ASTRONOMY

A S T R O N O M Y

IS a science which has been cultivated from the earliest ages, and is conversant about the most sublime objects of inquiry which can employ the mind of man. It has accordingly been treated at great length in the Encyclopædia 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 characterised 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.

balance in the ultimate effect alone which disposes us to employ the term in such cases: but we find no great inconvenience in the want of another term.

We farther find, that our exertion is necessary, not only for producing motion where there was none before, but also for producing any change of motion; and accurate observation shows us, that the same force is required for changing a motion by any given quantity, as for producing that quantity where there was none before.

Lastly, we are conscious of exerting force when we resist the exerted force of another; and that an exertion, perfectly similar to this, will prevent some very familiar tendencies to motion in the bodies around us: thus an exertion is necessary for carrying a weight, that is, for preventing the fall of that weight.

All these resemblances between the effects of our forcible exertions and the changes of motion which accompany the meeting, and sometimes the mere vicinity of other bodies, justify us in the use of this figurative language. The resemblance is found to be the more perfect as we observe it with more care, and, in short, appears to be without exception. Bodies are therefore said to act on each other, to resist each other, to resist a change of motion, &c.

Therefore, wherever we observe a change of motion, we infer the existence and exertion of a changing force; and we infer the direction of that exertion from the direction of the change; and the quantity of the exertion, or intensity of the force, from the quantity of the change.

The study of the causes of the celestial motions is therefore hardly different from the study of the motions themselves; since the agency, the kind, and the degree of the moving force, are immediate inferences from the existence, the kind, and the quantity of the change of motion.

Our notion of a moving power is that of a power which produces motion, that is, a successive change of place. Continuation of the motion produced is therefore involved in the very notion of the production of motion; therefore the continued agency of the moving power, or of any power, is not necessary for the continuation of the motion. Motion is considered as a state or condition of the body; there is not any exertion of power therefore in the continuation of motion: But every change is indicative of a changing cause; and when the change is the same, in all its circumstances, the cause is necessarily conceived to be the same or equal.

The condition of a body, in respect of motion, can differ from that of another equal body only in its direction and in its velocity. If the directions are the same, the difference of conditions can only be in the difference of velocity. One body has a determination, by which it would describe ten feet uniformly in a second, if nothing changed this determination; the other has a determination, by which it would describe twenty feet in a second. Each of these determinations are supposed to be the effects of forces acting similarly in every respect. Therefore these determinations are the only measures of these two forces; that is, moving forces are conceived by us as having the proportion of the velocities which they produce in a body by acting in a manner perfectly similar.

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 action. In such a motion, the changes of velocity are proportional to the times from the beginning of the action; 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 ostensible 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 of the acquired velocity, although there is not, in fact, any space observed to be uniformly described during any time whatever. In this motion we know that the 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, 80 feet were described in two seconds, and therefore 20 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 20 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.

7. Measure of the velocity 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, 2d, 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.

9. Measure of these forces obtained. 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, ed 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, and AD = $\backslash frac\{AC^2\}\{AE\}$. For similar reasons $ad\; =\; \backslash frac\{ac^2\}\{ae\}$. But AD and ad are respectively equal to BC and bc. Therefore BC = $\backslash frac\{AC^2\}\{AE\}$ and bc = $\backslash frac\{ac^2\}\{ae\}$. Therefore BC:bc = $\backslash frac\{AC^2\}\{AE\}\; :\; \backslash frac\{ac^2\}\{ae\}$, or BC:bc = $AC^2\; \backslash times\; ae\; :\; ac^2\; \backslash times\; AE$. 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 ae 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, $F:f\; =\; \backslash frac\{V^2\}\{C\}\; :\; \backslash frac\{v^2\}\{c\}$; or thus, in a proportional equation, $f\; =\; \backslash frac\{v^2\}\{c\}$.

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 $ae$ 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 $f\; =\; \backslash frac\{2v^2\}\{c\}$. 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 coincides 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.

(fig. 2.) 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 cord AE or the equicurve circle. It is the same 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 deflecting 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 earlier 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 characterises 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 suspicion or the discovery of the universal law of action, or the specific circumstance in the planetary force which distinguishes it from all others, and characterises 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 exact, 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 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 transverse 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 so 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 Encyclopaedia 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

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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 in 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 $AC^2\; \backslash times\; ae$ to $ae\; \backslash times\; AE$, which we may express thus: $F\; :\; f\; =\; V^2\; \backslash times\; c\; :\; v^2\; \backslash times\; C$. If we express the perpendiculars $SP,\; Sp$ (in fig. 3.) by the symbols $P,\; p$, we have $V^2\; :\; v^2\; =\; P^2\; :\; P^2$, and therefore $F\; :\; f\; =\; P^2\; \backslash times\; c\; :\; P^2\; \backslash times\; C$. 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 in those points from the centre of forces, and the inverse ratio of the deflective chords of the equicurve 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 as 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 deflective chords. Therefore, calling the aphelion and perihelion distances $D$ and $d$, the velocities in the aphelion and perihelion $V$ and $v$, let the common deflective chord be $C$. Then we have $F\; :\; f\; =\; V^2\; \backslash times\; C\; :\; v^2\; \backslash times\; C$, $=\; V^2\; :\; v^2$, $=\; D^2\; :\; d^2$. 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 sixteen 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 $ADBE$ (fig. 4.) be the elliptical orbit of a planet or comet, having 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 $PS$. Bisect $PO$ 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 deflective 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 $D$ is of the same constant magnitude, in whatever part of the circumference the point $P$ is taken.

It is evident that the triangles $NSP,\; RPQ$, and $QPO$, are all similar, by reason of the parallels $PN,\; QO$, and the right angles $SNP,\; PRQ,\; PQO$. Therefore we have $PR\; :\; PQ\; =\; PQ\; :\; PO$. Therefore $PR\; :\; PO\; =\; PR^2\; :\; PQ^2$, $=\; SN^2\; :\; SP^2$. Therefore $PR\; \backslash times\; SP^2\; =\; PO\; \backslash times\; SN^2$. But the latus rectum $L$ is equal to twice $PR$, and the deflective chord $C$ is equal to twice $PO$. Therefore $L\; \backslash times\; SP^2\; =\; C\; \backslash times\; SN^2$. 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 $C\; \backslash times\; SN^2$. Therefore, in the elliptical motion of the planets, the forces are inversely proportional to $L\; \backslash times\; SP^2$; and since $L$ is a constant quantity, the centripetal forces are inversely proportional to $SP^2$, or to the squares of the distances from the sun.

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 $\backslash frac\{v^2\}\{d\}$, are proportional to $\backslash frac\{v^2\}\{d^2\}$, 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 $v^2$, we may write $\backslash frac\{d^2\}\{t^2\}$, and then the forces will be proportional to $\backslash frac\{d^2\}\{t^2\; d^2\}$, or to $\backslash frac\{1\}\{t^2\}$; 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 $\backslash frac\{d^2\}\{d^3\}$, or to $\backslash frac\{1\}\{d^2\}$; 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 $s$. About $S$, with the radius $SD$, describe the circle $DFG$. Let $Dd$, $Dd\text{'}$ be equal small arches of the ellipse and the circle. Join

$dS$, $sS$. It is well known that $DS$ is half of the chord of the equicurve circle at $D$, and therefore $Ds$ 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 $Ds$. 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 $Ds$. Therefore the planet, if projected in the direction $Ds$, 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 $DSd$, $DSs$ are described in the same time. But the bases $Dd$, $Ds$ being equal, these areas are as their heights $Sn$ (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 $DSd$ to the area $DSs$ 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 radii vectors, 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, acting 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 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 for ever round the sun.

The 3d Keplerian law is also observed in the revolutions of the satellites of Jupiter, Saturn, and the lately discovered

discovered planet; and we must infer from it, that they are retained in their orbits round their respective primary planets, by forces whose intensity decreases according to the same law of the distances. Also the elliptical motion of the moon round the earth, shows that the force by which she is retained in her orbit varies in the same proportion of the distances. But when we compare the motion of a satellite of Jupiter with that of one of the satellites of the other two planets, we find that the proportion does not hold. We shall find, that, at equal distances from Jupiter and Saturn, the force toward Jupiter is almost thrice as great as the force toward Saturn. We shall also find that the force toward Jupiter is three hundred times greater than the force which retains the moon in its elliptical orbit round the earth, when acting at the same distance.

Since a force directed to the sun, and inversely as the square of the distance, is thus found to pervade all the planetary orbits, it is highly improbable that it will not affect the secondary planets also. The moon accompanies the earth in its motion round the sun. It may appear sufficient for this purpose, that the moon be retained in its orbit by a force directed to the earth. Were the moon connected with the earth by a rope or chain, this would be true; for the earth could get no motion without dragging the moon along with it: but it is quite otherwise with bodies moving in free space, without any material connections. When a body that is moving uniformly in a straight line is accompanied by another which describes around it areas proportional to the times, the force which continually deflects this satellite is always directed to the moving central body. This is easily seen; for whatever be the mutual action of two bodies, and their relative motions in consequence of this action, if the same velocity be impressed at once on both bodies in the same direction, their mutual actions and relative motions will be the same as they would have been without this common impulse. Thus every thing is done in a ship that is sailing steadily in the same manner as if she were at rest. If therefore the moon be observed to describe areas round the earth, which are precisely proportional to the times, while the earth moves in an orbit round the sun, we must infer that the moon receives, in every instant, an impulse the same in every respect with what the earth receives at the same instant; or that the moon is acted on by a force parallel to the earth's distance from the sun, and proportional to the square of that distance inversely. Now this is very nearly true of the lunar motions; and we must infer that the moon is subjected to this solar action, or this tendency to the sun. The same must be affirmed of the satellites of the other planets.

But a force inversely proportional to the square of the earth's distance from the sun is not what the universality of the law requires: It must be inversely as the square of the moon's distance from the sun: and it must not be parallel to the earth's distance from the sun, but must be directed toward the sun; and therefore, in the quadratures, it must converge to the earth's radius vector. Therefore, since a force having the above mentioned conditions will allow the description of areas round the earth exactly proportional to the times, a force acting on the moon, inversely proportional to the square of her distance from the sun, and directed exactly to the sun, is incompatible with the accurate elliptical

motion round the earth. At new moon, her tendency to the sun exceeds the earth's tendency to him; and this excess will diminish her tendency to the earth, and her motion will be less incurvated, so that she will retire a little from the earth. At full moon, the earth's tendency to the sun exceeds the moon's tendency to him, and the earth will separate a little from the moon, so that the relative orbit will again be less incurvated. In the quadratures, the impulse on the moon is indeed equal to that on the earth, but not parallel, and tends to make the moon approach the earth, and increase the curvature of her orbit. In other situations of the moon, this want of equality and parallelism of the forces acting on the earth and moon, must produce other disturbances of the regular elliptical motion.

Newton saw this at once; and, to his great delight, he saw that the great deviations from regular motion, which had been discovered by Ptolemy and Tycho Brahe, called the Annual Equation, the Variation, and the Evection, were such as most obviously resulted from the regular influence of the sun on the moon. The first deviation from the regular elliptical motion is occasioned by the increase of the sun's disturbing force as the earth approaches the perihelion; and it enlarges the lunar orbit, by diminishing the tendency to the earth, and increases the periodic time. The second arises from the direction of the disturbing force, by which it accelerates the moon's angular motion in the second and fourth quadrants of her orbit, and retards it in the first and third. The last affects the eccentricity of the orbit, by changing the ratio of the whole or compound tendency of the moon to the earth in her perigee and apogee. This success incited him to an accurate examination of the consequences of this influence. It is the boast of this discovery of the law of the planetary deflections, that all its effects may be calculated with the utmost precision. The part of the moon's deflection toward the sun, which is neither equal nor parallel to the simultaneous deflection of the earth, may be separated from the part which is equal and parallel to it, and it may be called the sun's disturbing force. Its proportion to the moon's deflection towards the earth may be accurately ascertained, and its inclination to the line of the moon's motion in every point of her orbit may be pointed out. This being done, the accumulated effect of this disturbing force after any given time, however variable, both in direction and intensity during this time, may be determined by the 39th and other propositions of the first book of the Mathematical Principles of Natural Philosophy. And thus may the moon's motion, when so disturbed, be determined and compared with her motion really observed.

All this has been done by Sir Isaac Newton with the most astonishing address and sagacity, sua mathesi facem preferens, partly in the PRINCIPIA, and partly in his LUNAR THEORIA. This investigation, whether we consider the complete originality of the whole process, or the ingenuity of the method, or the sagacity in seeing and clearly discriminating the different circumstances of the question, or the wonderful fertility of resource, or the new and most refined mathematical principles and methods that he employed—must ever be considered as the most brilliant specimen of human invention and reasoning that ever was exhibited to the world.

In this investigation Newton not only determined the quantity,

22
The satellites of all the planets subjected to this solar action.

23
May be calculated with precision.

quantity, the period, and the changes of those inequalities, which had been so considerable and remarkable as to be observed by former astronomers, and this with an exactness far surpassing what could ever be attained by mere observation; but he also pointed out several other periodical inequalities, which were too small, and too much implicated with the rest, ever to be discovered or to be separated from them. We do not say that he completed the theory of the lunar motions; but he pointed out the methods of investigation, and he furnished all the means of prosecuting it, by giving the world the elements of a new species of mathematics, without which it would have been in vain to attempt it. Both this new mathematics, and the methods of applying it to such questions, have been assiduously studied and improved by the great mathematicians of this century; and the lunar theory has been carried to such a degree of perfection, that we can compute her place in the heavens for any past age without deviating above one minute of a degree from the actual observation.

There is one empirical equation of the moon's motion which the comparison of ancient and modern eclipses obliges the astronomers to employ, without being able to deduce it, like the rest, a priori, from the theory of an universal force inversely proportional to the square of the distance. It has therefore been considered as a stumbling block in the Newtonian philosophy. This is what is called the secular equation of the moon's mean motion. The mean motion is deduced from a comparison of distant observations. The time between them, being divided by the number of intervening revolutions, gives the average time of one revolution, or the mean lunar period. When the ancient Chaldean observations are compared with those of Hipparchus, we obtain a certain period; when those of Hipparchus are compared with some in the 9th century, we obtain a period somewhat shorter; when the last are compared with those of Tycho Brahe, we obtain one still shorter; and when Brahe's are compared with those of our day, we obtain the shortest period of all—and thus the moon's mean motion appears to accelerate continually; and the accelerations appear to be in the duplicate ratio of the times. The acceleration for the century which ended in 1700 is about 9 seconds of a degree; that is to say, the whole motion of the moon during the 17th century must be increased 9 seconds, in order to obtain its motion during the 18th; and as much must be taken from it, or added to the computed longitude, to obtain its motion during the 16th; and the double of this must be taken from the motion during the 15th, to obtain its motion during the 14th, &c. Or it will be sufficient to calculate the moon's mean longitude for any time past or to come by the secular motion which obtains in the present century, and then to add to this longitude the product of 9 seconds, multiplied by the square of the number of centuries which intervene. Thus having found the mean longitude for the year 1200, add 9 seconds, multiplied by 36, for six centuries. By this method we shall make our calculation agree with the most ancient and all intermediate observations. If we neglect this correction, we shall differ more than a degree from the Chaldean observations of the moon's place in the heavens.

If mathematicians having succeeded so completely in deducing all the observed inequalities of the planets.

ry motions, from the single principle, that the deflecting forces diminished in the inverse duplicate ratio of the distances, were fretted by this exception, the reality of which they could not contest. Many opinions were formed about its cause. Some have attempted to deduce it from the action of the planets on the moon; others have deduced it from the oblate form of the earth, and the translation of the ocean by the tides; others have supposed it owing to the resistance of the ether in the celestial spaces; and others have imagined that the action of the deflecting force requires time for its propagation to a distance: But their deductions have been proved unsatisfactory, and have by no means the precision and evidence that have been attained in the other questions of physical astronomy. At last M. de la Place, of the Royal Academy of Sciences at Paris, has happily succeeded, and deduced the secular equation of the moon from the Newtonian law of planetary deflection. It is produced in the following manner.

Suppose the moon revolving round the earth, undisturbed by any deflection toward the sun, and that the time of her revolution is exactly ascertained. Now let the influence of the sun be added. This diminishes her tendency to the earth in opposition and conjunction, and increases it in the quadratures; but the diminutions exceed the augmentations both in quantity and duration; and the excess is equivalent to $\backslash frac\{1\}\{27\}$th of her tendency to the earth. Therefore this diminished tendency cannot retain the moon in the same orbit; she must retire farther from the earth, and describe an orbit which is less incurved by $\backslash frac\{1\}\{27\}$th part; and she must employ a longer time in a revolution. The period therefore which we observe, is not that which would have obtained had the moon been influenced by the earth alone. We should not have known that her natural period was increased, had the disturbing influence of the sun remained unchanged; but this varies in the inverse triplicate ratio of the earth's distance from the sun, and is therefore greater in our winter, when the earth is nearer to the sun. This is the source of the annual equation, by which the lunar period in January is made to exceed that in July nearly 24 minutes. The angular velocity of the moon is diminished in general $\backslash frac\{1\}\{27\}$th, and this numerical coefficient varies in the inverse ratio of the cube of the earth's distance from the sun. If we expand this inverse cube of the earth's distance into a series arranged according to the sines and cosines of the earth's mean motion, making the earth's mean distance unity, we shall find that the series contains a term equal to $\backslash frac\{1\}\{2\}$ of the square of the eccentricity of the earth's orbit. Therefore the expression of the diminution of the moon's angular velocity contains a term equal to $\backslash frac\{1\}\{27\}$th of this velocity, multiplied by $\backslash frac\{1\}\{2\}$ of the square of the earth's eccentricity; or equal to the product of the square of the eccentricity, multiplied by the moon's angular velocity, and divided by 119,33 ($\backslash frac\{1\}\{2\}$ of 179). Did this eccentricity remain constant, this product would also be constant, and would still be confounded with the general diminution, making a constant part of it: but the eccentricity of the earth's orbit is known to diminish, and its diminution is the result of the universality of the Newtonian law of the planetary deflections. Although this diminution is exceedingly small, its effect on the lunar motion becomes sensible by accumulation in the course of ages. The eccentricity diminishing, the diminution

nution of the moon's angular motion must also diminish, that is, the angular motion must increase.

During the 18th century, the square of the earth's eccentricity has diminished 0,000015325, the mean distance from the sun being = 1. This has increased the angular motion of the moon in that time 0,000001285. As this augmentation is gradual, we must multiply the angular motion during the century by the half of this quantity, in order to obtain its accumulated effect. This will be found to be 9' very nearly, which exceeds that deduced, from a most careful comparison of the motion of the last two centuries, only by a fraction of a second!

As long as the diminution of the square of the eccentricity of the earth's orbit can be supposed proportional to the time, this effect will be as the squares of the times. When this theory is compared with observations, the coincidence is wonderful indeed. The effect on the moon's motion is periodical, as the change of the solar eccentricity is, and its period includes millions of years. Its effect on the moon's longitude will amount to several degrees before the secular acceleration change to a retardation.

Those who are not familiar with the disquisitions of modern analysis, may conceive this question in the following manner.

Let the length of a lunar period be computed for the earth's distance from the sun for every day of the year. Add them into one sum, and divide this by their number, the quotient will be the mean lunar period. This will be found to be greater than the arithmetical medium between the greatest and the least. Then suppose the eccentricity of the earth's orbit to be greater, and make the same computation. The average period will be found still greater, while the medium between the greatest and least periods will hardly differ from the former. Something very like this may be observed without any calculation, in a case very similar. The angular velocity of the sun is inversely as the square of his distance. Look into the solar tables, and the greatest diurnal motion will be found 3673", and the least 3433". The mean of these is 3553", but the medium of the whole is 3548". Now make a similar observation in tables of the motion of the planet Mars, whose eccentricity is much greater. We shall find that the medium between the greatest and least exceeds the true medium of all in a much greater proportion.

Thus has the patient and assiduous cultivation of the Newtonian discoveries explained every phenomenon, and enabled us to foresee changes in them which no examination of the past appearances, unassisted by this theory, could have pointed out, and which must have exceedingly embarrassed future astronomers. This great but simple law of deflection represents every phenomenon of the system in the most minute circumstances. Far from fearing that future experience may overturn this law, we may rest assured that it will only confirm it more and more; and we may confide in its most remote consequences as if they were actually observed.

It is discovered by observation, that the deflection of the moon to the earth, and of the planets to the sun, are accompanied by an equal and opposite deflection of the earth to the moon, and of the sun to the planets.

The tendency of the earth to the moon is plainly indicated by the rise of the waters of the ocean under the

moon, and on the opposite side of the earth. Sir Isaac Newton tried what should be the result of a tendency of the water to the moon. His investigation of this question was very similar to that in his lunar theory. We may conceive the moon to be one of many millions of particles of a fluid, occupying a globe as big as the lunar orbit. Each will feel a similar disturbing force, which will diminish its tendency to the earth in the neighbourhood of the place of conjunction and opposition, and will increase it in the neighbourhood of the quadratures. They cannot therefore remain in equilibrium in their spherical form; they must sink in the quadratures, and rise in the conjunction and opposition, till their greater height compensates for the diminished weight of each particle. In like manner, the waters of the ocean must sink on those parts of the earth where the moon is seen in the horizon, and must rise in those which have the moon in the zenith or nadir. All these effects are not only to be seen in general, but they may all be calculated, and the very form pointed out which the surface of the ocean must assume: and thus a tendency of every particle of the ocean to the moon, inversely proportional to the square of its distance from it, gives us a theory of the ebbing and flowing of the sea. This is delivered in sufficient detail in the article TIDE of the Encyclopædia Britannica, and therefore need not be inserted on in this place. The same inference must be drawn from the precession of the equinoxes produced by the action of the moon on the protuberant matter of our equatorial regions. See PRECESSION in the Encycl.

But the mutual tendency of the earth and moon is clearly seen in a phenomenon that is much more simple. If we compute the sun's place in the heavens, on the supposition that the earth describes areas proportional to the times, we shall find it to agree with observation at every new and full moon: But at the first quarter the sun will be observed about 9 seconds too much advanced to the eastward; and at the last quarter he will be as much to the westward of his calculated place. In all intermediate positions, the deviation of the observed from the computed place of the sun will be 9 seconds, multiplied by the sine of the moon's distance from conjunction or opposition. In short, the appearances will be the same as if it were not the earth which described areas proportional to the times round the sun, but that a point, lying between the earth and moon, and very near the earth's surface, were describing the ellipse round the sun, while the earth and moon revolve round this point in the course of a lunation, having the point always in the line between them, in the same manner as if they were on the extremities of a rod which turns round this point, while the point itself revolves round the sun.

This then is the fact with respect to the motions; and the earth in a month describes an orbit round this common centre of the earth and moon. It cannot do this unless it be continually deflected from the tangent to this orbit; therefore it is continually deflected toward the moon: and the momentum of this deflection, that is, its quantity of motion, is the same with that of the moon's deflection, because their distances from the common centre are as their quantities of matter inversely.

Appearances perfectly similar to these oblige us to affirm

affirm that the sun is continually deflected toward the planets. Astronomical instruments, and the art of observing, have been prodigiously improved since Sir Isaac Newton's time; and the most scrupulous attention has been paid to the sun's motion, because it is to his place in the universe that continual reference is made in computing the place of all the planets. He is supposed at rest in the common focus of all their orbits; and the observed distance of a planet from the sun is always considered as the radius vector. If this be not the case, the orbital motions contained in our tables are not the absolute motions of the planets, nor the deflections from the tangents the real deflections from absolute rectilinear motion; and therefore the forces are not such as we infer from those mistaken deflections. Accordingly Sir Isaac Newton, induced by certain metaphysical considerations, assumed it as a law of motion, that every action of a body A on another body B, is accompanied by an equal and contrary action of B on A. We do not see the propriety of this assertion as a metaphysical axiom. It is perfectly conceivable that a piece of iron will always approach a magnet when in its neighbourhood; but we do not see that this obliges us to assert that therefore the magnet will also approach the iron. Those who explain the phenomena of magnetism by the impulse of a fluid, must certainly grant that there is no metaphysical necessity for another stream of fluid impelling the magnet toward the iron. And accordingly this, and the similar reciprocity in the phenomena of electricity, have always been considered as deductions of experimental philosophy; yet we observe the same reciprocity in all the actions of sublimary bodies; and Newton's third law of motion is received as true, and admitted as a principle of reasoning. But we apprehend that it was hasty in this great philosopher, and unlike his scrupulous caution, to extend it to the planetary motions. He did, however, extend it, and asserted, that as each planet was deflected toward the sun, the sun was equally (in respect of momentum) deflected toward each planet, and that his real motion was the composition of all those simultaneous deflections. He asserted, that there was a certain point round which the sun and his attending planets revolved; and that the orbit of a planet, which our measurements determined by continual reference to the sun as to a fixed body, was not the true orbit, but consisted of the contemporaneous orbits of that planet and of the sun round this fixed point. Any little sector of the apparent orbit was greater than the corresponding sector of the planet's true orbit in absolute space, and the apparent motion was compounded of the true motion of the planet, and the opposite to the true motion of the sun. After a most ingenious and refined investigation, he showed that, notwithstanding this great difference of the Keplerian laws from the truth, the inference, with respect to the law of planetary deflection, is just, and that not only the apparent deflections are in the inverse duplicate ratio of the distances from the sun, but that the real deflections vary in the same ratio of the distances from the fixed point, and also from the sun; for he showed that the distances from the sun were in a constant ratio to the distances from this point. He showed also that the same forces which produced the contemporaneous revolution of a planet and the sun round the centre of the system, would produce a revolution of

the planet in a similar orbit round the sun (supposed to be held fast in his place) at the same distance which really obtains between them, with this sole difference, that the periodic time will be longer, in the subduplicate ratio of the quantity of matter in the sun to the quantity of matter of the sun and planet together. Areas will be described proportional to the times, and the orbit will be elliptical; but the ratio of the squares of the periodic times will not be the same with the ratio of the cubes of the distances, unless all the planets are equal.

Thus was the attention of astronomers directed to a number of apparent irregularities in the motion of the earth, which must result from this derangement of the sun, which they had imagined to remain steadfast in his place. They were told what to expect, and on what positions of the planets the kind and quantity of every irregularity depended. This was a most inviting field of observation to a curious speculator; but it required the nicest and most expensive instruments, and an uninterrupted series of long continued observations, sufficient to occupy the whole of a man's time. Fortunately the accurate determination of the solar and lunar motions were of the utmost importance, nay, indispensably necessary for solving the famous problem of the longitude of a ship at sea; and thus the demands of commercial Europe came in aid of philosophical curiosity, and occasioned the erection of observatories, first at Greenwich, and soon after at Paris and other places, with establishments for astronomers, who should carefully watch the motions of the sun and moon, not neglecting the other planets.

The fortunate result of all this solicitude has been the complete establishment of the Newtonian conjecture (for by observation we must still think it), and the verification of Newton's assertion, that action was accompanied, through the whole solar system, by an equal and contrary reaction. All the inequalities of the solar motion predicted by Newton have been observed, although they are frequently so complicated that they could never have been detected, had not the Newtonian theory directed us when to find any of them pretty clear of complication, and how to ascertain the accumulated result of them all in any state of combination.

But in the course of this attention to the motions of the sun and moon, the planets came in for a share, and considerable deviations were found, from the supposition that all their deflections were directed to the sun, and were in the inverse duplicate ratio of their distances. The nice observation shewed, that the period of Jupiter was somewhat shorter than Kepler's law required.

A slight reflection shewed that this was no inconsistency; because the common centre of the conjoined orbits of the Sun and Jupiter was sensibly distant from the centre of the sun, namely, about the 1100th part of the radius vector; and therefore the real deflection was about a 2200th part less than was supposed. It was now plain that the distances to which the Keplerian law must be applied, are the distances, not from the sun, but from the fixed point round which the sun and planets revolve. This difference was too small to be observed in Kepler's time; but the seeming error is only a confirmation of the Newtonian philosophy.

But there are other irregularities which cannot be explained in this manner. The planetary orbits change their

their position; their aphelia advance, their nodes recede, their inclination to each other vary. The mean motions of Saturn and Jupiter are subject to considerable changes, which are periodical.

³⁴ Deflection of the planets towards each other. Sir Isaac Newton had no sooner discovered the universality and reciprocity of the deflections of the planets and the sun, than he also suspected that they were continually deflected towards each other. He immediately obtained a general notion of what should be the more general results of such a mutual action. They may be conceived in this way.

Plate VI. Let S (fig. 5.) represent the sun, E the earth, and I Jupiter, describing concentric orbits round the centre of the system. Make $IS\; :\; EA\; =\; EI^2\; :\; SI^2$. Then, if IS be taken to represent the deflection of the sun toward Jupiter, EA will represent the deflection of the Earth to Jupiter. Draw EB equal and parallel to SI, and complete the parallelogram EBAD. ED will represent the disturbing force of Jupiter. It may be resolved into EF, perpendicular to ES, and EG in the direction of SE. By the first of these the earth's angular motion round the sun is affected, and by the second its deflection toward him is diminished or increased.

³⁵ General result of such mutual action.

In consequence of this first part of the disturbing force, the angular motion is increased, while the earth approaches from quadrature to conjunction with Jupiter (which is the case represented in the figure), and is diminished from the time that Jupiter is in opposition till the earth is again in quadrature, westward of his opposition. The earth is then accelerated till Jupiter is in conjunction with the sun; after which it is retarded till the earth is again in quadrature.

The earth's tendency to the sun is diminished while Jupiter is in the neighbourhood of his opposition or conjunction, and increased while he is in the neighbourhood of his stationary positions. Jupiter being about 1000 times less than the sun, and 5 times more remote, IS must be considered as representing $\backslash frac\{1\}\{1000\}$ of the earth's deflection to the sun, and the forces ED and EG are to be measured on this scale.

In consequence of this change in the earth's tendency to the sun, the aphelion sometimes advances by the diminution, and sometimes retreats by the augmentation. It advances when Jupiter chances to be in opposition when the earth is in its aphelion; because this diminution of its deflection towards the sun makes it later before its path is brought from forming an obtuse angle with the radius vector, to form a right angle with it. Because the earth's tendency to the sun is, on the whole, more diminished by the disturbing force of Jupiter than it is increased, the aphelion of the earth's orbit advances on the whole.

In like manner the aphelia of the inferior planets advance by the disturbing forces of the superior: but the aphelion of a superior planet retreats; for these reasons, and because Jupiter and Saturn are larger and more powerful than the inferior planets, the aphelia of them all advance while that of Saturn retreats.

In consequence of the same disturbing forces, the node of the disturbed planet retreats on the orbit of the disturbing planet; therefore they all retreat on the ecliptic, except that of Jupiter, which advances by retreating on the orbit of Saturn, from which it suffers the greatest disturbance. This is owing to the particular position of the nodes and the inclinations of the orbits.

The inclination of a planetary orbit increases while the planet approaches the node, and diminishes while the planet retires from it.

M. de la Place has completed this deduction of the peculiar planetary inequalities, by explaining a peculiarity in the motions of Jupiter and Saturn, which has long employed the attention of astronomers. The accelerations and retardations of the planetary motions depend, as has and Saturn, been shown, on their configurations, or the relative quarters of the heavens in which they are. Those of Mercury, Venus, the Earth, and Mars, arising from their mutual deflections; and their more remarkable deflections to the great planets Jupiter and Saturn, nearly compensate each other, and no traces of them remain after a few revolutions: but the positions of the aphelia of Saturn and Jupiter are such, that the retardations of Saturn sensibly exceed the accelerations, and the anomalous period of Saturn increases almost a day every century; on the contrary, that of Jupiter diminishes. M. de la Place shows, that this proceeds from the position of the aphelia, and the almost perfect commensurability of their revolutions; five revolutions of Jupiter making 21,675 days, while two revolutions of Saturn make 21,538, differing only 137 days.

Supposing this relation to be exact, the theory shews, that the mutual action of these planets must produce mutual accelerations and retardations of their mean motions, and ascertains the periods and limits of the secular equations thence arising. These periods include several centuries. Again, because this relation is not precise, but the odd days nearly divide the periods already found, there must arise an equation of this secular equation, of which the period is immensely longer, and the maximum very minute. He shews that this retardation of Saturn is now at its maximum, and is diminishing again, and will, in the course of years, change to an acceleration.

This investigation of the small inequalities is the most intricate problem in mechanical philosophy, and has been completed only by very slow degrees, by the arduous efforts of the greatest mathematicians, of whom M. de la Grange is the most eminent. Some of his general results are very remarkable.

He demonstrates, that since the planets move in one direction, in orbits nearly circular, no mutual disturbances make any permanent change in the mean distances and mean periods of the planets, and that the periodic changes are confined within very narrow limits. The orbits can never deviate sensibly from circles. None of them ever has been or will be a comet moving in a very eccentric orbit. The ecliptic will never coincide with the equator, nor change its inclination above two degrees. In short, the solar planetary system oscillates, as it were, round a medium state, from which it never swerves very far.

This theory of the planetary inequalities, founded on the universal law of mutual deflection, has given to our tables a precision, and a coincidence with observation, that surpasses all expectation, and insures the legitimacy of the theory. The inequalities are most sensible in the motions of Jupiter and Saturn; and they present themselves in such a complicated state, and their periods are so long, that ages were necessary for discovering them by mere observation. In this respect, therefore, the theory has outstripped the observations on which it is founded.

38
authenticity of the Indian astronomy.
founded. It is very remarkable, that the periods which the Indians assign to these two planets, and which appeared so inaccurate that they hurt the credit of the science of those ancient astronomers, are now found precisely such as must have obtained about three thousand years before the Christian era; and thus they give an authenticity to that ancient astronomy. The periods which any nation of astronomers assign to those two planets would afford no contemptible mean for determining the age in which it was observed.

The following circumstance is remarkable: Suppose Jupiter and Saturn in conjunction in the first degree of Aries; twenty years after, it will happen in Sagittarius; and after another twenty years, it will happen in Leo. It will continue in these three signs for 200 years. In the next 200 it will happen in Taurus, Capricornus, and Virgo; in the next 200 years, it will happen in Gemini, Aquarius, and Libra; and in the next 200 years, it will happen in Cancer, Pisces, and Scorpio: then all begins again in Aries. It is highly probable that these remarkable periods of the oppositions of Jupiter and Saturn, progressive for 40 years, and oscillating during 160 more, occasioned the astrological division of the heavens into the four trigons, of fire, air, earth, and water. These relations of the signs, which compose a trigon, point out the repetitions of the chief irregularities of the solar system.

39
Origin of the astrological division of the heavens.
M. de la Place observes (in 1796), that the last discovered planet gives evident marks of the action of the rest; and that when these are computed and taken into the account of its bygone motions, they put it beyond doubt that it was seen by Flamsteed in 1690, by Mayer in 1756, and by Monnier in 1769.

40
Action of the comets.
We have hitherto overlooked the comets in our account of the mutual disturbances of the solar system. Their number is very great, and they go to all quarters of the universe: but we may conclude, from the wonderful regularity of the planetary motions, when all their own mutual actions are taken into account, that the quantity of matter in the comets is very inconsiderable. They remain but a short time in the neighbourhood of the planets, and they pass them with great rapidity. Some of them have come very near to Jupiter, but left no trace of their action in the motions of his satellites. They doubtless contribute, in general, to make the apses of the planetary orbits advance.

41
They are affected by the planets.
On the other hand, the comets may be considerably affected by the planets. The very important phenomenon of the return of the comet of 1682, which was to decide whether they were revolving planets describing ellipses, or bodies which came but once into the planetary regions, and then retired for ever, caused the astronomers to consider this matter with great care. Halley had shown, in a rough way, that this comet must have been considerably affected by Jupiter. Their motion near the aphelion must be so very slow, that a very small change of velocity or direction, while in the planetary regions, must considerably affect their periods. Halley thought that the action of Jupiter might change it half a year. Mr. Clairaut, by considering the disturbing forces of Jupiter and Saturn through the whole revolution, showed that the period then running would exceed the former nearly two years (618 days), and assigned the middle of April 1759 for the time of its perihelion. It really passed its perihelion on the 12th of March. This

was a wonderful precision, when we reflect that the comet had been seen but a very few days in its former apparitions.

A comet observed by Mr. Prosperin and others in 1771 has greatly puzzled the astronomers. Its motions appear to have been extremely irregular, and it certainly came so near Jupiter, that his momentary influence was at least equal to the sun's. It has not been recognized since that time, although there is a great probability that it is continually among the planets.

42
It is by no means impossible, nor highly improbable, Consequence of a comet and planet meeting.
that, in the course of ages, a comet may actually meet one of the planets. The effect of such a concourse must be dreadful; a change of the axis of diurnal rotation must result from it, and the sea must desert its former bed and overflow the new equatorial regions. The shock and the deluge must destroy all the works of man, and most of the race. The remainder, reduced to misery, must long struggle for existence, and all remembrance of former arts and events must be lost, and every thing must be invented anew. There are not wanting traces of such devastations in this globe: strata and things are now found on mountain tops which were certainly at the bottom of the ocean in former times; remains of tropical animals and plants are now dug up in the circumpolar regions. Tempora mutantur, et nos mutamur in illis.

It is plain, that when we know the direction and the intensity of the disturbing force, we can tell what will be the accumulated effect of its action for any time. The direction is easily determined by means of the distance: but how shall we determine the intensity? Since we see that the whole waters of the ocean are deflected toward the moon, and have such probable evidence that planetary deflection is mutual; it follows, that the moon is deflected towards every drop of water, and that all the matter in one body is deflected towards all the matter in another body; and therefore that the deflection towards the sun or a planet is greater or less in proportion to its quantity of matter. Newton indeed thought it unreasonable to suppose that a planet was deflected to the centre of the sun, which had no distinguishing physical property; and thought it more probable that the deflection of a planet to the sun was the accumulated deflection of every particle in the planet to every particle in the sun. But he was too scrupulous to take this for granted. He therefore endeavoured to discover what would be the sensible deflection of one sphere to another, when each consisted of matter, every particle of which was deflected to every particle of the other with an intensity inversely proportional to the square of the distance from it. By help of a most beautiful and simple process, he discovered, that the tendency of a particle of matter to a spherical surface, shell, or solid, of uniform density at equal distances from the centre, was the same as if all the particles in the surface, shell, or solid, were united in its centre: hence it legitimately followed, that the mutual tendency of spherical surfaces, shells, or solids, was proportional to the square quantities of matter in the attracting body, and inversely as the square of the distance of their centres; and thus the law of attraction, competent to every particle of planetary matter, was the same with that which was observed among spherical bodies consisting of such matter. And it is remarkable, that the inverse duplicate ratio.

tion of the distances is the only law that will hold, both with respect to single particles and to globes composed of such particles. He also demonstrated, that a particle placed within a sphere was not affected by all the shell, which was more distant than itself from the centre, being equally attracted on every side, and that it tended toward the centre of a homogeneous sphere, on the surface of which it was placed, with a force proportional to its distance from the centre.

Newton saw a case in which it was possible to discover whether the tendency of the matter of which the planets consisted was directed to a mathematical centre void of any physical properties, or whether it was the result of its united tendency to all the matter of the planet. He demonstrated, that if the earth consisted of matter which tended to the centre, it behaved it to assume the form of an elliptical spheroid, in consequence of the centrifugal force arising from its diurnal motion, and that the polar axis must be to its equatorial diameter as 577 to 578; but if every particle tends to every other particle in the inverse duplicate ratio of the distance from it, the form must still be elliptical, but more protuberant, and the polar axis must be to the equatorial diameter as 230 to 231. Then only will a column of water from the pole to the centre balance a column from the equator to the centre. He also shewed what should be the vibrations of pendulums in different latitudes, on both suppositions. Mathematicians were eager therefore to make those experiments on pendulums, and to determine the figure of the earth by the measurement of degrees of the meridian in different latitudes. The result of their endeavours has been decidedly in favour of the mutual tendency of all matter. This has been farther confirmed by the observations of the mathematicians who measured the degrees of the meridian in Peru, and by Dr Maskelyne in Britain, who found that a pendulum suspended in the neighbourhood of a great and solid mountain, sensibly deviated from the true vertical, and was deflected toward the mountain.

From a collective view of all these circumstances, Sir Isaac Newton concluded, with great confidence, that the deflection toward any planet was the united deflection toward every particle of matter contained in it. This enabled him to determine the intensity of the planetary disturbing forces, by previously ascertaining the proportions of their quantities of matter. This proportion, the discovery of which seems above our reach, is easily ascertained in all those bodies which have others revolving round them: for the deflection of the revolving body, being occasioned by all the matter in the central body, will be proportional (ceteris paribus) to the quantity of matter in the central body, and therefore will give us a measure of that quantity. Would we compare the quantity of matter in Jupiter with the quantity of matter in the sun, we have only to suppose that a planet revolves round the sun at the distance of Jupiter's fourth satellite. Kepler's third law will tell us the time of its revolution. The distances, in this case, being the same, the centripetal forces, and consequently the quantities of matter in the central bodies, will be inversely as the squares of the periodic times of the revolutions around them. In this way have the quantities of matter been determined for the Sun, the Earth, Jupiter, Saturn, and the last discovered pla-

net. If the quantity in the Earth be considered as the unit, we have,

The Earth	-	-	-	-	1
The newly discovered planet	-	-	-	-	17,75
Saturn	-	-	-	-	86,16
Jupiter	-	-	-	-	317,1
The Sun	-	-	-	-	338343

Thus we see that the sun is incomparably bigger than any planet, having more than a thousand times as much matter as Jupiter, the most massive of them all. There is a considerable uncertainty, however, in the proportion to the sun, because we do not know his distance nearer than within $\backslash frac\{1\}\{1000\}$th part. The proportions of the rest to each other are more accurate. The quantities of matter in Mercury and Mars can only be guessed at: the quantity in Mercury may be called 0,1, and Mars may be called 0,12. Venus is supposed nearly equal to the Earth. This is concluded from the effect which she produces on the precession of the equinoxes and the equation of the sun's motion. The moon is supposed to be about $\backslash frac\{1\}\{50\}$th of the earth, from the effect she produces on the tides and the precession of the equinoxes, compared with those produced by the sun.

When these quantities of matter are introduced into the computation of the planetary inequalities, and the intensity of the disturbing forces assumed accordingly, the results of the computations tally so exactly with observation, that we can now determine the sun's place for any moment within two or three seconds of a degree, and are certain of the transit of a planet within one beat of the clock!

Non dubios nulla caligine praeceat error ;
Quis superum penetrare domos atque ardua cali
Scendere sublimis genii concessit acumen.

HALLEY.

Sir Isaac Newton having already made the great discovery of an universal and mutual deflection of all the matter in the solar system, was one day speculating on this subject, and comparing it with other deflections which he observed among bodies, such as magnets, &c. He considered terrestrial gravity as a force of this kind. By the weight of terrestrial bodies they kept united with the earth. By its weight was the water of the ocean formed into a sphere. This force extended, with out any remarkable diminution, to the tops of the highest mountains. Might it not reach much farther? May it not operate even at the distance of the moon? In the same manner that the planetary force deflects the moon from the tangent to her orbit, and causes her to describe an ellipse, the weight of a cannon ball deflects it from the line of its direction, and makes it describe a parabola. What if the deflecting force which incurs her path towards the earth be the simple weight of the moon? If the weight of a body be the same with the general planetary force, it will diminish as the square of its distance from the earth increases. Therefore, said he to himself, since the distance of the moon from the centre of the earth is about 50 times greater than the distance of the stone which I throw from my hand, and which is deflected 16 feet in one second, the weight of this stone, if taken up to the height of the moon, should be reduced to the 2500th part, and should there deflect $\backslash frac\{1\}\{2500\}$th of 16 feet in a second; and the moon should deflect

deflect as much from the tangent in a second. Having the dimensions, as he thought, of the moon's orbit, he immediately computed the moon's deflection in a second; but he found it considerably different from what he wished it to be. He therefore concluded that the planetary force was not the weight of the planet. For some years he thought no more of it: but one day, in the Royal Society, he heard an account read of measurements of a degree of the meridian, which showed him that the radius of the earth and the distance of the moon were very different from what he had believed them to be. When he went home he repeated his computation, and found, that the deflection of a stone was to the simultaneous deflection of the moon as the square of the moon's distance from the centre of the earth to the square of the stone's distance. Therefore the moon is deflected by its weight; and the fall of a stone is just a particular instance of the exertion of the universal planetary force. This computation was but roughly made at first; but it was this coincidence that excited the philosopher to a more attentive review of the whole subject. When every circumstance which can affect the result is taken into account, the coincidence is found to be most accurate. The fall of the stone is not the full effect of its weight; for it is diminished by the rotation of the earth round its axis: It is also diminished by the weight of the air which it displaces: It is also diminished by its tendency to the moon. On the other hand, the moon does not revolve round the earth, but round a common centre of the earth and moon, and its period is about $\backslash frac\{1\}\{12\}$ shorter than if it revolved round the earth; and the moon's deflection is affected by the sun's disturbing force. But all these corrections can be accurately made, and the ratio of the full weight of the stone to the full deflection of the moon ascertained. This has been done.

Terrestrial gravity therefore, or that power by which bodies fall or press on their supports, is only a particular instance of that general tendency by which the planets are retained in their orbits. Bodies may be said to gravitate when they give indications of their being gravis or heavy, that is, when they fall or press on their supports; therefore the planets may be said to gravitate when they give similar indications of the same tendency by their curvilinear motions. The general fact, that the bodies of the solar system are mutually deflected toward each other, may be expressed by the verbal noun GRAVITATION. Gravitation does not express a quality, but an event, a deflection, or a pressure.

The weight of a terrestrial body, or its pressure on its support, is the effect of the accumulated gravitation of all its particles; for bodies of every kind of matter fall equally fast. This has been ascertained with the utmost accuracy by Sir Isaac Newton, by comparing the vibrations of pendulums made of every kind of matter. Therefore their united gravitation is proportional to their quantity of matter; and we have concluded, that every atom of terrestrial matter is heavy, and equally heavy. We extend this conclusion to the sun and planets, and say, that the observed gravitation of a planet is the united gravitation of every particle. Therefore Sir Isaac Newton inferred, from a collective view of all the phenomena, that all matter gravitates to all matter with a force in the inverse duplicate ratio of the distance.

But we do not think that this inference is absolutely

certain. We acknowledge that the experiments on pendulums, consisting of a vast variety of terrestrial matter, all of which performed their oscillations in equal times, demonstrate that the acceleration of gravity on those pendulums was proportional to their quantities of matter, and that equal gravitation may be affirmed of all terrestrial matter.

The elliptical motion of a planet is full proof that the accelerating power of its gravity varies in the inverse duplicate ratio of the distance; and the proportionality of the squares of the periods to the cubes of the distances, shows that the whole gravitations of the planets vary by the same law. But this third observation of Kepler might have been the same, although the gravitation of a particle of matter in Jupiter had been equal to that of a particle of terrestrial matter, provided that all the matter in Jupiter did not gravitate. If $\backslash frac\{1\}\{3\}$th of Jupiter had been such gravitating matter, his deflection from the tangent of his orbit would have been the same as at present, and the time of his revolution would have been what we observe. In order that the third law of Kepler may hold true of the planetary motions, no more is required than that the accumulated gravitation of the planet be proportional to its quantity of matter, and thus the matter which does not gravitate will be compensated by the superior gravitation of the rest.

But because we have no authority for saying that there is matter which gravitates differently from the rest, or which does not gravitate, we are entitled to suppose that gravity operates alike on all matter.

And this is the ultimatum of the Newtonian philosophy, that the solar system consists of bodies composed of matter, every particle of which is, in fact, continually deflected by its weight toward every other particle in the system; and that this deflection, or actual deviation, or actual pressure, tending to deviation from uniform rectilinear motion, is in the inverse duplicate ratio of the distance.

This doctrine has been called the system of universal gravitation; and it has been blamed as introducing an unphilosophical principle into science. Gravitation is said to be an occult quality; and therefore as unfit for the explanation of phenomena as any of the occult qualities of Aristotle. But this reproach is ungrounded; gravitation does not express any quality whatever, but a matter of fact, an event, an actual deflection, or an actual pressure, producing an actual deflection of the body pressed. These are not occult, but matters of continual observation. True, indeed, Newton does not deny, although he does not positively say, that this deflection, pressure, or gravitation, is an effect having a cause. Gravity is said to be this cause. Gravity is the being gravis or heavy, and gravitation is the giving indications of being heavy. Heaviness therefore is the word which expresses gravitas, and our notion of the cause of the planetary deflections is the same with our notion of heaviness. This may be indistinct and unsatisfactory to a mind fastidiously curious; but nothing can be more familiar. The planet is deflected, because it is heavy. We are supposed to explain the fall of a stone through water very satisfactorily, and without having recourse to any occult quality, when we say that it is heavier than the water; and we explain the rise of a piece of cork, when we say that it is not so heavy as the water.

The

The explanations of the mutual actions of the planets are equally satisfactory, founded on the same principles, and equally free from all sophistry or employment of occult causes. The weight of a body is not its heaviness, but the effect of its heaviness. It is a gravitation, an actual pressure, indicated by its balancing the supposed heaviness of another body, or by its balancing the known elasticity of a spring, or by balancing any other natural power. It is similar to the pressure which a magnet exerts on a piece of iron. This may perhaps be produced by the impulse of a stream of fluid; so may the weight of a heavy body. But we do not concern ourselves with this question. We gain a most extensive and important knowledge by our knowledge of this universal law; for we can now explain every phenomenon, by pointing out how it is contained in this law; and we can predict the whole events of the solar system with unerring exactness. This should satisfy the most inquisitive mind.

But, nihil in vetitum, semper enim usque negata. There seems to be a fatal and ruinous disposition in the human mind, a sort of præcipium of the understanding, that is irritated by every interdict of natural imperfection. We would take a microscope to look at light; we would know what knowing is, and we would weigh heaviness.

All who are acquainted with the writings of Aristotle have some notion of his whimsical opinions on this subject. He imagines that the planets are conducted in their orbits by a sort of intelligences, ἑσπὴρ ὄρχαί, which animate the orbs that wheel them round. Although this crude conception met with no favour in later times, another, not more reasonable, was maintained by Leibnitz, who called every particle of matter a monad, and gave it a perception of its situation in the universe, of its distance and direction from every other, and a power and will to move itself in conformity to this situation, by certain constant laws. This ἑσπὴρ ὄρχαί in the Monad is nothing but an awkward substitute for the principle of gravitation, which the learned infilled that Newton placed in every particle of matter as an innate power, and which they reprobated as unphilosophical. But in what respect this perception and active propensity is better, we do not perceive. It is more complex, and involves every notion that is reprehensible in the other; and it offers no better explanation of the phenomena.

But Newton is equally anxious with other philosophers not to ascribe gravity to matter as an innate inherent property. In a letter to Dr Bentley, he earnestly requests him not to charge him with such an absurd opinion. It is an avowed principle, that nothing can act on any thing that is at a distance; and this is considered as an intuitive axiom. But it is surely very obscure; for we cannot obtain, or at least convey, clear notions of the terms in which it is expressed. The word act is entirely figurative, borrowed from animal exertions; it is therefore unlike the expression of any thing intimated to the appellation of intuitive. If we try to express it without figure, we find our confidence in its certainty greatly diminished. Should we say that the condition of a body A cannot depend on another body B that is at distance from it, we believe that no person will say that he makes this assertion from perceiving the absurdity of the contrary proposition. In the demonstration,

as it is called, of the perseverance of a body in a state of rest, the only argument that is offered is, that no cause can be assigned why it should move in one direction rather than in another: but should any one say that another body is near it, to the right hand, and that this is a sufficient reason for its moving that way, we know no method by which this assertion can be shown to be false.

Such, however, has been the uniform opinion of philosophers. Nihil movetur (says Leibnitz) nisi a continuo et moto. The celebrated mathematician Euler having discovered, as he thought, the production of a pressure, like gravity, from motion, says, "as motion may arise from pressing powers, so we have seen that pressing powers may arise from motion. We see that both exist in the universe. It is the business of a philosopher to discover, by reason and observation, which is the origin of the other. It is incompatible with reason, that bodies should be possessed of inherent tendencies; much more that powers should exist independently. Farther, that philosopher must be reckoned to have assigned the true causes of phenomena, who demonstrates that they arise from motion; for motion, once existing, must be preserved for ever. In the present instance (a certain whimsical fact of a ball running round the inside of a hoop) we see how a pressing power may be derived from motion; but we cannot see how powers can exert themselves, or be preserved, without motion. Wherefore we may conclude that gravity, and all other powers, are derived from motion; and it is our business to investigate from what motions of what bodies each observed power derives its origin."

Accordingly many attempts have been made to trace the planetary deflections to their origin in the motion of some impelling matter; but these attempts could not be successful, because they are all built on hypotheses. It has been assumed, that there is a matter diffused through the celestial spaces; that this matter is in motion, and by its impulse moves the planets; but the only reason that can be given for the existence of this matter is the difficulty we find in explaining the planetary deflections without it. Even if the legitimate consequences of this hypothesis were consistent with the phenomena, we have not advanced in our knowledge, nor obtained any explanation. We have only learned, that the appearances are such as would have obtained had such a matter existed and acted in this manner. The observed laws of the phenomena are as extensive as those of the hypothesis; therefore it teaches us nothing but what we knew without it.

But this is not all that can be said against those attempts; their legitimate consequences are inconsistent with the phenomena. By legitimate consequences we mean of the laws of motion. These must be admitted, and are admitted, by the philosopher who attempts to explain the planetary motions by impulse. It would be ridiculous to suppose a matter to fill the heavens, having laws of impulse different from those that are observed by common matter, and which laws must be contrived so as to answer the purpose. It would be more simple at once to assign those pro re nata laws to the planets themselves.

Yet such was the explanation which the celebrated Descartes offered by his hypothesis of vortices, in which the planets were immersed and whirled round the sun.

53 Vortices of Delectio

It is astonishing that so crude a conception ever obtained any partisans; yet it long maintained its authority, and still has zealous defenders. Till Sir Isaac Newton saw the indispensable necessity of mathematical investigation in every question of matter in motion, no person had taken the trouble of giving any thing like a distinct description of those vortices, the circumstances of their motion, and the manner of their action; all determined with that precision that is required in the explanation: for this must always be kept in mind, that we want an explanation of the precise motions which have been observed, and which will enable us to predict those which are yet to happen. Men were contented with some vague notion of a sort of similarity between the effects of such vortices and the planetary motions in a few general circumstances; and were neither at the trouble to consider how these motions were produced, nor how far they tallied with the phenomena. Their account of things was only fit for careless chat, but unworthy of the attention of a naturalist. But since this explanation came from a person deservedly very eminent, it was respected by Newton, and he honoured it with a serious examination. It is to this examination alone that we are indebted for all the knowledge that we have of the constitution of a fluid vortex, of the motions of which it is susceptible, of the manner in which it can be produced, the laws of its circulation, and the effects which it can produce. We have this account in Sir Isaac Newton's Principles of Natural Philosophy; and it contains many very curious and interesting particulars, which have been found of great service in other branches of mechanical philosophy. But the result of the examination was fatal to the hypothesis; shewing that the motions which were possible in the vortices, and the effects which they must produce, are quite incompatible with the appearances in the heavens. We do not know one person who has acquired any reputation as a mechanician that now attempts to defend it; nor do we know of any other person besides Newton who has attempted to explain mathematically how the circulation of a fluid can produce the revolution of a planet, if we except Mr Leibnitz, the celebrated rival of the British philosopher.

54 Examined by Newton.

This gentleman published in the Leipic Review in 1689, three years after the publication of the Principia, an attempt to explain the elliptical motion of the planets, and the description of areas proportional to the times by the impulse of a vortex. It must not be passed over in this place, because it acquired great authority in Germany, and many of that country still affirm that Leibnitz is the discoverer of the law of planetary gravitation, and of the mechanical constitution of the solar system. We cannot help thinking this explanation the most faulty of any, and a most dillingenuous plagiarism from the writings of Newton.

55 Typ. th. fi. of Leibnitz.

Mr Leibnitz supposes a fluid, circulating round the sun in such a manner that the velocity of circulation in every part is inversely as its distance from the sun. (N. B. Newton had shown that such a circulation was possible, and that it was the only one which could be generated in a fluid by an action proceeding from the centre). Leibnitz calls this harmonical circulation. He supposes that the planet adopts this circulation in every part of its elliptical orbit, obeying without any resistance the motion of this fluid. He does not ascribe this to the impulse of the fluid, saying expressly that the pla-

net follows its motion, non abrepta tamen, sed tranquilliter quasi natante. The planet therefore has no tendency to persevere in its former state of motion. Why therefore does it not follow this harmonic motion exactly, and describe a circle tranquilliter natans? This is owing, says Leibnitz, to its centrifugal force, by which it perseveres in a state of rectilinear motion. It has no tendency to preserve its former velocity, but it perseveres in its former direction. The planet therefore is not like common matter, and has laws of motion peculiar to itself; it was needless therefore to employ any impulse to explain its motions. But to proceed: This centrifugal force must be counteracted in every point of the orbit. Leibnitz therefore supposes that it is also urged toward the centre by a solicitation like gravity or attraction. He calls it the paracentric force. He computes what must be its intensity in different parts of the orbit, in order to produce an elliptical motion, and he finds that it must be inversely as the square of the distance from the centre (for this reason he is frequently quoted by Bernoulli, Wolff, and others, as the discoverer of the law of gravitation). But Leibnitz arrives at this result by means of several mathematical blunders, either arising from his ignorance at that time of fluxionary geometry, or from his perceiving that an accurate procedure would lead him to a conclusion which he did not wish: for we have seen (and the demonstration is adopted by Leibnitz in all his posterior writings of this kind), that if the ordinary laws of motion are observed, a body, actuated by this paracentric force alone, will describe an ellipse, performing both its motion of harmonic circulation, and its motion of approach to and recede from the centre, without farther help. Therefore, if the harmonic circulation is produced by a vortex, a force inversely as the square of the distance from the centre, combined with the harmonic circulation, will produce a motion entirely different from the elliptical. It is demonstrated, that the force which is necessary for describing circles at different distances, with the angular velocity of the different parts of the orbit, is not in the inverse duplicate, but in the inverse triplicate, ratio of the distances. This must have been the nature of his paracentric force, in order to counteract the centrifugal force arising from the harmonic circulation. There-
56 Dillingenuous author.
Leibnitz has not arrived at his conclusion by just reasoning, nor can be said to have discovered it. He says, Video hanc propositionem inventisse viro celeberrimo Isaac Newtono, licet non possim judicare quomodo ad eam pervenerit. This is really somewhat like impudence. The Principia were published in 1686. They were reviewed at Leipic, and the Review published in 1687. Leibnitz was at that time the principal manager of that Review. When Newton published, Leibnitz was living at Hanover, and a copy was sent him, within two months of its publication, by Nicholas Facio, long before the Review. The language of the Review has several singularities, which are frequent in Leibnitz's own composition; and few doubt of its being his writing. Besides, this proposition in the Principia had been given to the Royal Society several years before, and was in the records before 1684. These were all seen by Leibnitz when in England, being lent him by his friend Collins.

We think that the opinion which a candid person must form of the whole is, that Leibnitz knew the proposition,

position, and attempted to demonstrate it in a way that would make it pass for his own discovery; or that he only knew the enunciation, without understanding the principles. His harmonic circulation is a clumsy way of explaining the proportionality of areas to the times; and even this circulation is borrowed from Newton's dissertation on the Cartesian vortices, which is also contained in the Leipzig Review above mentioned. Leibnitz was by this time a competitor with Newton for the honour of inventing the fluxionary mathematics, and was not guiltless of acts of disingenuity in asserting his claim. He published at the same time, in the same Review, an almost unintelligible dissertation on the resistance of fluids, which, when examined by one who has learned the subject by reading the Principia of Newton, affords an enigmatical description of the very theory published by Newton, as a necessary part of his great work.

But besides all the above objections to Leibnitz's theory of elliptical motion, we may ask, What is this paracentric force? He calls it like gravity. This is precisely Newton's doctrine. But Leibnitz supposes this also to be the impulse of a fluid. It would have been enough had he explained the action of this fluid, without the other circulating harmonically. He defers this explanation, however, to another opportunity. It must have very singular properties: it must impel the planet without disturbing the other fluid, or being disturbed by it. He also defers to another opportunity the explaining how the squares of the periodic times of different planets are proportional to the cubes of the mean distances; for this is quite incompatible with the harmonic circulation of his vortex. This would make the squares of the periods proportional to the distances. He has performed neither of these promises. Several years after this he made a correction of one of his mathematical blunders, by which he destroyed the whole of his demonstration. In short, the whole is such a heap of obscure, vague, inconsistent assumptions, and so replete with mathematical errors, that it is astonishing that he had the ignorance or the effrontery to publish it.

57
Hypothesis
of Le Sage.
There is another hypothesis that has acquired some reputation. M. le Sage of Geneva supposes, that there passes through every point of the universe a stream of fluid, in every direction, with astonishing velocity. He supposes that, in the densest bodies, the vacuity is incomparably more bulky than the solid matter; so that a solid body somewhat resembles a piece of wire cage-work. The quantity of fluid which passes through will be incomparably greater than that of the intercepted fluid; but the impulse of the intercepted fluid will be sensibly proportional to the quantity of solid matter of the body. A single body will be equally impelled in every direction, and will not be moved; but another body will intercept some fluid. Each will intercept some from the other; and the impulse on B, that is intercepted by A, will be nearly proportional to the matter in A, and inversely proportional to the square of its distance from B; and thus the two bodies will appear to tend toward each other by the law of gravitation.

M. le Sage published this in a work called Chimie Mechanique, and read lectures on this doctrine for many years in Geneva and Paris to crowded audiences. It is also published by Mr Prevost in the Berlin Memoirs,

under the name of Lucerne Newtonian; and there are many who consider it as a good explanation of gravitation: for our part, we think it inconceivable. The motions of the planets, with undiminished velocity, for more than four thousand years, appears incompatible with the impelling power of this fluid, be its velocity what it will. The absolute precision of the law of gravitation, which does not show the smallest error during that time, is incompatible with an impulse which cannot be exactly proportional to the quantity of matter, nor to the reciprocal of the square of the distance, nor the same on a body moving with the rapidity of the comet of 1680 in its perihelion, as on the planet Saturn, whose motion is almost incomparably slower. What is the origin of the motion of this fluid? Why does it not destroy itself by mutual impulse, since it is continually passing through every point? &c.

58
We have already observed that Newton expressed the same anxiety to avoid the supposition of action among bodies at a distance. He also seemed to show some disposition to account for gravitation by the action of a contiguous fluid. This is the subterfuge so much resorted to by precipitate speculatists, by the name of the ether of Sir Isaac Newton. He supposes it highly elastic, and much rarer in the pores of bodies and in their vicinity than at a distance; therefore exceedingly rare in the sun, and denser as we recede from him. Being highly elastic, and repelled by all bodies, it must impel them to that side on which it is most rare; therefore it must impel them toward the sun. This is enough of its general constitution to enable us to judge of its fitness for Newton's purpose. It is wholly unfit; for since it is fluid, unequally dense and elastic, its particles are not in contact. Particles that are elastic, and in a state of compression, and in contact, cannot be fluid; they must be like so many blown bladders compressed in a box; therefore they are not in contact; therefore they are elastic by mutual repulsion; that is, by acting on each other at a distance. It is indifferent whether this distance is a million of miles, or the millionth part of a hair's breadth; therefore this fluid does not free Newton from the supposition which he wishes to avoid. Nay, it can be demonstrated, that in order to form a fluid which shall vary in density from the sun to the extremity of the solar system, there must be a mutual repulsion extending to that distance. This is introducing millions of millions of the very difficulties which Newton wished to avoid; for each particle presents the same difficulty with a planet.

We would now ask these atomical philosophers, why they have, in all ages, been so anxious to trace the celestial motions to the effects of impulse? They imagine that they have a clear perception of the communication of motion by impulse, while their perception of the production of it in any other way is obscure. Seeing, in a very numerous and familiar collection of facts, that motion is communicated by impulse, they think that it is communicated in no other way, and that impulse is the only moving power in nature.

59
But is it true that our notion of impulse is more clear than that of gravitation? Its being more familiar is no argument. A cause may be real, though it has exerted itself but once since the beginning of time. In no case do we perceive the exertion of the cause; we only perceive the change of motion. The constitution of our mind

mind makes us consider this as an effect, indicating a cause which is inherent in that body which we always see associated with that change. Granting that our perception of the perseverance of matter in its state of motion is intuitive, it by no means follows that the body A in motion must move the body B by striking it. The moment it strikes B, all the metaphysical arguments for A's continuance in motion are at an end, and they are not in the least affected by the supposition that A and B should continue at rest after the stroke; and we may defy any person to give an argument which will prove that B will be moved; nay, the very existence of B may, for any thing we know to the contrary, be a sufficient reason for the cessation of the motion of A. The production of motion in B, by the impulse of A, must therefore stand on the same foundation with every other production of motion. It indicates a moving power in A; but this inherent power seems to have no dependence on the motion of A: (See what is contained in n° 81. of the article PHYSICS, and n° 67. of OPTICS of the Encycl.) We see there a motion produced in B without impulse, and taken from A, similar in every respect to every case of impulse; and we see that the motion of A is necessary for producing such a motion in B as is observed in all cases of impulse, merely in order that the moving power, which is inherent in A, whether it be in rest or in motion, may act during a sufficient time. Our confidence in the communication of motion, in the case mentioned there, is derived entirely from experience, which informs us that A possesses a moving power totally different from impulse. Our belief of the impelling power of matter therefore does not necessarily flow from our intuitive knowledge of the perseverance of matter, although it gives us the knowledge of this perseverance. It is like a mathematical demonstration, a road to the discovery of the property of figure, but not the cause of that property. The impulsion of matter is merely a fact, like its gravitation, and we know no more of the one than of the other.

It is not a clearer perception, therefore, which has procured this preference of impulsion as the ultimate explanation of motion, and has given rise to all the foolish hypotheses of planetary vortices, ethers, animal spirits, nervous fluids, and many other crude contrivances for explaining the absurd phenomena of nature.

Nor does it deserve any preference on account of its greater familiarity. Just the contrary: for one fact of undoubted impulse, we see millions where no impulse is observed. Consider the motion produced by the explosion of gunpowder. Where is the original impulse? Suppose the impulse of the first spark of fire to be immense, how comes it that a greater impulse is produced by a greater quantity of gunpowder, a greater quantity of quiescent matter? The ultimate impulse on the bullet should be less on this account. Here are plain exertions of moving powers, which are not reducible to impulse. Consider also the facts in animal motion. Reflect also, that there has been more motion, without any observed impulse, produced in the waters of a river since the beginning of the world, than by all the impulse that man has ever observed. Add to these, all the motions in magnetism, electricity, &c. Impulse is therefore a phenomenon which is comparatively rare.

Have we ever observed motion communicated by pure

impulse, without the action of forces at a distance? This appears to us very doubtful. Every one acquainted with Newton's discoveries in optics will grant, that the colours which appear between two object-glasses of long telescopes, when they are pressed together, demonstrate, that the glasses do not touch each other, except in the place where there is a black spot. It requires more than a thousand pounds to produce a square inch of this spot. Therefore every communication of motion between two pieces of glass, which can be produced by one of them striking the other, is produced without impulse, unless their mutual pressure has exceeded 1000 pounds on the square inch of the parts which act on each other. Nay, since we see that a black spot appears on the top of a soap bubble, in the middle of the coloured rings, we learn that there is a certain thickness at which light ceases to be visibly reflected; therefore the black spot between the glasses does not prove that they touch in that part; therefore we cannot say that any force whatever can make them touch. The ultimate repulsion may be insuperable. If this be the case, the production of motion by impulse is, in every instance, like the production of motion between the magnets in n° 81. of the article PHYSICS in the Encycl. and is of the same kind with the production of motion by gravity.

Therefore no explanation of gravitation can be derived from any hypothesis whatever of intervening fluids. They only substitute millions of bodies for one, and still leave the action e distant the same difficulty as before. It is not in the least necessary that we shall be able to conceive how a particle of matter can be influenced by another at a distance; if we have discovered in every instance the precise degree and direction of the effect of this influence, we have made a most important addition to our knowledge of nature; and our success in the case of the power of gravity should make us assiduous in our endeavours to discover, from the phenomena, the laws which regulate the other actions e distant, which observation is daily finding out. A knowledge equally accurate of the law of magnetic and electric action may enable us to give theories of magnetism and electricity equally exact with the Newtonian theory of gravitation.

Having, we hope, evinced the truth of this theory, by following out the investigations to which Newton was gradually led, we might proceed to consider, in order, the complicated and subordinate phenomena which depend on it. The lunar and planetary inequalities are the subjects that naturally come first in our way; but they have already been explained in all the detail that this concise account will admit, as they occurred to Newton as tests of the truth of his conjecture. If the law be such as he suspected, its consequences must be so and so; if the celestial motions do not agree with them, the law must be rejected. We shall not repeat anything therefore on this head, but confine our observations to such applications of the theory of universal gravitation as newly discovered objects, or the improvement of astronomical observation and of fluxionary analysis, have enabled us to make since the time of Newton.

The subfervency of the eclipses of Jupiter's satellites to geography and navigation had occasioned their motions to be very carefully observed, ever since these uses of them were first suggested by Galileo, and their

theory is as far advanced as that of the primary planets. It has peculiar difficulties. Being very near to Jupiter, the great deviation of his figure from perfect sphericity makes the relation between their distances from his centre and their gravitations toward it vastly complicated. But this only excited the mathematicians so much the more to improve their analysis; and they saw, in this little system of Jupiter and his attendants, an epitome of the solar system, where the great rapidity of the motions must bring about in a short time every variety of configuration or relative position, and thus give us an example of those mutual disturbances of the primary planets, which require thousands of years for the discovery of their periods and limits. We have derived some very remarkable and useful pieces of information from this investigation; and have been led to the discovery of the eternal durability of the solar system, a thing which Newton greatly doubted of.

Mr Pound had observed long ago, that the irregularities of the three interior satellites were repeated in a period of 427 days; and this observation is found to be just to this day.

247 revolutions of the first occupy	437 d.	3 h.	44'
123	second	437	3 42
61	third	437	3 36
26	fourth	435	14 16

This naturally led mathematicians to examine their motions, and see in what manner their relative positions or configurations, as they are called, corresponded to this period: and it is found, that the mean longitude of the first satellite, minus thrice the mean longitude of the second, plus twice the mean longitude of the third, always made 180 degrees. This requires that the mean motion of the first, added to twice that of the third, shall be equal to thrice the mean motion of the second. This correspondence of the mean motions is of itself a singular thing, and the odds against its probability seems infinitely great; and when we add to this the particular positions of the satellites in any one moment, which is necessary for the above constant relation of their longitudes, the improbability of the coincidence, as a thing quite fortuitous, becomes infinitely greater. Doubts were first entertained of the coincidence, because it was not indeed accurate to a second. The result of the investigation is curious. When we follow out the consequences of mutual gravitation, we find, that although neither the primitive motions of projection, nor the points of the orbit from which the satellites were projected, were precisely such as suited these observed relations of their revolutions and their contemporaneous longitudes; yet, if they differed from them only by very minute quantities, the mutual gravitations of the satellites would in time bring them into those positions, and those states of mean motion, that would induce the observed relations; and when they are once induced they will be continued for ever. There will indeed be a small equation, depending on the degree of unsuitableness of the first motions and positions; and this causes the whole system to oscillate, as it were a little, and but a very little way on each side of this exact and permanent state. The permanency of these relations will not be destroyed by any secular equations arising from external causes; such as the action of the fourth satellite, or of the sun, or of a resisting medium;

because their mutual actions will distribute this equation as it did the original error.

This curious result came into view only by degrees, as analysis improved and the mathematicians were enabled to manage more complicated formulas, including more terms of the infinite series that were employed to express the different quantities. It is to M. de la Grange that we are indebted for the completion of the discovery of the permanency of the system in a state very little different from what obtains in any period of its existence. Although this required all the knowledge and address of this great mathematician, in the management of the most complicated analysis, the evidence of its truth may be perceived by any person acquainted with the mere elements of fluxionary geometry. The law of the composition of forces enables us to express every action of the mutual forces of the sun and planets by the sines and cosines of circular arcs, which increase with an uniform motion, like the perpetual lapse of time. The nature of the circle shows, that the variations of the sines and cosines are proportional to the cosines and sines of the same arcs. The variations of their squares, cubes, or other powers, are proportional to the sines or cosines of the doubles or triples, or other multiples of the same arcs. Therefore since the infinite series which express those actions of forces, and their variations, include only sines and cosines, with their powers and fluxions, it follows, that all accumulated forces, and variations of forces, and variations of variations, through infinite orders, are still expressible by repeated sums of sines or cosines, corresponding to arcs which are generated by going round and round the circle. The analyst knows that these quantities become alternately positive and negative; and therefore, in whatever way they are compounded by addition of themselves, or their multiples, or both, we must always arrive at a period after which they will be repeated with all their intermediate variations. It may be extremely difficult, it may be impossible, in our present state of mathematical knowledge, to ascertain all those periods. It has required all the efforts of all the geniuses of Europe to manage the formulas which include terms containing the fourth and fifth powers of the eccentricities of the planetary orbits. Therefore the periods which we have already determined, and the limits to which the inequalities expressed by secular equations arrive, are still subjected to smaller corrections of incomparably longer periods, which arise from the terms neglected in our formulas. But the correction arising from any neglected term has a period and a limit; and thus it will happen that the system works itself into a state of permanency, containing many intervening apparent anomalies. The elliptical motion of the earth contains an anomaly or deviation from uniform circular motion; the action of Jupiter produces a deviation from this elliptical motion, which has a period depending on the configuration of the three bodies; Saturn introduces a deviation from this motion, which has also a period; and so on.

There is another accurate adjustment of motions which has attracted attention, as a thing in the highest degree improbable, in events wholly independent on each other. This is the exact coincidence of the period of the moon's revolution round the earth with that of her rotation round her own axis. The ellipticity or oval shape of the moon differs so insensibly from a sphere,

that if the original rotation had differed considerably from the period of revolution, the pendular tendency to the earth could never have operated a change: but if the difference between those two motions was so small, that the pendular tendency to the line joining the centres of the earth and moon was able to overcome it after some time, the pole of the lunar spheroid would deviate a little from the line joining the earth and moon, and then be brought back to it with an accelerated motion; would pass it as far on the other side, and then return again, vibrating perpetually to each side of the mean position of the radius vector. The extent of this vibration would depend on the original difference between the motion of rotation and the mean motion of revolution. This difference must have been very small, because this pendular vibration is not sensible from the earth. The observed LIBRATION of the moon is precisely what arises from the inequality of her orbital motion. For the same reasons, the effects of the secular equations of the moon (which would, in the course of ages, have brought her whole surface into our view, had her rotation been strictly uniform) are counteracted by her pendular tendency, which has a force sufficient to alter her rotation by nearly the same flow and insensible changes that obtain in her mean motions. The same causes also preserve the nodes of her equator and of her orbit in the same points of the ecliptic. The complete demonstration of this is perhaps the most delicate and elegant specimen that has been given of the modern analysis. We owe it to M. de la Grange: and he makes it appear that the figure of the moon is not that which a fluid sphere would acquire by its gravitation to the earth; it must be the effect of a more considerable ellipticity, or internal inequality of density.

63 This permanency of the system, within very narrow limits of deviation from its present state, depends entirely on the law of planetary deflection. Had it been directly or inversely as the distance, the deviations would have been such as to have quickly rendered it wholly unfit for its present purposes. They would have been very great, had the planetary orbits differed much from circles; nay, had some of them moved in the opposite direction. The selection of this law, and this form of the orbits, strikes the mind of a Newton, and indeed any heart possessed of sensibility to moral or intellectual excellence, as a mark of wisdom prompted by benevolence. But De la Place and others, infected with the Theophobia Gallica engendered by our licentious desires, are eager to point it out as a mark of fatalism. They say, that it is essential to all qualities that are diffused from a centre to diminish in the inverse duplicate ratio of the distance. But this is false, and very false: it is a mere geometrical conception. We indeed say, that the density of illumination decreases in this proportion; but who says that this is a quality? Whether it be considered as the emission of luminous corpuscles, or an undulation of an elastic fluid, it is not a quality emanating from a centre: and even in this estimation, it seems gratuitous, whether we shall consider the base of the luminous pyramid, or its whole contents, as the expression of the quantity. Nay, if all qualities must diminish at this rate, all action e distanti must do the same; for when the distances bear any great proportion to the diameters of the particles, their action deviates insensibly from this law, and is perceived only by the accumula-

tion of its effects after a long time. It is only thus that the effects of the oblate figure of Jupiter are perceived in the motion of his satellites. The boasted found philosophy which sees fatal necessity where the most successful students of nature saw moral excellence, has derived very little credit or title to the name of wisdom, by letting loose all those propensities of the human heart which are essentially destructive of social happiness. These propensities were always known to lurk in the 64 heart of man; and those surely were the wisest who laboured to keep them in check by the influence of moral principles, and particularly by cherishing that disposition of the human heart which prompts us to see contrivance wherever we see nice and refined adjustment of means to ends; and, from the admirable beauty of the solar system, to cry out,

"These are thy glorious works, Parent of good!
"Almighty, thine this universal frame,
"Thus wondrous fair; thyself how wondrous then!
"Unspeakable, who sitt'st above these heavens,
"To us invisible, or dimly seen
"In these thy lowly works; yet these declare
"Thy goodness beyond thought, and power divine."
Par. Lest, b. v.

"But wandering oft, with brute unconscious gaze,
"Man marks not THEE, marks not the mighty hand
"That, ever busy, wheels the silent spheres."
THOMSON.

The most important addition (in a philosophical view) that has been made to astronomical science since the discovery of the aberration of light and the nutation of the earth's axis, is that of the rotation of Saturn's ring. The ring itself is an object quite singular; and when it was discovered that all the bodies which had any immediate connection with a planet were heavy, or gravitated toward that planet, it became an interesting question, what was the nature of this ring? what supported this immense arch of heavy matter without its resting on the planet? what maintains it in perpetual concentricity with the body of Saturn, and maintains its surface in one invariable position?

The theory of universal gravitation tells us what things are possible in the solar system; and our conjectures about the nature of this ring must always be regulated by the circumstance of its gravitation to the planet. Philosophers had at first supposed it to be a luminous atmosphere, thrown out into that form by the great centrifugal force arising from a rotation: but its well defined edge, and, in particular, its being two very narrow rings, extremely near each other, yet perfectly separate, rendered this opinion of its constitution more improbable.

65 Dr Herschel's discovery of brighter spots on its surface, and that those spots were permanent during the whole time of his observation, seems to make it more probable that the parts of the ring have a solid connection. Mr Herschel has discovered, by the help of those spots, that the ring turns round its axis, and that this axis is also the axis of Saturn's rotation. The time of rotation is 10h. 32$\backslash frac\{1\}\{2\}$. But the other circumstances are not narrated with the precision sufficient for an accurate comparison with the theory of gravity. He informs us, that the radii of the four edges of the ring are 59$\backslash frac\{1\}\{2\}$, 75$\backslash frac\{1\}\{2\}$, 77$\backslash frac\{1\}\{2\}$, 83$\backslash frac\{1\}\{2\}$, of a certain scale, and that the angle

angle subtended by the ring at the mean distance from the earth is $46\backslash frac\{1\}\{2\}\text{'}\text{'}$. Therefore its elongation is $23\backslash frac\{1\}\{2\}\text{'}\text{'}$. The elongation of the second Cassinian satellite is $56\text{'}\text{'}$, and its revolution is 2d. 17h. 44'. This should give, by the third law of Kepler, 17h. 10' for the revolution of the outer edge of the ring, or rather of an atom of that edge, in order that it may maintain itself in equilibrio. The same calculation applied to the outer edge of the inner ring gives about 13h. 36'; and we obtain 11h. 16' for the inner edge of this ring. Such varieties are inconsistent with the permanent appearance of a spot. We may suppose the ring to be a luminous fluid or vapour, each particle of which maintains its situation by the law of planetary revolution. In such a state, it would consist of concentric strata, revolving more slowly as they were more remote from the planet, like the concentric strata of a vortex, and therefore having a relative motion incompatible with the permanency of any spot. Besides, the rotation observed by Herschel is too rapid even for the innermost part of the ring. We think therefore that it consists of cohering matter, and of considerable tenacity, at least equal to that of a very clammy fluid, such as melted glass.

We can tell the figure which a fluid ring must have, so that it may maintain its form by the mutual gravitation of its particles to each other, and their gravitation to the planet. Suppose it cut by a meridian. It may be in equilibrio if the section is an ellipse, of which the longer axis is directed to the centre of the planet, and very small in comparison with its distance from the centre of the planet, and having the revolution of its middle round Saturn, such as agrees with the Keplerian law. These circumstances are not very consistent with the dimensions of Saturn's inner ring. The distance between the middle of its breadth and the centre of Saturn is 670, and its breadth is 161', nearly one-fourth of the distance from the centre of Saturn. De la Place says, that the revolution of the inner ring observed by Herschel is very nearly that required by Kepler's law: but we cannot see the grounds of this assertion. The above comparisons with the second Cassinian satellite show the contrary. The elongation of that satellite is taken from Bradley's observations, as is also its periodic time. A ring of detached particles revolving in 10h. 32' must be of much smaller diameter than even the inner edge of Saturn's ring. Indeed, the quantity of matter in it might be such as to increase the gravitation considerably; but this would be seen by its disturbing the seventh and sixth satellites, which are exceedingly near it. We cannot help thinking therefore that it consists of matter which has very considerable tenacity. An equatorial zone of matter, tenacious like melted glass, and whirled briskly round, might be thrown off, and, retaining its great velocity, would stretch out while whirling, enlarging in diameter and diminishing in thickness or breadth, or both, till the centrifugal force was balanced by the united force of gravity and tenacity. We find that the equilibrium will not be sensibly disturbed by considerable deviations, such as unequal breadth, or even want of flatness. Such inequalities appear on this ring at the time of its disparition, when its edge is turned to the sun or to us. The appearances of its different sides are then considerably different.

Such a ring or rings must have an oscillatory motion round the centre of Saturn, in consequence of their mu-

tual action, and the action of the sun, and their own irregularities: but there will be a certain position which they have a tendency to maintain, and to which they will be brought back, after deviating from it, by the ellipticity of Saturn, which is very great. The sun will occasion a nutation of Saturn's axis and a precession of his equinoxes, and this will drag along with it both the rings and the neighbouring satellites.

The atmosphere which surrounds a whirling planet cannot have all its parts circulating according to the third law of Kepler. The mutual attrition of the planet, and of the different strata, arising from their different velocities, must accelerate the slowly moving strata, and retard the rapid, till all acquire a velocity proportional to their distance from the axis of rotation; and this will be such that the momentum of rotation of the planet and its atmosphere remains always the same. It will swell out at the equator, and sink at the poles, till the centrifugal force at the equator balances the weight of a superficial particle. The greatest ratio which the equatorial diameter can acquire to the polar axis is that of four to three, unless a cohesive force keeps the particles united, so that it constitutes a liquid, and not an elastic fluid like air; and an elastic fluid cannot form an atmosphere bounded in its dimensions, unless there be a certain rarity which takes away all elasticity. If the equator swells beyond the dimension which makes the gravitation balance the centrifugal force, it must immediately dissipate.

If we suppose that the atmosphere has extended to this limit, and then condenses by cold, or any chemical or other cause different from gravity, its rotation necessarily augments, preferring its former momentum, and the limit will approach the axis; because a greater velocity produces a greater centrifugal force, and requires a greater gravitation to balance it. Such an atmosphere may therefore desert, in succession, zones of its own matter in the plane of its equator, and leave them revolving in the form of rings. It is not unlikely that the rings of Saturn may have been furnished in this very way; and the zones, having acquired a common velocity in their different strata, will preserve it; and they are susceptible of irregularities arising from local causes at the time of their separation, which may afford permanent spots.

We think that the rotation of Saturn's ring affords some hopes of deciding a very important question about the nature of light. If light be the propagation of elastic undulations, its velocity depends entirely on the elasticity and density of the fluid; but if it be the emission of corpuscles, their velocity may be affected by other causes. The velocity of Saturn's ring is $\backslash frac\{1\}\{2\}$ of that of the earth in its orbit, and therefore about $\backslash frac\{1\}\{2\}$ of the velocity of light. The western extremity (to us in the northern regions) is moving from us, and the eastern is moving toward us. If light, by which we see it, be reflected like an elastic ball from an elastic body, there will be an excess in the velocity of the light by which we see the eastern limb above the velocity of the light by which we see the western limb. This excess will be $\backslash frac\{1\}\{2\}$ of the mean velocity of light. This should be discovered by a difference in the refraction of the two lights. If an acromatic prism could be made to refract fourteen degrees, and if Saturn be viewed thro' a telescope with this prism placed before it, there should be

Fig. 2.

Fig. 5.

Fig. 1.

Fig. 4.

Fig. 3.

Fig. 6.

This image shows a blank, aged page from a book. The paper has a light beige or cream color with a slightly textured appearance. A faint, thin rectangular border is visible around the perimeter of the page. In the center of the page, there is a large, faint, circular watermark or impression, which appears to be a stylized floral or geometric design. There are also some minor blemishes and discolorations on the paper, particularly a small, faint brownish stain on the left side.

be a change of shape amounting to sixteen seconds; if the axis of the prism be parallel to the longer axis of the ring, it will distort it prodigiously, and give it an oblique position.

A similar effect will be produced by placing the prism between the eye-glass and the image in the focus of the object-glass.

Our expectation is founded on this unquestionable principle in dynamics, that when a particle of light passes through the active stratum of a transparent body which refracts light toward the perpendicular, the addition made to the square of its velocity by the refracting forces is equal to the square of the velocity which those forces would communicate to a particle at rest on the surface of this refracting stratum of the transparent body. Therefore if the velocity of the incident light be increased, the ratio of the sine of incidence to the sine of refraction will be diminished. It is consonant to common sense, that when the incident light has a greater velocity, it passes more rapidly through the attracting stratum, and a smaller addition is made to the velocity. When the velocity of the incident light is 10000 times greater than that of the earth's annual motion, the sine of incidence is to the sine of refraction in glass as 20 to 31, or as 10000 to 15500. If this be increased $\backslash times\; 1000$, making it 10004, the ratio will be that of 10004 to 15502.62, or of 10000 to 15496.4. The difference between the refractions of the light from the eastern and western extremities of the ring will be, to all sense, the same, if the velocity of the one be diminished to 9998, and the other increased to 10002.

We may just add here, by the way, that the action of another body may considerably change the constitution of this atmosphere. Thus, supposing that the moon had originally an atmosphere, the limit will be that distance from the moon where the centrifugal force, arising from the moon's rotation, added to the gravitation to the earth, balances the gravitation to the moon. If the moon be $\backslash frac\{1\}\{2\}$th of the earth, this limit will be about $\backslash frac\{1\}\{2\}$th of the moon's distance from the earth. If at this distance the elasticity of the atmosphere is not annihilated by its rarefaction, it will be all taken off by the earth, and accumulate round it. This may be the reason why we see no atmosphere about the moon.

What has been said in the article TIDE (Encycl.), will explain the trade-winds on the earth and in Jupiter and Saturn. On the earth they are increased by the expansion of the air by heat. This causes it to rise

in the parts warmed by the sun, and flow off toward the poles, where it is again cooled and condensed. The under stratum of colder and denser air is continually flowing in from the poles. This having less velocity of circulation than the equatorial parts of the earth, must have a relative motion contrary to that of the earth, or from east to west, and this must augment the current produced by gravitation.

Thus we see that all the mechanical phenomena of the solar system, whether relating to the revolutions round the various centres of gravitation, or to the figure of the planets and the oscillations of the fluids which cover them, or to the rotations round their respective axes—are necessary consequences of one simple principle of a gravitation in every particle, decreasing in the reciprocal duplicate ratio of the distance. We see that ⁷⁷ All the mechanical phenomena of the solar system flow from one simple principle described that conic surface which forms the precession of the equinoxes. One impulse, not passing through the centre of the earth, nor in the plane of the ecliptic, will produce the two first motions, and the protuberant matter produced by the rotation will generate the third motion, by the tendency of its parts to the other heavenly bodies. Without this principle, the elliptic motion of the planets and comets, their various inequalities, secular or periodical, those of the moon and of the satellites of Jupiter, the precession of the equinoxes, the nutation of the earth's axis, the figure of the earth, the undulations of its ocean—all would have been imperfectly known, as matters of fact, wholly different from each other, and solitary and unconnected. It is truly deserving admiration, that such an immense variety of important phenomena flow so palpably from one principle, of such simplicity, and such universality, that no phenomenon is now left out unexplained, and predicted with a certainty almost equal to actual observation.

Quæ toties animos veterum torfere sophorum,
Quæque scholas hodie ræuo certamine versant
Obvia conspicimus, nubem pellente Mathesi,
Surgite mortales, terrenas mittite curas,
Atque hinc caliginæ vires dignoscite mentis,
A pecudum vitâ longe lateque remota,

AST

ASY

ASTROTHEMATA, the places or positions of the stars, in an astrological scheme of the heavens.

ASTROTHESIA, is used by some for a constellation or collection of stars in the heavens.

ASTRUM, or ASTRON, a constellation or assemblage of stars: in which sense it is distinguished from Aster, which denotes a single star. Some apply the term, in a more particular sense, to the Great Dog, or rather to the large bright star in his mouth.

ASYMMETRY, the want of proportion, otherwise

called incommensurability, or the relation of two quantities which have no common measure, as between 1 and $\backslash sqrt\{2\}$, or the side and diagonal of a square.

ASYMPTOTES (see Encycl.) are, by some, distinguished into various orders. The asymptote is said to be of the first order, when it coincides with the base of the curvilinear figure; of the second order, when it is a right line parallel to the base; of the third order, when it is a right line oblique to the base; of the fourth order, when it is the common parabola, having its ^axis