a name given by the Greeks to two constellations of the northern hemisphere, by the Latins called URSA MAJOR and MINOR, and by us the Greater and Lesser Bear.
BINARY ARITHMETIC. See Binary Arithmetic, Encycl.
Duodecimal Arithmetic, is that which proceeds from 12 to 12, or by a continual subdivision according to 12. This is greatly used by most artificers in calculating the quantity of their work; as bricklayers, carpenters, painters, tilers, &c.
Harmonical Arithmetic, is so much of the doctrine of numbers as relates to the making the comparisons, reductions, &c. of musical intervals.
Arithmetic of Infinites, is the method of summing up a series of numbers, of which the number of terms is infinite. This method was first invented by Dr Wallis, as appears by his treatise on that subject; where he shows its use in geometry, in finding the areas of superficies, the contents of solids, &c. But the method of fluxions, which is a kind of universal arithmetic of infinites, performs all these more easily, as well as a great many other things, which the former will not reach.
Logistical Arithmetic, a name sometimes employed for the arithmetic of sexagesimal fractions, used in astronomical computations. Shakerly, in his Tabula Britannica, has a table of logarithms adapted to sexagesimal fractions, which he calls logistical logarithms; and the expeditious arithmetic, obtained by means of them, he calls logistical arithmetic. The term logistical arith-
Suppl. Vol. I. Part I. insects, and he no longer tried to excel in these branches. We marched together as equals in lithology, and the history of quadrupeds. When one of us made an observation, he communicated it to the other; scarce a day passed in which one did not learn from the other some new and interesting particular. Thus emulation excited our industry, and mutual assistance aided our efforts. In spite of the distance of our lodgings, we saw each other every day. At last I set out for Lapland; he went to London. He bequeathed to me his manuscripts and his books.
"In 1735 I went to Leyden, where I found Artedi. I recounted my adventures; he communicated his to me. He was not rich, and therefore was unable to be at the expense of taking his degrees in physic. I recommended him to Seba, who engaged him to publish his work on fishes. Artedi went to join him at Amsterdam.
"Scarcely had I finished my Fundamenta Botanica. I communicated it to him; he let me see his Philosophia Ichthyologica. He proposed to finish as quickly as possible the work of Seba, and to put the last hand to it. He showed me all his manuscripts which I had not seen; I was pressed in point of time, and began to be impatient as being detained so long. Alas! if I had known this was the last time I should see him, how should I have prolonged it!
"Some days after, as he returned to sup with Seba, the night being dark, he fell into the canal. Nobody perceived it, and he perished. Thus died, by water, this great ichthyologist, who had ever delighted in that element."
Of the works of this eminent naturalist there have been two editions, of which the former was published by Linnæus in 1738, and the latter by Dr Walbaum of Lubeck, in the years 1788, 1789, and 1792. This edition, which is by much the most valuable, is in three volumes 4to; of which the first contains the history of the science of ichthyology, commencing several years before the Christian era, and coming down to the present times. The second presents to the reader the Philosophia Ichthyologica of Artedi, improved by Walbaum, who was benefited by the writings of Monro, Camper, Kaestleter, and others. Here also are added tables containing the system of fishes by Ray, Dale, Schaeffer, Linnæus, Gowas, Scopola, Klein, and Gronovius. The third volume, which completes the collection of Artedi's works, contains the technical definitions of the science. After the generic and individual characters, come the names and Latin phrases of Artedi; the synonyms of the best naturalists; the vulgar names in English, German, Swedish, Russian, Danish, Norwegian, Dutch, and Samoyed; the season and the countries where every kind is found, their varieties, their description, and observations. The modern discoveries, even to our own times, are added; so that in this part is collected the observations of Gronovius, Brunich, Penant, Forster, Klein, Bloch, Gmelin, Halfequill, Broussonet, Leake, Builh, Linnæus, and other great examiners of nature.
ASTRONOMY
Is a science which has been cultivated from the earliest ages, and is concerned about the most sublime objects of inquiry which can employ the mind of man. It has accordingly been treated at great length in the Encyclopaedia Britannica; but, in the opinion of some of the most judicious readers of that work, the compiler of the system which is there delivered has failed in his attempt to give a perspicuous and connected view of the science in its present state of improvement. This defect it is our duty to remedy. Our object, therefore, in this supplementary article, will be to bring into one point of view the physical science which may be derived from the consideration of the celestial motions; that is, to deduce from the general laws of those motions the inferences with respect to their supposed causes, which constitute the philosophy of the astronomer.
The causes of all phenomena are not only inferred from the phenomena, but are characterized by them; and we can form no notion of their nature but what we conceive as competent to the phenomena themselves. The astronomical phenomena are assumed to be the motions of the bodies, which we call the sun, the planets, the comets, &c. The notion which we express by the word body in the present case, is supposed to be the same with that which we form of other objects around us, to which we give the same name; such as stones, sticks, the bodies of animals, &c. Therefore the notion which we have of the causes of the celestial motions must be the same with that which we have of the causes of motion in those more familiar bodies. All men seem to have agreed in giving the name forces, or moving forces, to the causes of those familiar motions. This is a figurative or metaphorical term. The true and original meaning of it is, the exertion which we are conscious of making when we ourselves put other bodies in motion. Force, when used without figure, always signifies the exertion of a living and acting thing. We are more interested in those productions of motion than in any other, and our recollections of them are more numerous. Hence it has happened that we use the same term to express the cause of bodily motion in general, and say that a magnet has force, that a spring has force, that a moving body has force.
Our own force is always exerted by the intervention of our own body; and we find that the same exertion by which we move a stone, enables us to move another man; therefore we conceive his body to resemble a stone in this respect, and that it also requires the exertion of force to put it in motion. But when we reflect on our employment of force for producing motion in a body, we find ourselves puzzled how to account for the motion of our own bodies. Here we perceive no intervening exertion but that of willing to do it; yet we find that we cannot move it as we please. We also find that a greater motion requires a greater exertion. It is therefore to this exertion that the reflecting man restrains the term force; and he acknowledges that every other use of it is metaphorical, and that it is a resemblance. We can conceive a force acting equally or unequally. If we suppose it to act equally or uniformly, we suppose that in equal times it produces equal effects; that is, equal determinations, or equal changes of determination. We have no other notion of equality or uniformity of action. Therefore it must produce equal augmentations or diminutions of velocity in equal times; therefore it must produce an uniformly accelerated or retarded motion. Uniformly accelerated or retarded motion is, therefore, the mark of uniform or unvaried acceleration. In such a motion, the changes of velocity are proportional to the times from the beginning of the acceleration; and if the motion has begun from rest, the whole acquired velocities are proportional to the times from the beginning of the motion. In this case, the spaces described are as the squares of the times from the beginning of the motion; and thus we arrive at an intelligible mark of the unvaried action of a moving force, viz., spaces increasing in the duplicate ratio of the times; for space and time are all that we can immediately observe in any motion that is continually varying; the velocity or determination is only an inference, on the supposition that the motion continues unchanged for some time, or that all action ceases for some time.
This abstract reasoning is perfectly agreeable to every phenomenon that we can observe with distinctness. Thus we cannot, or at least we do not, conceive the weight of a body to vary its action during the fall. We consider this weight as the cause of the fall—as the moving force—and we conceive it to act uniformly. And, in fact, a body falling freely, describes spaces which are proportional, not to the times, but to the squares of the times, and the fall is a motion uniformly accelerated. In like manner, the motion of a body rising in the air, in opposition to gravity, is uniformly retarded.
This kind of motion also gives us a certain measure and gives of the acquired velocity, although there is not, in fact, a measure of any space observed to be uniformly described during any time whatever. In this motion we know that the velocity, final determination, produced by the accumulated or continued action of the unvaried force, is such that the body would describe uniformly twice the space which it has described with the accelerated motion.
And it is by this method that we obtain the simplest measure of any moving force, and can compare it with another. If we observe that by the action of one force (known to be uniform by the spaces being proportional to the squares of the times) ten feet have been described in a second, and that by the uniform action of another force eighty feet are described in two seconds, we know that the last force is double of the first; for in the second motion, so feet were described in two seconds, and therefore so feet of this were described in the first second (because the motion is uniformly accelerated); and at the end of a second, the first body had a determination by which it would describe so feet uniformly in a second; and the second body had acquired a determination by which it would have described 40 feet uniformly in the next second, had not the moving force continued to act on it, and made it really describe 60 feet with an accelerated motion.
Because halves have the same proportions with the units of which they are the halves, it is plain that we may take the spaces, described in equal times with mo- tions uniformly accelerated, as measures of the forces which have produced those motions. The velocities generated are, however, the best measures.
When the actions of forces are not uniform, it is more difficult to learn what is the measure of the velocity produced by their accumulated action. But it can be determined with equal accuracy; that is, we can determine what is the velocity which would have been produced by the uniform action of the force during the same time, and therefore we obtain a measure of the force. Mathematicians are further able to demonstrate, that if forces vary their continued action in any manner whatever, the proportion of the spaces described by two bodies in equal times approaches nearer and nearer to the proportion of the spaces which they would describe in those times by the uniform action of the forces, as the times themselves are smaller; and therefore whenever we can point out the ultimate ratio of the spaces described in equal times, these times being diminished without end, we obtain the ratio of the forces.
Motions may be changed, not only in quantity, by acceleration or retardation, but also in direction, by deflecting a body from its former direction. When a body, moving uniformly in the direction AB (fig. 1.), has its motion changed in the point B, and, instead of describing BC uniformly in the next moment with the former velocity, describes BD uniformly in that moment, it is plain that the motion BD will be the same, whether the body had begun to move in A, or in F, or in G, or in B, provided only that its determination to move, or its velocity, be the same in all those points.
Complete the parallelogram BCDE. It is well known, that if one force act on the body which would make it describe BC, and another which would make it describe BE, the body will describe BD. Hence we learn, that when a body has the motion BC changed into the motion BD, it has been acted on in the point B by a force which would have caused a body at rest in B to describe BE. Thus we can discover the intensity and direction of the transverse force which produces any deflection from the former direction. In general, the force is that which would have produced in a body at rest that motion BE, which, when compounded with the former motion BC, produces the new motion BD.
These two principles, viz. 1st, that forces are proportional to the velocities which they produce in the same circumstances, and, 2nd, the composition of motion or forces, will serve for all the physical investigations in astronomy. All the celestial motions are curvilinear, and therefore are instances of continual deflection, and of the continual action of transverse or deflecting forces. We must therefore endeavour to obtain a general measure of such continual deflecting forces.
Let two bodies A and a (fig. 2.) describe in the same time the arches AC, ac of two circles. They are deflected from the tangents AB, ab. Let us suppose that the direction of the deflecting forces is known to be that of the chords AE, ae of these circles. Let these be called the deflective chords. Draw CB, cb parallel to AE, ae, and CD, cd parallel to AB, ab. Join AC, ac, CE, and ce. It is plain that the angle BAC is equal to the angle CEA in the alternate segment. Therefore ACD is also equal to it; and, because the angle CAD is common to the two triangles CAD and EAC, these two triangles are similar, and
\[ AD : AC = AC : AE, \text{ and } AD = \frac{AC^3}{AE}. \]
For similar reasons \( ad = \frac{ac^3}{ae} \). But AD and ad are respectively equal to BC and bc. Therefore BC = \(\frac{AC^3}{AE}\) and \( bc = \frac{ac^3}{ae} \). Therefore BC : bc = \(\frac{AC^3}{AE} : \frac{ac^3}{ae} \) or BC : bc = \( AC^3 \times ae : ac^3 \times AE \). But BC and bc being respectively equal to AD and ad, are equal to the spaces through which the deflecting forces would have impelled the bodies from a state of rest in the time of describing the arches AC, ac. Therefore, when these times are diminished without end, the ultimate ratio of AD and ad is the ratio of the forces which deflect the bodies in the points A and a. But it is evident that the ultimate ratio of AC to ac is the ratio of the velocity in the point A to the velocity in the point a; because these arches are supposed to be described in the same or equal times. Therefore the deflecting forces, by which bodies are made to describe arches of circles, are to each other as the squares of the velocities directly, and as the deflective chords of those circles inversely. This ratio may be expressed symbolically thus,
\[ F : f = \frac{V^2}{C} : \frac{v^2}{c}; \text{ or thus, in a proportional equation, } f = \frac{v^2}{c}. \]
It is easy to see that in this last formula \( f \) expresses directly the line bc, or the space through which the body is actually made to deviate from rectilinear motion in the time of describing the arch ac. It is a third proportional to ac the deflective chord, and ac the arch of the circumference described in a small moment of time. This is the measure afforded immediately by observation. We have observed the arch AC that is described, and know the direction and the length of AE from some circumstances of the case. The formula which comes to us, when treating this question by the help of fluxions, is \( f = \frac{2v^2}{c} \). This is perhaps a more proper expression of the physical fact; for it expresses twice the line bc, or the measure of the velocity which the deflecting force would have generated in the body by acting on it during the time of its describing the arch ac. But it is indifferent which measure we take, provided we always take the same measure. The first mathematicians, however, have committed mistakes by mixing them.
The planets, however, do not describe circles; but all the curves which can be described by the action of finite deflecting forces are of such a nature, that we can describe a circle through any point, having the same tangent, and the same curvature which the planetary curve has in that point, and which therefore ultimately coalesces with it. This being the case, it is plain that the planet, while passing through a point of the curve, and describing an indefinitely small arch of it, is in the same condition as if describing the coincident arch of the equicurve circle. Hence we obtain this most general proposition, that the transverse force by which a planet is made to describe any curve, is directly as the square of its velocity, and inversely as the deflective chord of the equicurve circle.
Farther: The velocity of a body in any point A (fig. of the curve, is equal to that which the deflective force in that point would generate in the body by acting uniformly on it along AF, one-fourth part of the deflective chord AE of the equicurve circle. It is the time which the body would acquire at F, after a uniformly accelerated motion along AF.
For it is certain that there is some length AF, such that the velocity acquired at F is the same with the velocity in the point A of the curve. Draw FG parallel to the tangent, and join AG. Make the arch ACI = 2AF. Then, because the space described with a uniformly accelerated motion is one half of the space which would be uniformly described with the final velocity, the arch ACI would be uniformly described with the velocity which the body has at A in the time that AF is described with the uniformly accelerated motion; and the arch AB will be to the arch AI as the time of describing AB to that of describing AI; that is, as the time of falling through AD to that of falling through AF. But the motion along AF being uniformly accelerated, the spaces are as the squares of the times. Therefore AD is to AF as the square of the arch AC to the square of the arch AI. But AD is to AF as the square of the chord AC is to the square of the chord AG. Therefore the arch AC is to the chord AC as the arch AI is to the chord AG. But the arch and chord AC are ultimately in the ratio of equality. Therefore the chord AG is equal to the arch AI. Therefore AG is double of AF. But because the triangles FAG and GAE are similar, AF is to AG as AG to AE; and therefore AE is double of AG and quadruple of AF. Therefore the velocity at A in the curve is that which would be produced by the uniform impulse of the deflective force along the fourth part of the deflective chord of the equicurve circle.
These two affections or properties of curvilinear motions are of the most extensive use, and give an easier solution of most questions than we obtain by the more usual methods, and deserve to be kept in remembrance by such as engage much in the discussion of questions of this kind.
Thus the investigation of the forces which regulate the planetary motions, is reduced to the task of discovering the velocity of the planet in the different points of its orbit, and the curvature in those points, and the position of the deflective chords.
The physical science of astronomy must consist in the discovery of the general laws which can be affirmed with respect to the exertion of those forces, whether with respect to their direction or the intensity of their action. If the mechanician can do more than this, and show that every motion that is observed is an immediate or remote consequence of those general laws, he will have completed the science, and explained every appearance.
This has accordingly been done by Sir Isaac Newton and his followers. Sir Isaac Newton has discovered the general laws which regulate the exertions of those forces which produce the planetary motions, by reasoning from general phenomena which had been observed with a certain precision before his time; and has also shown that certain considerable deviations from the generality which he supposed to be perfect were necessary consequences of the very universality of the physical law, although the phenomenon was not so general as was at first imagined. He has gone farther, and has pointed out some other minute deviations which must result from the physical law, but which the art of observation was not then sufficiently advanced to discover in the phenomena. This excited the efforts of men of science to improve the art of astronomical observation; and not only have the intimations of Newton been verified by modern observation, but other deviations have been discovered, and, in processes of time, have also been shown to be consequences of the same general law of agency: And, at this present day, there is not a single anomaly of the planetary motions which has not been shown to be a modification of one general law which regulates the action; and therefore characterizes the nature of that single force which actuates the whole system of the sun, and his attending planets and comets.
It was a most fortunate circumstance that the constitution of the solar system was such that the deviations from the general law are not very considerable. The case might have been far otherwise, although the law, or nature of the planetary force, were the same, and the system had been equally harmonious and beautiful. Had two or three of the planets been vastly larger than they are, it would have been extremely difficult to discover any laws of their motion sufficiently general to have led to the supposition or the discovery of the universal law of action, or the specific circumstance in the planetary force which distinguishes it from all others, and characterizes its nature. But the three laws of the planetary motions discovered by Kepler were so nearly true, at least with respect to the primary planets, that the deviations could not be observed, and they were thought to be exact. It was on the supposition that they were exactly that Newton affirmed that they were only modifications of one law still more general, nay universal.
We shall follow in order the steps of this investigation.
Sir Isaac Newton took it for granted, that the sun and planets consisted of matter which resembled those by which bodies which we daily handle, at least in respect of their mobility; and that the forces which agitate them, considered merely as moving forces, but without considering or attending to their mode of operation, were to be inferred, both as to their direction and as to their intensity, from the changes of motion which were ascribed to their agency. He first endeavoured to discover the direction of that transferre force by which the planets are made to describe curve lines. Kepler's first law furnished him with ample means for this discovery. Kepler had discovered, that the right line joining the sun and any planet described areas proportional to the times. Newton demonstrated, that if a body was carried round a fixed point situated in the plane of its motion, that the right line joining it with that point described areas proportional to the times, the force which deflected it from an uniform rectilinear motion was continually directed to that fixed point. This makes the 2d proposition of his immortal work "The Mathematical Principles of Natural Philosophy," and it is given in the article "Astronomy" of the Encyclopedia Britannica, § 260.
Hence Sir Isaac Newton inferred, that the primary planets were retained in their orbits by a force continually directed to the sun; and, because Kepler's law of motion was also observed by the secondary planets... in their revolutions round their respective primary planets, this inference was extended to them.
From the circumstance that the planetary deflecting forces in the different points of the orbit are always directed toward one point as to a centre, they have been called centripetal forces.
From this proposition may be deduced a corollary which establishes a general law of the motion of any planet in the different parts of its orbit, namely, that the velocity which a planet has in the different points of its path are inversely proportional to the perpendiculars drawn from the sun on the tangents to the orbit at those points respectively. For, let \(AB\), \(ab\) (fig. 3.) be two arches (extremely small), described in equal times; these arches must be ultimately proportional to the velocities with which they are described. Let \(SP\), \(Sp\) be perpendicular to the tangents \(AP\), \(ap\). The triangles \(ASB\), \(aSB\) are equal, because equal areas are described by the radii vectors \(SA\), \(Sa\), in equal times; but in equal triangles, the bases \(AB\), \(ab\) are reciprocally as their heights \(SP\), \(Sp\), or \(AB : ab = SP : Sp\).
This corollary gives us another expression of the ratio of the centripetal forces in different points \(A\) and \(a\) of a curve. We saw by a former proposition, that the force at \(A\) (fig. 2.) is to the force at \(a\) as \(AC \times ae\) to \(ae \times AE\), which we may express thus: \(F : f = V^2 \times c : v^2 \times C\). If we express the perpendiculars \(SP\), \(Sp\) (in fig. 3.) by the symbols \(Pp\), \(Pp\), we have \(V^2 : v^2 = P^2 : P^2\), and therefore \(F : f = P^2 \times c : P^2 \times C\). The centripetal forces in different points of an orbit are in the ratio compounded of the inverse duplicate ratio of the perpendiculars drawn to the tangents at those points from the centre of forces, and the inverse ratio of the defective chords of the equicurves circles.
We are now in a condition to determine the law of action of the centripetal force by which a planet is retained in its orbit round the sun, or the relation which subsists between the intensity of its action and the distance of the planet from the sun; for we know the elliptical figure of the orbit, and we can draw a tangent to it in any point, and a perpendicular from the sun to that tangent.
Kepler's second law or observation of the planetary motions was, that each primary planet described an ellipse, having the sun in one focus. It is easy to show, even without any knowledge of the geometrical properties of the ellipse, what is the proportion of the intensities of the deflecting force at the aphelion and perihelion (see fig. 4.). At those two points of the orbit, the motion of the planet is at right angles to the line joining it with the sun. Therefore, since the areas described in equal times are equal, the arches described in equal times must be inversely at the distances from the sun; or the velocities must be inversely as the distances from the sun. But the curvature in the aphelion and perihelion is the same; and therefore the diameters of the equicurve circles in those points are equal. But those diameters are, in this particular case, what we called the defective chords. Therefore, calling the aphelion and perihelion distances \(D\) and \(d\), the velocities in the aphelion and perihelion \(V\) and \(v\), let the common defective chord be \(C\). Then we have \(F : f = V^2 \times C : v^2 \times C\), \(= V^2 : v^2 = d^2 : D^2\). That is, the forces which deflect the planet in the aphelion and perihelion are inversely as the squares of the distances from the sun. A person almost ignorant of mathematics may see the truth of this by looking into a table of natural versed sines. He will observe, that the versed sine of one degree is quadruple the versed sine of half a degree, and fifteen times the versed sine of a quarter of a degree; in short, that the versed sines of small arches are in the proportion of the squares of the arches. Now since the arches described in equal times are inversely as the distances, their versed sines are inversely as the squares of the distances. But these versed sines are the spaces through which the centripetal forces at the aphelion and perihelion deflect the planet from the tangent. Therefore, &c.
Thus we have found, that in the aphelion and perihelion the centripetal force acts with an intensity that is proportional to the squares of the distances inversely. As these are the extreme situations of a planet, and as the proportion of the aphelion and perihelion distances are considerably different in the different planets, and yet this law of action is observed in them all, it is reasonable to imagine that it holds true, not in those situations only, but in every intermediate situation. But a conjecture, however probable, is not sufficient, when we aim at accurate science, and it is necessary to examine whether this law of action is really observed in every point of the elliptical orbit.
For this purpose it is necessary to mention some geometrical properties of the ellipse. Therefore let \(ABDE\) be the elliptical orbit of a planet or comet, \(S\) being the sun in the focus \(S\). Let \(AB\) be the transverse axis, and \(DE\) the conjugate axis, and \(C\) the centre. Let \(P\) be any point of the ellipse. Draw \(PS\) through the focus. Draw the tangent \(PN\), and \(SN\) from the focus, perpendicular to \(PN\). Draw \(PQ\) perpendicular to \(PN\), meeting the transverse axis in \(Q\). Draw \(QO\) parallel to \(PN\), meeting \(PS\) in \(O\). Also draw \(QR\) perpendicular to \(PS\). Bisection \(PO\) in \(T\).
It is demonstrated in the treatises of conic sections, that \(PO\) is one half of the chord of the equicurve or osculating circle drawn through the point \(P\). Therefore \(PO\) is one half of the defective chord of the planetary orbit. It is also demonstrated, that \(PR\) is one half of the parameter or \(latus rectum\) of the transverse axis \(AB\), or that it is the third proportional to \(AC\) and \(DC\). Therefore \(PR\) or \(Dr\) is of the same constant magnitude, in whatever part of the circumference the point \(P\) is taken.
It is evident that the triangles \(NSP\), \(RPO\), and \(QPO\), are all similar, by reason of the parallels \(PN\), \(QO\), and the right angles \(SNP\), \(PRQ\), \(PQO\). Therefore we have \(PR : PQ = PO : PO\). Therefore \(PR : PO = PR : PQ = SN : SP\). Therefore \(PR \times SP = PO \times SN\). But the \(latus rectum\) \(L\) is equal to twice \(PR\), and the defective chord \(C\) is equal to twice \(PO\). Therefore \(L \times SP = C \times SN\). But we have seen, that when a curve is described by means of a centripetal force, so that areas are described proportional to the times, and therefore the velocities are reciprocally proportional to the perpendiculars drawn from the centre of forces to the tangents, the forces are inversely proportional to \(C \times SN\). Therefore, in the elliptical motion of the planets, the forces are inversely proportional to \(L \times SP\); and since \(L\) is a constant quantity, the centripetal forces are inversely proportional to \(SP\), or to the squares of the distances from the sun.
Thus Thus it appears that, with respect to any individual planet, the centripetal force which continually deflects it from the tangent to its orbit diminishes in the inverse duplicate ratio of the distance from the sun. The same thing is observed to be very nearly true in the moon's motion round the earth, and in the motion of such satellites of Jupiter and Saturn as describe orbits which are sensibly elliptical. It is also observed in the motion of the comets, at least in that which appeared in 1682 and in 1759.
It was therefore very natural for Sir Isaac Newton to examine whether the like diminution of force obtained in the action of this force on different planets; that is, whether the deflection of the earth from the tangent of its orbit was to the simultaneous deflection of Jupiter as the square of Jupiter's distance from the sun to the square of the earth's distance. This was very probable, but by no means certain. Its probability is very great indeed, when we know that a comet moves so in its orbit that its deflections in equal times are inversely as the squares of its distances from the sun, and that the comet passes through the orbits of all the planets; and when at the same distance from the sun as any one of them, it suffers the same deflection with it. Newton therefore calculated the actual simultaneous deflections of the different planets, and found them agreeable to this law. But it was desirable to obtain a demonstration of this important proposition in general terms. This was supplied by Kepler's third general observation of the motions, viz., that the squares of the periodic times of the different planets were proportional to the cubes of their mean distances from the sun. The orbits of the planets are so nearly circular, that we may suppose them exactly so in the present question, without any remarkable error. In this case, then, the deflective chords are the diameters of the orbits (for DS is equal to AC), and are proportional to the distances, which are their halves. The centripetal forces, being proportional to $\frac{d^3}{S^2}$, are proportional to $\frac{d^3}{S^2}$, when $d$ is the radius of the orbit, or the mean distance from the sun. But the velocity in a circular orbit is as the circumference directly, and as the time of a revolution inversely. Therefore, instead of $\frac{d^3}{S^2}$, we may write $\frac{d^3}{S^2}$, and then the forces will be proportional to $\frac{d^3}{S^2}$, or to $\frac{d^3}{S^2}$; that is, directly as the distances, and inversely as the squares of the times of revolution. But, by Kepler's observation, $t^2$ is proportional to $d^3$. Therefore the centripetal forces are proportional to $\frac{d^3}{S^2}$, or to $\frac{d^3}{S^2}$; that is, inversely as the squares of the mean distances from the sun.
But since the orbits of the planets are not accurate circles, this determination is but an approximation to the truth, and therefore insufficient for the foundation of so important a proposition; at any rate, it will not apply to the comets, whose orbits are very far from being circular. We must obtain a more accurate demonstration.
Therefore draw SD (fig. 4.) to the extremity of the conjugate axis, and bisect it in $t$. About $S$, with the radius SD, describe the circle DFG. Let $D_1$, $D_2$ be equal small arches of the ellipse and the circle. Join $dS$, $tS$. It is well known that DS is half of the chord of the equicurve circle at D, and therefore $D_1$ is one fourth part of it. It has been demonstrated, that the velocity in any point D of a curve, described by means of a deflecting force, is that which the force in that point would communicate to it by uniformly impelling it along the fourth part of the deflective chord, that is, along $D_1$. But if a body revolved round S in a circle DFG, its velocity in that circle would be that which the deflecting force would communicate to it by uniformly impelling it along one-fourth of the diameter, that is, along $D_1$. Therefore the planet, if projected in the direction $D_1$, with the velocity which it has in the point D of the ellipse, would describe the circle DFG by the action of the centripetal force. Farther, it would describe it in the same time that it describes the ellipse; for because the velocities are equal, the areas $DS_1$, $DS_2$ are described in the same time. But the halves $D_1$, $D_2$ being equal, these areas are as their heights $S_1$ (or CD), and $SD$ (or CA). But because the diameter of the circle is equal to AB, the area of the whole ellipse is to the area of the circle as CD is to CA; that is, as the area $DS_1$ to the area $DS_2$ described in the same time. Therefore the elliptical and circular areas are similar portions of the ellipse and circle; and therefore the times of describing them are similar portions of the whole revolutions in the ellipse and in the circle. Therefore these revolutions are performed in equal times.
And thus it follows, that if all the planets and comets were projected, when at their mean distances from the sun, perpendicularly to the radius vector, they would describe circles round the sun, and the squares of their periodic times would be proportional to the cubes of their mean distances from the sun, as Kepler has observed; and therefore the centripetal forces would be inversely as the squares of their distances from the sun.
They are not different forces therefore which retain the different planets in their respective orbits, but one force acts retaining by the same law upon them all. We may either conceive it as an attractive force, exerted by the sun, or as a tendency in each planet; nay, nothing hinders us from conceiving it as a force external, both to sun and same force, planets, impelling them towards the sun. It may be the impulse of a stream of fluid moving continually toward the sun. Sir Isaac Newton did not concern himself with this question, but contented himself with the discovery of the law according to which its action was exerted. The steps of this investigation showed him, that a body, projected in any direction whatever, and with any velocity whatever, and subjected to the action of a force directed to the sun, and inversely proportional to the square of the distance from the sun, will necessarily describe a conic section, having the sun in the focus. This will be a parabola, if the velocity of projection be that which the centripetal force in that place would communicate to the body by acting on it uniformly along a line equal to half its distance from the sun. If the velocity be greater than this, the path will be a hyperbola; if the velocity be less than this, the path will be an elliptical orbit, in which the body will revolve forever round the sun.
The 3rd Keplerian law is also observed in the revolutions of the satellites of Jupiter, Saturn, and the lately discovered...