to their annual parallax (for it came out contrary thereto) but to the aberration of their light, which can result from no known cause besides that of the earth's annual motion; and as it agrees so exactly therewith, it proves, beyond dispute, that the earth has such a motion: For this aberration completes all its various phenomena every year; and proves that the velocity of star-light is such as carries it through a space equal to the sun's distance from us in 8 minutes 13 seconds of time. Hence, the velocity of light is 10 thousand 210 times as great as the earth's velocity in its orbit; which velocity (from what we know already of the earth's distance from the sun) may be asserted to be at least between 57 and 58 thousand miles every hour: And supposing it to be 58000, this number, multiplied by the above 10210, gives 592 million 180 thousand miles for the hourly motion of light; which last number, divided by 3600, the number of seconds in an hour, shews that light flies at the rate of more than a hundred and sixty-four thousand miles every second of time, or swing of a common clock pendulum. CHAP. VIII. The different Lengths of Days and Nights, and the Vicissitudes of Seasons, explained. The Explanation of the Phenomena of Saturn's Ring concluded.

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 abcd, which being viewed obliquely, appears elliptical, Plate XLII. fig. 3. 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 unwilling 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 abcd, 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 unwilling 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 shews, 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. But now, define the person who holds the wire, to hold it obliquely in the position ABCD, raising the side F just as much as he depresses the side 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 r, 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 shewn 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; shewing 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 part 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, shewing 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 thro' 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 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 no day or night for several 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. 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. And if this book be held so that the circle \(ABCD\) be parallel to the horizon, both the circle abcd, and the thread or axis \(K\), will be inclined to it. But if the book or plate be held so that the thread be perpendicular to the horizon, then the orbit \(ABCD\) will be inclined to the thread, and the orbit abcd perpendicular to it, and parallel to the horizon. We generally consider the earth's annual orbit as having no inclination, and the orbits of all the other planets as inclined to it.

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 \(23^\circ 47'\) degrees 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.

In Plate XLI. fig. 4., let \(abcdefg\) be the earth in eight different parts of its orbit, equidistant from one another, \(N\) its axis, \(N\) the north pole, \(S\) the 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 \(abcd\), &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 toward 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. 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 \(ABCDEFGH\) (Plate XLII., fig. 1.). Let us suppose this circle to be divided into 12 equal parts, called signs, having their names affixed to them; and each sign into 30 equal parts, called degrees, numbered 10, 20, 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 (as at \(g\) in Plate XLI. fig. 4.), 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 \(AE\) \(TL\) 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 show 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 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 \(c\) 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 \(a\) in Plate XLI. fig. 4.; 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 show, that as the earth advances from Capricorn towards Aries, and the sun appears to move from Cancer towards Libra, the north pole recedes towards the dark, which causes the days to decrease, and the nights to increase in length, till... till the earth comes to the beginning of Aries, and then they are equal as before; for the boundary of light and darkness cut 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 half way 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 $23\frac{1}{2}$ 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. It is needless to enlarge farther on this subject, as we shall have occasion to mention the seasons again in describing the orrery. Only this must be noted, that 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.

As Saturn goes round the sun, his obliquely posted 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 round edgewise to the sun, at which time it is also edgewise to the earth, and therefore it disappears once in every fifteen 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.

The earth's orbit being elliptical, and the sun constantly keeping in its lower focus, which is 1,377,000 miles from the middle point of the longer axis, the earth comes twice so much, or 2,754,000 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. 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 million 377 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 so 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 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.

**Chap. IX. The Method of finding the Longitude by the Eclipses of Jupiter's Satellites: The amazing Velocity of Light demonstrated by these Eclipses.**

Geographers arbitrarily choose to call the meridian of some remarkable place the first meridian. There they begin their reckoning; and just so many degrees and minutes as any other place is to the eastward or westward of that meridian, so much east or west longitude they say it has. A degree is the 360th part of a circle, be it great or small; and a minute the 60th part of a degree. The English geographers reckon the longitude from the meridian of the Royal Observatory at Greenwich, and the French from the meridian of Paris.

If we imagine 12 great circles, (Plate XLII. fig. 1.) one of which is the meridian of any given place, to intersect each other in the two poles of the earth, and to cut the equator $E$ at every 15th degree, they will be divided by the poles into 24 semicircles which divide the equator into 24 equal parts; and as the earth turns on its axis, the planes of these semicircles come successively after one another every hour to the sun. As in an hour of time there is a revolution of 15 degrees of the equator, in a minute of time there will be a revolution of 15 minutes of the equator, and in a second of time a revolution of 15 seconds.

Because the sun enlightens only one half of the earth at once, as it turns round its axis, he rises to some places at the same moments of absolute time that he sets to others; and when it is mid-day to some places, it is midnight to others. The XII on the middle of the earth's enlightened side, next the sun, stands for mid-day; and the opposite XII on the middle of the dark side, for midnight. If we suppose this circle of hours to be fixed in the plane of the equinoctial, and the earth to turn round within it, any particular meridian will come to the different hours so as to show the true time of the day or night at all places on that meridian. Therefore,

To every place 15 degrees eastward from any given meridian, it is noon an hour sooner than on that meridian, because their meridian comes to the sun an hour sooner; and to all places 15 degrees westward, it is noon an hour later, because their meridian comes an hour later to the sun, and so on; every 15 degrees of motion causing an hour's difference in time. Therefore, they who have noon an hour later than we, have their meridian, that is, their longitude, 15 degrees westward from us; and they who have noon an hour sooner than we, have their meridian 15 degrees eastward from ours; and so for every hour's difference of time 15 degrees differ- A S T R O N O M Y.

rence of longitude. Consequently, if the beginning or ending of a lunar eclipse be observed, suppose at London, to be exactly at midnight, and in some other place at 11 at night, that place is 15 degrees westward from the meridian of London; if the same eclipse be observed at 1 in the morning at another place, that place is 15 degrees eastward from the said meridian.

But as it is not easy to determine the exact moment either of the beginning or ending of a lunar eclipse, because the earth's shadow, through which the moon passes, is faint and ill-defined about the edges, we have recourse to the eclipses of Jupiter's satellites, which disappear so instantaneously as they enter Jupiter's shadow, and emerge so suddenly out of it, that we may fix the phenomenon to half a second of time. The first or nearest satellite to Jupiter is the most advantageous for this purpose, because its motion is quicker than the motion of any of the rest, and therefore its immersions and emersions are more frequent.

The English astronomers have calculated tables for shewing the times of the eclipses of Jupiter's satellites to great precision, for the meridian of Greenwich. Now, 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, as above; 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 telescopical observations.

To explain this by a figure, in Plate XLII, 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 its 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 can then 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, only the immersions of Jupiter's satellites into his shadow are generally seen; and their emersions out of it while the earth goes from G to B. Indeed, both 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 emersions 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 in true proportion to his diameter; but, in proportion to the earth's orbit, they are drawn 81 times 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 forwarder as Jupiter has moved in that time, to overtake him again; just like the minute-hand of a watch, which must, from any conjunction with the hour-hand, go 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 16½ minutes of time to go through a space equal to the diameter of the earth's orbit, which is 162 millions of miles in length; and consequently the particles of light fly about 164 thousand 494 miles every second of time, which is above a million of times swifter than the motion of a cannon-bullet. And as light is 16½ minutes in travelling across the earth's orbit, it must be 8½ minutes in coming from the sun to us; therefore if the sun were annihilated, we should see him for 8½ minutes after; and if he were again created, he would be 8½ minutes old before we could see him.

To illustrate this progressive motion of light, (Plate XLII, fig. 2.), let A and B be the earth in two different parts of its orbit, whose distance from each other is 81 millions 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 8½ minutes sooner when the earth is at B, than when it is at A. And so, as

Vol. I. Numb. 19. 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.

**Chap. X. Of Solar and Sydereal Time.**

The fixed 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 miles, 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 sydereal.

The diameter of the earth's orbit is but a physical point in proportion to the distance of the stars; for which reason, and the earth's uniform motion on its axis, any given meridian will revolve from any star to the same star again in every absolute turn of the earth on its axis, without the least perceptible difference of time shewn by a clock which goes exactly true.

If the earth had only a diurnal motion, without an annual, any given meridian would revolve from the sun to the star 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 sydereal days would be just four minutes shorter than the solar.

In Plate XLII. fig. 3, 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 the same way from the sun to the sun again 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 \( NBn \) 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 2 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 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 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 planets 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 they set out. So, if there were two earths revolving equably on their axes, and if one remained at A until the other travelled 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; 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.

**Chap. XI. Of the Equation of Time.**

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, shewn by a well-regulated clock; and at other times in somewhat more: So that the time shewn by an equal going clock and a true sun-dial is never the same but on the 1st of April, the 16th of June, the 31st of August, and the 24th of December. The clock, if it 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.

The easiest and most expeditious way of drawing a meridian line is this: 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 ciment 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 ciment 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; and so regulate your clock for the day of observation by that line and any good equation table.

As the equation of time, or difference between the time shewn by a well-regulated clock and a true sun-dial, 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 either from the beginning 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 run 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... enters Libra; and the 21st of December, when he enters Capricorn. But, as there is only one sun, and his apparent motion is always on the ecliptic, let us henceforth call him the real sun; and the other, which is supposed to move in the equator, the fictitious; to which last, the motion of a well-regulated clock always answers.

In Plate XLII, fig. 4, let \( Z \) be the earth, \( ZF \) its axis, \( abcd \) etc., the equator, \( ABCDE \) etc., the northern half of the ecliptic from \( \gamma \) or \( \omega \) on the side of the globe next the eye; and \( MNO \) etc., the southern half on the opposite side from \( \omega \) to \( \gamma \). Let the points at \( ABCDEF \) etc., quite round from \( \gamma \) to \( \gamma \) again bound equal portions of the ecliptic, gone through in equal times by the real sun; and those at \( abcdef \) etc., equal portions of the equator, described in equal times by the fictitious sun; and let \( Z \gamma z \) be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun directly in the equator, with respect to the meridian; a degree, or any number of degrees, between \( \gamma \) and \( F \) on the ecliptic, must be nearer the meridian \( Z \gamma z \) than a degree, or any corresponding number of degrees on the equator from \( \gamma \) 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 \( \gamma F \), than the fictitious sun does in the quadrant \( \gamma F \); 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 \( F \) to \( \omega \), he comes later to the meridian every day, than the fictitious sun moving through the second quadrant of the equator from \( F \) to \( \omega \); for the points at \( GHIK \) and \( L \), being farther from the meridian than their corresponding points at \( ghik \) and \( l \), they must be later of coming to it: And as both suns come at the same moment to the point \( \omega \), they come to the meridian at the moment of noon by the clock.

In departing from Libra, through the third quadrants, the real sun going through \( MNO \) towards \( r \) at \( R \), and the fictitious sun through \( mnoq \) towards \( r \), the former comes to the meridian every day sooner than the latter, until the real sun comes to \( r \), and the fictitious to \( r \), and then they both come to the meridian at the same time.

Lastly, as the real sun moves equably through \( STUVW \), from \( r \) towards \( \gamma \); and the fictitious sun thro' \( stuvw \), from \( r \) towards \( \gamma \); the former comes later every day to the meridian than the latter, until they both arrive at the point \( \gamma \), and then they make noon at the same time with the clock.

This part of the equation of time may perhaps be somewhat difficult to understand by a figure, because both halves of the ecliptic seem to be on the same side of the globe; but it may be made very easy to any person who has a real globe before him, by putting final patches on every tenth or fifteenth degree, both of the equator and ecliptic, beginning at Aries \( \gamma \); and then, turning the ball slowly round westward, he will see all the patches from Aries to Cancer come to the brazen meridian sooner than the corresponding patches on the equator; all those from Cancer to Libra will come latter to the meridian than their corresponding patches on the equator; those from Libra to Capricorn sooner, and those from Capricorn to Aries latter: And the patches at the beginnings of Aries, Cancer, Libra, and Capricorn, being either on, or even with those on the equator, shew that the two suns either meet there, or are even with one another, and so come to the meridian at the same moment.

Let us suppose that there are two little balls moving equably round a celestial globe by clock-work, one always keeping in the ecliptic, and gilt with gold, to represent the real sun; and the other keeping in the equator, and silvered, to represent the fictitious sun: And that whilst these balls move once round the globe, according to the order of signs, the clock turns the globe 366 times round its axis westward. The stars will make 366 diurnal revolutions from the brazen meridian to it again; and the two balls representing the real and fictitious sun always going farther eastward from any given star, will come later than it to the meridian every following day; and each ball will make 365 revolutions to the meridian; coming equally to it at the beginnings of Aries, Cancer, Libra, and Capricorn: But in every other point of the ecliptic, the gilt ball will come either sooner or latter to the meridian than the silver ball, like the patches above mentioned.

This would be a pretty enough way of shewing the reason why any given star, which, on a certain day of the year, comes to the meridian with the sun, passes over it so much sooner every following day, as on that day twelvemonth to come to the meridian with the sun again; and also to show the reason why the real sun comes to the meridian sometimes sooner, sometimes later, than it is noon by the clock; and, on four days of the year, at the same time; whilst the fictitious sun always comes to the meridian when it is twelve at noon by the clock. This would be no difficult task for an artist to perform; for the gold ball might be carried round the ecliptic by a wire from its north pole, and the silver ball round the equator by a wire from its south pole, by means of a few wheels to each.

It is plain, that if the ecliptic were more obliquely fitted to the equator, as the dotted circle \( \gamma X \omega \), the equal divisions from \( \gamma \) to \( X \) would come till sooner to the meridian \( Z \gamma \) than those marked \( ABCD \) and \( E \) do; for two divisions containing 30 degrees, from \( \gamma \) to the second dot, a little short of the figure 1, come sooner to the meridian than one division containing only 15 degrees from \( \gamma \) to \( A \) does, as the ecliptic now stands; and those of the second quadrant from \( X \) to \( \omega \) would be so much later. The third quadrant would be as the first, and the fourth as the second. And it is likewise plain, that where the ecliptic is most oblique, namely, about Aries and Libra, the difference would be greatest; and least about Cancer and Capricorn, where the obliquity is least.

Having explained one cause of the difference of time shewn by a well-regulated clock and a true sun-dial; 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 the summer, when the sun is farthest from the earth, and quickest 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 a 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 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 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 latter 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. In Plate XLIII. fig. 1, let \(ABCD\) be the ecliptic or orbit in which the real sun moves, and the dotted circle \(abcd\) 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 \(f\), through a larger arc. Therefore, the meridian \(EH\) will revolve sooner from \(H\) to \(b\) under the real sun at \(F\), than from \(H\) to \(k\) under the fictitious sun at \(f\); and consequently it will then be noon, by the sun-dial 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 increasing 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 through 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 come 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 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 these points to the other, is called the line of the apsides.

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 24 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, than 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.

The obliquity of the ecliptic to the equator, which is the first mentioned cause of the equation of time, would make the sun and clocks agree on four days of the year; which are, when the sun enters Aries, Cancer, Libra, and Capricorn: But the other cause, now explained, would make the sun and clocks equal only twice a year; that is, when the sun is in his apogee and perigee. Consequently, when these two points fall in the beginnings of Cancer and Capricorn, or of Aries and Libra, they concur in making the sun and clocks equal in these points. But the apogee at present is in the 9th degree of Cancer, and the perigee in the 9th degree of Capricorn, and therefore the sun and clocks cannot be equal about the beginning of these signs, nor at any time of the year, except when the swiftness or slowness of equation resulting from one cause just balances the slowness or swiftness arising from the other.

**CHAP. XII. Of the Precession of the Equinoxes.**

It is a known fact, that there is a greater quantity of matter accumulated all round the equatorial parts of the earth than anywhere else.

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. $17\frac{1}{2}$ 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 $14\frac{1}{2}$ seconds. The sidereal year is therefore 20 minutes $17\frac{1}{2}$ seconds longer than the solar or tropical year; and 9 minutes $14\frac{1}{2}$ seconds longer than the Julian or civil year, which we state at 365 days 6 hours: So that the civil year is almost a mean betwixt the sidereal and tropical.

As the sun describes the whole ecliptic, or 360 degrees, in a tropical year, he moves 59 minutes 8 seconds of a degree very day at a mean rate; and consequently 50 seconds of a degree in 20 minutes $17\frac{1}{2}$ seconds of time: Therefore, he will arrive at the same equinox or solstice when he is 50 seconds of a degree short of the same star or fixed point in the heavens from which he set out in 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 degrees 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, (Plate XLIII. fig. 1.) let the sun be in conjunction with a fixed star at S, suppose in the 30th degree of $\alpha$ on the 21st of May 1756. Then, making 2160 revolutions through the ecliptic $VWX$, at the 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 degrees of Taurus at $T$, which has receded back from $S$ to $T$ in that time, by the precession of the equinoctial points $\gamma$ Aries and $\alpha$ Libra. The arc $ST$ will be equal to the amount of the precession of the equinox in 2160 years, at the rate of 50 seconds of a degree, or 20 minutes $17\frac{1}{2}$ seconds of time, annually: This, in so many years, makes 30 days $10\frac{1}{2}$ 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 minutes 3 seconds, or 16 days 13 hours 48 minutes, which is the difference between 2160 Julian and tropical years. A Table shewing the Precession of the Equinoctial Points in the Heavens, both in Motion and Time; and the Anticipation of the Equinoxes on Earth.

| Julian years | Procession of the Equinoctial Points in the Heavens | Anticipation of the Equinoxes on the Earth | |--------------|-----------------------------------------------------|-------------------------------------------| | | Motion. | D. H. M. S. | | | s o ' " | Days. H. M. S. | | 1 | 0 0 50 | 0 0 20 17½ | | 2 | 0 0 140 | 0 0 40 35 | | 3 | 0 0 230 | 0 1 0 52½ | | 4 | 0 0 320 | 0 1 21 10 | | 5 | 0 0 410 | 0 1 41 27½ | | 6 | 0 0 50 | 0 2 1 45 | | 7 | 0 0 550 | 0 2 22 2½ | | 8 | 0 0 640 | 0 2 42 20 | | 9 | 0 0 730 | 0 3 2 37½ | | 10 | 0 0 820 | 0 3 22 55 | | 20 | 0 0 1640 | 0 6 45 50 | | 30 | 0 0 250 | 0 10 8 45 | | 40 | 0 0 3320 | 0 13 31 40 | | 50 | 0 0 4140 | 0 16 54 35 | | 60 | 0 0 500 | 0 20 17 30 | | 70 | 0 0 5820 | 0 23 40 25 | | 80 | 0 1 640 | 0 1 3 3 20 | | 90 | 0 1 150 | 0 1 6 26 15 | | 100 | 0 1 2320 | 0 1 9 49 10 | | 200 | 0 2 4640 | 0 2 19 38 20 | | 300 | 0 4 100 | 0 4 5 27 30 | | 400 | 0 5 3320 | 0 5 15 16 40 | | 500 | 0 6 5640 | 0 7 1 5 50 | | 600 | 0 8 200 | 0 8 10 55 0 | | 700 | 0 9 4320 | 0 9 20 44 10 | | 800 | 0 11 640 | 0 11 6 33 20 | | 900 | 0 12 300 | 0 12 16 22 30 | | 1000 | 0 13 5320 | 0 14 2 11 40 | | 2000 | 0 27 4640 | 0 28 4 23 20 | | 3000 | 1 11 400 | 0 42 6 35 0 | | 4000 | 1 25 3320 | 0 56 8 46 40 | | 5000 | 2 9 2640 | 0 70 10 58 20 | | 6000 | 2 2320 | 0 84 13 10 0 | | 7000 | 3 7 1320 | 0 98 15 21 40 | | 8000 | 3 21 640 | 0 112 17 33 20 | | 9000 | 4 500 | 0 126 19 45 0 | | 10000 | 4 18 5320 | 0 140 21 56 40 | | 20000 | 9 7 4640 | 0 281 19 53 20 | | 25020 | 12 00 | 0 365 6 0 |

From... 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 Hefiod, Eudoxius, Virgil, Pliny, &c., by no means answer at this time to their descriptions. The preceding table shews the quantity of this shifting both in the heavens and on the earth, for any number of years to 25,920, which compleats the grand celestial period; within which any number and its quantity is easily found, as in the following example, for 5763 years; which, at the autumnal equinox, A.D. 1756, is thought to be the age of the world. So that with regard to the fixed stars, the equinoctial points in the heavens have receded 2° 26' 2" 30" since the creation; which is as much as the sun moves in 81° 5' 0" 52". And since that time, or in 5763 years, the equinoxes with us have fallen back 44° 5' 21" 9"; hence, reckoning from the time of the Julian equinox, A.D. 1756, viz. Sept. 11th, it appears, that the autumnal equinox at the creation was on the 25th of October.

| Precelion of the Equinoctial Points in the Heavens | Anticipation of the Equinoxes on the Earth | |---|---| | Julian years | Motion | Time | D. H. M. S. | D. H. M. S. | | 5000 | 2° 9' 26" 40" | 7° 10' 58" 20" | 3° 8' 50" 0" | | 700 | 9' 43" 20" | 9' 20' 44" 10" | 5' 8' 55" 0" | | 60 | 50' 0" | 20' 17' 30" | 11' 3" 0" | | 3 | 2' 30" | 1' 52" | 0' 33" 9" | | 5763 | 2° 20' 2" 30" | 8° 1' 5" 52" | 44' 5' 21" 9" |

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 feasts 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 bisectional 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 bisectional 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 already 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 describe the phenomena by a diagram.

In Plate XLIII. fig. 2, 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, TQZ the tropic of Cancer, and VTrs 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 NOm and SOs, 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 ABGDA in the starry heavens, round the pole of the ecliptic, which keeps immovable in the centre of that circle. The earth's axis being 23¼ degrees inclined to the axis of the ecliptic, the circle ADGDA, 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 deserted 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 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 EQQ, will then be changed into eOq; the tropic of Cancer TΣZ, into VtΣ; and the tropic of Capricorn VTr, into trZ; 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 present time, to bring the north pole N quite round, so as to be directed towards 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.

**CHAP. XIII. The moon's surface mountainous: Her phases described: Her path and the paths of Jupiter's moons delineated: The proportions of the diameters of their orbits, and those of Saturn's moons, to each other, and to the diameter of the Sun.**

By looking at the moon with an ordinary telescope, we perceive that her surface is diversified with long tracts of prodigious high mountains and deep cavities. Some of her mountains, by comparing their height with her diameter (which is 2180 miles) are found to be three times higher than the highest hills on our earth. This ruggedness of the moon's surface 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 show 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 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 side next the eye, especially if the sun or a candle shines obliquely on that side, yet the line terminating the visible part still appears smooth and even.

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 the earth being surrounded by an atmosphere, and the moon having none, we have twilight after the sun sets; but the lunar inhabitants have an immediate transition from the brightest sun-shine to the blackest darkness. For, (Plate XLIII. fig. 3.) let throw be the earth, and ABCDEFGH the moon in eight different parts of 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 S 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 shows us one half of her enlightened side, as at c, and we say, she is a quarter old. At D 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 or 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 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, wherever 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 sun 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 inclining 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 nonagefimal 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.

The inclination of that part of the ecliptic to the horizon in which the moon is at any time when horned, may be known by the position of her horns; for a right line touching their points is perpendicular to the ecliptic. And as the angle that the moon's orbit makes with the ecliptic can never raise her above, nor depress her below the ecliptic, more than two minutes of a degree, as seen from the sun, it can have no sensible effect upon the position of her horns. Therefore, if a quadrant be held up, so as one of its edges may seem to touch the moon's horns, the graduated side being kept towards the eye, and as far from the eye as it can be conveniently held, the arc between the plumb-line and that edge of the quadrant which seems to touch the moon's horns, will show the inclination of that part of the ecliptic to the horizon. And the arc between the other edge of the quadrant and plumb-line will show the inclination of the moon's horns to the horizon.

The moon generally appears as large as the sun; for the angle \( \theta kA \) (Plate XLIII. fig. 3.) under which the moon is seen from the earth, is the same with the angle \( LkM \), under which the sun is seen from it. And therefore the moon may hide the sun's whole disk from us, as sometimes does in solar eclipses. The reason why she does not eclipse the sun at every change shall be explained afterwards. If the moon were farther from the earth, as at \( a \), she could never hide the whole of the sun from us; for then she would appear under the angle \( NkO \), eclipsing only that part of the sun which lies between \( N \) and \( O \): were she still further from the earth, as at \( X \), she would appear under the small angle \( TKW \), like a spot on the sun, hiding only the part \( TW \) from our sight.

The moon turns round her axis in the time that she goes round her orbit; which is evident from hence, that 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 the 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 compleat 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 compleating a solar day, as the earth has gone forward in its orbit during that time, i.e., almost a twelfth part of a circle.

The moon's periodical and synodical revolution may be familiarly represented by the motions of the hour and minute-hands of a watch round its dial-plate, which is divided into 12 equal parts or hours, as the ecliptic is divided into 12 signs, and the year into 12 months. Let us suppose these 12 hours to be 12 signs, the hour-hand the sun, and the minute-hand the moon; then will the former go round once in a year, and the latter once in a month; but the moon, or minute-hand, must go more than round from any point of the circle where it was last conjoined with the sun, or hour-hand, to overtake it again: For the hour-hand being in motion, can never be overtaken by the minute-hand at that point from which they started at their last conjunction.

If the earth had no annual motion, the moon's motion round the earth, and her track in absolute space, would be always the same. 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 anything over.

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... 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 nail 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 of the circle which the pin in the nave describes round the axle, as \(337\frac{1}{2}\) to 1; 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 \(abcde\). 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 tracks 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 pin 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 \(ABCD\), 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 \(abcde\) show the moon's orbit in due proportion to the earth's; and the smallest curve \(abcdef\) represents the line of the moon's path in the heavens for 32 days, accounted from any particular new moon at \(a\). The machine, Plate XLIX, fig. 2, is for delineating the moon's path, and will be described, with the rest of the astronomical machinery, in the last chapter. 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 bears 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 \(337\frac{1}{2}\).

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 \(ab\); 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 \(bc\); 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 \(cd\); and is in her third quarter at \(d\) when the earth is at \(D\). Once more, whilst the earth describes the curve \(D\) 23 24 25 26 27 28 29, the moon describes the curve \(de\), 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 \(abcde\), 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 shew 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 thousand 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 \(e\), 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 to, 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 \(Gg\), let down from the earth's place upon the straight line \(bd\) 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 thousand miles; and the moon, when new, only approaching nearer to the sun by 240 thousand 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 shews the concavity of that part of her path, will be about 400 thousand 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 almost 230 thousand times as great as 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 about 70 thousand miles nearer the earth than the moon is at her change. It may now 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. 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 all; like bodies in the cabin of a ship, which 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 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. In Plate XLIII. fig. 3, let ABCDE, &c. be as much of Jupiter's orbit as he describes in 18 days from A to T; and the curves abcd 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 a, his second at b, his third at c, and his fourth at d. At the end 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 shew 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, shew 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 b, 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 c moving still slower, and having described the curve 1 2 3 4 5 6 7, comes to an angle at 7 in conjunction with the sun at the end of 7 days 4 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 d 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. In order to have a tolerably good figure of the paths of these satellites, take the following method.

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.

In Plate XLIV. fig. 1, we have the proportions of the orbits of Saturn's five satellites, and of Jupiter's four, to one another, to our moon's orbit, and to the disk of the sun. S is the sun; M m the moon's orbit, (the earth supposed to be at E); J Jupiter; 1 2 3 4 the orbits of his four moons or satellites; Sat Saturn; and 1 2 3 4 5 the orbits of his five moons. Hence it appears, that the sun would much more than fill the whole orbit of the moon; for the sun's diameter is 763,000 miles, and the diameter of the moon's orbit only 480,000. In proportion to all these orbits of the satellites, the radius of Saturn's annual orbit would be 21½ yards, of Jupiter's orbit 11½, and of the earth's 2½, taking them in round numbers.

**CHAP. XIV. The Phenomena of the Harvest-moon explained by a common Globe: The years in which the Harvest-moons are least and most beneficial from 1751, to 1861. The long Duration of Moon-light at the Poles in Winter.**

It is generally believed that the moon rises about 48 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; with which farmers were better acquainted than astronomers till of late; and gratefully ascribed the early rising of the full moon at that time of the year to the goodness of God, in ordering it so on purpose to give them an immediate supply of moon-light after sun-set for their greater convenience in reaping the fruits of the earth. And indeed, in this instance of the harvest-moon, as in many others discoverable by astronomy, the wisdom and beneficence of the Deity is conspicuous, who really ordered the course of the moon so, as to bestow more or less light on all parts of the earth as their several circumstances and seasons render it more or less serviceable. About the equator, where there is no variety of seasons, and the weather changes seldom, and at stated times, moonlight is not necessary for gathering in the produce of the ground; ground; and there the moon rises about 48 minutes later every day or night than on the former. At considerable distances from the equator, where the weather and seasons are more uncertain, 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.

It is soon said that all these phenomena are owing to the different angles made by the horizon and different parts of the moon's orbit; and that the moon can be full but once or twice in a year in those parts of her orbit which rise with the least angles. But to explain this subject intelligibly, we must dwell much longer upon it.

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 from the sun every day, as it is in her orbit, she would rise and set 48 minutes later every day than on the preceding: for 12 degrees of the equinoctial rise or set in 48 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 the greatest difference in those signs which rise or set with the greatest angles.

But, because all who read this treatise may not be provided with globes, though in this case it is requisite to know how to use them, we shall substitute the figure of a globe; (Plate XLIV. fig. 2.) in which FUP is the axis, STR the tropic of Cancer, LTR the tropic of Capricorn, EUR the ecliptic touching both the tropics, which are 47 degrees from each other, and AB 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 V 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 NL above the horizon, making the angle NUL 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 rs Capricorn comes to the meridian and V 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 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, the differs but two hours in rising for six 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.

All these things will be made plain by putting small patches on the ecliptic of a globe, as far from one another as the moon moves from any point of the celestial ecliptic in 24 hours, which at a mean rate is 13½ degrees; and then in turning the globe round, observe the rising and setting of the patches in the horizon, as the index points out the different times in the hour-circle. A few of these patches are represented by dots at o 1 2 3, &c. on the ecliptic, which has the position LUI when Aries rises in the east; and by the dots o 1 2 3, &c., when Libra rises in the east; at which time the ecliptic has the position EUR; making an angle of 62 degrees with the horizon in the latter case, and an angle of no more than 15 degrees with it in the former; supposing the globe rectified to the latitude of London.

Having rectified the globe, turn it until the patch at o, about the beginning of X Pisces in the half LUI of the ecliptic, comes to the eastern side of the horizon; and then keeping the ball steady, set the hour-index to XII, because that hour may perhaps be more easily remembered than any other. Then turn the globe round westward, and in that time, suppose the patch o to have moved thence to r, 13½ degrees, whilst the earth turns once round its axis, and you will see that r rises only about 20 minutes later than o did on the day before. Turn the globe round again, and in that time suppose the same patch to have moved from r to 2; and it will rise only 20 minutes later by the hour-index than it did at o on the day or turn before. At the end of the next turn, suppose the patch to have gone from 2 to 3 at U, and it will rise 20 minutes later than it did at 2. And so on for six turns, in which time there will scarce be two hours difference: nor would there have been so much if the 6 degrees of the sun's motion in that time had been allowed for. At the first turn the patch rises south of the east, at the middle turn due east, and at the last turn north of the east. But these patches will be 9 hours of setting on the western side of the horizon, which shows that the moon will be so much later of setting in that week in which she moves through these two signs. The cause of this difference is evident; for Pisces and Aries make only an angle of 15 degrees with the horizon when they rise; but they make an angle of 62 degrees with it when they set. As the signs Taurus, Gemini, Cancer, Leo, Virgo, and Libra, rise successively, the angle increases gradually which they make with the horizon; and decreases in the same proportion as they set. And for that reason, the moon differs gradually more in the time of her rising every day whilst she is in these signs; and less in her setting; after which, through the other six signs, viz. Scorpio, Sagittary, Capricorn, Aquarius, Pisces, and Aries, the rising difference becomes less every day, until it be at the least of all, namely, in Pisces and Aries.

The moon goes round the ecliptic in 27 days 8 hours; but not from change to change in less than 29 days 12 hours: so that she is in Pisces and Aries at least once in every lunation, and in some lunations twice.

If the earth had no annual motion, the sun would never appear to shift his place in the ecliptic. And then every new moon would fall in the same sign and degree of the ecliptic, and every full moon in the opposite; for the moon would go precisely round the ecliptic from change to change. So that if the moon was once full in Pisces or Aries, she would always be full when she came round to the same sign and degree again. And as the full moon rises at sun-set (because when any point of the ecliptic sets, the opposite point rises) she would constantly rise within two hours of sun-set, on the parallel of London, during the week in which she were full. But in the time that the moon goes round the ecliptic from any conjunction or opposition, the earth goes almost a sign forward; and therefore the sun will seem to go as far forward in that time, namely, 27\(\frac{1}{3}\) degrees; so that the moon must go 27\(\frac{1}{3}\) degrees more than round, and as much farther as the sun advances in that interval, which is 27\(\frac{1}{3}\) degrees, before she can be in conjunction with, or opposite to, the sun again. Hence it is evident, that there can be but one conjunction or opposition of the sun and moon in a year in any particular part of the ecliptic. This may be familiarly exemplified by the hour and minute-hands of a watch, which are never in conjunction or opposition in that part of the dial-plate where they were so last before. And indeed, if we compare the twelve hours on the dial-plate to the twelve signs of the ecliptic, the hour-hand to the sun, and the minute-hand to the moon, we shall have a tolerably near resemblance in miniature to the motions of our great celestial luminaries. The only difference is, that whilst the sun goes once round the ecliptic, the moon makes 12\(\frac{1}{3}\) conjunctions with him: but whilst the hour-hand goes round the dial-plate, the minute-hand makes only 11 conjunctions with it; because the minute-hand moves slower in respect of the hour-hand than the moon does with regard to the sun.

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 so near the time of sun-set, for a week together, as above mentioned. 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, since 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 at all the fore-mentioned patches rises and sets nearly at equal angles with the horizon all the year round, and about 48 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 angle gradually diminishes 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 proportional 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. places between the polar circles. But, during the above fourteen days, the moon is 24 sidereal hours later in setting; for the fix signs, which rise all at once on the eastern side of the horizon, are 24 hours in setting on the western side of it; as any one may see by making chalk-marks at the beginning of Capricorn and of Cancer, and then, having elevated the pole 66½ degrees, turn the globe slowly round its axis, and observe the rising and setting of the ecliptic. As the beginning of Aries is equally distant from the beginning of Cancer and of Capricorn, it is in the middle of that half of the ecliptic which rises all at once. And when the sun is at the beginning of Libra, he is in the middle of the other half. Therefore, when the sun is in Libra, and the moon in Capricorn, the moon is a quarter of a circle before the sun; opposite to him, and consequently full in Aries, and a quarter of a circle behind him, when in Cancer. But when Libra rises, Aries sets, and all that half of the ecliptic of which Aries is the middle; and therefore, at that time of the year, the moon rises at full-moon from her first to her third quarter.

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 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 5½ 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 fourteen 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 whilst 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 19½ 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 5½ 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 there would be if she kept in the ecliptic. But in 9 years and 112 days afterward, the descending node comes to Aries; and then the moon's orbit makes an angle 5½ degrees greater with the horizon when Aries rises, that 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 9½ degrees on the parallel of London when Aries rises. But when the descending node comes to Aries, the angle is 20½ degrees; this occasions as great a difference of the moon's rising in the same signs every 9 years, as there would be on two parallels to 10½ degrees from one another, if the moon's course were in the ecliptic.

As there is a complete revolution of the nodes in 18½ 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. The following table shews in what years the harvest-moons are least beneficial as to the times of their rising, and in what years most, from 1751 to 1861. The column of years under the letter L are those in which the harvest-moons are least of all beneficial, because they fall about the descending node; and those under M are the most of all beneficial, because they fall about the ascending node. In all the columns from N to S, the harvest-moons descend gradually in the lunar orbit, and rise to less heights above the horizon. From S to N they ascend in the same proportion, and rise to greater heights above the horizon. In both the columns under S, the harvest-moons are in the lowest part of the moon's orbit; that is, farthest south of the ecliptic; and therefore stay shortest of all above the horizon; in the columns under N, just the reverse. And, in both cases, their rising, though not at the same times, are nearly the same with regard to difference of time, as if the moon's orbit were coincident with the ecliptic. 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 moon-light: In winter, they go high, and stay long above the horizon, when the nights are long, and we want the greatest quantity of moon-light.

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 all 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 the moon falls 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 the 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 light 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 (Plate XLIV. fig. 3.) will make this 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-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, $V\mathcal{E}C\mathcal{E}rs$ is the moon’s orbit, in which she goes round the earth according to the order of the letters $abcd$, $ABCD$. 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 towards him. Thus, in our summer, the moon is above the horizon of the north pole. pole whilst she describes the northern half of the ecliptic \( \varphi \) or, from her third quarter to her first; and below the horizon during her progress through the southern half \( \omega \); 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 23\(\frac{1}{2}\) 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 \( F \). \( S \) and \( S_r \) 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.

**Chap. XV. Of the Ebbing and Flowing of the Sea.**

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 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, 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.

The power of gravity diminishes as the square of the distance increases; and therefore the waters (Plate XLIV., fig. 4.) 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 \( ABCDEFGH \), 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 \( ZIBLnKFNZ \), 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 bodies, so as to form a fluid sphere round \( G \); then, as the whole moves toward \( M \), 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 24\(\frac{1}{2}\) 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 toward 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 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. In Plate XLV., fig. 1., let \( M \) be the moon, \( TW \) 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 \( ged \) round \( C \), to which circle \( gak \) 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 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 with the distance of 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, so far as we have explained it, 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 \(M\) (Plate XLIV. fig. 4.) is generally past the north and south meridians, 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 increment 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 she approaches nearer to the earth than her mean distance, and recedes farther from it, in every lunar month. 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 \(H\) (Plate XLV. fig. 2.) 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, strictly speaking, 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 48 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 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 toward 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 toward 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. In Plate XLV. let \(NESQ\) be the earth, \(NGS\) its axis, \(EQ\) the equator, \(T\&\) the tropic of Cancer, \(tr\) the tropic of Capricorn, \(ab\) the arctic circle, \(cd\) 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 \(d\) (fig. 3.), at \(N\) and \(S\) (fig. 4.), and at \(b\) and \(c\) (fig. 5.), 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 \(tr\); and these two elevations describe the tropics. picks by the earth's diurnal rotation. All places in the northern hemisphere ENQ have the highest tides when they come into the position bEQ, under the moon; and the lowest tides when the earth's diurnal rotation carries them into the position aTE, on the side opposite to the moon; the reverse happens at the same time in the southern hemisphere ESQ, as is evident by sight.

The axis of the tides aCd 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 farther description.

In Plate XLV, fig. 3, 4, 5, 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. In the same Plate fig. 2, let HZON 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 ETP6 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 1 (under D) revolving as formerly, goes sooner from 1 to 11, (under F), than from 11 to 1, because the parallel it describes is cut into unequal segments by the high-water circle HCO; but the points 1 and 11 being equidistant from the pole of the tides at C, which is directly under the pole of the moon's orbit MG, 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 23½ degrees to the earth's axis at a mean rate; 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 Plate XLV, fig. 4, when the moon is vertical to the equator EQ, the poles of the tides seem to fall in with the poles of the world N and S: but when we consider that FHG 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 N, 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 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 unequal 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 is 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 suffer 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.

**CHAP. XVI. Of Eclipses: Their Number and Periods. A large Catalogue of ancient and modern 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 shews 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 millions of 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 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 $5\frac{1}{2}$ 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 deg. 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 thro' 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 XLVI. fig. 1.) let $ABCD$ be the ecliptic, $RSTU$ a circle lying in the same plane with the ecliptic, and $VWXYZ$ 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 $a$, 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 $I$, because she is then near the other node; yet, in the time that she goes round the earth to her next change, according to the order of the letters $XTVW$, 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 $o$ (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 90 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 newmoon $l$ 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 $5\frac{1}{2}$ 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 backward, or contrary to the earth's annual motion, $19\frac{2}{3}$ deg. every year; and therefore the same node comes round to 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 rGs; the ascending node having gone backward, that is, contrary to the order of signs, from X to r, and the descending node from V to s; each $19\frac{2}{3}$ 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 225 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 223 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, or 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 12492 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 south 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, even since the creation, till A.D. 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 1962, July 18th old style, at 10 h. 36 m. 21 sec. p.m. when the same eclipse will have returned 38 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 12th 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 12492 years afterward.—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 ex- And as this 26 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 12492 years. For, as 36 is to 1388, so is 324 to 12492.

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 30th of April, it began to touch the south 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 sunrise, and observed both at Würtemberg 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 expanse, though 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 thousand 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 through a circuit of one year and ten months, every Chaldean period being ten or eleven 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 expanse 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 expanse, and let 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 expanse on the south side of the earth.

The eclipses therefore which happened about the creation are little more than half way yet of their ethereal circuit; and will be 4000 years before they enter the earth any more. This grand revolution seems to have been entirely unknown to the ancients.

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 new regulation and adjustment of their original motions.

* There are two ancient eclipses of the moon, recorded by Ptolemy from Hipparchus, which afford an undeniable proof of the moon's acceleration. The first of these was observed at Babylon, Decem. 22d, in the year before Christ 383; when the moon began to be eclipsed, about half an hour before the sun rose, and the eclipse was not 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 58th year before Christ; when accordingly there happened a very signal eclipse of the sun, on the 28th of May, answering to the present 10th 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 fartherst that even the Caroline tables carry it; and consequently make it invisible to any part of Asia, in the total character; though 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 set 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 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 obviated 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 backward every year, the sun will come to either of them 173 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 173 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 above mentioned, 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 moon 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 above mentioned.

A List of Eclipses, and historical Events, which happened about the same times, from Ricciolus.

| Before Christ | Month | Day | |---------------|-------|-----| | 754 | July | 5 | | 721 | March | 19 |

But, according to an old calendar, this eclipse of the sun was on the 21st of April, on which day the foundations of Rome were laid; if we may believe Taruntius Firmianus.

A total eclipse of the moon. The Assyrian empire at an end; the Babylonian established.

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not over before the moon set: But, by most of our astronomical tables, the moon was set at Babylon half an hour before the eclipse began; in which case, there could have been no possibility of observing it. The second eclipse was observed at Alexandria, Septem. 22d, the year before Christ 201; where the moon rose so much eclipsed, that the eclipse must have begun about half an hour before she rose: Whereas, by most of our tables, the beginning of this eclipse was not till about 10 minutes after the moon rose at Alexandria. Had these eclipses begun and ended while the sun was below the horizon, we might have imagined, that as the ancients had no certain way of measuring time, they might have been so far mistaken in the hours, that we could not have laid any stress on the accounts given by them. But as, in the first eclipse, the moon was set, and consequently the sun risen, before it was over; and in the second eclipse the sun was set, and the moon not risen, till some time after it began; these are such circumstances as the observers could not possibly be mistaken in. Mr Struyk, in the following catalogue, notwithstanding the express words of Ptolemy, puts down these two eclipses as observed at Athens; where they might have been seen as above, without any acceleration of the moon’s motion, Athens being 20 degrees west of Babylon, and 7 degrees west of Alexandria. An eclipse of the sun foretold by Thales, by which a peace was brought about between the Medes and Lydians.

An eclipse of the moon, which was followed by the death of Cambyses.

An eclipse of the moon, which was followed by the slaughter of the Sabines, and death of Valerius Publicola.

An eclipse of the sun. The Persian war, and the falling off of the Persians from the Egyptians.

An eclipse of the moon, which was followed by a great famine at Rome; and the beginning of the Peloponnesian war.

A total eclipse of the sun. A comet and plague at Athens.

A total eclipse of the moon. Nicias with his ship destroyed at Syracuse.

An eclipse of the sun. The Persians beat by Conon in a sea-engagement.

A total eclipse of the moon. The next day Perseus, king of Macedonia, was conquered by Paulus Emilius.

An eclipse of the sun. This is reckoned among the prodigies, on account of the murder of Agrippinus by Nero.

A total eclipse of the sun. A sign that the reign of the Gordiani would not continue long. A sixth persecution of the Christians.

An eclipse of the sun. The stars were seen, and the emperor Constans died.

A dreadful eclipse of the sun. And Lewis the Pious died within six months after it.

An eclipse of the sun. And Jerusalem taken by the Saracens.

A terrible eclipse of the sun. The stars were seen. A schism in the church, occasioned by there being three Popes at once.

We have not enumerated one half of Ricciolus's list of portentous eclipses; and for the same reason that he declines giving any more of them than what that list contains, namely, that it is most disagreeable to dwell any longer on such nonsense: the superstition of the ancients may be seen by the few here copied.

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 fociuses, 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 eclipseth 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 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 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 slowest; less, if she be at her least distance, because of her quicker motion.

To make several of the above and other phenomena plainer, (Plate XLVI. fig. 3.), let S be the sun, E the earth, M the moon, and AMP the moon's orbit. Draw the right line Wc 12 from the western side of the sun at W, touching the western side of the moon at c, and the earth at 12; draw also the right line Vd 12 from the eastern side of the sun at V, touching the eastern side of the moon at d, and the earth at 12: the dark space ce12d included between those lines is the moon's shadow, ending in a point at 12, 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 about e, and therefore the sun would have appeared like a luminous ring all around the moon. Draw the right lines WXdb and VXcg, touching the contrary sides of the sun and moon, and ending on the earth at a and b; draw also the right line SXM12, from the centre of the sun's disk, through the moon's centre, to the earth at 12; and suppose the two former lines WXdb and VXcg to revolve on the line SXM12 as an axis, and their points a and b will describe the limits of the penumbra T'T' on the earth's surface, including the large space aob12a; within which the sun appears more or less eclipsed, as the places are more or less distant from the verge of the penumbra aob.

Draw the right line y12 across the sun's disk, perpendicular to SXM the axis of the penumbra: then divide the line y12 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 12 (on the earth's surface YY') to its edge aob, draw twelve concentric circles, as 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 b, 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 XLVII. fig. 1, at the left hand; but, at the same moment of absolute time to an observer at a in Plate XLVI. fig. 3, the western edge of the moon at c leaves the eastern edge of the sun at V', and the eclipse ends, as at the right hand C, Plate XLVII. fig. 1. At the very same instant, to all those who live on the circle marked 1 on the earth E, in Plate XLVI. 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 XLVII. fig. 1.: to those who live on the circle marked 2 in Plate XLVI. fig. 3, the moon cuts off two twelfths parts of the sun, as at 2 in Plate XLVII. fig. 1.; to those on the circle 3, three parts; and so on to the centre at 12 in Plate XLVI. fig. 3, where the sun is centrally eclipsed, as at B in the middle of fig. 1., Plate XLVII.; under which figure there is a scale of hours and minutes, to shew at a mean state 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 XLVI. 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 aob, 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 XLVII. 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, EQ the equator, T the tropic of Cancer, t the tropic of Capricorn, 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 EQ. 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° degrees towards the sun, falls so many degrees within the earth's enlightened disk, because the sun is then vertical to D, 23° degrees north of the equator A Q; 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° 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 A Q. 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 NVS; when its south pole S inclining 23° degrees toward the sun, falls 23° degrees within the enlightened disk, as seen from the sun or new moon, which are then vertical to the tropic of Capricorn t, 23° degrees south of the equator A Q; 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 sidewise 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, 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 XLVII., fig. 2.), let FG be part of the ecliptic, and IK, ik, ik, ik, part of the moon's orbit; both seen edgewise, 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 forementioned 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 (iDk 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, (iDk 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 A Q, and describing the line nBq, (similar to the former line Aom in open space), goes obliquely northward over the earth, and leaves it at q, 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 penumbra's 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 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 P 18 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 P 12 falls on the northern parts of the earth at the middle of the general eclipse: Had the change 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 expatrium, or open space. When the moon changes 6 degrees from the node, almost the whole penumbra P 6 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... 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 c; but whilst the penumbra moves eastward over c, 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 c on its 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, seems 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 XLVI.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 fghi, concentric to the earth, include the atmosphere whose refractive power vanishes at the heights f and i; so that the rays Wfaw and Vivi go on straight without suffering the least refraction: But all those rays which enter the atmosphere between f and k, and between i and l, on opposite sides of the earth, are gradually more bent inward as they go through a greater portion of the atmosphere, until the rays Wk and VI touching the earth at m and n, are bent so much as to meet at g, a little short of the moon; and therefore the dark shadow of the earth is contained in the space moqn, 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 those 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 po, she would be invisible during her stay in it; but visible before and after in the fainter shadow RR.

When the moon goes through 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 state 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 XLVII.fig.3, 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, AB the earth's shadow, and M the moon. When the 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, though 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, thro' it: And such an eclipse being both total and central is of the longest duration, namely, 3 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 farthest from the earth, she moves slowest; and when nearest to it, quickest.

The moon's diameter, as well as the sun's, is supposed to be divided into twelve 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, shew 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 scarcely 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 broke on that part; and the moment he 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 overlaid at least all the land of Judea.

With regard to the method of calculating and projecting eclipses, we must refer the reader to the astronomical tables of Mr Ferguson and others. When the principles are explained, the application and use of the tables is a matter of small difficulty, and easily acquired by a little practice.

**Chap. XVII. Of the fixed Stars.**

The stars 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 a thousand 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 surprised 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 three thousand, in both hemispheres.

As we have incomparably more light from the moon than from all the stars together, it were the greatest absurdity furdity to imagine that the stars were made for no other purpose than to cast a faint light upon the earth; 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 $32,000,000,000,000$ of miles from us, which is farther than a cannon-bullet would fly in $7,000,000$ of years. Hence it is easy to prove, that the sun, seen 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, though 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.

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 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 telestic 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 Sevex'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 parts. 1. The Zodiak (Ζωδιακός) from ζωδίαξ, zodiac, an animal, because most of the constellations in it, which are twelve in number, are the figures of animals: As Aries the ram, Taurus the bull, Gemini the twins, Cancer the crab, Leo the lion, Virgo the virgin, 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 twenty-one constellations. And, 3. That on the south side, containing fifteen.

The ancients divided the zodiac into the above twelve 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 distil drop by drop into another vessel set beneath to receive it; beginning at the moment when some star rose, and continuing until it rose the next following night. The water fallen 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 though this could not be done in one night, yet in many they observed the rising of twelve stars or points, by which they divided the zodiac into twelve parts.

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

| Constellation | 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, Artiphilax | 23 | 18 | 52 | 54 | | Corona Borealis | 8 | 8 | 8 | 21 | | Hercules, Engonasa | 29 | 28 | 45 | 113 | | Lyra | 10 | 11 | 17 | 21 | | Cygnus, Gallina | 19 | 18 | 47 | 81 | | Carthocea | 13 | 26 | 37 | 55 | | Perseus | 29 | 29 | 46 | 59 | | Auriga | 14 | 9 | 40 | 66 | | Serpentarius, Ophiuchus| 29 | 15 | 40 | 74 |

Serpens ### The ancient Constellations.

| Constellation | Ptolemy | Tycho | Hevelius | Flamsteed | |------------------------|---------|-------|----------|-----------| | Serpens | 18 | 13 | 22 | 64 | | Sagitta | 5 | 5 | 5 | 18 | | Aquila, Vultur | 15 | 12 | 23 | 71 | | Antinous | 3 | 3 | 19 | | | Delphinus | 10 | 10 | 14 | 18 | | Equinus, Equi sedio | 4 | 4 | 6 | 10 | | Pegasus, Equus | 20 | 19 | 38 | 89 | | Andromeda | 23 | 23 | 47 | 66 | | Triangulum | 4 | 4 | 12 | 16 | | Aries | 18 | 21 | 27 | 66 | | Taurus | 44 | 43 | 51 | 141 | | Gemini | 25 | 25 | 38 | 85 | | Cancer | 23 | 15 | 29 | 83 | | Leo | 30 | 49 | 95 | | | Coma Berenices | 35 | 14 | 21 | 43 | | Virgo | 32 | 33 | 50 | 110 | | Libra, Chelae | 17 | 10 | 20 | 51 | | Scorpius | 24 | 10 | 20 | 44 | | Sagittarius | 31 | 14 | 22 | 69 | | Capricornus | 28 | 28 | 29 | 51 | | Aquarius | 45 | 41 | 47 | 108 | | Pisces | 38 | 36 | 39 | 113 | | Cetus | 22 | 21 | 45 | 97 | | Orion | 38 | 42 | 62 | 78 | | Eridanus, Fluvius | 34 | 10 | 27 | 84 | | Lepus | 12 | 13 | 16 | 19 | | Canis major | 29 | 13 | 21 | 31 | | Canis minor | 2 | 2 | 13 | 14 | | Argo Navis | 45 | 3 | 4 | 64 | | Hydra | 27 | 19 | 31 | 60 | | Crater | 7 | 3 | 10 | 31 | | Corvus | 7 | 4 | | 9 | | Centaurus | 37 | | | 35 | | Lupus | 19 | | | 24 | | Ara | 7 | | | 9 | | Corona Australis | 13 | | | 12 | | Pifcis Australis | 18 | | | 24 |

### The new Southern Constellations.

| Constellation | Asterion & Chara | Hevelius | Flamsteed | |------------------------|-------------------|----------|-----------| | Columba Noachi | Noah’s Dove | Cerberus | 23 | | Robur Carolinum | The Royal Oak | Vulpecula & Anser | 27 | | Grus | The Crane | Scutum Sobieski | 7 | | Phoenix | The Phenix | Lacerta | 10 | | Indus | The Indian | Camelopardalus | 32 | | Pavo | The Peacock | Monocorns | 19 | | Apus, Avis Indica | The Bird of Paradise | Sextans | 11 | | Apis, Musca | The Bee or Fly | | | | Chameleon | The Chameleon | | | | Triangulum Australis | The South Triangle| | | | Pifcis volans, Paffer | The Flying Fish | | | | Dorado, Xiphias | The Sword Fish | | | | Toucan | The American Goose| | | | Hydrus | The Water Snake | | |

There is a remarkable track round the heavens, called the Milky Way, from its peculiar whiteness, which was formerly thought to be owing to a vast number of very small stars therein; but the telescope shews it to be quite otherwise; and therefore its whiteness must be owing to some other cause. This track appears single in some parts, in others double.

There are several little whitish spots in the heavens, which appear magnified, and more luminous when seen through telescopes; yet without any stars in them. One of these is in Andromeda's girdle, and was first observed A.D. 1612, by Simon Marius: it has some whitish rays.

rays near its middle, is liable to several changes, and is sometimes invisible. Another is near the ecliptic; between the head and bow of Sagittarius: it is small, but very luminous. A third is on the back of the Centaur, which is too far south to be seen in Britain. A fourth, of a smaller size, is before Antinous's right foot; having a star in it, which makes it appear more bright. A fifth is in the constellation of Hercules, between the stars ζ and η, which spot, though but small, is visible to the bare eye, if the sky be clear and the moon absent.

Cloudy stars are so called from their misty appearance. They look like dim stars to the naked eye; but through a telescope they appear broad illuminated parts of the sky; in some of which is one star, in others more. Five of these are mentioned by Ptolemy. 1. One at the extremity of the right hand of Perseus. 2. One in the middle of the Crab. 3. One unformed, near the sting of the Scorpion. 4. The eye of Sagittarius. 5. One in the head of Orion. In the first of these appear more stars through the telescope than in any of the rest, although 21 have been counted in the head of Orion, and above 40 in that of the Crab. Two are visible in the eye of Sagittarius without a telescope, and several more with it. Flamsteed observed a cloudy star in the bow of Sagittarius, containing many small stars; and the star above Sagittarius's right shoulder is encompassed with several more. Both Cassini and Flamsteed discovered one between the Great and Little Dog, which is very full of stars visible only by the telescope. The two whitish spots near the south pole, called the Magellanic Clouds by sailors, which to the bare eye resemble part of the Milky Way, appear through telescopes to be a mixture of small clouds and stars. But the most remarkable of all the cloudy stars is that in the middle of Orion's Sword, where seven stars (of which three are very close together) seem to shine through a cloud, very lucid near the middle, but faint and ill defined about the edges. It looks like a gap in the sky, through which one may see (as it were) part of a much brighter region. Although most of these spaces are but a few minutes of a degree in breadth, yet, since they are among the fixed stars, they must be spaces larger than what is occupied by our solar system; and in which there seems to be a perpetual uninterrupted day among numberless worlds, which no human art ever can discover.

Several stars are mentioned by ancient astronomers, which are not now to be found; and others are now visible to the bare eye which are not recorded in the ancient catalogues. Hipparchus observed a new star about 170 years before Christ; but he has not mentioned in what part of the heaven it was seen, although it occasioned his making a catalogue of the stars; which is the most ancient that we have.

The first new star that we have any good account of, was discovered by Cornelius Gemma on the 8th of November A.D. 1572, in the chair of Cassiopea. It surpassed Sirius in brightness and magnitude; and was seen for 16 months successively. At first it appeared bigger than Jupiter to some eyes, by which it was seen at midday; afterwards it decayed gradually both in magnitude and lustre, until March 1573, when it became invisible.

On the 13th of August 1596, David Fabricius observed the Stella Mira, or wonderful star, in the neck of the Whale; which has been since found to appear and disappear periodically, seven times in six years, continuing in the greatest lustre for 15 days together; and is never quite extinguished.

In the year 1600, William Janssenius discovered a changeable star in the neck of the Swan; which, in time, became so small as to be thought to disappear entirely, till the years 1657, 1658, and 1659, when it recovered its former lustre and magnitude; but soon decayed, and is now of the smallest size.

In the year 1604 Kepler and several of his friends saw a new star near the heel of the right foot of Serpentarius, so bright and sparkling, that it exceeded anything they had ever seen before; and took notice that it was every moment changing into some of the colours of the rainbow, except when it was near the horizon, at which time it was generally white. It surpassed Jupiter in magnitude, which was near it all the month of October, but easily distinguished from Jupiter, by the steady light of Jupiter. It disappeared between October 1605 and the February following, and has not been seen since that time.

In the year 1670, July 15, Hevelius discovered a new star, which in October was so decayed as to be scarcely perceptible. In April following it regained its lustre, but wholly disappeared in August. In March 1672 it was seen again, but very small; and has not been visible since.

In the year 1686 a new star was discovered by Kirch, which returns periodically in 404 days.

In the year 1672, Cassini saw a star in the neck of the Bull, which he thought was not visible in Tycho's time, nor when Bayer made his figures.

Many stars, besides those above mentioned, have been observed to change their magnitudes: and as none of them could ever be perceived to have tails, it is plain they could not be comets; especially as they had no parallax, even when largest and brightest. It would seem, that the periodical stars have vast clusters of dark spots, and very slow rotations on their axes; by which means, they must disappear when the side covered with spots is turned towards us. And as for those which break out all of a sudden with such lustre, it is by no means improbable that they are suns whose fuel is almost spent, and again supplied by some of their comets falling upon them, and occasioning an uncommon blaze and splendor for some time; which indeed appears to be the greatest use of the cometary part of any system.

* M. Maupertuis, 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 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, though the stars themselves were really immoveable. On the 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 motion is for our perception.

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 incorrectness 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 answers 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 between those planes. Nor is it less probable that the mutual attractions of all the planets should have a tendency to bring their orbits to a coincidence: but this change is too small to become sensible in many ages.

**Chap. XVIII. 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, 57 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, as from any fixed star to the same star again, is called the Sidereal year; which contains 365 days 6 hours 9 minutes 14½ seconds; and is 20 minutes 17½ seconds longer than the true solar year.

The time measured by twelve 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 0 minutes 21 seconds shorter than the solar year. This is the foundation of the exact.

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 bisextile 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 23d of February, which in the Roman calendar was the fifth of the kalends of March, that fifth day was twice reckoned, or the 23d and 24th were reckoned as one day, and was called bis sextus dies; and thence came the name bisextile for that year. But in our common almanacks 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 embolimic 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 3½ days short of the solar year in that time.

The Romans also used the lunar embolimic year at first, as it was settled by Romulus their first king, who made it to consist only of ten 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.

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. But Julius Cæsar, 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 120 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 XIIIth, 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 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 $27^d\ 7^h\ 43^m$. 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 $29^d\ 12^h\ 44^m$.

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 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.

| No | The Jewish year | Days | |----|----------------|------| | 1 | Tifri | Aug.—Sept. 30 | | 2 | Marcheshvan | Sept.—Oct. 29 | | 3 | Casleu | 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 | Jiar | 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 embolimic year after Adar they added a month called Ve-Adar of 30 days.

| No | The Egyptian year | Days | |----|------------------|------| | 1 | Thoth | August 29 30 | | 2 | Paophi | Septemb. 28 30 | | 3 | Athir | October 28 30 | | 4 | Chojac | Novemb. 27 30 | | 5 | Tybi | Decemb. 27 30 | | 6 | Mechir | January 26 30 | | 7 | Phamenoth | February 25 30 | | 8 | Parmuthi | March 27 30 | | 9 | Pachon | April 26 30 | | 10 | Payni | May 26 30 | | 11 | Epiphi | June 25 30 | | 12 | Mefori | July 25 30 |

Epagomenæ or days added = 5

Days in the year = 365 | No | The Arabic and Turkish year | Days | No | The ancient Grecian year | Days | |----|---------------------------|------|----|-------------------------|------| | 1 | Muharram | July | 16 | 30 | Hecatombaion | June—July | 30 | | 2 | Safar | August | 15 | 29 | Metagitnion | July—Aug. | 29 | | 3 | Rabia I. | Septemb. | 13 | 30 | Boedromion | Aug.—Sept. | 30 | | 4 | Rabia II. | October | 13 | 29 | Pyanephius | Sept.—Oct. | 29 | | 5 | Jomada I. | Novemb. | 11 | 30 | Maimaetorion | Oct.—Nov. | 30 | | 6 | Jomada II. | Decemb. | 11 | 29 | Pofideon | Nov.—Dec. | 29 | | 7 | Rajab | January | 9 | 30 | Gamelion | Dec.—Jan. | 30 | | 8 | Shaiban | February | 8 | 29 | Anthefteron | Jan.—Feb. | 29 | | 9 | Ramadam | March | 9 | 30 | Elaphebolion | Feb.—Mar. | 30 | | 10 | Shawal | April | 8 | 29 | Munichion | Mar.—Apr. | 29 | | 11 | Dulhaadah | May | 7 | 30 | Thargelion | Apr.—May | 30 | | 12 | Dulheggia | June | 5 | 29 | Schirrophorion | May—June | 29 |

Days in the year 354

The Arabians add 11 days at the end of every year, which keep the same months to the same seasons.

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, Bohemians, Silesians, 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 shown 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 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. The indiction 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 our 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 In the above 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 chapter shews 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, is a revolution of 532 years, found by multiplying the solar cycle 28 by the lunar cycle 19. If the new 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 22d 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 almanacks, 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 Astronomy

(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 fifth 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 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 indication of 15 years, arises the great Julian period, consisting of 7980 years, which had its beginning 764 years before Strachius'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 84th before his birth) was the 4129th year of the said period.

Lastly, to find the cycles of the sun, moon, and indication 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 indication 4; the solar cycle having run through 168 courses, the lunar 248, and the indication 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 as 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 Jefdegird: All which, together with several others of less note, have their beginnings to 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.

### Table of remarkable Eras and Events

| Era | Julian Year of the Period | Before World | Christ | |----------------------------|---------------------------|--------------|--------| | 1. The creation of the world| 706 | 0 | 4007 | | 2. The deluge, or Noah's flood| 2362 | 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 |

Vol. I. No. 21. | Event | Julian Period | Y.of the World | Before Christ | |----------------------------------------------------------------------|---------------|----------------|---------------| | The beginning of king David's reign | 3050 | 2944 | 1063 | | The foundation of Solomon's temple | 3701 | 2995 | 1012 | | The Argonautic expedition | 3776 | 3070 | 927 | | Lycurgus forms his excellent laws | 3829 | 3103 | 884 | | Arbaces, the first king of the Medes | 3838 | 3132 | 875 | | Mandaucus, the second | 3865 | 3159 | 848 | | Solarmus, the third | 3915 | 3209 | 798 | | The beginning of the Olympiads | 3938 | 3232 | 775 | | Artica, the fourth king of the Medes | 3945 | 3239 | 768 | | The Catonian epocha of the building of Rome | 3961 | 3255 | 752 | | The era of Nabonassar | 3967 | 3261 | 746 | | The destruction of Samaria by Salmanefer | 3992 | 3286 | 721 | | The first eclipse of the moon on record | 3993 | 3287 | 720 | | Cardicea, the fifth king of the Medes | 3996 | 3290 | 717 | | Phraortes, the sixth | 4058 | 3352 | 655 | | Cyaxares, the seventh | 4080 | 3374 | 633 | | The first Babylonish captivity by Nebuchadnezzar | 4107 | 3401 | 606 | | The long war ended between the Medes and Lydians | 4111 | 3405 | 602 | | The second Babylonish captivity, and birth of Cyrus | 4114 | 3408 | 599 | | The destruction of Solomon's temple | 4125 | 3419 | 588 | | Nebuchadnezzar struck with madness | 4144 | 3438 | 569 | | Daniel's vision of the four monarchies | 4158 | 3452 | 555 | | Cyrus begins to reign in the Persian empire | 4177 | 3471 | 536 | | The battle of Marathon | 4223 | 3517 | 490 | | Artaxerxes Longimanus begins to reign | 4249 | 3543 | 404 | | The beginning of Daniel's seventy weeks of years | 4256 | 3550 | 457 | | The beginning of the Peloponnesian war | 4282 | 3576 | 431 | | Alexander's victory at Arbela | 4382 | 3677 | 330 | | The death of Alexander | 4390 | 3684 | 323 | | The captivity of 100000 Jews by king Ptolemy | 4393 | 3687 | 320 | | The Colossus of Rhodes thrown down by an earthquake | 4491 | 3875 | 222 | | Antiochus defeated by Ptolemy Philopater | 4496 | 3790 | 217 | | The famous Archimedes murdered at Syracuse | 4506 | 3800 | 207 | | Jason butchers the inhabitants of Jerusalem | 4543 | 3837 | 170 | | Corinth plundered and burnt by consul Mummius | 4567 | 3861 | 146 | | Julius Caesar invades Britain | 4659 | 3953 | 54 | | He corrects the calendar | 4677 | 3991 | 46 | | Is killed in the Senate-house | 4671 | 3965 | 42 | | Herod made king of Judea | 4673 | 3967 | 40 | | Anthony defeated at the Battle of Actium | 4683 | 3977 | 30 | | Agrippa builds the Pantheon at Rome | 4688 | 3982 | 25 | | The true era of Christ's birth | 4709 | 4003 | 4 | | The death of Herod | 4710 | 4004 | 3 | | The Dionysian, or vulgar era of Christ's birth | 4713 | 4007 | 0 | | The true year of his crucifixion | 4746 | 4040 | 33 | | The destruction of Jerusalem | 4783 | 4077 | 70 | | Adrian builds the long wall in Britain | 4833 | 4127 | 120 | | Constantius defeats the Picts in Britain | 5019 | 4313 | 306 | | The council of Nice | 5038 | 4332 | 325 | | The death of Constantine the great | 5050 | 4344 | 337 | | The Saxons invited into Britain | 5158 | 4452 | 445 | | The Arabian Hegira | 5335 | 4629 | 622 | | The death of Mohammed the pretended prophet | 5343 | 4637 | 620 | | The Persian Yefdegird | 5344 | 4638 | 631 | | The sun, moon, and all the planets in Libra; Sep. 14, as seen from the earth | 5899 | 5193 | 1186 | | The art of printing discovered | 6153 | 5447 | 1440 | | The reformation begun by Martin Luther | 6230 | 5524 | 1517 | In fixing the year of the creation to the 76th 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, shewing 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 | Years less than an hundred | |-------------------|---------------------------| | 0 | 1900 | | 1 | 2000 | | 2 | 2100 | | 3 | 2200 | | 4 | 2300 | | 5 | 2400 | | 6 | 2500 | | 7 | 2600 | | 8 | 2700 | | 9 | 2800 | | 10 | 2900 | | 11 | 3000 | | 12 | 3100 | | 13 | 3200 | | 14 | 3300 | | 15 | 3400 | | 16 | 3500 | | 17 | 3600 | | 18 | 3700 |

**Chap. XIX. A Description of the Astronomical Machinery serving to explain and illustrate the foregoing part of this Treatise.**

The Orrery, (Plate XLVII. fig. 4.) 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 winch; 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.

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 $25\frac{1}{4}$ days on its axis, of which the north pole inclines toward the 8th degree of Pisces in the great ecliptic, (No. 11.), whereon 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 $87\frac{3}{4}$ 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 $24\frac{1}{3}$ diurnal rotations of the earth. Her axis inclines 75 degrees from the axis of the ecliptic, and her north pole inclines towards... towards the 20th degree of Aquarius, according to the observations of Bianchini. She shews all the phenomena described in Chap. I.

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 4 seconds, of mean solar time; but from the sun to the sun again, in 24 hours of the same time. No. 6. is a sydereal dial-plate under the earth, and No. 7. a solar dial-plate on the cover of the machine. The index of the former shews sydereal, 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 sydereal days, or apparent revolutions of the stars. In the time that the earth makes 365 1/4 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 23° 27' 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: 3 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 on 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 above mentioned 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 backward through all its lines 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 engraved 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 fall, the moon falls between either of the little moons 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 XLVIII. 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 XLVII.) 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 XLVIII. fig. 2.) This curious 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 Desaguliers.

The dark elliptical groove round the letters \(abcde\text{fg}hijklm\) 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 \(b\), 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 \(nopqrs\), 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 \(b\), and it appears to describe the large space \(m\) 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, 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 XLVIII. 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 focusses, 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 \(b\). On \(H\), the axis of the handle or winch \(N\) in fig. 2. is an endless screw in fig. 3. 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 \(FF\) 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 \(b\) of the elliptical plate \(FF\) being farther from its axis \(E\) than the opposite end \(I\) is, must describe a circle so much 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 instance in two points, if the distance \(Kk\) be four, five, or six times as great as the distance \(Kh\), 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 \(Kh\).

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 \(abcd\), &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 true motion of the moon about the earth, or of any planet about the sun, by making the elliptical plates of the same eccentricity, 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 a day about such motions and equations.

The Improved Celestial Globe, (Plate XLIV. fig. 2.) On the north pole of the axis, above the hour-circle, is fixed an arch MKH of $23\frac{1}{2}$ 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 $29\frac{1}{2}$ 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 $5\frac{1}{2}$ 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 LI. fig. 2. (by mistake omitted to be inserted in the proper plate) is a small piece of pasteboard, of which the curved edge at S is to be set 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 $5\frac{1}{2}$ degrees of the moon's latitude on both sides of the ecliptic; and when this piece is set 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 brass meridian, and set the hour-index to XII at noon, that is to the upper XII on the hour-circle; keeping the globe in that situation, slacken the screw G, and set 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 show 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 shows the times of their rising and setting; and likewise the time of the moon's palling 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 fore-mentioned 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 appulses 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 XLIX. 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 brass meridian is grooved round the outer edge; and in this groove is a slender semi-circle of brass, the ends of which are fixed to the horizon in its north and south points: this semi-circle slides in the groove as the horizon is moved in rectifying it for different latitudes. To the middle of this semi-circle 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 center. 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 $23\frac{1}{2}$ 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 brafen 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 XLIX. fig. 2. This machine is for delineating the paths of the earth and moon, shewing what sort of curves they make in the etherial regions. S is the sun, and E the earth, whose centres are 81 inches distant from each other; every inch answering to a million of miles. M is the moon, whose centre is \( \frac{2}{3} \) parts of an inch from the earth's in this machine, this being in just proportion to the moon's distance from the earth. AA' is a bar of wood, to be moved by hand round the axis g which is fixed in the wheel T. 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' croiling 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 T has the months and days of the year all round its limb; and in the bar AA' 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 T, 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, cross-wise to the bar; which done, move the bar slowly round the axis g of the wheel T; 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 pushed a very little off, as if from the pencil m, to about \( \frac{1}{10} \) part of their distance, and the pencil m pushed 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, o line of the moon's path. This is an ocular proof of the moon's turning round her axis. The Tide-Dial, Plate L, fig. 1. The outside parts of this machine consist of:

1. An eight-sided box, on the top of which the corners is shown the phases of the moon at the octants, quarters, and full. Within these is a circle of 29 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 compasses 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 I and K, which show the times of high water, in the hour circle, at all these places.

4. 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, show 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 show 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 midnight: 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 show 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 show 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 show 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 show 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 show 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 marks towards the sun, and on the opposite side, showing 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 P 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.