Proposition 45, Problem 31
It is required to find the motions of the apsides of orbits that differ very little from circles.

This problem is solved arithmetically by taking the orbit that is described in an immobile plane by a body revolving in a mobile ellipse (as in prop. 44, corol. 2 or 3) and making it approach the form of the orbit whose apsides are required, and by seeking the apsides of the orbit which that body describes in an immobile plane. Orbits will acquire the same shape if the centripetal forces with which those orbits are described, when compared with each other, are made proportional at equal heights. Let point V be the upper apsis, and write T for the greatest height CV, A for any other height CP or Cp, and X for the difference CV — CP of the heights; then the force by which a body moves in an ellipse revolving about its own focus C (as in corol. 2)—and which in corol. 2 was as , that is, as will, when T − X is substituted for A, be as . Any other centripetal force is similarly to be reduced to a fraction whose denominator is A³; and the numerators are to be made analogous [i.e., made proportional in the same degree] by bringing together homologous terms [i.e., corresponding terms, or terms of the same degree]. All of this will be clarified by the following examples.

EXAMPLE 1. Let us suppose the centripetal force to be uniform and thus as , or (writing T − X for A in the numerator) as ; and by bringing together the corresponding [or homologous] terms of the numerators (namely, given ones with given ones, and ones not given with ones not given), RG² − RF² + TF² to T³ will come to be as -F²X to −3T²X + 3TX² − X³ or as −F² to −3T² + 3TX − X². Now, since the orbit is supposed to differ very little from a circle, let the orbit come to coincide with a circle; and because R and T become equal and X is diminished indefinitely, the ultimate ratios will be RG² to T³ as −F² to −3T², or G² to T² as F² to 3T², and by alternation [or alternando] G² to F² as T² to 3T², that is, as 1 to 3; and therefore G is to F, that is, the angle VCp is to the angle VCP, as 1 to √3. Therefore, since a body in an immobile ellipse, in descending from the upper apsis to the lower apsis, completes the angle VCP (so to speak) of 180 degrees, another body in the mobile ellipse (and hence in the immobile orbit with which we are dealing) will, in descending from the upper apsis to the lower apsis, complete the angle VCp of degrees; this is so because of the similarity of this orbit, which the body describes under the action of a uniform centripetal force, to the orbit which a body completing its revolutions in a revolving ellipse describes in a plane at rest. By the above collation of terms, these orbits are made similar, not universally but at the time when they very nearly approach a circular form. Therefore a body revolving with uniform centripetal force in a very nearly circular orbit will always complete an angle of degrees between the upper apsis and the lower apsis, or 103°55′23″ at the center, arriving at the lower apsis from the upper apsis when it has completed this angle once, and returning from the lower to the upper apsis when it has completed the same angle again, and so on without end.

EXAMPLE 2. Let us suppose the centripetal force to be as the height A raised to any power, as Aⁿ⁻³ (that is, ), where n − 3 and n signify any indices of powers whatsoever—integral or fractional, rational or irrational, positive or negative. On reducing the numerator Aⁿ = (T − X)ⁿ to an indeterminate series by our method of converging series, the result is Tⁿ − nXTⁿ⁻¹ + X²Tⁿ⁻² · · · . And by collating the terms of this with the terms of the other numerator RG² − RF² + TF² − F²X, the result is that RG² − RF² + TF² is to Tⁿ as −F² to −nTⁿ⁻¹ + XTⁿ⁻² · · ·. And after taking the ultimate ratios that result when the orbits approach circular form, RG² will be to Tⁿ as −F² to −nTⁿ⁻¹, or G² to Tⁿ⁻¹ as F² to nTⁿ⁻¹, and by alternation [or alternando] G² is to F² as Tⁿ⁻¹ to nTⁿ⁻¹, that is, as 1 to n; and therefore G is to F, that is, the angle VCp is to the angle VCP as 1 to √n. Therefore, since the angle VCP, completed in the descent of a body from the upper apsis to the lower apsis in an ellipse, is 180 degrees, the angle VCp, completed in the descent of a body from the upper apsis to the lower apsis in the very nearly circular orbit which any body describes under the action of a centripetal force proportional to Aⁿ⁻³, will be equal to an angle of degrees; and when this angle is repeated, the body will return from the lower apsis to the upper apsis, and so on without end.

For example, if the centripetal force is as the distance of the body from the center, that is, as A or , n will be equal to 4 and √n will be equal to 2; and therefore the angle between the upper apsis and the lower apsis will be equal to or 90°. Therefore, at the completion of a quarter of a revolution the body will arrive at the lower apsis, and at the completion of another quarter the body will arrive at the upper apsis, and so on by turns without end. This is also manifest from prop. 10. For a body urged by this centripetal force will revolve in an immobile ellipse whose center is in the center of forces. But if the centripetal force is inversely as the distance, that is, directly as or , n will be equal to 2, and thus the angle between the upper and the lower apsis will be degrees, or 127°16′45″, and therefore a body revolving under the action of such a force will—by the continual repetition of this angle—go alternately from the upper apsis to the lower and from the lower to the upper forever. Further, if the centripetal force is inversely as the fourth root of the eleventh power of the height, that is, inversely as A^11/4 and thus directly as or as , n will be equal to ¼, and will be equal to 360°; and therefore a body, setting out from the upper apsis and continually descending from then on, will arrive at the lower apsis when it has completed an entire revolution, and then, completing another entire revolution by continually ascending, will return to the upper apsis; and so on by turns forever.

EXAMPLE 3. Let m and n be any indices of powers of the height, and let b and c be any given numbers, and let us suppose that the centripetal force is as , that is, as or (again by our method of converging series) as then, if the terms of the numerators are collated, the result will be RG² − RF² + TF² to bT^m + cTⁿ as −F² to −mbT^m−1 − ncTⁿ⁻¹ + bXT^m−2 + cXTⁿ⁻² · · · . And after taking the ultimate ratios that result when the orbits approach circular form, G² will be to bT^m−1 + cTⁿ⁻¹ as F² to mbT^m−1 + ncTⁿ⁻¹, and by alternation [or alternando] G² will be to F² as bT^m−1 + cTⁿ⁻¹ to mbTⁿ⁻¹ + ncTⁿ⁻¹. This proportion, if the greatest height CV or T is expressed arithmetically by unity, becomes G² to F² as b + c to mb + nc and thus as 1 to . Hence G is to F, that is, the angle VCp is to the angle VCP, as 1 to . And therefore, since the angle VCP between the upper apsis and the lower apsis in the immobile ellipse is 180 degrees, the angle VCp between the same apsides, in the orbit described by a body under the action of a centripetal force proportional to the quantity , will be equal to an angle of degrees. And by the same argument, if the centripetal force is as , the angle between the apsides will be found to be degrees. And the problem will be resolved in just the same way in more difficult cases. The quantity to which the centripetal force is proportional must always be resolved into converging series having the denominator A³. Then the ratio of the given part of the numerator (resulting from that operation) to its other part, which is not given, is to be made the same as the ratio of the given part of this numerator RG² − RF² + TF² − F²X to its other part, which is not given; and when the superfluous quantities are taken away and unity is written for T, the proportion of G to F will be obtained.

COROLLARY 1. Hence, if the centripetal force is as some power of the height, that power can be found from the motion of the apsides, and conversely. That is, if the total angular motion with which the body returns to the same apsis is to the angular motion of one revolution, or 360 degrees, as some number m to another number n, and the height is called A, the force will be as the power of the height whose index is . This is manifest by the instances in ex. 2. Hence it is clear that the force, in receding from the center, cannot decrease in a ratio greater than that of the cube of the height; if a body revolving under the action of such a force and setting out from an apsis begins to descend, it will never reach the lower apsis or minimum height but will descend all the way to the center, describing that curved line which we treated in prop. 41, corol. 3. But if the body, on setting out from an apsis, begins to ascend even the least bit, it will ascend indefinitely and will never reach the upper apsis. For it will describe the curved line treated in the above-mentioned corol. 3 and in prop. 44, corol. 6. So also, when the force, in receding from the center, decreases in a ratio greater than that of the cube of the height, a body setting out from an apsis (depending on whether it begins to descend or to ascend) either will descend all the way to the center or will ascend indefinitely. But if the force, in receding from the center, either decreases in a ratio less than that of the cube of the height or increases in any ratio of the height whatever, the body will never descend all the way to the center, but will at some time reach a lower apsis; and conversely, if a body descending and ascending alternately from apsis to apsis never gets to the center, either the force in receding from the center will be increased or it will decrease in a ratio less than that of the cube of the height; and the more swiftly the body returns from apsis to apsis, the farther the ratio of the forces will recede from that of the cube.

For example, if by alternate descent and ascent a body returns from upper apsis to upper apsis in 8 or 4 or 2 or 1½ revolutions, that is, if m is to n as 8 or 4 or 2 or 1½ to 1, and therefore has the value ¹/64 − 3 or ¹/16 − 3 or ¼ − 3 or ⁴/9 − 3, the force will be as A^1/64−3 or A^1/16−3 or A^¼−3 or A^4/9−3, that is, inversely as A^3−1/64 or A^3−1/16 or A^3−¼ or A^3−4/9. If the body returns in each revolution to the same unmoving apsis, m will be to n as 1 to 1, and thus will be equal to A⁻² or ; and therefore the decrease in force will be as the square of the height, as has been demonstrated in the preceding propositions. If the body returns to the same apsis in three-quarters or two-thirds or one-third or one-quarter of a single revolution, m will be to n as ¾ or ⅔ or ⅓ or ¼ to 1, and so will be equal to A^16/9−3 or A^9/4−3 or A⁹⁻³ or A¹⁶⁻³; and therefore the force will be either inversely as A^11/9 or A^¾, or directly as A⁶ or A¹³. Finally, if the body in proceeding from upper apsis to upper apsis completes an entire revolution and an additional three degrees (and therefore, during each revolution of the body, that apsis moves three degrees forward [or in consequential]), m will be to n as 363° to 360° or as 121 to 120, and thus will be equal to , and therefore the centripetal force will be inversely as or inversely as approximately. Therefore the centripetal force decreases in a ratio a little greater than that of the square, but 59¾ times closer to that of the square than to that of the cube.

COROLLARY 2. Hence also if a body, under the action of a centripetal force that is inversely as the square of the height, revolves in an ellipse having a focus in the center of forces, and any other extraneous force is added to or taken away from this centripetal force, the motion of the apsides that will arise from that extraneous force can be found out (by instances in ex. 3), and conversely. For example, if the force under the action of which the body revolves in the ellipse is as and the extraneous force which has been taken away is as cA, and hence the remaining force is as , then (as in ex. 3) b will be equal to 1, m will be equal to 1, and n will be equal to 4, and therefore the angle of the revolution between apsides will be equal to an angle of degrees. Let us suppose the extraneous force to be 357.45 times less than the other force under the action of which the body revolves in the ellipse, that is, let us suppose c to be , A or T being equal to 1, and then will come to be , or 180.7623, that is, 180°45′44″. Therefore a body, setting out from the upper apsis, will reach the lower apsis by an angular motion of 180°45′44″ and will return to upper apsis if this angular motion is doubled; and thus in each revolution the upper apsis will move forward through 1°31′28″. ^aThe [advance of the] the apsis of the moon is about twice as swift.^a

So much concerning the motion of bodies in orbits whose planes pass through the center of forces. It remains for us to determine additionally those motions which occur in planes that do not pass through the center of forces. For writers who deal with the motion of heavy bodies are wont to consider the oblique ascents and descents of weights in any given planes as well as perpendicular ascents and descents, and there is equal justification for considering here the motion of bodies that tend to centers under the action of any forces whatever and are supported by eccentric planes. We suppose, however, that these planes are highly polished and absolutely slippery, so as not to retard the bodies. Further, in these demonstrations, in place of the planes on which bodies rest and which they touch by resting on them, we even make use of planes parallel to them, in which the centers of the bodies move and by so moving describe orbits. And by the same principle we then determine the motions of bodies performed in curved surfaces.

SECTION 10

The motion of bodies on given surfaces and the oscillating motion of ^asimple pendulums^a

Proposition 46, Problem 32
Suppose a centripetal force of any kind, and let there be given both the center of force and any plane in which a body revolves, and grant the quadratures of curvilinear figures; it is required to find the motion of a body setting out from a given place with a given velocity along a given straight line in that plane.

Let S be the center of force, SC the least distance of this center from the given plane, P a body setting out from place P along the straight line PZ, Q the same body revolving in its trajectory, and PQR the required trajectory described in the given plane. Join CQ and also QS, and if SV is taken in QS and is proportional to the centripetal force by which the body is drawn toward the center S, and VT is drawn parallel to CQ and meeting SC in T, then the force SV will be resolved (by corol. 2 of the laws) into the forces ST and TV, of which ST, by drawing the body along a line perpendicular to the plane, does not at all change the body’s motion in this plane. But the other force TV, by acting along the position of the plane, draws the body directly toward [i.e., along a line directed toward] the given point C in the plane and thus causes the body to move in this plane just as if the force ST were removed and as if the body revolved in free space about the center C under the action of the force TV alone. But, given the centripetal force TV under the action of which the body Q revolves in free space about the given center C, there are also given (by prop. 42) not only the trajectory PQR described by the body, but also the place Q in which the body will be at any given time, and finally the velocity of the body in that place Q; and conversely. Q.E.I.

Proposition 47, Theorem 15
Suppose that a centripetal force is proportional to the distance of a body from a center; then all bodies revolving in any planes whatever will describe ellipses and will make their revolutions in equal times; and bodies that move in straight lines, by oscillating to and fro, will complete in equal times their respective periods of going and returning.

For, under the same conditions as in prop. 46, the force SV, by which the body Q revolving in any plane PQR is drawn toward the center S, is as the distance SQ; and thus—because SV and SQ, TV and CQ are proportional—the force TV, by which the body is drawn toward the given point C in the plane of the orbit, is as the distance CQ. Therefore, the forces by which bodies that are in the plane PQR are drawn toward point C are, in proportion to the distances, equal to the forces by which bodies are drawn from all directions toward the center S; and thus in the same times the bodies will move in the same figures in any plane PQR about the point C as they would move in free spaces about the center S; and hence (by prop. 10, corol. 2, and prop. 38, corol. 2) in times which are always equal, they will either describe ellipses [i.e., complete a whole revolution in such ellipses] in that plane about the center C or will complete periods of oscillating to and fro in straight lines drawn through the center C in that plane. Q.E.D.

Scholium
The ascents and descents of bodies in curved surfaces are very closely related to the motions just discussed. Imagine that curved lines are described in a plane, that they then revolve around any given axes passing through the center of force and describe curved surfaces by this revolution, and then that bodies move in such a way that their centers are always found in these surfaces. If those bodies, in ascending and descending obliquely, oscillate to and fro, their motions will be made in planes passing through the axis and hence in curved lines by whose revolution those curved surfaces were generated. In these cases, therefore, it is sufficient to consider the motion in those curved lines.

Proposition 48, Theorem 16
If a wheel stands upon the outer surface of a globe at right angles to that surface and, rolling as wheels do, moves forward in a great circle [in the globe’s surface], the length of the curvilinear path traced out by any given point in the perimeter [or rim] of the wheel from the time when that point touched the globe (a curve which may be called a cycloid or epicycloid) will be to twice the versed sine of half the arc [of the rim of the wheel] which during the time of rolling has been in contact with the globe’s surface as the sum of the diameters of the globe and wheel is to the semidiameter of the globe.

Proposition 49, Theorem 17
If a wheel stands upon the inner surface of a hollow globe at right angles to that surface and, rolling as wheels do, moves forward in a great circle [in the globes surface], the length of the curvilinear path traced out by any given point in the perimeter [or rim] of the wheel from the time when that point touched the globe will be to twice the versed sine of half the arc [of the rim of the wheel] which during the time of rolling has been in contact with the globe’s surface as the difference of the diameters of the globe and wheel is to the semidiameter of the globe.

Let ABL be the globe, C its center, BPV the wheel standing upon it, E the center of the wheel, B the point of contact, and P the given point in the perimeter of the wheel. Imagine that this wheel proceeds in the great circle ABL from A through B toward L and, while rolling, rotates in such a way that the arcs AB and PB are always equal to each other and that the given point P in the perimeter of the wheel is meanwhile describing the curvilinear path AP. Now, let AP be the whole curvilinear path described since the wheel was in contact with the globe at A, and the length AP of this path will be to twice the versed sine of the arc ½PB as 2CE to CB. For let the straight line CE (produced if need be) meet the wheel in V, and join CP, BP, EP, VP, and drop the normal VF to CP produced. Let PH and VH, meeting in H, touch the circle in P and V, and let PH cut VF in G, and drop the normals GI and HK to VP. With the same center C and with any radius whatever describe the circle nom cutting the straight line CP in n, the wheel’s perimeter BP in o, and the curvilinear path AP in m; and with center V and radius Vo describe a circle cutting VP produced in q.

Since the wheel, in rolling, always revolves about the point of contact B, it is manifest that the straight line BP is perpendicular to the curved line AP described by the wheel’s point P, and therefore that the straight line VP will touch this curve in point P. Let the radius of the circle nom be gradually increased or decreased, and so at last become equal to the distance CP; then, because the evanescent figure Pnomq and the figure PFGVI are similar, the ultimate ratio of the evanescent line-elements Pm, Pn, Po, and Pq, that is, the ratio of the instantaneous changes of the curve AP, the straight line CP, the circular arc BP, and the straight line VP, will be the same as that of the lines PV, PF, PG, and PI respectively. But since VF is perpendicular to CF, and VH is perpendicular to CV, and the angles HVG and VCF are therefore equal, and the angle VHG is equal to the angle CEP (because the angles of the quadrilateral HVEP are right angles at V and P), the triangles VHG and CEP will be similar; and hence it will come about that EP is to CE as HG to HV or HP and as KI to KP, and by composition [or componendo] or by separation [or dividendo] CB is to CE as PI to PK, and—by doubling of the consequents—CB is to 2CE as PI to PV and as Pq to Pm. Therefore the decrement of the line VP, that is, the increment of the line BV − VP, is to the increment of the curved line AP in the given ratio of CB to 2CE, and therefore (by lem. 4, corol.) the lengths BV − VP and AP, generated by those increments, are in the same ratio. But since BV is the radius, VP is the cosine of the angle BVP or ½BEP, and therefore BV − VP is the versed sine of the same angle; and therefore in this wheel, whose radius is ½BV, BV − VP will be twice the versed sine of the arc ½BP. And thus AP is to twice the versed sine of the arc ½BP as 2CE to CB. Q.E.D.

For the sake of distinction, we shall call the curved line AP in prop. 48 a cycloid outside the globe, and the curved line AP in prop. 49 a cycloid inside the globe.

COROLLARY 1. Hence, if an entire cycloid ASL is described and is bisected in S, the length of the part PS will be to the length VP (which is twice the sine of the angle VBP, where EB is the radius) as 2CE to CB, and thus in a given ratio.

COROLLARY 2. And the length of the semiperimeter AS of the cycloid will be equal to a straight line that is to the diameter BV of the wheel as 2CE to CB.

Proposition 50, Problem 33
To make a pendulum bob oscillate in a given cycloid.

Within a globe QVS described with center C, let the cycloid QRS be given, bisected in R and with its end-points Q and S meeting the surface of the globe on the two sides. Draw CR bisecting the arc QS in O, and produce CR to A, so that CA is to CO as CO to CR. Describe an outer globe DAF with center C and radius CA; and inside this globe let two half-cycloids AQ and AS be described by means of a wheel whose diameter is AO, and let these two half-cycloids touch the inner globe at Q and S and meet the outer globe in A. Let a body T hang from the point A by a thread APT equal to the length AR, and let this body T oscillate between the two half-cycloids AQ and AS in such a way that each time the pendulum departs from the perpendicular AR, the upper part AP of the thread comes into contact with that half-cycloid APS toward which the motion is directed, and is bent around it as an obstacle, while the other part PT of the thread, to which the half-cycloid is not yet exposed, stretches out in a straight line; then the weight T will oscillate in the given cycloid QRS. Q.E.F.

For let the thread PT meet the cycloid QRS in T and the circle QOS in V, and draw CV; and from the end-points P and T of the straight part PT of the thread, erect BP and TW perpendicular to PT, meeting the straight line CV in B and W. It is evident, from the construction and the generation of the similar figures AS and SR, that the perpendiculars PB and TW cut off from CV the lengths VB and VW equal respectively to OA and OR, the diameters of the wheels. Therefore, TP is to VP (which is twice the sine of the angle VBP, where ½BV is the radius) as BW to BV, or AO + OR to AO, that is (since CA is proportional to CO, CO to CR, and by separation [or dividendo] AO to OR), as CA + CO to CA, or, if BV is bisected in E, as 2CE to CB. Accordingly (by prop. 49, corol. 1), the length of the straight part PT of the thread is always equal to the arc PS of the cycloid, and the whole thread APT is always equal to the half-arc APS of the cycloid, that is (by prop. 49, corol. 2), to the length AR. And therefore, conversely, if the thread always remains equal to the length AR, point T will move in the given cycloid QRS. Q.E.D.

COROLLARY. The thread AR is equal to the half-cycloid AS and thus has the same ratio to the semidiameter AC of the outer globe that the half-cycloid SR, similar to it, has to the semidiameter CO of the inner globe.

Proposition 51, Theorem 18
If a centripetal force tending from all directions to the center C of a globe is in each individual place as the distance of that place from the center; and if, under the action of this force alone, the body T oscillates (in the way just described) in the perimeter of the cycloid QRS; then I say that the times of the oscillations, however unequal the oscillations may be, will themselves be equal.

For let the perpendicular CX fall to the indefinitely produced tangent TW of the cycloid and join CT. Now the centripetal force by which the body T is impelled toward C is as the distance CT, and CT may be resolved (by corol. 2 of the laws) into the components CX and TX, of which CX (by impelling the body directly from P) stretches the thread PT and is wholly nullified by the resistance of the thread and produces no other effect, while the other component TX (by urging the body transversely or toward X) directly accelerates the motion of the body in the cycloid; hence it is manifest that the body’s acceleration, which is proportional to this accelerative force, is at each individual moment as the length TX, that is (because CV and WV—and TX and TW, proportional to them—are given), as the length TW, that is (by prop. 49, corol. 1), as the length of the arc of the cycloid TR. Therefore, if the two pendulums APT and Apt are drawn back unequally from the perpendicular [or vertical] AR and are let go simultaneously, their accelerations will always be as the respective arcs to be described TR and tR. But the parts of these arcs described at the beginning of the motion are as the accelerations, that is, as the whole arcs to be described at the beginning, and therefore the parts that remain to be described and the subsequent accelerations proportional to these parts are also as the whole arcs, and so on. Therefore the accelerations—and hence the velocities generated, the parts of the arcs described with these velocities, and the parts to be so described—are always as the whole arcs; and therefore the parts to be described, preserving a given ratio to one another, will vanish simultaneously, that is, the two oscillating bodies will arrive at the same time at the perpendicular [or vertical] AR. And since, conversely, the ascents of the pendulums, made from the lowest place R through the same cycloidal arcs with a reverse motion, are retarded in individual places by the same forces by which their descents were accelerated, it is evident that the velocities of the ascents and descents made through the same arcs are equal and hence occur in equal times; and therefore, since the two parts RS and RQ of the cycloid, each lying on a different side of the perpendicular [or vertical], are similar and equal, the two pendulums will always make their whole oscillations as well as their half-oscillations in the same times. Q.E.D.

COROLLARY. The force by which body T is accelerated or retarded in any place T of the cycloid is to the total weight of body T in the highest place S or Q as the arc TR of the cycloid to its arc SR or QR.

Proposition 52, Problem 34
To determine both the velocities of pendulums in individual places and the times in which complete oscillations, as well as the separate parts of oscillations, are completed.

With any center G and with a radius GH equal to the arc RS of the cycloid, describe the semicircle HKM bisected by the semidiameter GK. And if a centripetal force proportional to the distances of places from the center tends toward that center G, and if in the perimeter HIK that force is equal to the centripetal force in the perimeter of the globe QOS tending toward its center, and if, at the same time that the pendulum T is let go from its highest place S, some other body L falls from H to G; then, since the forces by which the bodies are urged are equal at the beginning of the motion, and are always proportional to the spaces TR and LG which are to be described, and are therefore equal in the places T and L if TR and LG are equal, it is evident that the two bodies describe the equal spaces ST and HL at the beginning of the motion and thus will proceed thereafter to be equally urged and to describe equal spaces. Therefore (by prop. 38), the time in which the body describes the arc ST is to the time of one oscillation as the arc HI (the time in which the body H will reach L) to the semiperiphery HKM (the time in which the body H will reach M). And the velocity of the pendulum bob at the place T is to its velocity at the lowest place R (that is, the velocity of body H in the place L to its velocity in the place G, or the instantaneous increment of the line HL to the instantaneous increment of the line HG, where the arcs HI and HK increase with a uniform flow) as the ordinate LI to the radius GK, or as √(SR² − TR²) to SR. Hence, since in unequal oscillations arcs proportional to the total arcs of the oscillations are described in equal times, both the velocities and the arcs described in all oscillations universally can be found from the given times. As was first to be found.

Now let simple pendulums oscillate in different cycloids described within different globes, whose absolute forces are also different; and if the absolute force of any globe QOS is called V, the accelerative force by which the pendulum is urged in the circumference of this globe, when it begins to move directly toward its center, will be jointly as the distance of the pendulum bob from that center and the absolute force of the globe, that is, as CO × V. Therefore the line-element HY (which is as this accelerative force CO × V) will be described in a given time; and if the normal YZ is erected so as to meet the circumference in Z, the nascent arc HZ will denote that given time. But this nascent arc HZ is as the square root of the rectangle GH × HY, and thus as √(GH × CO × V). Hence the time of a complete oscillation in the cycloid QRS (since it is directly as the semiperiphery HKM, which denotes that complete oscillation, and inversely as the arc HZ, which similarly denotes the given time) will turn out to be as GH directly and √(GH × CO × V) inversely, that is, because GH and SR are equal, as , or (by prop. 50, corol.) as . Therefore the oscillations in all globes and cycloids, made with any absolute forces whatever, are as the square root of the length of the thread directly and as the square root of the distance between the point of suspension and the center of the globe inversely and also as the square root of the absolute force of the globe inversely. Q.E.I.

COROLLARY 1. Hence also the times of bodies oscillating, falling, and revolving can be compared one with another. For if the diameter of the wheel by which a cycloid is described within a globe is made equal to the semidiameter of the globe, the cycloid will turn out to be a straight line passing through the center of the globe, and the oscillation will now be a descent and subsequently an ascent in this straight line. Hence the time of the descent from any place to the center is given, as well as the time (equal to that time of descent) in which a body, by revolving uniformly about the center of the globe at any distance, describes a quadrantal arc. For this time (by the second case [that is, according to the second paragraph above]) is to the time of a half-oscillation in any cycloid QRS as 1 to .

COROLLARY 2. Hence also there follows what Wren and Huygens discovered about the common cycloid. For if the diameter of the globe is increased indefinitely, its spherical surface will be changed into a plane, and the centripetal force will act uniformly along lines perpendicular to this plane, and our cycloid will turn into a common cycloid. But in that case the length of the arc of the cycloid between that plane and the describing point will come out equal to four times the versed sine of half of the arc of the wheel between that same plane and the describing point, as Wren discovered; and a pendulum between two cycloids of this sort will oscillate in a similar and equal cycloid in equal times, as Huygens demonstrated. But also the descent of heavy bodies during the time of one oscillation will be the descent which Huygens indicated.

Moreover, the propositions that we have demonstrated fit the true constitution of the earth, insofar as wheels, moving in the earth’s great circles, describe cycloids outside this globe by the motion of nails fastened in their perimeters; and pendulums suspended lower down in mines and caverns of the earth must oscillate in cycloids within globes in order that all their oscillations may be isochronous. For gravity (as will be shown in book 3) decreases in going upward from the surface of the earth as the square of the distance from the earth’s center, and in going downward from the surface is as the distance from that center.

Proposition 53, Problem 35
Granting the quadratures of curvilinear figures, it is required to find the forces by whose action bodies moving in given curved lines will make oscillations that are always isochronous.

Let a body T oscillate in any curved line STRQ whose axis is AR passing through the center of forces C. Draw TX touching that curve in any place T of the body, and on this tangent TX take TY equal to the arc TR. [This may be done] since the length of that arc can be known from the quadratures of figures by commonly used methods. From point Y draw the straight line YZ perpendicular to the tangent. Draw CT meeting the perpendicular in Z, and the centripetal force will be proportional to the straight line TZ. Q.E.I.

For if the force by which the body is drawn from T toward C is represented by the straight line TZ taken proportional to it, this will be resolved into the forces TY and YZ, of which YZ, by drawing the body along the length of the thread PT, does not change its motion at all, while the other force TY directly accelerates or directly retards its motion in the curve STRQ. Accordingly, since this force is as the projection TR to be described, the body’s accelerations or retardations in describing proportional parts of two oscillations (a greater and a lesser oscillation) will always be as those parts, and will therefore cause those parts to be described simultaneously. And bodies that in the same time describe parts always proportional to the wholes will describe the wholes simultaneously. Q.E.D.

COROLLARY 1. Hence, if body T, hanging by a rectilinear thread AT from the center A, describes the circular arc STRQ and meanwhile is urged downward along parallel lines by some force that is to the uniform force of gravity as the arc TR to its sine TN, the times of any single oscillations will be equal. For, because TZ and AR are parallel, the triangles ATN and ZTY will be similar; and therefore TZ will be to AT as TY to TN; that is, if the uniform force of gravity is represented by the given length AT, the force TZ, by the action of which the oscillations will turn out to be isochronous, will be to the force of gravity AT as the arc TR (equal to TY) to the sine TN of that arc.

COROLLARY 2. And therefore in [pendulum] clocks, if the forces impressed by the mechanism upon the pendulum to maintain the motion can be compounded with the force of gravity in such a way that the total force downward is always as the line that arises from dividing the rectangle of the arc TR and the radius AR by the sine TN, all the oscillations will be isochronous.

Proposition 54, Problem 36
Granting the quadratures of curvilinear figures, to find the times in which bodies under the action of any centripetal force will descend and ascend in any curved lines described in a plane passing through the center of forces.

Let a body descend from any place S through any curved line STtR given in a plane passing through the center of forces C. Join CS and divide it into innumerable equal parts, and let Dd be some one of those parts. With center C and radii CD and Cd, describe the circles DT and dt, meeting the curved line STtR in T and t. Then, since both the law of centripetal force and the height CS from which the body has fallen are given, the velocity of the body at any other height CT will be given (by prop. 39). Moreover, the time in which the body describes the line-element Tt is as the length of this line-element (that is, as the secant of the angle tTC) directly and as the velocity inversely. Let the ordinate DN be proportional to this time and perpendicular to the straight line CS through point D; then, because Dd is given, the rectangle Dd × DN, that is, the area DNnd, will be proportional to that same time. Therefore if PNn is the curved line that point N continually traces out, ^aand its asymptote is the straight line SQ standing perpendicularly upon the straight line CS,^a the area SQPND will be proportional to the time in which the body, by descending, has described the line ST; and accordingly, when that area has been found, the time will be given. Q.E.I.

Proposition 55, Theorem 19
If a body moves in any curved surface whose axis passes through a center of forces, and a perpendicular is dropped from the body to the axis, and a straight line parallel and equal to the perpendicular is drawn from any given point of the axis; I say that the parallel will describe an area proportional to the time.

Let BKL be the curved surface, T the body revolving in it, STR the trajectory which the body describes in it, S the beginning of the trajectory, OMK the axis of the curved surface, TN the perpendicular straight line dropped from the body to the axis; and let OP be the straight line parallel and equal to TN and drawn from a point O that is given in the axis, AP the path described by point P in the plane AOP of the revolving line OP, A the beginning of the projection (corresponding to point S); and let TC be a straight line drawn from the body to the center, TG the part of TC that is proportional to the centripetal force by which the body is urged toward the center C, TM a straight line perpendicular to the curved surface, TI the part of TM proportional to the force of pressure by which the body urges the surface and is in turn urged by the surface toward M; and let PTF be a straight line parallel to the axis and passing through the body, and GF and IH straight lines dropped perpendicularly from the points G and I to the parallel PHTF. I say now that the area AOP, described by the radius OP from the beginning of the motion, is proportional to the time. For the force TG (by corol. 2 of the laws) is resolved into the forces TF and FG, and the force TI into the forces TH and HI. But the forces TF and TH, by acting along the line PF perpendicular to the plane AOP, change the body’s motion only insofar as it is perpendicular to this plane. And therefore the body’s motion, insofar as it takes place in the position of the plane—that is, the motion of point P, by which the projection AP of the trajectory is described in this plane—is the same as if the forces TF and TH were taken away and the body were acted on by the forces FG and HI alone; that is, it is the same as if the body were to describe the curve AP in the plane AOP under the action of a centripetal force tending toward the center O and equal to the sum of the forces FG and HI. But by the action of such a force the area AOP is (by prop. 1) described proportional to the time.^a Q.E.D.

COROLLARY. By the same argument, if a body, acted on by forces tending toward two or more centers in any one given straight line CO, described any curved line ST in free space, the area AOP would always be proportional to the time.

Proposition 56, Problem 37
Granting the quadratures of curvilinear figures, and given both the law of centripetal force tending toward a given center and a curved surface whose axis passes through that center, it is required to find the trajectory that a body will describe in that same surface when it has set out from a given place with a given velocity, in a given direction in that surface.

Assuming the same constructions as in prop. 55, let body T go forth from the given place S, along a straight line given in position, in the required trajectory STR, and let the projection of this trajectory in the plane BLO be AP. And since the velocity of the body is given at the height SC, its velocity at any other height TC will be given. With this velocity, let the body in a given minimally small time describe the particle Tt of its trajectory, and let Pp be its projection described in the plane AOP. Join Op, and let the projection (in the plane AOP) of the little circle described with center T and radius Tt in the curved surface be the ellipse pQ. Then, because the little circle Tt is given in magnitude, and its distance TN or PO from the axis CO is given, the ellipse pQ will be given in species and in magnitude, as well as in its position with respect to the straight line PO. And since the area POp is proportional to the time and therefore given because the time is given, the angle POp will be given. And hence the common intersection p of the ellipse and the straight line Op will be given, along with the angle OPp in which the projection APp of the trajectory cuts the line OP. And accordingly (by consulting prop. 41 with its corol. 2) the way of determining the curve APp is readily apparent. Then, erecting perpendiculars to the plane AOP one by one, from the points P of the projection, so as to meet the curved surface in T, the points T of the trajectory will be given one by one. Q.E.I.

SECTION 11

The motion of bodies drawn to one another by centripetal forces

Up to this point, I have been setting forth the motions of bodies attracted toward an immovable center, such as, however, hardly exists in the natural world. For attractions are always directed toward bodies, and—by the third law—the actions of attracting and attracted bodies are always mutual and equal; so that if there are two bodies, neither the attracting nor the attracted body can be at rest, but both (by corol. 4 of the laws) revolve about a common center of gravity as if by a mutual attraction; and if there are more than two bodies that either are all attracted by and attract a single body or all attract one another, these bodies must move with respect to one another in such a way that the common center of gravity either is at rest or moves uniformly straight forward. For this reason I now go on to set forth the motion of bodies that attract one another, considering centripetal forces as attractions, although perhaps—if we speak in the language of physics—they might more truly be called impulses. For here we are concerned with mathematics; and therefore, putting aside any debates concerning physics, we are using familiar language so as to be more easily understood by mathematical readers.

Proposition 57, Theorem 20
Two bodies that attract each other describe similar figures about their common center of gravity and also about each other.

For the distances of these bodies from their common center of gravity are inversely proportional to the masses of the bodies and therefore in a given ratio to each other and, by composition [or componendo], in a given ratio to the total distance between the bodies. These distances, moreover, rotate about their common end-point with an equal angular motion because, since they always lie in the same straight line, they do not change their inclination toward each other. And straight lines that are in a given ratio to each other and that rotate about their end-points with an equal angular motion describe entirely similar figures about the end-points in planes that, along with these end-points, either are at rest or move with any motion that is not angular. Accordingly, the figures described by the rotation of these distances are similar. Q.E.D.

Proposition 58, Theorem 21
If two bodies attract each other with any forces whatever and at the same time revolve about their common center of gravity, I say that by the action of the same forces there can be described around either body if unmoved a figure similar and equal to the figures that the bodies so moving describe around each other.

Let bodies S and P revolve about their common center of gravity C, going from S to T and from P to Q. From a given point s let sp and sq be drawn always equal and parallel to SP and TQ; then the curve pqv, which the point p describes by revolving around the motionless point s, will be similar and equal to the curves that bodies S and P describe around each other; and accordingly (by prop. 57) this curve pqv will be similar to the curves ST and PQV, which the same bodies describe around their common center of gravity C; and this is so because the proportions of the lines SC, CP, and SP or sp to one another are given.

CASE 1. The common center of gravity C (by corol. 4 of the laws) either is at rest or moves uniformly straight forward. Let us suppose first that it is at rest, and at s and p let two bodies be placed, a motionless one at s and a moving one at p, similar and equal to bodies S and P. Then let the straight lines PR and pr touch the curves PQ and pq in P and p, and let CQ and sq be produced to R and r. Then, because the figures CPRQ and sprq are similar, RQ will be to rq as CP to sp and thus in a given ratio. Accordingly, if the force with which body P is attracted toward body S, and therefore toward the intermediate center C, were in that same given ratio to the force with which body p is attracted toward center s, then in equal times these forces would always attract the bodies from the tangents PR and pr to the arcs PQ and pq through the distances RQ and rq proportional to them; and therefore the latter force would cause body p to revolve in orbit in the curve pqv, which would be similar to the curve PQV, in which the former force causes body P to revolve in orbit, and the revolutions would be completed in the same times. But those forces are not to each other in the ratio CP to sp but are equal to each other (because bodies S and s, P and p are similar and equal, and distances SP and sp are equal); therefore, the bodies will in equal times be equally drawn away from the tangents; and therefore, for the second body p to be attracted through the greater distance rq, a greater time is required, which is as the square root of the distances, because (by lem. 10) the spaces described at the very beginning of the motion are as the squares of the times. Therefore, let the velocity of body p be supposed to be to the velocity of body P as the square root of the ratio of the distance sp to the distance CP, so that the arcs pq and PQ, which are in a simple ratio, are described in times which are as the square roots of the distances. Then bodies P and p, being always attracted by equal forces, will describe around the centers C and s at rest the similar figures PQV and pqv, of which pqv is similar and equal to the figure that body P describes around the moving body S. Q.E.D.

CASE 2. Let us suppose now that the common center of gravity, along with the space in which the bodies are moving with respect to each other, is moving uniformly straight forward; then (by corol. 6 of the laws) all motions in this space will occur as in case 1. Hence the bodies will describe around each other figures which are the same as before and which therefore will be similar and equal to the figure pqv. Q.E.D.

COROLLARY 1. Hence (by prop. 10) two bodies, attracting each other with forces proportional to their distance, describe concentric ellipses, both around their common center of gravity and also around each other; and, conversely, if such figures are described, the forces are proportional to the distance.

COROLLARY 2. And (by props. 11, 12, and 13) two bodies, under the action of forces inversely proportional to the square of the distance, describe—around their common center of gravity and also around each other—conics having their focus in that center about which the figures are described. And, conversely, if such figures are described, the centripetal forces are inversely proportional to the square of the distance.

COROLLARY 3. Any two bodies revolving in orbit around a common center of gravity describe areas proportional to the times, by radii drawn to that center and also to each other.

Proposition 59, Theorem 22
The periodic time of two bodies S and P revolving about their common center of gravity C is to the periodic time of one of the two bodies P, revolving in orbit about the other body S which is without motion, and describing a figure similar and equal to the figures that the bodies describe around each other, as the square root of the ratio of the mass of the second body S to the sum of the masses of the bodies S + P.

For, from the proof of prop. 58, the times in which any similar arcs PQ and pq are described are as the square roots of the distances CP and SP or sp, that is, as the square root of the ratio of body S to the sum of the bodies S + P [or, as √S to √(S + P)]. And by composition [or componendo] the sums of the times in which all the similar arcs PQ and pq are described, that is, the whole times in which the whole similar figures are described, are in that same ratio. Q.E.D.

Proposition 60, Theorem 23
If two bodies S and P, attracting each other with forces inversely proportional to the square of the distance, revolve about a common center of gravity, I say that the principal axis of the ellipse which one of the bodies P describes by this motion about the other body S will be to the principal axis of the ellipse which the same body P would be able to describe in the same periodic time about the other body S at rest as the sum of the masses of the two bodies S + P is to the first of two mean proportionals between this sum and the mass of the other body S.^a

For if the ellipses so described were equal to each other, the periodic times would (by prop. 59) be as the square root of the mass of body S is to the square root of the sum of the masses of the bodies S + P. Let the periodic time in the second ellipse be decreased in this same ratio, and then the periodic times will become equal; but the principal axis of the second ellipse (by prop. 15) will be decreased as the ³/2 power of the former ratio, that is, in the ratio of which the ratio S to S + P is the cube; and therefore the principal axis of the second ellipse will be to the principal axis of the first ellipse as the first of two mean proportionals between S + P and S to S + P. And inversely, the principal axis of the ellipse described about the body in motion will be to the principal axis of the ellipse described about the body not in motion as S + P to the first of two mean proportionals between S + P and S. Q.E.D.

Proposition 61, Theorem 24
If two bodies, attracting each other with any kind of forces and not otherwise acted on or impeded, move in any way whatever, their motions will be the same as if they were not attracting each other but were each being attracted with the same forces by a third body set in their common center of gravity. And the law of the attracting forces will be the same with respect to the distance of the bodies from that common center and with respect to the total distance between the bodies.

For the forces with which the bodies attract each other, in tending toward the bodies, tend toward a common center of gravity between them and therefore are the same as if they were emanating from a body between them. Q.E.D.

And since there is given the ratio of the distance of either of the two bodies from that common center to the distance between the bodies, there will also be given the ratio of any power of one such distance to the same power of the other distance, as well as the ratio that any quantity derived in any manner from one such distance together with given quantities has to another quantity derived in the same manner from the other distance together with the same number of given quantities having that given ratio of distances to the former ones. Accordingly, if the force with which one body is attracted by the other is directly or inversely as the distance of the bodies from each other or as any power of this distance or finally as any quantity derived in any manner from this distance and given quantities, the same force with which the same body is attracted to the common center of gravity will be likewise directly or inversely as the distance of the attracted body from that common center or as the same power of this distance or finally as a quantity derived in the same manner from this distance and analogous given quantities. That is, the law of the attracting force will be the same with respect to either of the distances. Q.E.D.

Proposition 62, Problem 38
To determine the motions of two bodies that attract each other with forces inversely proportional to the square of the distance and are let go from given places.

These bodies will (by prop. 61) move just as if they were being attracted by a third body set in their common center of gravity; and by hypothesis, that center will be at rest at the very beginning of the motion and therefore (by corol. 4 of the laws) will always be at rest. Accordingly, the motions of the bodies are (by prop. 36) to be determined just as if they were being urged by forces tending toward that center, and the motions of the bodies attracting each other will then be known. Q.E.I.

Proposition 63, Problem 39
To determine the motions of two bodies that attract each other with forces inversely proportional to the square of the distance and that set out from given places with given velocities along given straight lines.

From the given motions of the bodies at the beginning the uniform motion of the common center of gravity is given, as well as the motion of the space that moves along with this center uniformly straight forward, and also the initial motions of the bodies with respect to this space. Now (by corol. 5 of the laws and prop. 61), the subsequent motions take place in this space just as if the space itself, along with that common center of gravity, were at rest, and as if the bodies were not attracting each other but were being attracted by a third body situated in that center. Therefore the motion of either body in this moving space, setting out from a given place with a given velocity along a given straight line and pulled by a centripetal force tending toward that center, is to be determined (by props. 17 and 37), and at the same time the motion of the other body about the same center will be known. This motion is to be compounded with that uniform progressive motion (found above) of the system of the space and bodies revolving in it, and the absolute motion of the bodies in an unmoving space will be known. Q.E.I.

Proposition 64, Problem 40
If the forces with which bodies attract one another increase in the simple ratio of the distances from the centers, it is required to find the motions of more than two bodies in relation to one another.

Suppose first that two bodies T and L have a common center of gravity D. These bodies will (by prop. 58, corol. 1) describe ellipses that have their centers at D and that have magnitudes which become known by prop. 10.

Now let a third body S attract the first two bodies T and L with accelerative forces ST and SL, and let it be attracted by those bodies in turn. The force ST (by corol. 2 of the laws) is resolved into forces SD and DT, and the force SL into forces SD and DL. Moreover, the forces DT and DL, which are as their sum TL and therefore as the accelerative forces with which bodies T and L attract each other, when added respectively to those forces of bodies T and L, compose forces proportional to the distances DT and DL, as before, but greater than those former forces, and therefore (by prop. 10, corol. 1, and prop. 4, corols. 1 and 8) they cause those bodies to describe ellipses as before, but with a swifter motion. The remaining accelerative forces, each of which is SD, by attracting those bodies T and L equally and along lines TI and LK (which are parallel to DS) with motive actions SD × T and SD × L (which are as the bodies), do not at all change the situations of those bodies in relation to one another, but make them equally approach line IK, which is to be conceived as drawn through the middle of body S, perpendicular to the line DS. That approach to line IK, however, will be impeded by causing the system of bodies T and L on one side and body S on the other to revolve in orbit with just the right velocities about a common center of gravity C. Body S describes an ellipse about that same point C with such a motion, because the sum of the motive forces SD × T and SD × L, which are proportional to the distance CS, tends toward the center C; and because CS and CD are proportional, point D will describe a similar ellipse directly opposite. But bodies T and L, being attracted respectively by motive forces SD × T and SD × L equally and along the parallel lines TI and LK (as has been said), will (by corols. 5 and 6 of the laws) proceed to describe their own ellipses about the moving center D, as before. Q.E.I.

Now let a fourth body V be added, and by a similar argument it will be concluded that this point and point C describe ellipses about B, the common center of gravity of all the bodies, while the motions of the former bodies T, L, and S about centers D and C remain the same as before, but accelerated. And by the same method it will be possible to add more bodies. Q.E.I.

These things are so, even if bodies T and L attract each other with accelerative forces that are greater or less than those by which they attract the rest of the bodies in proportion to the distance. Let the mutual accelerative attractions of all the bodies to one another be as the distances multiplied by the attracting bodies; then, from what has gone before, it will be easily deduced that all the bodies describe different ellipses in equal periodic times about B, the common center of gravity of them all, in a motionless plane. Q.E.I.

Proposition 65, Theorem 25
More than two bodies whose forces decrease as the squares of the distances from their centers are able to move with respect to one another in ellipses and, by radii drawn to the foci, are able to describe areas proportional to the times very nearly.

In prop. 64 the case was demonstrated in which the several motions occur exactly in ellipses. The more the law of force departs from the law there supposed, the more the bodies will perturb their mutual motions; nor can it happen that bodies will move exactly in ellipses while attracting one another according to the law here supposed, except by maintaining a fixed proportion of distances one from another. In the following cases, however, the orbits will not be very different from ellipses.

CASE 1. Suppose that several lesser bodies revolve about some very much greater one at various distances from it, and that absolute forces proportional to these bodies [i.e., their masses] tend toward each and every one of them. Then, since the common center of gravity of them all (by corol. 4 of the laws) either is at rest or moves uniformly straight forward, let us imagine that the lesser bodies are so small that the greater body never is sensibly distant from this center. In this case, the greater body will—without any sensible error—either be at rest or move uniformly straight forward, while the lesser ones will revolve about this greater one in ellipses and by radii drawn to it will describe areas proportional to the times, except insofar as there are errors introduced either by a departure of the greater body from that common center of gravity or by the mutual actions of the lesser bodies on one another. The lesser bodies, however, can be diminished until that departure and the mutual actions are less than any assigned values, and therefore until the orbits square with ellipses and the areas correspond to the times without any error that is not less than any assigned value. Q.E.O.

CASE 2. Let us now imagine a system of lesser bodies revolving in the way just described around a much greater one, or any other system of two bodies revolving around each other, to be moving uniformly straight forward and at the same time to be urged sideways by the force of another very much greater body, situated at a great distance. Then, since the equal accelerative forces by which the bodies are urged along parallel lines do not change the situations of the bodies in relation to one another, but cause the whole system to be transferred simultaneously, while the motions of the parts with respect to one another are maintained; it is manifest that no change whatsoever of the motion of the bodies attracted among themselves will result from their attractions toward the greater body, unless such a change comes either from the inequality of the accelerative attractions or from the inclination to one another of the lines along which the attractions take place. Suppose, therefore, that all the accelerative attractions toward the greater body are with respect to one another inversely as the squares of the distances; then by increasing the distance of the greater body until the differences (with respect to their length) among the straight lines drawn from this body to the other bodies and their inclinations with respect to one another are less than any assigned values, the motions of the parts of the system with respect to one another will persevere without any errors that are not less than any assigned values. And since, because of the slight distance of those parts from one another, the whole system is attracted as if it were one body, that system will be moved by this attraction as if it were one body; that is, by its center of gravity it will describe about the greater body some conic (namely, a hyperbola or parabola if the attraction is weak, an ellipse if the attraction is stronger) and by a radius drawn to the greater body will describe areas proportional to the times without any errors except the ones that may be produced by the distances between the parts, and these are admittedly slight and may be diminished at will. Q.E.O.

By a similar argument one can go on to more complex cases indefinitely.

COROLLARY 1. In case 2, the closer the greater body approaches to the system of two or more bodies, the more the motions of the parts of the system with respect to one another will be perturbed, because the inclinations to one another of the lines drawn from this great body to those parts are now greater, and the inequality of the proportion is likewise greater.

COROLLARY 2. But these perturbations will be greatest if the accelerative attractions of the parts of the system toward the greater body are not to one another inversely as the squares of the distances from that greater body, especially if the inequality of this proportion is greater than the inequality of the proportion of the distances from the greater body. For if the accelerative force, acting equally and along parallel lines, in no way perturbs the motions of the parts of the system with respect to one another, it will necessarily cause a perturbation to arise when there is an inequality in its action, and such perturbation will be greater or less according as this inequality is greater or less. The excess of the greater impulses acting on some bodies, but not acting on others, will necessarily change the situation of the bodies with respect to one another. And this perturbation, added to the perturbation that arises from the inclination and inequality of the lines, will make the total perturbation greater.

COROLLARY 3. Hence, if the parts of this system—without any significant perturbation—move in ellipses or circles, it is manifest that these parts either are not urged at all (except to a very slight degree indeed) by accelerative forces tending toward other bodies, or are all urged equally and very nearly along parallel lines.

Proposition 66^a, Theorem 26
Let three bodies—whose forces decrease as the squares of the distances—attract one another, and let the accelerative attractions of any two toward the third be to each other inversely as the squares of the distances, and let the two lesser ones revolve about the greatest. Then I say that if that greatest body is moved by these attractions, the inner body [of the two revolving bodies] will describe about the innermost and greatest body, by radii drawn to it, areas more nearly proportional to the times and a figure more closely approaching the shape of an ellipse (having its focus in the meeting point of the radii) than would be the case if that greatest body were not attracted by the smaller ones and were at rest, or if it were much less or much more attracted and were acted on either much less or much more.

This is sufficiently clear from the demonstration of the second corollary of prop. 65, but it is proved as follows by a more lucid and more generally convincing argument.

CASE 1. Let the lesser bodies P and S revolve in the same plane about a greatest body T, and let P describe the inner orbit PAB, and S the outer orbit ESE. Let SK be the mean distance between bodies P and S, and let the accelerative attraction of body P toward S at that mean distance be represented by that same line SK. Let SL be taken to SK as SK² to SP², and SL will be the accelerative attraction of body P toward S at any distance SP. Join PT, and parallel to it draw LM meeting ST in M; then the attraction SL will be resolved (by corol. 2 of the laws) into attractions SM and LM. And thus body P will be urged by a threefold accelerative force. One such force tends toward T and arises from the mutual attraction of bodies T and P. By this force alone (whether T is motionless or is moved by this attraction), body P must, by a radius PT, describe around body T areas proportional to the times and must also describe an ellipse whose focus is in the center of body T. This is clear from prop. 11 and prop. 58, corols. 2 and 3.

The second force is that of the attraction LM, which (since it tends from P to T) will, when added to the first of these forces, coincide with it and will thus cause areas to be described that are still proportional to the times, by prop. 58, corol. 3. But since this force is not inversely proportional to the square of the distance PT, it will, together with the first force, compose a force differing from this proportion—and the more so, the greater the proportion of this force is to that first force, other things being equal. Accordingly, since (by prop. 11 and by prop. 58, corol. 2) the force by which an ellipse is described about the focus T must tend toward that focus and be inversely proportional to the square of the distance PT, that composite force, by differing from this proportion, will cause the orbit PAB to deviate from the shape of an ellipse having its focus in T, and the more so the greater the difference from this proportion; and the difference from this proportion will be greater according as the proportion of the second force LM to the first force is greater, other things being equal.

But now the third force SM, by attracting body P along a line parallel to ST, will, together with the former forces, compose a force which is no longer directed from P to T and which deviates from this direction the more, the greater the proportion of this third force is to the former forces, other things being equal; and this compound force therefore will make body P describe, by a radius TP, areas no longer proportional to the times and will make the divergence from this proportionality be the greater, the greater the proportion of this third force is to the other forces. This third force will increase the deviation of the orbit PAB from the aforesaid elliptical shape for two reasons: not only is this force not directed from P to T, but also it is not inversely proportional to the square of the distance PT. Once these things have been understood, it is manifest that the areas will be most nearly proportional to the times when this third force is least, the other forces remaining the same as they were; and that the orbit PAB approaches closest to the aforesaid elliptical shape when both the second force and the third (but especially the third force) are least, the first force remaining the same as it was.

Let the accelerative attraction of body T toward S be represented by line SN; and if the accelerative attractions SM and SN were equal, they would, by attracting bodies T and P equally and along parallel lines, not at all change the situation of those two bodies with respect to each other. In this case, their motions with respect to each other would (by corol. 6 of the laws) be the same as it would be without these attractions. And for the same reason, if the attraction SN were smaller than the attraction SM, it would take away the part SN of the attraction SM, and only the part MN would remain, by which the proportionality of the times and areas and the elliptical shape of the orbit would be perturbed. And similarly, if the attraction SN were greater than the attraction SM, the perturbation of the proportionality and of the orbit would arise from the difference MN alone. Thus SM, the third attraction above, is always reduced by the attraction SN to the attraction MN, the first and second attractions remaining completely unchanged; and therefore the areas and times approach closest to proportionality, and the orbit PAB approaches closest to the aforesaid elliptical shape, when the attraction MN is either null or the least possible—that is, when the accelerative attractions of bodies P and T toward body S approach as nearly as possible to equality, in other words, when the attraction SN is neither null nor less than the least of all the attractions SM, but is a kind of mean between the maximum and minimum of all those attractions SM, that is, not much greater and not much smaller than the attraction SK. Q.E.D.

CASE 2. Now let the lesser bodies P and S revolve about the greatest body T in different planes; then the force LM, acting along a line PT situated in the plane of orbit PAB, will have the same effect as before, and will not draw body P away from the plane of its orbit. But the second force NM, acting along a line that is parallel to ST (and therefore, when body S is outside the line of the nodes, is inclined to the plane of orbit PAB), besides the perturbation of its motion in longitude, already set forth above, will introduce a perturbation of the motion in latitude, by attracting body P out of the plane of its orbit. And this perturbation, in any given situation of bodies P and T with respect to each other, will be as the generating force MN, and therefore becomes least when MN is least, that is (as I have already explained), when the attraction SN is not much greater and not much smaller than the attraction SK. Q.E.D.

COROLLARY 1. Hence it is easily gathered that if several lesser bodies P, S, R, . . . revolve about a greatest body T, the motion of the innermost body P will be least perturbed by the attractions of the outer bodies when the greatest body T is attracted and acted on as much by the other bodies (according to the ratio of the accelerative forces) as the other bodies are by one another.

COROLLARY 2. In a system of three bodies T, P, and S, if the accelerative attractions of any two toward the third are to each other inversely as the squares of the distances, body P will describe, by a radius PT, an area about body T more swiftly near their conjunction A and their opposition B than near the quadratures C and D. For every force by which body P is urged and body T is not, and which does not act along line PT, accelerates or retards the description of areas, according as its direction is forward and direct [or in consequentia] or retrograde [or in antecedentia]. Such is the force NM. In the passage of body P from C to A, this force is directed forward [or in consequentia] and accelerates the motion; afterward, as far as D, it is retrograde [or in antecedentia] and retards the motion; then forward up to B, and finally retrograde in passing from B to C.

COROLLARY 3. And by the same argument it is evident that body P, other things being the same, moves more swiftly in conjunction and opposition than in the quadratures.

COROLLARY 4. The orbit of body P, other things being the same, is more curved in the quadratures than in conjunction and opposition. For swifter bodies are deflected less from a straight path. And besides, in conjunction and opposition the force KL, or NM, is opposite to the force with which body T attracts body P and therefore diminishes that force, while body P will be deflected less from a straight path when it is less urged toward body T.

COROLLARY 5. Accordingly, body P, other things being the same, will recede further from body T in the quadratures than in conjunction and opposition. These things are so if the motion of [i.e., change in] eccentricity is neglected. For if the orbit of body P is eccentric, its eccentricity (as will shortly be shown in corol. 9 of this proposition) will come out greatest when the apsides are in the syzygies; and thus it can happen that body P, arriving at the upper apsis, may be further away from body T in the syzygies than in the quadratures.

COROLLARY 6. Since the centripetal force of the central body T, which keeps body P in its orbit, is increased in the quadratures by the addition of the force LM and is diminished in the syzygies by the subtraction of the force KL and, because of the magnitude of the force KL [which is greater than LM], is more diminished than increased; and since that centripetal force (by prop. 4, corol. 2) is in a ratio compounded of the simple ratio of the radius TP directly and the squared ratio of the periodic time inversely [i.e., the force is directly as the radius and inversely as the square of the periodic time], it is evident that this compound ratio is diminished by the action of the force KL, and therefore that the periodic time (assuming the radius TP of the orbit to remain unchanged) is increased as the square root of the ratio in which that centripetal force is diminished. It is therefore further evident that, assuming this radius to be increased or diminished, the periodic time is increased more or diminished less than as the ³/2 power of this radius, by prop. 4, corol. 6. If the force of the central body were gradually to weaken, body P, attracted always less and less, would continually recede further and further from the center T; and on the contrary, if the force were increased, body P would approach nearer and nearer. Therefore, if the action of the distant body S, whereby the force is diminished, is alternately increased and diminished, radius TP will at the same time also be alternately increased and diminished, and the periodic time will be increased and diminished in a ratio compounded of the ³/2 power of the ratio of the radius and the square root of the ratio in which the centripetal force of the central body T is diminished or increased by the increase or decrease of the action of the distant body S.

COROLLARY 7. From what has gone before, it follows also that with respect to angular motion the axis of the ellipse described by body P, or the line of the apsides, advances and regresses alternately, but nevertheless advances more than it regresses and is carried forward [or in consequentia] by the excess of its direct forward motion. For the force whereby body P is urged toward body T in the quadratures, when the force MN vanishes, is compounded of the force LM and the centripetal force with which body T attracts body P. If the distance PT is increased, the first force LM is increased in about the same ratio as this distance, and the latter force is decreased as the square of that ratio, and so the sum of these forces is decreased in a less than squared ratio of the distance PT, and therefore (by prop. 45, corol. 1) causes the auge, or upper apsis, to regress. But in conjunction and opposition the force whereby body P is urged toward body T is the difference between the force by which body T attracts body P and the force KL; and that difference, because the force KL is increased very nearly in the ratio of the distance PT, decreases in a ratio of the distance PT that is greater than the square of the distance PT, and so (by prop. 45, corol. 1) causes the upper apsis to advance. In places between the syzygies and quadratures the motion of the upper apsis depends on both of these causes jointly, so that according to the excess of the one or the other it advances or regresses. Accordingly, since the force KL in the syzygies is roughly twice as large as the force LM in the quadratures, the excess will have the same sense as the force KL and will carry the upper apsis forward [or in consequentia]. The truth of this corollary and its predecessor will be easily understood by supposing that a system of two bodies T and P is surrounded on all sides by more bodies S, S, S, . . . that are in an orbit ESE. For by the actions of these bodies, the action of T will be diminished on all sides and will decrease in a ratio greater than the square of the distance.

COROLLARY 8. Since, however, the advance or retrogression of the apsides depends on the decrease of the centripetal force, a decrease occurring in a ratio of the distance TP that is either greater or less than the square of the ratio of the distance TP, in the passage of the body from the lower to the upper apsis, and also depends on a similar increase in its return to the lower apsis, and therefore is greatest when the proportion of the force in the upper apsis to the force in the lower apsis differs most from the ratio of the inverse squares of the distances, it is manifest that KL or NM − LM, the force that subtracts, will cause the apsides to advance more swiftly in their syzygies and that LM, the force that adds, will cause them to recede more slowly in their quadratures. And because of the length of time in which the swiftness of the advance or slowness of the retrogression is continued, this inequality becomes by far the greatest.

COROLLARY 9. If a body, by the action of a force inversely proportional to the square of its distance from a center, were to revolve about this center in an ellipse, and if then, in its descent from the upper apsis or auge to the lower apsis, that force—because of the continual addition of a new force—were increased in a ratio that is greater than the square of the diminished distance, it is manifest that that body, being always impelled toward the center by the continual addition of that new force, would incline toward this center more than if it were urged only by a force increasing as the square of the diminished distance, and therefore would describe an orbit inside the elliptical orbit and in its lower apsis would approach nearer to the center than before. Therefore by the addition of this new force, the eccentricity of the orbit will be increased. Now if, during the receding of the body from the lower to the upper apsis, the force were to decrease by the same degrees by which it had previously increased, the body would return to its former distance; and so, if the force decreases in a greater ratio, the body, now attracted less, will ascend to a greater distance, and thus the eccentricity of its orbit will be increased still more. And therefore, if the ratio of the increase and decrease of the centripetal force is increased in each revolution, the eccentricity will always be increased; and contrariwise, the eccentricity will be diminished if that ratio decreases.

Now, in the system of bodies T, P, and S, when the apsides of the orbit PAB are in the quadratures, this ratio of the increase and decrease is least, and it becomes greatest when the apsides are in the syzygies. If the apsides are in the quadratures, the ratio near the apsides is smaller and near the syzygies is greater than the squared ratio of the distances, and from that greater ratio arises the forward or direct motion of the upper apsis, as has already been stated. But if one considers the ratio of the total increase or decrease in the forward motion between the apsides, this ratio is smaller than the squared ratio of the distances. The force in the lower apsis is to the force in the upper apsis in a ratio that is less than the squared ratio of the distance of the upper apsis from the focus of the ellipse to the distance of the lower apsis from that same focus; and conversely, when the apsides are in the syzygies, the force in the lower apsis is to the force in the upper apsis in a ratio greater than that of the squares of the distances.

For the forces LM in the quadratures, added to the forces of body T, compose forces in a smaller ratio, and the forces KL in the syzygies, subtracted from the forces of body T, leave forces in a greater ratio. Therefore, the ratio of the total decrease and increase during the passage between apsides is least in the quadratures and greatest in the syzygies; and therefore, during the passage of the apsides from quadratures to syzygies, this ratio is continually increased and it increases the eccentricity of the ellipse; and in the passage from syzygies to quadratures, this ratio is continually diminished and it diminishes the eccentricity.

COROLLARY 10. To give an account of the errors in latitude, let us imagine that the plane of the orbit EST remains motionless; then from the cause of errors just expounded, it is manifest that of the forces NM and ML (which are the entire cause of these errors) the force ML, always acting in the plane of the orbit PAB, never perturbs the motions in latitude. It is likewise manifest that when the nodes are in the syzygies, the force NM, also acting in the same plane of the orbit, does not perturb these motions; but when the nodes are in the quadratures, this force perturbs those motions to the greatest extent, and—by continually attracting body P away from the plane of its orbit—diminishes the inclination of the plane during the passage of the body from quadratures to syzygies and increases that inclination in turn during the passage from syzygies to quadratures. Hence it happens that when the body is in the syzygies the inclination turns out to be least of all, and it returns approximately to its former magnitude when the body comes to the next node. But if the nodes are situated in the octants after the quadratures, that is, between C and A, or D and B, it will be understood from what has just been explained that in the passage of body P from either node to a position 90 degrees from there, the inclination of the plane is continually diminished; then, in its passage through the next 45 degrees to the next quadrature, the inclination is increased; and afterward, in its next passage through another 45 degrees to the next node, it is diminished. Therefore, the inclination is diminished more than it is increased, and hence it is always less in each successive node than in the immediately preceding one. And by a similar reasoning, it follows that the inclination is increased more than it is diminished when the nodes are in the other octants between A and B, or B and C. Thus, when the nodes are in the syzygies, the inclination is greatest of all. In the passage of the nodes from syzygies to quadratures, the inclination is diminished in each appulse of the body to the nodes, and it becomes least of all when the nodes are in the quadratures and the body is in the syzygies; then it increases by the same degrees by which it had previously decreased, and at the appulse of the nodes to the nearest syzygies it returns to its original magnitude.

COROLLARY 11. When the nodes are in the quadratures, the body P is continually attracted away from the plane of its orbit in the direction toward S, during its passage from the node C through the conjunction A to the node D, and in the opposite direction in its passage from node D through opposition B to node C; hence it is manifest that the body, in its motion from node C, continually recedes from the first plane CD of its orbit until it has reached the next node; and therefore at this node, being at the greatest distance from that first plane CD, it passes through EST, the plane of the orbit, not in the other node D of that plane but in a point that is closer to body S and which accordingly is a new place of the node, behind its former place. By a similar argument the nodes will continue to recede in the passage of the body from this node to the next node. Hence the nodes, when situated in the quadratures, continually recede; in the syzygies, when the motion in latitude is not at all perturbed, the nodes are at rest; in the intermediate places, since they share in both conditions, they recede more slowly; and therefore, since the nodes always either have a retrograde motion or are stationary, they are carried backward [or in antecedentia] in each revolution.

COROLLARY 12. All the errors described in these corollaries are slightly greater in the conjunction of bodies P and S than in their opposition; and this occurs because then the generating forces NM and ML are greater.

COROLLARY 13. And since the proportions in these corollaries do not depend on the magnitude of the body S, all the preceding statements are valid when the magnitude of body S is assumed to be so great that the system of two bodies T and P will revolve about it. And from this increase of body S, and consequently the increase of its centripetal force (from which the errors of body P arise), all those errors will—at equal distances—come out greater in this case than in the other, in which body S revolves around the system of bodies P and T.

COROLLARY 14.^b When body S is extremely far away, the forces NM and ML are very nearly as the force SK and the ratio of PT to ST jointly (that is, if both the distance PT and the absolute force of body S are given, as ST³ inversely), and those forces NM and ML are the causes of all the errors and effects that have been dealt with in the preceding corollaries; hence it is manifest that all these effects—if the system of bodies T and P stays the same and only the distance ST and the absolute force of body S are changed—are very nearly in a ratio compounded of the direct ratio of the absolute force of body S and the inverse ratio of the cube of the distance ST. Accordingly, if the system of bodies T and P revolves about the distant body S, those forces NM and ML and their effects will (by prop. 4, corols. 2 and 6) be inversely as the square of the periodic time. And hence also, if the magnitude of body S is proportional to its absolute force, those forces NM and ML and their effects will be directly as the cube of the apparent diameter of the distant body S when looked at from body T, and conversely. For these ratios are the same as the above-mentioned compounded ratio.

COROLLARY 15. If the magnitudes of the orbits ESE and PAB are changed, while their forms and their proportions and inclinations to each other remain the same, and if the forces of bodies S and T either remain the same or are changed in any given ratio, then these forces (that is, the force of body T, by whose action body P is compelled to deflect from a straight path into an orbit PAB; and the force of body S, by whose action that same body P is compelled to deviate from that orbit) will always act in the same way and in the same proportion; thus it will necessarily be the case that all the effects will be similar and proportional and that the times for these effects will be proportional as well—that is, all the linear errors will be as the diameters of the orbits, the angular errors will be the same as before, and the times of similar linear errors or of equal angular errors will be as the periodic times of the orbits.

COROLLARY 16. And hence, if the forms of the orbits and their inclination to each other are given, and the magnitudes, forces, and distances of the bodies are changed in any way, then from the given errors and given times of errors in one case there can be found the errors and times of errors in any other case very nearly. This may be done more briefly, however, by the following method. The forces NM and ML, other things remaining the same, are as the radius TP, and their periodic effects are (by lem. 10, corol. 2) jointly as the forces and the square of the periodic time of body P. These are the linear errors of body P, and hence the angular errors as seen from the center T (that is, the motions of the upper apsis and of the nodes, as well as all the apparent errors in longitude and latitude) are in any revolution of body P very nearly as the square of the time of revolution. Let these ratios be compounded with the ratios of corol. 14; then in any system of bodies T, P, and S, in which P revolves around T which is near to it and T revolves around a distant S, the angular errors of body P, as seen from the center T, will—in each revolution of that body P—be as the square of the periodic time of body P directly and the square of the periodic time of body T inversely. And thus the mean motion of the upper apsis will be in a given ratio to the mean motion of the nodes, and each of the two motions will be as the periodic time of body P directly and the square of the periodic time of body T inversely. By increasing or decreasing the eccentricity and inclination of the orbit PAB, the motions of the upper apsis and of the nodes are not changed sensibly, except when the eccentricity and inclination are too great.

COROLLARY 17. Since, however, the line LM is sometimes greater and sometimes less than the radius PT, let the mean force LM be represented by that radius PT; then this force will be to the mean force SK or SN (which can be represented by ST) as the length PT to the length ST. But the mean force SN or ST by which body T is kept in its orbit around S is to the force by which body P is kept in its orbit around T in a ratio compounded of the ratio of the radius ST to the radius PT and the square of the ratio of the periodic time of body P around T to the periodic time of body T around S. And from the equality of the ratios [or ex aequo] the mean force LN is to the force by which a body P is kept in its orbit around T (or by which the same body P could revolve in the same periodic time around any immobile point T at a distance PT) in the same squared ratio of the periodic times. Therefore, if the periodic times are given, along with the distance PT, the mean force LM is also given; and if the force LM is given, the force MN is also given very nearly by the proportion of lines PT and MN.

COROLLARY 18. Let us imagine many fluid bodies to move around body T at equal distances from it according to the same laws by which body P revolves around the same body T; then let a ring—fluid, round, and concentric to body T—be produced by making these individual fluid bodies come into contact with one another; these individual parts of the ring, carrying out all their motions according to the law of body P, will approach closer to body T and will move more swiftly in the conjunction and opposition of themselves and body S than in the quadratures. The nodes of this ring, or its intersections with the plane of the orbit of body S or T, will be at rest in the syzygies, but outside the syzygies they will move backward [or in antecedentia], and do so most swiftly in the quadratures and more slowly in other places. The inclination of the ring will also vary, and its axis will oscillate in each revolution; and when a revolution has been completed, it will return to its original position except insofar as it is carried around by the precession of the nodes.

COROLLARY 19. Now imagine the globe T, which consists of nonfluid matter, to be so enlarged as to extend out to this ring, and to have a channel to contain water dug out around its whole circumference; and imagine this new globe to revolve uniformly about its axis with the same periodic motion. This water, being alternately accelerated and retarded (as in the previous corollary), will be swifter in the syzygies and slower in the quadratures than the surface of the globe itself, and thus will ebb and flow in the channel just as the sea does. If the attraction of body S is taken away, the water—now revolving about the quiescent center of the globe—will acquire no motion of ebb and flow. This is likewise the case for a globe advancing uniformly straight forward and meanwhile revolving about its own center (by corol. 5 of the laws), and also for a globe uniformly attracted away from a rectilinear path (by corol. 6 of the laws). But let body S now draw near, and by its nonuniform attraction of the water, the water will soon be disturbed. For its attraction of the nearer water will be greater and that of the more distant water will be smaller. Moreover, the force LM will attract the water downward in the quadratures and will make it descend as far as the syzygies, and the force KL will attract this same water upward in the syzygies and will prevent its further descent and will make it ascend as far as the quadratures, except insofar as the motion of ebb and flow is directed by the channel of water and is somewhat retarded by friction.

COROLLARY 20. If the ring now becomes hard and the globe is diminished, the motion of ebb and flow will cease; but the oscillatory motion of the inclination and the precession of the nodes will remain. Let the globe have the same axis as the ring and complete its revolutions in the same times, and let its surface touch the inside of the ring and adhere to it; then, with the globe participating in the motion of the ring, the structure of the two will oscillate and the nodes will regress. For the globe, as will be shown presently, is susceptible to all impressions equally. The greatest angle of inclination of the ring alone, with the globe removed, occurs when the nodes are in the syzygies. From there in the forward motion of the nodes to the quadratures it endeavors to diminish its inclination and by that endeavor impresses a motion upon the whole globe. The globe keeps this impressed motion until the ring removes this motion by an opposite endeavor and impresses a new motion in the opposite direction; and in this way the greatest motion of the decreasing inclination occurs when the nodes are in the quadratures, and the least angle of inclination occurs in the octants after the quadratures; and the greatest motion of reclination occurs in the syzygies, and the greatest angle in the next octants. And this is likewise the case for a globe which has no such ring and which in the regions of the equator is either a little higher than near the poles or consists of matter a little denser. For that excess of matter in the regions of the equator takes the place of a ring. And although, by increasing the centripetal force of this globe in any way whatever, all its parts are supposed to tend downward, as the gravitating parts of the earth do, nevertheless the phenomena of this corollary and of corol. 19 will scarcely be changed on that account, except that the places of the greatest and least height of the water will be different. For the water is now sustained and remains in its orbit not by its own centrifugal force but by the channel in which it is flowing. And besides, the force LM attracts the water downward to the greatest degree in the quadratures, and the force KL or NM − LM attracts the same water upward to the greatest degree in the syzygies. And these forces conjoined cease to attract the water downward and begin to attract the water upward in the octants before the syzygies, and they cease to attract the water upward and begin to attract the water downward in the octants after the syzygies. As a result, the greatest height of the water can occur very nearly in the octants after the syzygies, and the least height can occur very nearly in the octants after the quadratures, except insofar as the motion of ascent or descent impressed on the water by these forces either perseveres a little longer because of the inherent force of the water or is stopped a little more swiftly because of the impediments of the channel.

COROLLARY 21. In the same way that the excess matter of a globe near its equator makes the nodes regress (and thus the retrogression is increased by increase of equatorial matter and is diminished by its diminution and is removed by its removal), it follows that if more than the excess matter is removed, that is, if the globe near the equator is made either more depressed or more rare than near the poles, there will arise a motion of the nodes forward [or in consequentia].

COROLLARY 22. And thus, in turn, from the motion of the nodes the constitution of a globe can be found. That is to say, if a globe constantly preserves the same poles and there occurs a motion backward [or in antecedentia], there is an excess of matter near the equator; if there occurs a motion forward [or in consequentia], there is a deficiency. Suppose that a uniform and perfectly spherical globe is at first at rest in free space; then is propelled by any impetus whatever delivered obliquely upon its surface, from which it takes on a motion that is partly circular [i.e., rotational] and partly straight forward. Because the globe is indifferent to all axes passing through its center and does not have a greater tendency to turn around any one axis or an axis at any particular inclination, it is clear that the globe, by its own force alone, will never change its axis and the inclination of the axis. Now let the globe be impelled obliquely by any new impulse whatever, delivered to that same part of the surface as before; then, since the effect of an impulse is in no way changed by its being delivered sooner or later, it is manifest that the same motion will be produced by these two impulses being successively impressed as if they had been impressed simultaneously, that is, the resultant motion will be the same as if the globe had been impelled by a simple force compounded of these two (by corol. 2 of the laws), and hence will be a simple motion about an axis of a given inclination. This is likewise the case for a second impulse impressed in any other place on the equator of the first motion; and also for a first impulse impressed in any place on the equator of the motion which the second impulse would generate without the first, and hence for both impulses impressed in any places whatever. These two impulses will generate the same circular motion as if they had been impressed together and all at once in the place of intersection of the equators of the motions which each of them would generate separately. Therefore a homogeneous and perfect globe does not retain several distinct motions but compounds all the motions impressed on it and reduces them to one; and insofar as it can in and of itself, it always rotates with a simple and uniform motion about a single axis of a given and always invariable inclination. A centripetal force cannot change either this inclination of the axis or the velocity of rotation.

If a globe is thought of as divided into two hemispheres by any plane passing through the center of the globe and the center toward which a force is directed, that force will always urge both hemispheres equally and therefore will not cause the globe—as regards its motion of rotation—to incline in any direction. Let some new matter, heaped up in the shape of a mountain, be added to the globe anywhere between the pole and the equator; then this matter, by its continual endeavor to recede from the center of its motion, will disturb the motion of the globe and will make its poles wander over its surface and continually describe circles about themselves and the point opposite to them. And this tremendous wandering of the poles will not be corrected, save by placing the mountain either in one of the two poles, in which case (by corol. 21) the nodes of the equator will advance, or on the equator, in which case (by corol. 20) the nodes will regress, or finally by placing on the other side of the axis some additional matter by which the mountain is balanced in its motion, and in this way the nodes will either advance or regress, according as the mountain and this new matter are closer to a pole or to the equator.

Proposition 67, Theorem 27
With the same laws of attraction being supposed, I say that with respect to the common center of gravity O of the inner bodies P and T, the outer body S—by radii drawn to that center—describes areas more nearly proportional to the times, and an orbit more closely approaching the shape of an ellipse having its focus in that same center, than it can describe about the innermost and greatest body T by radii drawn to that body.

For the attractions of body S toward T and P compose its absolute attraction, which is directed more toward the common center of gravity O of bodies T and P than toward the greatest body T, and which is more nearly inversely proportional to the square of the distance SO than to the square of the distance ST, as will easily be seen by anyone carefully considering the matter.

Proposition 68, Theorem 28
With the same laws of attraction being supposed, I say that with respect to the common center of gravity O of the inner bodies P and T, the outer body S—by radii drawn to that center—describes areas more nearly proportional to the times, and an orbit more closely approaching the shape of an ellipse having its focus in the same center, if the innermost and greatest body is acted on by these attractions just as the others are, than would be the case if it is either not attracted and is at rest or is much more or much less attracted or much more or much less moved.

This is demonstrated in almost the same way as prop. 66, but the proof is more prolix and I therefore omit it. The following considerations should suffice.

From the demonstration of the last proposition it is apparent that the center toward which body S is urged by both forces combined is very near to the common center of gravity of the other bodies P and T. If this center were to coincide with the common center of those two bodies, and the common center of gravity of all three bodies were to be at rest, body S on the one hand and the common center of the other two bodies on the other would describe exact ellipses about the common center of them all which is at rest. This is clear from the second corollary of prop. 58 compared with what is demonstrated in props. 64 and 65. Such an exact elliptical motion is perturbed somewhat by the distance of the center of the two bodies from the center toward which the third body S is attracted. Let a motion be given, in addition, to the common center of the three, and the perturbation will be increased. Accordingly, the perturbation is least when the common center of the three is at rest, that is, when the innermost and greatest body T is attracted by the very same law as the others; and it always becomes greater when the common center of the three bodies, by a diminution of the motion of body T, begins to be moved and thereupon acted on more and more.

COROLLARY. And hence, if several lesser bodies revolve about a greatest one, it can be found that the orbits described will approach closer to elliptical orbits, and the descriptions of areas will become more uniform, if all the bodies attract and act on one another by accelerative forces that are directly as their absolute forces and inversely as the squares of the distances, and if the focus of each orbit is located in the common center of gravity of all the inner bodies (that is to say, with the focus of the first and innermost orbit in the center of gravity of the greatest and innermost body; the focus of the second orbit in the common center of gravity of the two innermost bodies; the focus of the third in the common center of gravity of the three inner bodies; and so on), than if the innermost body is at rest and is set at the common focus of all the orbits.

Proposition 69, Theorem 29
If, in a system of several bodies A, B, C, D, . . . , some body A attracts all the others, B, C, D, . . . , by accelerative forces that are inversely as the squares of the distances from the attracting body; and if another body B also attracts the rest of the bodies A, C, D, . . . , by forces that are inversely as the squares of the distances from the attracting body; then the absolute forces of the attracting bodies A and B will be to each other in the same ratio as the bodies [i.e., the masses] A and B themselves to which those forces belong.

For, at equal distances, the accelerative attractions of all the bodies B, C, D, . . . toward A are equal to one another by hypothesis; and similarly, at equal distances, the accelerative attractions of all the bodies toward B are equal to one another. Moreover, at equal distances, the absolute attractive force of body A is to the absolute attractive force of body B as the accelerative attraction of all the bodies toward A is to the accelerative attraction of all the bodies toward B at equal distances; and the accelerative attraction of body B toward A is also in the same proportion to the accelerative attraction of body A toward B. But the accelerative attraction of body B toward A is to the accelerative attraction of body A toward B as the mass of body A is to the mass of body B, because the motive forces—which (by defs. 2, 7, and 8) are as the accelerative forces and the attracted bodies jointly—are in this case (by the third law of motion) equal to each other. Therefore the absolute attractive force of body A is to the absolute attractive force of body B as the mass of body A is to the mass of body B. Q.E.D.

COROLLARY 1. Hence if each of the individual bodies of the system A, B, C, D, . . . , considered separately, attracts all the others by accelerative forces that are inversely as the squares of the distances from the attracting body, the absolute forces of all those bodies will be to one another in the ratios of the bodies [i.e., the masses] themselves.

COROLLARY 2. By the same argument, if each of the individual bodies of the system A, B, C, D, . . . , considered separately, attracts all the others by accelerative forces that are either inversely or directly as any powers whatever of the distances from the attracting body, or that are defined in terms of the distances from each one of the attracting bodies according to any law common to all these bodies; then it is evident that the absolute forces of those bodies are as the bodies [i.e., the masses].

COROLLARY 3. If, in a system of bodies whose forces decrease in the squared ratio of the distances [i.e., vary inversely as the squares of the distances], the lesser bodies revolve about the greatest one in ellipses as exact as they can be, having their common focus in the center of that greatest body, and—by radii drawn to the greatest body—describe areas as nearly as possible proportional to the times, then the absolute forces of those bodies will be to one another, either exactly or very nearly, as the bodies, and conversely. This is clear from the corollary of prop. 68 compared with corol. 1 of this proposition.

Scholium
By these propositions we are directed to the analogy between centripetal forces and the central bodies toward which those forces tend. For it is reasonable that forces directed toward bodies depend on the nature and the quantity of matter of such bodies, as happens in the case of magnetic bodies. And whenever cases of this sort occur, the attractions of the bodies must be reckoned by assigning proper forces to their individual particles and then taking the sums of these forces.

I use the word “attraction” here in a general sense for any endeavor whatever of bodies to approach one another, whether that endeavor occurs as a result of the action of the bodies either drawn toward one another or acting on one another by means of spirits emitted or whether it arises from the action of aether or of air or of any medium whatsoever—whether corporeal or incorporeal—in any way impelling toward one another the bodies floating therein. I use the word “impulse” in the same general sense, considering in this treatise not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions.

Mathematics requires an investigation of those quantities of forces and their proportions that follow from any conditions that may be supposed. Then, coming down to physics, these proportions must be compared with the phenomena, so that it may be found out which conditions [or laws] of forces apply to each kind of attracting bodies. And then, finally, it will be possible to argue more securely concerning the physical species, physical causes, and physical proportions of these forces. Let us see, therefore, what the forces are by which spherical bodies, consisting of particles that attract in the way already set forth, must act upon one another, and what sorts of motions result from such forces.

SECTION 12

The attractive forces of spherical bodies

Proposition 70, Theorem 30
If toward each of the separate points of a spherical surface there tend equal centripetal forces decreasing as the squares of the distances from the point, I say that a corpuscle placed inside the surface will not be attracted by these forces in any direction.

Let HIKL be the spherical surface, and P the corpuscle placed inside. Through P draw to this surface the two lines HK and IL intercepting minimally small arcs HI and KL; and because triangles HPI and LPK are similar (by lem. 7, corol. 3), those arcs will be proportional to the distances HP and LP; and any particles of the spherical surface at HI and KL, terminated everywhere by straight lines passing through point P, will be in that proportion squared. Therefore the forces exerted on body P by these particles of surface are equal to one another. For they are as the particles directly and the squares of the distances inversely. And these two ratios, when compounded, give the ratio of equality. The attractions, therefore, being made equally in opposite directions, annul each other. And by a similar argument, all the attractions throughout the whole spherical surface are annulled by opposite attractions. Accordingly, body P is not impelled by these attractions in any direction. Q.E.D.

Proposition 71, Theorem 31
With the same conditions being supposed as in prop. 70, I say that a corpuscle placed outside the spherical surface is attracted to the center of the sphere by a force inversely proportional to the square of its distance from that same center.

Let AHKB and ahkb be two equal spherical surfaces, described about centers S and s with diameters AB and ab, and let P and p be corpuscles located outside those spheres in those diameters produced. From the corpuscles draw lines PHK, PIL, phk, and pil, so as to cut off from the great circles AHB and ahb the equal arcs HK and hk, and IL and il. And onto these lines drop perpendiculars SD and sd, SE and se, IR and ir, of which SD and sd cut PL and pl at F and f. Also drop perpendiculars IQ and iq onto the diameters. Let angles DPE and dpe vanish; then, because DS and ds, ES and es are equal, lines PE, PF and pe, pf and the line-elements DF and df may be considered to be equal, inasmuch as their ultimate ratio, when angles DPE and dpe vanish simultaneously, is the ratio of equality.

On the basis of these things, therefore, PI will be to PF as RI to DF, and pf to pi as df or DF to ri, and from the equality of the ratios [or ex aequo] PI × pf will be to PF × pi as RI to ri, that is (by lem. 7, corol. 3), as the arc IH to the arc ih. Again, PI will be to PS as IQ to SE, and ps will be to pi as se or SE to iq; and from the equality of the ratios [or ex aequo] PI × ps will be to PS × pi as IQ to iq. And by compounding these ratios, PI² × pf × ps will be to pi² × PF × PS as IH × IQ to ih × iq; that is, as the circular surface that the arc IH will describe by the revolution of the semicircle AKB about the diameter AB to the circular surface that the arc ih will describe by the revolution of the semicircle akb about the diameter ab. And the forces by which these surfaces attract the corpuscles P and p (along lines tending to these same surfaces) are (by hypothesis) as these surfaces themselves directly and the squares of the distances of these surfaces from the bodies inversely, that is, as pf × ps to PF × PS.

Now (once the resolution of the forces has been made according to corol. 2 of the laws), these forces are to their oblique parts, which tend along the lines PS and ps toward the centers, as PI to PQ and pi to pq; that is (because the triangles PIQ and PSF, piq and psf are similar), the forces are to their oblique parts as PS to PF and ps to pf. Hence, from the equality of the ratios [or ex aequo] the attraction of this corpuscle P toward S becomes to the attraction of the corpuscle p toward s as to , that is, as ps² to PS². And by a similar argument, the forces by which the surface described by the revolution of the arcs KL and kl attract the corpuscles will be as ps² to PS². And the same ratio will hold for the forces of all the spherical surfaces into which each of the two spherical surfaces can be divided by taking sd always equal to SD and se equal to SE. And by composition [or componendo] the forces of the total spherical surfaces exercised upon the corpuscles will be in the same ratio. Q.E.D.

Proposition 72, Theorem 32
If toward each of the separate points of any sphere there tend equal centripetal forces, decreasing in the squared ratio of the distances from those points, and there are given both the density of the sphere and the ratio of the diameter of the sphere to the distance of the corpuscle from the center of the sphere, I say that the force by which the corpuscle is attracted will be proportional to the semidiameter of the sphere.

For imagine that two corpuscles are attracted separately by two spheres, one corpuscle by one sphere, and the other corpuscle by the other sphere, and that their distances from the centers of the spheres are respectively proportional to the diameters of the spheres, and that the two spheres are resolved into particles that are similar and similarly placed with respect to the corpuscles. Then the attractions of the first corpuscle, made toward each of the separate particles of the first sphere, will be to the attractions of the second toward as many analogous particles of the second sphere in a ratio compounded of the direct ratio of the particles and the inverse squared ratio of the distances [i.e., the attractions will be to one another as the particles directly and the squares of the distances inversely]. But the particles are as the spheres, that is, they are in the cubed ratio of the diameters, and the distances are as the diameters; and thus the first of these ratios directly combined with the second ratio taken twice inversely becomes the ratio of diameter to diameter. Q.E.D.

COROLLARY 1. Hence, if corpuscles revolve in circles about spheres consisting of equally attractive matter, and their distances from the centers of the spheres are proportional to the diameters of the spheres, the periodic times will be equal.

COROLLARY 2. And conversely, if the periodic times are equal, the distances will be proportional to the diameters. These two corollaries are evident from prop. 4, corol. 3.

COROLLARY 3. If toward each of the separate points of any two similar and equally dense solids there tend equal centripetal forces decreasing in the squared ratio of the distances from those points, the forces by which corpuscles will be attracted by those two solids, if they are similarly situated with regard to them, will be to each other as the diameters of the solids.

Proposition 73, Theorem 33
If toward each of the separate points of any given sphere there tend equal centripetal forces decreasing in the squared ratio of the distances from those points; I say that a corpuscle placed inside the sphere is attracted by a force proportional to the distance of the corpuscle from the center of the sphere.

Let a corpuscle P be placed inside the sphere ABCD, described about center S; and about the same center S with radius SP, suppose that an inner sphere PEQF is described. It is manifest (by prop. 70) that the concentric spherical surfaces of which the difference AEBF of the spheres is composed do not act at all upon body P, their attractions having been annulled by opposite attractions. There remains only the attraction of the inner sphere PEQF. And (by prop. 72) this is as the distance PS. Q.E.D.

Scholium
The surfaces of which the solids are composed are here not purely mathematical, but orbs [or spherical shells] so extremely thin that their thickness is as null: namely, evanescent orbs of which the sphere ultimately consists when the number of those orbs is increased and their thickness diminished indefinitely. Similarly, when lines, surfaces, and solids are said to be composed of points, such points are to be understood as equal particles of a magnitude so small that it can be ignored.

Proposition 74, Theorem 34
With the same things being supposed as in prop. 73, I say that a corpuscle placed outside a sphere is attracted by a force inversely proportional to the square of the distance of the corpuscle from the center of the sphere.

For let the sphere be divided into innumerable concentric spherical surfaces; then the attractions of the corpuscle that arise from each of the surfaces will be inversely proportional to the square of the distance of the corpuscle from the center (by prop. 71). And by composition [or componendo] the sum of the attractions (that is, the attraction of the corpuscle toward the total sphere) will come out in the same ratio. Q.E.D.

COROLLARY 1. Hence at equal distances from the centers of homogeneous spheres the attractions are as the spheres themselves. For (by prop. 72) if the distances are proportional to the diameters of the spheres, the forces will be as the diameters. Let the greater distance be diminished in that ratio; and, the distances having now become equal, the attraction will be increased in that ratio squared, and thus will be to the other attraction in that ratio cubed, that is, in the ratio of the spheres.

COROLLARY 2. At any distances the attractions are as the spheres divided by the squares of the distances.

COROLLARY 3. If a corpuscle placed outside a homogeneous sphere is attracted by a force inversely proportional to the square of the distance of the corpuscle from the center of the sphere, and the sphere consists of attracting particles, the force of each particle will decrease in the squared ratio of the distance from the particle.

Proposition 75, Theorem 35
If toward each of the points of a given sphere there tend equal centripetal forces decreasing in the squared ratio of the distances from the points, I say that this sphere will attract any other homogeneous sphere with a force inversely proportional to the square of the distance between the centers.^a

For the attraction of any particle is inversely as the square of its distance from the center of the attracting sphere (by prop. 74), and therefore is the same as if the total attracting force emanated from one single corpuscle situated in the center of this sphere. Moreover, this attraction is as great as the attraction of the same corpuscle would be if, in turn, it were attracted by each of the individual particles of the attracted sphere with the same force by which it attracts them. And that attraction of the corpuscle (by prop. 74) would be inversely proportional to the square of its distance from the center of the sphere; and therefore the sphere’s attraction, which is equal to the attraction of the corpuscle, is in the same ratio. Q.E.D.

COROLLARY 1. The attractions of spheres toward other homogeneous spheres are as the attracting spheres [i.e., as the masses of the attracting spheres] divided by the squares of the distances of their own centers from the centers of those that they attract.

COROLLARY 2. The same is true when the attracted sphere also attracts. For its individual points will attract the individual points of the other with the same force by which they are in turn attracted by them; and thus, since in every attraction the attracting point is as much urged (by law 3) as the attracted point, the force of the mutual attraction will be duplicated, the proportions remaining the same.

COROLLARY 3. Everything that has been demonstrated above concerning the motion of bodies about the focus of conics is valid when an attracting sphere is placed in the focus and the bodies move outside the sphere.

COROLLARY 4. And whatever concerns the motion of bodies around the center of conics applies when the motions are performed inside the sphere.

Proposition 76, Theorem 36
If spheres are in any way nonhomogeneous (as to the density of their matter and their attractive force) going from the center to the circumference, but are uniform throughout in every spherical shell at any given distance from the center, and the attractive force of each point decreases in the squared ratio of the distance of the attracted body, I say that the total force by which one sphere of this sort attracts another is inversely proportional to the square of the distance between their centers.

^aLet there be any number of concentric homogeneous spheres [i.e., hollow spheres, or spherical shells or surfaces] AB, CD, EF, . . . ; and suppose that the addition of one or more inner ones to the outer one or ones forms a sphere composed of matter more dense, or the taking away leaves it less dense, toward the center than at the circumference. Then these spheres will together (by prop. 75) attract any number of other concentric homogeneous spheres GH, IK, LM, . . . , each sphere of one set attracting every one of the other set with forces inversely proportional to the square of the distance SP. And by adding up these forces (or by the reverse process when spheres are taken away) the sum of all those forces (or the excess of any one—or of some—of them above the others); that is, the force with which the whole sphere AB, composed of any concentric spheres (or the difference between some concentric spheres and others which have been taken away), attracts the whole sphere GH, composed of any concentric spheres (or the differences between some such concentric spheres and others)—will be in the same inverse ratio of the square of the distance SP. Let the number of concentric spheres be increased indefinitely, in such a way that the density of the matter, together with the force of attraction, may—on going from the circumference to the center—increase or decrease according to any law whatever; and by the addition of non-attracting matter, let the deficiencies in density be supplied wherever needed so that the spheres may acquire any desired form; then the force with which one of these spheres attracts the other will still be, by the former argument, in the same inverse ratio of the square of the distance.^a Q.E.D.

COROLLARY 1. Hence, if many spheres of this sort, similar to one another in all respects, attract one another, the accelerative attraction of any one to any other of them, at any equal distances between the centers, will be as the attracting spheres.

COROLLARY 2. And at any unequal distances, as the attracting sphere divided by the square of the distances between the centers.

COROLLARY 3. And the motive attractions, or the weights of spheres toward other spheres, will—at equal distances from the centers—be as the attracting and the attracted spheres jointly, that is, as the products produced by multiplying the spheres by each other.

COROLLARY 4. And at unequal distances, as those products directly and the squares of the distances between the centers inversely.

COROLLARY 5. These results are valid when the attraction arises from each sphere’s force of attraction being mutually exerted upon the other sphere. For the attraction is duplicated by both forces acting, the proportion remaining the same.

COROLLARY 6. If some spheres of this sort revolve about others at rest, one sphere revolving about each sphere at rest, and the distances between the centers of the revolving spheres and those at rest are proportional to the diameters of those at rest, the periodic times will be equal.

COROLLARY 7. And conversely, if the periodic times are equal, the distances will be proportional to those diameters.

COROLLARY 8. Everything that has been demonstrated above about the motion of bodies around the foci of conics holds when the attracting sphere, of any form and condition that has already been described, is placed in the focus.

COROLLARY 9. As also when the bodies revolving in orbit are also attracting spheres of any condition that has already been described.

Proposition 77, Theorem 37
If toward each of the individual points of spheres there tend centripetal forces proportional to the distances of the points from attracted bodies, I say that the composite force by which two spheres will attract each other is as the distance between the centers of the spheres.

CASE 1. Let AEBF be a sphere, S its center, P an attracted exterior corpuscle, PASB that axis of the sphere which passes through the center of the corpuscle, EF and ef two planes by which the sphere is cut and which are perpendicular to this axis and equally distant on both sides from the center of the sphere, G and g the intersections of the planes and the axis, and H any point in the plane EF. The centripetal force of point H upon the corpuscle P, exerted along the line PH, is as the distance PH; and (by corol. 2 of the laws) along the line PG, or toward the center S, as the length PG. Therefore the force of all the points in the plane EF (that is, of the total plane) by which the corpuscle P is attracted toward the center S is as the distance PG multiplied by the number of such points, that is, as the solid contained by that plane EF itself and the distance PG [i.e., as the product of the plane EF and the distance PG]. And similarly the force of the plane ef, by which the corpuscle P is attracted toward the center S, is as that plane multiplied by its distance Pg, or as the plane EF equal thereto multiplied by that distance Pg; and the sum of the forces of both planes is as the plane EF multiplied by the sum of the distances PG + Pg; that is, as that plane multiplied by twice the distance PS between the center S and the corpuscle P; that is, as twice the plane EF multiplied by the distance PS, or as the sum of the equal planes EF + ef multiplied by that same distance. And by a similar argument, the forces of all the planes in the whole sphere, equally distant on both sides from the center of the sphere, are as the sum of those planes multiplied by the distance PS, that is, as the whole sphere and the distance PS jointly. Q.E.D.

CASE 2. Now let the corpuscle P attract the sphere AEBF. Then by the same argument it can be proved that the force by which that sphere is attracted will be as the distance PS. Q.E.D.

CASE 3. Now let a second sphere be composed of innumerable corpuscles P; then, since the force by which any one corpuscle is attracted is as the distance of the corpuscle from the center of the first sphere and as that same sphere jointly, and thus is the same as if all the force came from one single corpuscle in the center of the sphere, the total force by which all the corpuscles in the second sphere are attracted (that is, by which that whole sphere is attracted) will be the same as if that sphere were attracted by a force coming from one single corpuscle in the center of the first sphere, and therefore is proportional to the distance between the centers of the spheres. Q.E.D.

CASE 4. Let the spheres attract each other mutually; then the force, now duplicated, will keep the former proportion. Q.E.D.

CASE 5. Now let a corpuscle p be placed inside the sphere AEBF. Then, since the force of the plane ef upon the corpuscle is as the solid contained by [or the product of] that plane and the distance pg; and the opposite force of the plane EF is as the solid contained by [or the product of] that plane and the distance pG; the force compounded of the two will be as the difference of the solids [or the products], that is, as the sum of the equal planes multiplied by half of the difference of the distances, that is, as that sum multiplied by pS, the distance of the corpuscle from the center of the sphere. And by a similar argument, the attraction of all the planes EF and ef in the whole sphere (that is, the attraction of the whole sphere) is jointly as the sum of all the planes (or the whole sphere) and as pS, the distance of the corpuscle from the center of the sphere. Q.E.D.

CASE 6. And if from innumerable corpuscles p a new sphere is composed, placed inside the former sphere AEBF, then it can be proved as above that the attraction, whether the simple attraction of one sphere toward the other, or a mutual attraction of both toward each other, will be as the distance pS between the centers. Q.E.D.

Proposition 78, Theorem 38
If spheres, on going from the center to the circumference, are in any way nonhomogeneous and nonuniform, but in every concentric spherical shell at any given distance from the center are homogeneous throughout; and the attracting force of each point is as the distance of the body attracted; then I say that the total force by which two spheres of this sort attract each other is proportional to the distance between the centers of the spheres.

This is demonstrated from prop. 77 in the same way that prop. 76 was demonstrated from prop. 75.

COROLLARY. Whatever was demonstrated above in props. 10 and 64 on the motion of bodies about the centers of conics is valid when all the attractions take place by the force of spherical bodies of the condition already described, and when the attracted bodies are spheres of the same condition.

Scholium
I have now given explanations of the two major cases of attractions, namely, when the centripetal forces decrease in the squared ratio of the distances or increase in the simple ratio of the distances, causing bodies in both cases to revolve in conics, and composing centripetal forces of spherical bodies that decrease or increase in proportion to the distance from the center according to the same law—which is worthy of note. It would be tedious to go one by one through the other cases which lead to less elegant conclusions. I prefer to comprehend and determine all the cases simultaneously under a general method as follows.

Lemma 29^a
If any circle AEB is described with center S; and then two circles EF and ef are described with center P, cutting the first circle in E and e, and cutting the line PS in F and f; and if the perpendiculars ED and ed are dropped to PS; then I say that if the distance between the arcs EF and ef is supposed to be diminished indefinitely, the ultimate ratio of the evanescent line Dd to the evanescent line Ff is the same as that of line PE to line PS.

For if line Pe cuts arc EF in q, and the straight line Ee, which coincides with the evanescent arc Ee, when produced meets the straight line PS in T, and the normal SG is dropped from S to PE; then because the triangles DTE, dTe, and DES are similar, Dd will be to Ee as DT to TE, or DE to ES; and because the triangles Eeq and ESG (by sec. 1, lem. 8 and lem. 7, corol. 3) are similar, Ee will be to eq or Ff as ES to SG; and from the equality of the ratios [or ex aequo] Dd will be to Ff as DE to SG—that is (because the triangles PDE and PGS are similar), as PE to PS. Q.E.D.

Proposition 79, Theorem 39
If the surface EFfe, just now vanishing because its width has been indefinitely diminished, describes by its revolution about the axis PS a concavo-convex spherical solid, toward each of whose individual equal particles there tend equal centripetal forces; then I say that the force by which that solid attracts an exterior corpuscle located in P is in a ratio compounded of the ratio of the solid [or product] DE² × Ff and the ratio of the force by which a given particle at the place Ff would attract the same corpuscle.

For if we first consider the force of the spherical surface FE, which is generated by the revolution of the arc FE and is cut anywhere by the line de in r, the annular part of this surface generated by the revolution of the arc rE will be as the line-element Dd, the radius PE of the sphere remaining the same (as Archimedes demonstrated in his book on the Sphere and Cylinder). And the force of that surface, exerted along lines PE or Pr, placed everywhere in the surface of a cone, will be as this annular part of the surface—that is, as the line-element Dd or, what comes to the same thing, as the rectangle of the given radius PE of the sphere and that line-element Dd; but along the line PS tending toward the center S, this force will be smaller in the ratio of PD to PE, and hence this force will be as PD × Dd. Now suppose the line DF to be divided into innumerable equal particles, and let each of them be called Dd; then the surface FE will be divided into the same number of equal rings, whose total forces will be as the sum of all the products PD × Dd, that is, as ½PF² − ½PD², and thus as DE². Now multiply the surface FE by the altitude Ff, and the force of the solid EFfe exerted upon the corpuscle P will become as DE² × Ff, if there is given the force that some given particle Ff exerts on the corpuscle P at the distance PF. But if that force is not given, the force of the solid EFfe will become as the solid DE² × Ff and that non-given force jointly. Q.E.D.

Proposition 80, Theorem 40
If equal centripetal forces tend toward each of the individual equal particles of some sphere ABE, described about a center S; and if from each of the individual points D to the axis AB of the sphere, in which some corpuscle P is located, there are erected the perpendiculars DE, meeting the sphere in the points E; and if on these perpendiculars, the lengths DN are taken, which are jointly as the quantity and as the force that a particle of the sphere, located on the axis, exerts at the distance PE upon the corpuscle P; then I say that the total force with which the corpuscle P is attracted toward the sphere is as the area ANB comprehended by the axis AB of the sphere and the curved line ANB, which the point N traces out.

For, keeping the same constructions as in lem. 29 and prop. 79, suppose the axis AB of the sphere to be divided into innumerable equal particles Dd, and the whole sphere to be divided into as many spherical concavo-convex laminae EFfe; and erect the perpendicular dn. By prop. 79, the force with which the lamina EFfe attracts the corpuscle P is jointly as DE² × Ff and the force of one particle exerted at the distance PE or PF. But Dd is to Ff (by lem. 29) as PE to PS, and hence Ff is equal to , and DE² × Ff is equal to Dd; and therefore the force of the lamina EFfe is jointly as Dd and the force of a particle exerted at the distance PF; that is (by hypothesis) as DN × Dd, or as the evanescent area DNnd. Therefore the forces upon body P exerted by all the laminae are as all the areas DNnd, that is, the total force of the sphere is as the total area ANB. Q.E.D.

COROLLARY 1. Hence, if the centripetal force tending toward each of the individual particles always remains the same at all distances, and DN is taken proportional to , the total force by which the corpuscle P is attracted by the sphere will be as the area ANB.

COROLLARY 2. If the centripetal force of the particles is inversely as the distance of the attracted corpuscle, and DN is taken proportional to , the force by which the corpuscle P is attracted by the whole sphere will be as the area ANB.

COROLLARY 3. If the centripetal force of the particles is inversely as the cube of the distance of the attracted corpuscle, and DN is taken proportional to , the force by which the corpuscle is attracted by the whole sphere will be as the area ANB.

COROLLARY 4. And universally, if the centripetal force tending toward each of the individual particles of a sphere is supposed to be inversely as the quantity V, and DN is taken proportional to , the force by which a corpuscle is attracted by the whole sphere will be as the area ANB.

Proposition 81, Problem 41
Under the same conditions as before, it is required to measure the area ANB.

From point P draw the straight line PH touching the sphere in H; and, having dropped the normal HI to the axis PAB, bisect PI in L; then (by book 2, prop. 12, of Euclid’s Elements) PE² will be equal to PS²+SE²+2(PS × SD). Moreover, SE² or SH² (because the triangles SPH and SHI are similar) is equal to the rectangle PS × SI. Therefore PE² is equal to the rectangle of PS and PS + SI + 2SD, that is, of PS and 2LS + 2SD, that is, of PS and 2LD. Further, DE² is equal to SE² − SD², or SE² − LS² +2(SL × LD) − LD², that is, 2(SL × LD) − LD² − AL × LB. For LS² − SE² or LS² − SA² (by book 2, prop. 6, of the Elements) is equal to the rectangle AL × LB. Write, therefore, 2(SL × LD) − LD² − AL × LB for DE²; and the quantity , which (according to corol. 4 of the preceding prop. 80) is as the length of the ordinate DN, will resolve itself into the three parts : where, if for V we write the inverse ratio of the centripetal force, and for PE the mean proportional between PS and 2LD, those three parts will become ordinates of as many curved lines, whose areas can be found by ordinary methods. Q.E.F.

EXAMPLE 1. If the centripetal force tending toward each of the individual particles of the sphere is inversely as the distance, write the distance PE in place of V, and then 2PS × LD in place of PE², and DN will become as SL − ½LD − . Suppose DN equal to its double 2SL − LD − ; and the given part 2SL of that ordinate multiplied by the length AB will describe a rectangular area 2SL × AB, and the indefinite part LD multiplied perpendicularly by the same length AB in a continual motion (according to the rule that, while moving, either by increasing or decreasing, it is always equal to the length LD) will describe an area , that is, the area SL × AB, which, subtracted from the first area 2SL × AB, leaves the area SL × AB. Now the third part , likewise multiplied perpendicularly by the same length AB in a local [i.e., continual] motion, will describe a hyperbolic area, which subtracted from the area SL × AB will leave the required area ANB. Hence, there arises the following construction of the problem.

At points L, A, and B erect perpendiculars Ll, Aa, and Bb, of which Aa is equal to LB, and Bb to LA. With asymptotes Ll and LB, through points a and b describe the hyperbola ab. Then the chord ba, when drawn, will enclose the area aba equal to the required area ANB.

EXAMPLE 2. If the centripetal force tending toward each of the individual particles of the sphere is inversely as the cube of the distance, or (which comes to the same thing) as that cube divided by any given plane, write for V, and then 2PS × LD for PE², and DN will become as , that is (because PS, AS, and SI are continually proportional [or PS is to AS as AS to SI]), as . If the three parts of this quantity are multiplied by the length AB, the first, , will generate a hyperbolic area; the second, ½SI, will generate the area ½AB × SI; the third, , will generate the area , that is, ½AB × SI. From the first subtract the sum of the second and third, and the required area ANB will remain.

Hence there arises the following construction of the problem. At the points L, A, S, and B erect the perpendiculars Ll, Aa, Ss, and Bb, of which Ss is equal to SI; and through the point s, with asymptotes Ll and LB, describe the hyperbola asb meeting the perpendiculars Aa and Bb in a and b; then the rectangle 2AS × SI subtracted from the hyperbolic area AasbB will leave the requirea area ANB.

EXAMPLE 3. If the centripetal force tending toward each of the individual particles of the sphere decreases as the fourth power of the distance from those particles, write for V, and then √(2PS × LD) for PE, and DN will become as . Those three parts, multiplied by the length AB, produce three areas, namely multiplied by ; multiplied by (√LB − √LA); and multiplied by . And these, after the due reduction, become , SI², and . And when the latter two quantities are subtracted from the first one, the result comes out to be . Accordingly, the total force by which the corpuscle P is attracted to the center of the sphere is as , that is, inversely as PS³ × PI. Q.E.I.

The attraction of a corpuscle located inside a sphere can be determined by the same method, but more expeditiously by means of the following proposition.

Proposition 82, Theorem 41
If—in a sphere described about center S with radius SA−SI, SA, and SP are taken continually proportional [i.e., SI to SA as SA to SP], I say that the attraction of a corpuscle inside the sphere at any place I is to its attraction outside the sphere at place P in a ratio compounded of the square root of the ratio of the distances IS and PS from the center, and the square root of the ratio of the centripetal forces, tending at those places P and I toward the center.

If, for example, the centripetal forces of the particles of the sphere are inversely as the distances of the corpuscle attracted by them, the force by which the corpuscle situated at I is attracted by the total sphere will be to the force by which it is attracted at P in a ratio compounded of the square root of the ratio of the distance SI to the distance SP and the square root of the ratio of the centripetal force at place I arising from some particle in the center to the centripetal force at place P arising from the same particle in the center, which is the square root of the ratio of the distances SI and SP to each other inversely. Compounding these two square roots of ratios gives the ratio of equality, and therefore the attractions produced at I and P by the whole sphere are equal. By a similar computation, if the forces of the particles of the sphere are inversely in the squared ratio of the distances, it will be seen that the attraction at I is to the attraction at P as the distance SP is to the semidiameter SA of the sphere. If those forces are inversely in the cubed ratio of the distances, the attractions at I and P will be to each other as SP² to SA²; if as the inverse fourth power, as SP³ to SA³. Hence, since—in this last case [of the inverse fourth power, as in the final ex. 3 of prop. 81]—the attraction at P was found to be inversely as PS³ × PI, the attraction at I will be inversely as SA³ × PI, that is (because SA³ is given), inversely as PI. And the progression goes on in the same way indefinitely. Moreover, the theorem is demonstrated as follows.

With the same construction and with the corpuscle being in any place P, the ordinate DN was found to be as . Therefore, if IE is drawn, that ordinate for any other place I of the corpuscle will—mutatis mutandis [i.e., by substituting I for P in the considerations and arguments that have previously been applied to P]—come out as . Suppose the centripetal forces emanating from any point E of the sphere to be to each other at the distances IE and PE as PEⁿ to IEⁿ (where let the number n designate the index of the powers of PE and IE); then those ordinates will become as and , whose ratio to each other is as PS × IE × IEⁿ to IS × PE × PEⁿ. Because SI, SE, and SP are continually proportional, the triangles SPE and SEI are similar, and hence IE becomes to PE as IS to SE or SA; for the ratio of IE to PE, write the ratio of IS to SA, and the ratio of the ordinates will come out PS × IEⁿ to SA × PEⁿ. But PS to SA is the square root of the ratio of the distances PS and SI, and IEⁿ to PEⁿ (because IE is to PE as IS to SA) is the square root of the ratio of the forces at the distances PS and IS. Therefore the ordinates, and consequently the areas that the ordinates describe and the attractions proportional to them, are in a ratio compounded of the foregoing square-root ratios. Q.E.D.

Proposition 83, Problem 42
To find the force by which a corpuscle located in the center of a sphere is attracted toward any segment of it whatever.

Let P be the corpuscle in the center of the sphere, and RBSD a segment of the sphere contained by the plane RDS and the spherical surface RBS. Let DB be cut at F by the spherical surface EFG described about the center P, and divide that segment into the parts BREFGS and FEDG. But let that surface be taken to be not purely mathematical, but physical, having a minimally small thickness. Call that thickness O, and this surface (by what Archimedes has demonstrated), will be as PF × DF × O. Let us suppose, additionally, the attractive forces of the particles of the sphere to be inversely as that power of the distances whose index is n; then the force by which the surface EFG attracts the body P will be (by prop. 79) as , that is, as . Let the perpendicular FN drawn in [the thickness] O be proportional to this quantity; then the curvilinear area BDI, as described by the ordinate FN, drawn in a continual motion, applied to the length DB, will be as the whole force by which the whole segment RBSD attracts the corpuscle P. Q.E.I.

Proposition 84, Problem 43
To find the force with which a corpuscle is attracted by a segment of a sphere when it is located on the axis of the segment beyond the center of the sphere.

Let corpuscle P, located on the axis ADB of the segment EBK, be attracted by that segment. About center P and with radius PE describe the spherical surface EFK, which divides the segment into two parts EBKFE and EFKDE. Find the force of the first part by prop. 81 and the force of the second part by prop. 83, and the sum of these two forces will be the force of the whole segment EBKDE. Q.E.I.

Scholium
Now that the attractions of spherical bodies have been explained, it would be possible to go on to the laws of the attractions of certain other bodies similarly consisting of attracting particles, but to treat these in particular cases is not essential to my design. It will be enough to subjoin certain more general propositions concerning the forces of bodies of this sort and the motions that arise from such forces, because these propositions are of some use in philosophical questions [i.e., questions of natural philosophy, or physical science].

SECTION 13

The attractive forces of nonspherical bodies

Proposition 85, Theorem 42
If the attraction of an attracted body is far stronger when it is contiguous to the attracting body than when the bodies are separated from each other by even a very small distance, then the forces of the particles of the attracting body decrease, as the attracted body recedes, in a more than squared ratio of the distances from the particles.

For if the forces decrease in the squared ratio of the distances from the particles, the attraction toward a spherical body will not be sensibly increased by contact, because (by prop. 74) it is inversely as the square of the distance of the attracted body from the center of the sphere; and still less will it be increased by contact, if the attraction decreases in a smaller ratio as the attracted body recedes. Therefore, this proposition is evident in the case of attracting spheres. It is the same for concave spherical orbs^a attracting external bodies. And it is much more established in the case of orbs attracting bodies placed inside of them, since the attractions spreading out through the concavities of the orbs are annulled by opposite attractions (by prop. 70), and therefore the attracting forces are null, even in contact. But if any parts remote from the place of contact are taken away from these spheres and spherical orbs, and new parts are added anywhere away from the place of contact, the shapes of these attracting bodies can be changed at will; and yet the parts added or subtracted will not notably increase the excess of attraction that arises from contact, since they are remote from the place of contact. Therefore the proposition is established concerning bodies of all shapes. Q.E.D.

Proposition 86, Theorem 43
If the forces of the particles composing an attracting body decrease, as an attracted body recedes, in the cubed or more than cubed ratio of the distances from the particles, the attraction will be far stronger in contact than when the attracting body and attracted body are separated from each other by even a very small distance.

For by the solution of prop. 81 given in exx. 2 and 3, it is established that the attraction is increased indefinitely in the approach of an attracted corpuscle to an attracting sphere of this sort. By the combination of those examples and prop. 82, the same result is easily inferred concerning the attractions of bodies toward concavo-convex orbs whether the attracted bodies are placed outside those orbs or in the cavities inside the orbs. But the proposition will also be established concerning all bodies universally by adding some attractive matter to these spheres and orbs, or taking some away from them, anywhere away from the place of contact, so that the attracting bodies take on any desired shape. Q.E.D.

Proposition 87, Theorem 44
If two bodies, similar to each other and consisting of equally attracting matter, separately attract corpuscles proportional to those bodies and similarly placed with respect to them, then the accelerative attractions of the corpuscles toward the whole bodies will be as the accelerative attractions of those corpuscles toward particles of those bodies proportional to the wholes and similarly situated in those whole bodies.

For if the bodies are divided into particles that are proportional to the whole bodies and similarly placed in those whole bodies, then the attraction toward an individual particle of the first body will be to the attraction toward the corresponding individual particle of the second body as the attractions toward any given particles of the first body are to the attractions toward the corresponding particles of the second body, and by compounding, the attraction toward the whole first body will be to the attraction toward the whole second body in that same ratio. Q.E.D.

COROLLARY 1. Therefore, if the attracting forces of the particles, on increasing the distances of the attracted corpuscles, decrease in the ratio of any power of those distances, the accelerative attractions toward the whole bodies will be as the bodies directly and those powers of the distances inversely. For example, if the forces of the particles decrease in the squared ratio of the distances from the attracted corpuscles, and the bodies are as A³ and B³, and thus both the cube roots of the bodies and the distances of the attracted corpuscles from the bodies are as A and B, the accelerative attractions toward the bodies will be as and , that is, as those cube roots A and B of the bodies. If the forces of the particles decrease in the cubed ratio of the distances from the attracted corpuscles, the accelerative attractions toward the whole bodies will be as and , that is, will be equal. If the forces decrease in the fourth power of the distance, the attractions toward the bodies will be as and , that is, inversely as the cube roots A and B. And so on.

COROLLARY 2. Hence, on the other hand, from the forces with which similar bodies attract corpuscles similarly placed with respect to such bodies, there can be gathered the ratio of the decrease of the forces of the attracting particles, as the attracted corpuscle recedes, so long as that decrease is directly or inversely in some ratio of the distances.

Proposition 88, Theorem 45
If the attracting forces of equal particles of any body are as the distances of places from the particles, the force of the whole body will tend toward its center of gravity, and will be the same as the force of a globe consisting of entirely similar and equal matter and having its center in that center of gravity.

Let the particles A and B of the body RSTV attract some corpuscle Z by forces which, if the particles are equal to each other, are as the distances AZ and BZ: but if the particles are supposed unequal, are as these particles and their distances AZ and BZ jointly, or (so to speak) as these particles multiplied respectively by their distances AZ and BZ. And let the forces be represented by those solids [or products] A × AZ and B × BZ. Join AB, and let it be cut in G so that AG is to BG as the particle B to the particle A; then G will be the common center of gravity of the particles A and B. The force A × AZ (by corol. 2 of the laws) is resolved into the forces A × GZ and A × AG, and the force B × BZ into the forces B × GZ and B × BG. But the forces A × AG and B × BG are equal (because A is to B as BG to AG); and therefore, since they tend in opposite directions, they nullify each other. There remain the forces A × GZ and B × GZ. These tend from Z toward the center G and compose the force (A + B) × GZ—that is, the same force as if the attracting particles A and B were situated in their common center of gravity G and there composed a globe.

By the same argument, if a third particle C is added, and its force is compounded with the force (A + B) × GZ tending toward the center G, the force thence arising will tend toward the common center of gravity of the globe (at G) and the particle C (that is, toward the common center of gravity of the three particles A, B, and C), and will be the same as if the globe and the particle C were situated in their common center, there composing a greater globe. And so on indefinitely. Therefore the whole force of all the particles of any body RSTV is the same as if that body, while maintaining the same center of gravity, were to assume the shape of a globe. Q.E.D.

COROLLARY. Hence the motion of the attracted body Z will be the same as if the attracting body RSTV were spherical; and therefore, if that attracting body either is at rest or progresses uniformly straight forward, the attracted body will move in an ellipse having its center in the center of gravity of the attracting body.

Proposition 89, Theorem 46
If there are several bodies consisting of equal particles whose forces are as the distances of places from each individual particle, the force—compounded of the forces of all these particles—by which any corpuscle is attracted will tend toward the common center of gravity of the attracting bodies and will be the same as if those attracting bodies, while maintaining their common center of gravity, were united together and were formed into a globe.

This is demonstrated in the same way as the preceding proposition.

COROLLARY. Therefore the motion of an attracted body will be the same as if the attracting bodies, while maintaining their common center of gravity, came together and were formed into a globe. And hence, if the common center of gravity of the attracting body either is at rest or progresses uniformly in a straight line, the attracted body will move in an ellipse having its center in the common center of gravity of the attracting bodies.

Proposition 90, Problem 44
If equal centripetal forces, increasing or decreasing in any ratio of the distances, tend toward each of the individual points of any circle, it is required to find the force by which a corpuscle is attracted when placed anywhere on the straight line that stands perpendicularly upon the plane of the circle at its center.

Suppose a circle to be described with center A and any radius AD in a plane to which the straight line AP is perpendicular; then it is required to find the force by which any corpuscle P is attracted toward the circle. From any point E of the circle, draw the straight line PE to the attracted corpuscle P. In the straight line PA take PF equal to PE, and erect the normal FK so that it will be as the force by which the point E attracts the corpuscle P. And let IKL be the curved line that the point K traces out. Let that line meet the plane of the circle in L. In PA take PH equal to PD, and erect the perpendicular HI meeting the aforesaid curve at I, and the attraction of the corpuscle P toward the circle will be as the area AHIL multiplied by the altitude AP. Q.E.I.

For on AE take the minimally small line Ee. Join Pe, and in PE and PA take PC and Pf equal to Pe. And since the force by which any point E of the ring described with center A and radius AE in the aforesaid plane attracts body [i.e., corpuscle] P toward itself has been supposed to be as FK, and hence the force by which that point attracts body P toward A is as ; and the force by which the whole ring attracts body P toward A is as the ring and jointly; and that ring is as the rectangle of the radius AE and the width Ee, and this rectangle (because PE is to AE as Ee to CE) is equal to the rectangle PE × CE or PE × Ff; it follows that the force by which that ring attracts body P toward A will be as PE × Ff and jointly, that is, as the solid [or product] Ff × FK × AP, or as the area FKkf multiplied by AP. And therefore the sum of the forces by which all the rings in the circle that is described with center A and radius AD attract body P toward A is as the whole area AHIKL multiplied by AP. Q.E.D.

COROLLARY 1. Hence, if the forces of the points decrease in the squared ratio of the distances, that is, if FK is as and thus the area AHIKL is as , the attraction of the corpuscle P toward the circle will be as 1 − , that is, as .

COROLLARY 2. And universally, if the forces of the points at the distances D are inversely as any power Dⁿ of the distances that is, if FK is a , and hence the area AHIKL is as , the attraction of the corpuscle P toward the circle will be as .

COROLLARY 3. And if the diameter of the circle is increased indefinitely and the number n is greater than unity, the attraction of the corpuscle P toward the whole indefinitely extended plane will be inversely as PAⁿ⁻², because the other term, , will vanish.

Proposition 91, Problem 45
To find the attraction of a corpuscle placed in the axis of a round solid, to each of whose individual points there tend equal centripetal forces decreasing in any ratio of the distances.

Let corpuscle P, placed in the axis AB of the solid DECG, be attracted toward that same solid. Let this solid be cut by any circle RFS perpendicular to this axis, and in its semidiameter FS, in a plane PALKB passing through the axis, take (according to prop. 90) the length FK proportional to the force by which the corpuscle P is attracted toward that circle. Let point K touch the curved line LKI meeting the planes of the outermost circles AL and BI at L and I, and the attraction of the corpuscle P toward the solid will be as the area LABI. Q.E.I.

COROLLARY 1. Hence, if the solid is a cylinder described by parallelogram ADEB revolving about the axis AB, and the centripetal forces tending toward each of its individual points are inversely as the squares of the distances from the points, the attraction of the corpuscle P toward this cylinder will be as AB − PE + PD. For the ordinate FK (by prop. 90, corol. 1) will be as 1 − . The unit part of this or the quantity 1 in 1 − multiplied by the length AB describes the area 1 × AB, and the other part multiplied by the length PB describes the area 1 × (PE − AD), which can easily be shown from the quadrature of the curve LKI; and similarly the same part multiplied by the length PA describes the area 1 × (PD − AD), and multiplied by the difference AB of PB and PA describes the difference of the areas 1 × (PE − PD). From the first product 1 × AB take away the last product 1 × (PE − PD), and there will remain the area LABI equal to 1 × (AB − PE + PD). Therefore the force proportional to this area is as AB − PE + PD.

COROLLARY 2. Hence also the force becomes known by which a spheroid AGBC attracts any body P, situated outside the spheroid in its axis AB. Let NKRM be a conic whose ordinate ER, perpendicular to PE, is always equal to the length of the line PD, which is drawn to the point D in which the ordinate cuts the spheroid. From the vertices A and B of the spheroid, erect AK and BM perpendicular to the axis AB of the spheroid and equal respectively to AP and BP, and therefore meeting the conic in K and M; and join KM cutting off the segment KMRK from the conic. Let the center of the spheroid be S, and its greatest semidiameter SC. Then the force by which the spheroid attracts the body P will be to the force by which a sphere described with diameter AB attracts the same body as to . And by the same mode of computation it is possible to find the forces of the segments of the spheroid.

COROLLARY 3. But if the corpuscle is located inside the spheroid and in its axis, the attraction will be as its distance from the center. This is seen more easily by the following argument, whether the particle is in the axis or in any other given diameter. Let AGOF be the attracting spheroid, S its center, and P the attracted body. Through that body P draw both the semidiameter SPA and any two straight lines DE and FG meeting the spheroid in D and F on one side and in E and G on the other; and let PCM and HLN be the surfaces of two inner spheroids, similar to and concentric with the outer spheroid; and let the first of these pass through the body P and cut the straight lines DE and FG in B and C, and let the latter cut the same straight lines in H, I and K, L. Let all the spheroids have a common axis, and the parts of the straight lines intercepted on the two sides, DP and BE, FP and CG, DH and IE, FK and LG will be equal to one another, because the straight lines DE, PB, and HI are bisected in the same point, as are also the straight lines FG, PC, and KL. Now suppose that DPF and EPG designate opposite cones described with the infinitely small vertical angles DPF and EPG, and that the lines DH and EI also are infinitely small; then the particles of the cones—that is, the particles DHKF and GLIE—cut off by the surfaces of the spheroids will (because of the equality of the lines DH and EI) be to each other as the squares of their distances from the corpuscle P, and therefore will attract the corpuscle equally. And by a like reasoning, if the spaces DPF and EGCB are divided into particles by the surfaces of innumerable similar concentric spheroids, having a common axis, then all of these particles will attract the body P in opposite directions equally on both sides. Therefore the forces of the cone DPF and of the conical segment [or truncated cone] EGCB are equal, and—being opposite—annul each other. And it is the same with regard to the forces of all the matter outside the innermost spheroid PCBM. Therefore the body P is attracted only by the innermost spheroid PCBM, and accordingly (by prop. 72, corol. 3) its attraction is to the force by which the body A is attracted by the whole spheroid AGOD as the distance PS to the distance AS. Q.E.D.

Proposition 92, Problem 46
Given an attracting body, it is required to find the ratio by which the centripetal forces tending toward each of its individual points decrease [i.e., decrease as a function of distance].

From the given body a sphere or cylinder or other regular figure is to be formed, whose law of attraction—corresponding to any ratio of decrease [in relation to distance]—can be found by props. 80, 81, and 91. Then, by making experiments, the force of attraction at different distances is to be found; and the law of attraction toward the whole that is thus revealed will give the ratio of the decrease of the forces of the individual parts, which was required to be found.

Proposition 93, Theorem 47
If a solid, plane on one side but infinitely extended on the other sides, consists of equal and equally attracting particles, whose forces—in receding from the solid—decrease in the ratio of any power of the distances that is more than the square; and if a corpuscle set on either side of the plane is attracted by the force of the whole solid; then I say that that force of attraction of the solid in receding from its plane surface will decrease in the ratio of the distance of the corpuscle from the plane raised to a power whose index is less by 3 units than that of the power of the distances in the law of attractive force [lit. will decrease in the ratio of the power whose base is the distance of the corpuscle from the plane and whose index is less by 3 than the index of the power of the distances].

CASE 1. Let LGl be the plane by which the solid is terminated. Let the solid lie on the side of this plane toward I, and let it be resolved into innumerable planes mHM, nIN, oKO, . . . parallel to GL. And first let the attracted body C be placed outside the solid. Draw CGHI perpendicular to those innumerable planes, and let the forces of attraction of the points of the solid decrease in the ratio of a power of the distances whose index is the number n not smaller than 3. Therefore (by prop. 90, corol. 3) the force by which any plane mHM attracts the point C is inversely as CHⁿ⁻². In the plane mHM take the length HM inversely proportional to CHⁿ⁻², and that force will be as HM. Similarly, on each of the individual planes lGL, nIN, oKO, . . . , take the lengths GL, IN, KO, . . . inversely proportional to CGⁿ⁻², CIⁿ⁻², CKⁿ⁻², . . . ; then the forces of these same planes will be as the lengths taken, and thus the sum of the forces will be as the sum of the lengths; that is, the force of the whole solid will be as the area GLOK produced infinitely in the direction OK. But that area (by the well-known methods of quadratures) is inversely as CGⁿ⁻³, and therefore the force of the whole solid is inversely as CGⁿ⁻³. Q.E.D.

CASE 2. Now let the corpuscle C be placed on the side of the plane lGL inside the solid, and take the distance CK equal to the distance CG. Then the part LGloKO of this solid, terminated by the parallel planes lGL and oKO, will not attract the corpuscle C (situated in the middle) in any direction, the opposite actions of opposite points annulling each other because of their equality. Accordingly, corpuscle C is attracted only by the force of the solid situated beyond the plane OK. But this force (by case 1) is inversely as CKⁿ⁻³, that is (because CG and CK are equal), inversely as CGⁿ⁻³. Q.E.D.

COROLLARY 1. Hence, if the solid LGIN is terminated on both sides by two infinitely extended and parallel planes LG and IN, its force of attraction becomes known by subtracting from the force of attraction of the whole infinitely extended solid LGKO the force of attraction of the further part NIKO produced infinitely in the direction KO.

COROLLARY 2. If the more distant part of this infinitely extended solid is ignored, since its attraction compared with the attraction of the nearer part is of almost no moment, then the attraction of that nearer part, with an increase of the distance, will decrease very nearly in the ratio of the power CGⁿ⁻³.

COROLLARY 3. And hence, if any body that is finite and plane on one side attracts a corpuscle directly opposite the middle of that plane, and the distance between the corpuscle and the plane is exceedingly small compared with the dimensions of the attracting body, and the attracting body consists of homogeneous particles whose forces of attraction decrease in the ratio of any power of the distances that is more than the fourth; the force of attraction of the whole body will decrease very nearly in the ratio of a power of that exceedingly small distance, whose index is less by 3 than the index of the stated power. This assertion is not valid for a body consisting of particles whose forces of attraction decrease in the ratio of the third power of the distances, because in this case the attraction of the more distant part of the infinitely extended body in corol. 2 is always infinitely greater than the attraction of the nearer part.

Scholium
If a body is attracted perpendicularly toward a given plane, and the motion of the body is required to be found from the given law of attraction, the problem will be solved by seeking (by prop. 39) the motion of the body descending directly to this plane and by compounding this motion (according to corol. 2 of the laws) with a uniform motion performed along lines parallel to the same plane. And conversely, if it is required to find the law of an attraction made toward the plane along perpendicular lines, under the condition that the attracted body moves in any given curved line whatever, the problem will be solved by the operations used in the third problem [i.e., prop. 8].

The procedure can be shortened by resolving the ordinates into converging series. For example, if B is the ordinate to the base A at any given angle, and is as any power of that base, and the force is required by which a body that is either attracted toward the base or repelled away from the base (according to the position of the ordinate) can move in a curved line that the upper end of the ordinate traces out; I suppose the base to be increased by a minimally small part O, and I resolve the ordinate into the infinite series and I suppose the force to be proportional to the term of this series in which O is of two dimensions, that is, to the term . Therefore the required force is as , or, which is the same, as . For example, if the ordinate traces out a parabola, where m = 2 and n = 1, the force will become as the given quantity 2B°, and thus will be given. Therefore with a given [i.e., constant] force the body will move in a parabola, as Galileo demonstrated. But if the ordinate traces out a hyperbola, where m = 0 − 1 and n = 1, the force will become as 2A⁻³ or 2B³; and therefore with a force that is as the cube of the ordinate, the body will move in a hyperbola. But putting aside propositions of this sort, I go on to certain others on motion which I have not as yet considered.

SECTION 14

The motion of minimally small bodies that are acted on by centripetal forces tending toward each of the individual parts of some great body

Proposition 94, Theorem 48
If two homogeneous mediums are separated from each other by a space terminated on the two sides by parallel planes, and a body passing through this space is attracted or impelled perpendicularly toward either medium and is not acted on or impeded by any other force, and the attraction at equal distances from each plane (taken on the same side of that plane) is the same everywhere; then I say that the sine of the angle of incidence onto either plane will be to the sine of the angle of emergence from the other plane in a given ratio.

CASE 1. Let Aa and Bb be the two parallel planes. Let the body be incident upon the first plane Aa along line GH, and in all its passage through the intermediate space let it be attracted or impelled toward the medium of incidence, and by this action let it describe the curved line HI and emerge along the line IK. To the plane of emergence Bb erect the perpendicular IM meeting the line of incidence GH produced in M and the plane of incidence Aa in R; and let the line of emergence KI produced meet HM in L. With center L and radius LI describe a circle cutting HM in P and Q, as well as MI produced in N. Then first, if the attraction or impulse is supposed uniform, the curve HI (from what Galileo demonstrated) will be a parabola, of which this is a property: that the rectangle of its given latus rectum and the line IM is equal to HM squared; but also the line HM will be bisected in L. Hence, if the perpendicular LO is dropped to MI, MO and OR will be equal; and when the equals ON and OI have been added to these quantities, the totals MN and IR will become equal. Accordingly, since IR is given, MN is also given; and the rectangle NM × MI is to the rectangle of the latus rectum and IM (that is, to HM²) in a given ratio. But the rectangle NM × MI is equal to the rectangle PM × MQ, that is, to the difference of the squares ML² and PL² or LI²; and HM² has a given ratio to its fourth part ML²: therefore the ratio of ML² − LI² to ML² is given, and by conversion [or convertendo] the ratio LI² to ML² is given, and also the square root of that ratio, LI to ML. But in every triangle LMI, the sines of the angles are proportional to the opposite sides. Therefore the ratio of the sine of the angle of incidence LMR to the sine of the angle of emergence LIR is given. Q.E.D.

CASE 2. Now let the body pass successively through several spaces terminated by parallel planes, AabB, BbcC, . . . , and be acted on by a force that is uniform in each of the individual spaces considered separately but is different in each of the different spaces. Then by what has just been demonstrated, the sine of the angle of incidence upon the first plane Aa will be to the sine of the angle of emergence from the second plane Bb in a given ratio; and this sine, which is the sine of the angle of incidence upon the second plane Bb, will be to the sine of the angle of emergence from the third plane Cc in a given ratio; and this sine will be in a given ratio to the sine of the angle of emergence from the fourth plane Dd; and so on indefinitely. And from the equality of the ratios [or ex aequo] the sine of the angle of incidence upon the first plane will be in a given ratio to the sine of the angle of emergence from the last plane. Now let the distances between the planes be diminished and their number increased indefinitely, so that the action of attraction or of impulse, according to any assigned law whatever, becomes continuous; then the ratio of the sine of the angle of incidence upon the first plane to the sine of the angle of emergence from the last plane, being always given, will still be given now. Q.E.D.

Proposition 95, Theorem 49
With the same suppositions as in prop. 94, I say that the velocity of the body before incidence is to its velocity after emergence as the sine of the angle of emergence to the sine of the angle of incidence.

Let AH be taken equal to Id, and erect the perpendiculars AG and dK meeting the lines of incidence and emergence GH and IK in G and K. In GH take TH equal to IK, and drop Tv perpendicular to the plane Aa. And (by corol. 2 of the laws) resolve the motion of the body into two motions, one perpendicular, the other parallel, to the planes Aa, Bb, Cc, . . . . The [component of the] force of attraction or of impulse acting along perpendicular lines does not at all change the motion in the direction of the parallels; and therefore the body, by this latter motion, will in equal times pass through equal distances along parallels between the line AG and the point H, and between the point I and the line dK, that is, it will describe the lines GH and IK in equal times. Accordingly, the velocity before incidence is to the velocity after emergence as GH to IK or TH; that is, as AH or Id to vH, that is (with respect to the radius TH or IK), as the sine of the angle of emergence to the sine of the angle of incidence. Q.E.D.

Proposition 96, Theorem 50
With the same suppositions, and supposing also that the motion before incidence is faster than afterward, I say that as a result of ^achanging the inclination^a of the line of incidence, the body will at last be reflected, and the angle of reflection will become equal to the angle of incidence.

For suppose the body to describe parabolic arcs between the parallel planes Aa, Bb, Cc, . . ., as before; and let those arcs be HP, PQ, QR, . . . . And let the obliquity of the line of incidence GH to the first plane Aa be such that the sine of the angle of incidence is to the radius of the circle whose sine it is in the ratio which that same sine of the angle of incidence has to the sine of the angle of emergence from the plane Dd into the space DdeE; then, because the sine of the angle of emergence will now have become equal to the radius, the angle of emergence will be a right angle, and hence the line of emergence will coincide with the plane Dd. Let the body arrive at this plane at the point R; and since the line of emergence coincides with that same plane, it is obvious that the body cannot go any further toward the plane Ee. But neither can it go on in the line of emergence Rd, because it is continually attracted or impelled toward the medium of incidence. Therefore, this body will be turned back between the planes Cc and Dd, describing an arc of the parabola QRq, whose principal vertex (according to what Galileo demonstrated) is at R, and will cut the plane Cc in the same angle at q as formerly at Q; and then, proceeding in the parabolic arcs qp, ph, . . . , similar and equal to the former arcs QP and PH, this body will cut the remaining planes in the same angles at p, h, . . ., as formerly at P, H, . . ., and will finally emerge at h with the same obliquity with which it was incident upon the plane at H. Now suppose the distances between the planes Aa, Bb, Cc, Dd, Ee, . . . to be diminished and their number increased indefinitely, so that the action of attraction or impulse, according to any assigned law whatever, is made to be continuous; then the angle of emergence, being always equal to the angle of incidence, will still remain equal to it now. Q.E.D.

Scholium
These attractions are very similar to the reflections and refractions of light made according to a given ratio of the secants, as Snel discovered, and consequently according to a given ratio of the sines, as Descartes set forth. For the fact that light is propagated successively [i.e., in time and not instantaneously] and comes from the sun to the earth in about seven or eight minutes is now established by means of the phenomena of the satellites of Jupiter, confirmed by the observations of various astronomers. Moreover, the rays of light that are in the air (as Grimaldi recently discovered, on admitting light into a dark room through a small hole—something I myself have also tried) in their passing near the edges of bodies, whether opaque or transparent (such as are the circular-rectangular edges of coins minted from gold, silver, and bronze, and the sharp edges of knives, stones, or broken glass), are inflected around the bodies, as if attracted toward them; and those of the rays that in such passing approach closer to the bodies are inflected the more, as if more attracted, as I myself have also diligently observed. And those that pass at greater distances are less inflected, and at still greater distances are inflected somewhat in the opposite direction and form three bands of colors.

In the figure, s designates the sharp edge of a knife or of any wedge AsB, and gowog, fnunf, emtme, and dlsld are rays, inflected in the arcs owo, nun, mtm, and lsl toward the knife, more so or less so according to their distance from the knife. Moreover, since such an inflection of the rays takes place in the air outside the knife, the rays which are incident upon the knife must also be inflected in the air before they reach it. And the case is the same for those rays incident upon glass. Therefore refraction takes place not at the point of incidence, but gradually by a continual inflection of the rays, made partly in the air before the rays touch the glass, and partly (if I am not mistaken) within the glass after they have entered it, as has been delineated in the rays ckzc, biyb, and ahxa incident upon the glass at r, q, and p, and inflected between k and z, i and y, h and x. Therefore because of the analogy that exists between the propagation of rays of light and the motion of bodies, I have decided to subjoin the following propositions for optical uses, meanwhile not arguing at all about the nature of the rays (that is, whether they are bodies or not), but only determining the trajectories of bodies, which are very similar to the trajectories of rays.

Proposition 97, Problem 47
Supposing that the sine of the angle of incidence upon some surface is to the sine of the angle of emergence in a given ratio, and that the inflection of the paths of bodies in close proximity to that surface takes place in a very short space, which can be considered to be a point; it is required to determine the surface that may make all the corpuscles emanating successively from a given place converge to another given place.

Let A be the place from which the corpuscles diverge, B the place to which they should converge, CDE the curved line that—by revolving about the axis AB—describes the required surface, D and E any two points of that curve, and EF and EG perpendiculars dropped to the paths AD and DB of the body. Let point D approach point E; then the ultimate ratio of the line DF (by which AD is increased) to the line DG (by which DB is decreased) will be the same as that of the sine of the angle of incidence to the sine of the angle of emergence. Therefore the ratio of the increase of the line AD to the decrease of the line DB is given; and as a result, if a point C is taken anywhere on the axis AB, this being a point through which the curve CDE should pass, and the increase CM of AC is taken in that given ratio to the decrease CN of BC, and if two circles are described with centers A and B and radii AM and BN and cut each other at D, that point D will touch the required curve CDE, and by touching it anywhere whatever will determine that curve. Q.E.I.

COROLLARY 1. But by making point A or B in one case go off indefinitely, in another case move to the other side of point C, all the curves which Descartes exhibited with respect to refractions in his treatises on optics and geometry will be traced out. Since Descartes concealed the methods of finding these, I have decided to reveal them by this proposition.

COROLLARY 2. If a body, incident upon any surface CD along the straight line AD drawn according to any law, emerges along any other straight line DK; and if from point C the curved lines CP and CQ, always perpendicular to AD and DK, are understood to be drawn; then the increments of the lines PD and QD, and hence the lines themselves PD and QD generated by those increments, will be as the sines of the angles of incidence and emergence to each other, and conversely.

Proposition 98, Problem 48
The same conditions being supposed as in prop. 97, and supposing that there is described about the axis AB any attracting surface CD, regular or irregular, through which the bodies coming out from a given place A must pass; it is required to find a second attracting surface EF that will make the bodies converge to a given place B.

Join AB and let it cut the first surface in C and the second in E, point D being taken in any way whatever. And supposing that the sine of the angle of incidence upon the first surface is to the sine of the angle of emergence from that first surface, and that the sine of the angle of emergence from the second surface is to the sine of the angle of incidence upon the second surface, as some given quantity M is to another given quantity N; produce AB to G so that BG is to CE as M − N to N, and produce AD to H so that AH is equal to AG, and also produce DF to K so that DK is to DH as N to M. Join KB, and with center D and radius DH describe a circle meeting KB produced in L, and draw BF parallel to DL; then the point F will touch the line EF, which—on being revolved about the axis AB—will describe the required surface. Q.E.F

Now suppose the lines CP and CQ to be everywhere perpendicular to AD and DF respectively, and the lines ER and ES to be similarly perpendicular to FB and FD, with the result that QS is always equal to CE; then (by prop. 97, corol. 2) PD will be to QD as M to N, and therefore as DL to DK or FB to FK; and by separation [or dividendo] as DL − FB or PH − PD — FB to FD or FQ — QD, and by composition [or componendo] as PH − FB to FQ, that is (because PH and CG, QS and CE are equal), as CE + BG − FR to CE − FS. But (because BG is proportional to CE and M − N is proportional to N) CE + BG is also to CE as M to N, and thus by separation [or dividendo] FR is to FS as M to N; and therefore (by prop. 97, corol. 2) the surface EF compels a body incident upon it along the line DF to go on in the line FR to the place B. Q.E.D.

Scholium
It would be possible to use the same method for three surfaces or more. But for optical uses spherical shapes are most suitable. If the objective lenses of telescopes are made of two lenses that are spherically shaped and water is enclosed between them, it can happen that errors of the refractions that take place in the extreme surfaces of the lenses are accurately enough corrected by the refractions of the water. Such objective lenses are to be preferred to elliptical and hyperbolical lenses, not only because they can be formed more easily and more accurately but also because they more accurately refract the pencils of rays situated outside the axis of the glass. Nevertheless, the differing refrangibility of different rays [i.e., of rays of different colors] prevents optics from being perfected by spherical or any other shapes. Unless the errors arising from this source can be corrected, all labor spent in correcting the other errors will be of no avail.