# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 2

*Of the invention of centripetal forces.*

Proposition I. Theorem I.

*The areas, which revolving bodies describe by radii drawn to an immoveable centre of force, do lie in the same immovable planes, and are proportional to the times in which they are described. Pl. 2. Fig. 5.*

OR suppose the time to be divided into equal parts, and in the first part of that time, let the body by its innate force describe the right line *AB*. In the second part of that time, the same would, (by law 1.) if not hinder'd, proceed directly to *c*, along the line *Bc* equal to *AB*; so that by the radii *AS*, *BS*, *cS* drawn to the centre, the equal areas *ASB*, *BSc*, would be described. But when the body is arrived at *B*, suppose that a centripetal force act at once with a great impulse, and turning aside the body from the right line *Bc*, compells it afterwards to continue its motion along the right line *BC*. Draw *cC* parallel to *BS* meeting *BC* in *C*; and at the end of the second part of the time, the body (by Cor. 1. of the laws) will be found in *C*, in the same plane with the triangle *ASB*. Joyn *SC*, and, because *SB* and *Cc* are parallel, the triangle SBC will be equal to the triangle *SBc*, and therefore also to the triangle *SAB*. By the like argument, if the centripetal force acts successively in *C*, *D*, *E*, &c. and makes the body in each single particle of time, to describe the right lines *CD*, *DE*, *EF*, &c. they will all lye in the same plane; and the triangle *SCD* will be equal to the triangle *SBC*, and *SDE* to *SCD*, and *SEP* to *SDE*. And therefore in equal times, equal areas are describ'd in one immovable plane: and, by composition, any sums *SADS*, *SAFS*, of those areas, are one to the other, as the times in which they are describ'd. Now let the number of those triangles be augmented, and their breadth dimnished *in infinitum*; and (by cor. 4. lem. 5.) their ultimate perimeter *ADF* will be a curve line: and therefore the centripetal force, by which the body is perpetually drawn back from the tangent of this curve, will act continually; and any describ'd areas *SADS*, *SAFS*, which are always proportional to the times of description, will, in this case also, be proportional to those times. *Q. E. D.*

Cor. 1. The velocity of a body attracted towards an immovable centre, in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the velocities in those places *A*, *B*, *C*, *D*, *E* are as the bases *AB*, *BC*, *CD*, *DE*, *EF*, of equal triangles; and these bases are reciprocally as the perpendiculars let fall upon them.

Cor. 2. If the chords *AB*, *BC* of two arcs, successively described in equal times, by the same body, in spaces void of resistance, are compleated into a parallelogram *ABCB* and the diagonal *BV* of this parallelogram, in the position which it ultimately acquires when those arcs are diminished *in infinitum*, is produced both ways, it will pass through the centre of force.

Cor. 3. If the chords *AB*, *BC*, and *DE*, *EF*, of arcs describ'd in equal times, in spaces void of resistance, are compleated into the parallelograms *ABCD*, *DEFZ*; the forces in *B* and *E* are one to the other in the ultimate ratio of the diagonals *BV*, *EZ*, when those arcs are diminished *in infinitum*. For the motions *B*; and *EF* of the body (by cor. 1. of the laws) are compounded of the motions *Bc*, *BV* and *Ef*, *EZ*: but *BV* and *EZ*, which are equal to *Cc* and *Ff* in the demonstration of this proposition, were generated by the impulses of the centripetal force in *B* and *E*, and are therefore proportional to those impulses.

Cor. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the vers'd sines of arcs described in equal times; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such vers'd sines are the halfs of the diagonals mentioned in cor. 3.

Cor. 5. And therefore those forces are to the force of gravity, as the said vers'd sines to the vers'd sines perpendicular to the horizon of those parabolic arcs which projectiles describe in the same time.

Cor. 6. And the same things do all hold good (by cor. 5. of the laws) when the planes in which the bodies are mov'd, together with the centres of force which are placed in those planes, are not at rest but move uniformly forward in right lines.

Proposition II. Theorem II.

*Every body, that moves in any curve line described in a plane, and by a radius, drawn to a point either immoveable, or moving forward with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.*

Case 1. For every body that moves in a curve line, is (by law 1.) turned aside from its rectilinear course by the action of some force that impels it. And that force by which the body is turned off from its rectilinear course, and is made to describe, in equal times, the equal least triangles *SAB*, *SBG*, *SCD*, &c. about the immovable point *S*, (by prop. 40. book 1. elem. and law 2.) acts in the place *B*, according to the direction of a line parallel to *cC*, that is, in the direction of the line *BS*; and in the place *C*, according to the direction of a line parallel to *dD*, that is, in the direction of the line *CS*, &c. And therefore acts always in the direction of lines tending to the immovable point *S*. *Q. E. D.*

Case 2. And (by cor. 5. of the laws) it is indifferent whether the superficies in which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure describ'd, and its point *S*, uniformly forwards in right lines.

Cor. 1. In non-resisting spaces or mediums, if the areas are not proportional to the times, the forces are not directed to the point in which the radii meet; but deviate therefrom *in consequantia*, or towards the parts to which the motion is directed. if the description of the areas is accelerated; but *in antecedentia*, if retarded.

Cor. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.

Scholium

A body may be urged by a centripetal force compounded or several forces. In which case the meaning of the proposition is, that the force which results out of all, tends to the point *S*. But if any force, acts perpetually in the direction of lines perpendicular to the describ'd surface; this force will make the body to deviate from the plane of its motion: but will neither augment nor diminish the quantity of the described surface, and is therefore to be neglected in the composition of forces.

Proposition III. Theorem III.

*Every body, that, by a radius drawn to the centre of another body howsover moved, described areas about that centre proportional to the times, is urged by a force compounded out of the centripetal force tending to that other body, and of all the accelerative force by which that other body is impelled.*

Let *L* represent the one, and *T* the other body; and (by Cor. 6 of the laws) if both bodies are urged in the direction of parallel lines, by a new force equal and contrary to that by which the second body *T* is urged, the first body *L* will go on to describe about the other body *T*, the same areas as before: but the force, by which that other body *T* was urged, will be now destroyed by an equal and contrary force; and therefore (by Law 1.) that other body *T*, now left to it self, will either rest, or move uniformly forward in a right line: and the first body *L* impell'd by the difference of the forces, that is, by the force remaining, will go on to describe about the other body *T*; areas proportional to the times. And therefore (by Theor. 2.) the difference of the forces is directed to the other body *T* as its centre. *Q. E. D.*

Cor. 1. Hence if the one body *L*, by a radius drawn to the other body *T*, describes areas proportional to the times; and from the whole force, by which the first body *L* is urged (whether that force is simple, or, according to cor. 2. of the laws, compounded out of several forces) we subduct (by the same cor.) that whole accelerative force, by which the other body is urged; the whole remaining force by which the first body is urged, will tend to the other body *T*, as its centre.

Cor. 2. And, if these areas are proportional to the times nearly, the remaining force will tend to the other body *T* nearly.

Cor. 3. And *vice versa*, if the remaining force tends nearly to the other body *T*, those areas will be nearly proportional to the times.

Cor. 4. If the body *L*, by a radius drawn to the other body *T*, describes areas, which compared with the times, are very unequal; and that other body *T* be either at rest or moves uniformly forward in a right line: the action of the centripetal force tending to that other body *T*, is either none at all, or it is mix'd and compounded with very powerful actions of other forces: and the whole force compounded of them all, if they are many, is directed to another (immovable or moveable) centre. The same thing obtains, when the other body is moved by any motion whatsoever; provided that centripetal force is taken, which remains after subducting that whole force acting upon that other body *T*.

Scholium

Because the equable description of areas indicates that a centre respected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit; why may we not be allowed in the following discourse, to use the equable description of areas as an indication of a centre, about which all circular motion is performed in free spaces?

Proposition IV. Theorem IV.

*The centripetal forces of bodies, which by equoble motions describe different circles, tend to the centres of the same circles; and are one to the other, as the squares of* the arcs described in equal times applied to the radii applied the circles.

These forces tend to the centres of the circles (by prop. 2. and cor. 2. prop. 1) and are one to another as the versed sines of the least arcs described in equal times (by cor. 4. prop. 1.) that is, as the squares of the same arcs applied to the diameters of the circles, (by lem. 7.) and therefore since those arcs are as arcs described in any equal times, and the diameters are as the radii; the forces will be as the squares of an arcs described in the same time applied to the radii of the circles. *Q. E. D.*

Cor. 1. Therefore, since those arcs are as the velocities of the bodies, the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the simple ratio of the radii inversely.

Cor. 2. And, since the periodic times are in a ratio compounded of the ratio of the radii directly; and the ratio of the velocities inversely; the centripetal forces are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely.

Cor. 3. Whence if the periodic times are equal, and the velocities therefore as the radii; the centripetal forces will be also as the radii; and the contrary.

Cor. 4. If the periodic times and the velocities are both in the subduplicate ratio of the radii; the centripetal forces will be equal among themselves: and the contrary.

Cor. 5. If the periodic times are as the radii, and therefore the velocities equal; the centripetal forces will be reciprocally as the radii: and the contrary.

Cor. 6. If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii; the centripetal forces will be in the duplicate ratio of the radii inversely: and the contrary.

Cor. 7. And universally, if the periodic time is as any power of the radius *R*, and therefore the velocity reciprocally as the power of the radius; the centripetal force will be reciprocally as the power of the radius: and the contrary.

Cor. 8. The same things all hold concerning the times, the velocities, and forces by which bodies describe the similar parts of any similar figures, that have their centres in a similar position within those figures; as appears by applying the demonstration of the preceding cases to those. And the application is easy by only substituting the equable description of areas in the place of equable motion and using the distances of the bodies from the centres instead of the radii.

Cor. 9. From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given force, describe sin any time, is a mean proportioanl between the diameter of the circle, and the space which the same body falling by the same given space would descend thro' in the same given time.

Scholium.

The case of the 6th corollary obtains in the celestial bodies, (as Sir *Christopher Wren*, Dr. *Hooke*, Dr. *Halley* have severally observed) and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres. Moreover, by means of the preceding proposition and its corollaries, we may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the Earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given, (by cor. 9. of this prop.) And by such propositions, Mr. *Huygens*, in his excellent book *De Horlogie Oscillatorio*, has compared the force of gravity with the centripetal forces of revolving bodies.

The preceding proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflectedfrom the circle at the several angular points; the force, with which at every direction it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflexions conjunctly; that is, (if the species of the polygon be given) as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of the circle; that is, as the square of that length applied to the ratios: and therefore if the polygon, by having its sides diminished is incresead, coincides with the circle, as the square of the arc described in a given time applied to the radius. This is the centrifugal force, with which the body impells the circle; and to which the contrary force, wherewith the circle continually repells the body towards the centre, is equal.

Proposition V. Problem I.

*There being given in any places, the velocity with which a body desribes a given figure, by means of forces directed to some common centre; to find that centre.* Pl. 3. Fig. 1.

Let the three right lines *PT*, *TQV*, *VR* touch the figure described in as many points *P*, *Q*, *R*, and meet in *T* and *V*. On the tangents erects the perpendiculars *PA*, *QB*, *RC*, reciprocally proportional to the velocities of the body in the points *P*, *Q*, *R*, from which the perpendiculars were raised, that is, so that *PA* may be to *QB* as the velocity in *Q* to the velocity in *P*, and *QB* to *RC* as the velocity in *R* to the velocity in *Q*; Thro' the ends *A*, *B*, *C*, of the perpendiculars draw *AD*, *DBE*, *EC*, at right angles, meeting in *D* and *E*: And the right lines *TD*, *VE* produced, will meet in *S* the centre required.

For the perpendiculars let fall from the centre *S* on the tangents *PT*, *QT*, are reciprocally as the velocities of the bodies in the points *P* and *Q* (by cor. 1. prop. 1.) and therefore, by consruction, as the perpendiculars *AP*, *BQ* directly; that is, as the perpendiculars let fall from the point *D* on the tangents. Whence it is easy to infer, that the points *S*, *D*, *T*, are in one right line. And by the like argument the points *S*, *E*, *V* are also in one right line; and therefore the centre *S* is in the point where the right lines *TD*, *VE* meet. *Q. E. D.*

Proposition VI. Theorem V.

*In a space void of resisŧance, if a body revolves in any orbit about an immoveable centre, and in the least time describes any arc just then nascent; and the versed sine of that arc is suppofed to be drawn, bisecting the chord, and produced passing through the centre of force: the centripetal force in the middle of the arc, will be as the versed sine directly and the square of the time inversely.*

For the versed sine in a given time is as the force (by cor. 4. prop. 1.) and augmenting the time in any ratio, because the arc will be augmented in the same ratio, the versed sine will be augmented in the duplicate of that ratio, (by cor. 2 and 3. lem. 2.) and therefore is as the force and the square of the time. Subduct on both sides the duplicate ratio of the time, and the force will be as the versed line directly and the square of the time inversely. Q. E. D.

And the same thing may also be easily demonstrated by corol. 4. lem. 10.

Cor. 1. If a body P revolving about the centre *S*, (Pl. 3. *Fig.* 2.) describes a curve line *APQ* which a right line *ZPR* touches in any point *P*; and from any other point *Q* of the curve. *QR* is drawn parallel to the distance *SP*, meeting the tangent in *R*; and *QT* is drawn perpendicular to the distance *SP*: the centripetal force will be reciprocally as the solid , if the solid be taken of that magnitude which it ultimately acquires when the points *P* and *Q* coincide. For *QR* is equal to the versed sine of double the arc *QP*, whose middle is *P*: and double the triangle *SQP*, or *SP* x *QT* is proportional to the time, in which that double arc is described; and therefore may be used for the exponent of the time.

Cor. 2. By a like reasoning, the centripetal force is reciprocally as the solid if *ST* is a perpendicular from the centre of force on *PR* the tangent of the orbit. For the rectangles *ST* x *QP* and *SP* x *QT* are equal.

Cor. 3. If the orbit is either a circle, or touches or cuts a circle concentrically, that is contains with a circle the least angle of contact or section, having the same curvature and the same radius of curvature at the point *P*; and if *P*, *V* be a chord of this circle, drawn from the body through the centre of force; the centripetal force will be reciprocally as the solid . For *PV* is

Cor. 4. The same things being supposed the centripetal force is as the square of the velocity directly. and that chord inversely. For the velocity is reciprocally as the perpendicular *ST*; by cor. 1. prop. 1.

Cor. 5. Hence if any curvilinear figure *APQ* is given; and therein a point *S* is also given to which a centripetal force is perpetually directed; that law of centripetal force may be found, by which the body *P* will be continually drawn back from a rectlinear course, and being detained in the perimeter of that figure. will describe the same by a perpetual revolution. That is, we are to find by computation, either the solid . of the solid , reciprocally proportional to this force. Examples of this we shall give in the following problems.

Proposition VII. Problem II.

*If a body revolves in the circumference of a circle; it is proposed to find the law of centripetal force directed to any given point.* Pl. 3. Fig. 3.

Let *VQPA* be the circumference of the circle; *S* the given point to which as to a centre the force tends; *P* the body moving in the circumference; *Q* the next place into which it is to move; and *PRZ* the tangent of the circle at the preceding place. Through the point *S* draw the chord *PV*, and the diameter *VA* of the circle, join *AP*, and draw, *QT* perpendicular to *SP*, which produced, may meet the tangent tangent *PR* in *Z*; and lastly, thro' the point *Q* draw *LR* parallel to *SP*, meeting the circle in *L*, and the tangent *PZ* in *R*. And, because of the {{ls]]imilar triangles *ZQR*, *ZTP*, *VPA*, we shall have , that is, *QRL*, to , as to . And therefore is equal to . Multiply those equals by and the points *P* and *Q* coinciding, for *RL* write *PV*; then we shall have . And therefore (by cor. 1. and 5. prop. 6.) the centripetal force is reciprocally as , that is (because is given) reciprocally as the square of the distance of altitude *SP*, and the cube of the chord *PV* conjunctly. *Q. E. I.*

On the tangent *PR* produced, let fall the perpendicular *ST*: and (because of the similar triangles *STP*, *VPA*) we shall have *AV* to *PV* as *SP* to *ST*, and therefore , and , that is, (because *AV* is given) reciprocally as . *Q. E. I.*

Cor. 1. Hence if the given point *S*, to which the centripetal force always tends, is placed in the circumference of the circle, as at *V*; the force will be reciprocally as the quadrato-cube (or fifth power) of the altitude *SP*.

Cor. 2. The force by which the body *P* in the circle *APTV* (Pl. 3. *Fig.* 4.) revolves about the centre of force *S* is to the force by which the same body *P* may revolve in the same circle and in the same periodic time about any other centre of force *R*, as to the cube of the right line *SG*, which from the first centre of force *S*, is drawn parallel to the distance *PR* of the body from the second centre of force *R*, meeting the tangent *PG* of the orbit in *G*. For by the construction of this proposition, the former force is to the latter as to ; that is, as to or, (because of the similar triangles *PSG*, *TPV*) to .

Cor. 3. The force by which the body *P* in any orbit revolves about the centre of force *S*, is to the force by which the same body may revolve in the same orbit, and in the same periodic time about any other centre of force *R*, as the solid , contained under the distance of the body from the first centre of force *S*, and the square of its distance from the second centre of force *R*, to the cube of the right line *SG*, drawn from the first centre of force *S*, parallel to the distance RP of the body from the second centre of force *R*, meeting the tangent *PG* of the orbit in *G*. For the force in this orbit at any point *P* is the same; as in a circle of the same curvature

Proposition VIII. Problem III.

*If a body moves in the semi-circumference* PQA; *it is proposed to find the law of the centripetal force tending to a point* S, *so remote, that all the lines* PS, RS *drawn thereto, may be taken for parallels*. Pl. 3. Fig. 5.

From *C* the centre of the semi-circle, let the semidiameter *CA* be drawn, cutting the parallels at right angles in *M* and *M* and join *CP*. Because of the similar triangles *CPM*, *PZT* and *RZQ* we shall have to as to ; and from the nature of the circle, is equal to the rectangle , or the points *P*, *Q* coinciding, to the rectangle . Therefore is to as to ; and and that is, (neglecting the given ration reciprocally as . *Q. E. I.*

And the same thing is likewise easly inferred from the preceding Proposition.

Scholium.

And by a like reasoning, a body will be moved in an ellipsis, or even in an hyperbole, or parabola, by a centriperal force which is reciprocally as the cube of the ordinate directed to an infinitely remote centre of force.

Proposition IX. Problem IV.

*If a body revolves in a final* PS*, cutting all the radii* SP*,* SQ*, &c. in a given angle: it is proposed to fund the law of the centripetal force tending to the centre of that sspiral.* Pl. 3. Fig. 6.

Suppose the indefinitely small angle *PSQ* to be given; because then all the angles are given, the figure *SPRQT* will be given in specie. Therefore the ratio is also given, and is as *QT* that is (because the figure is given in specie) as *SP*. But if the angle *PSQ* is any way changed, the right line *QR*, subtending the angle of contact *QPR*, (by lem. 11.) will be changed in the duplicate ratio of *PR* or *QT*. Therefore the ratio remains the same as before, that is a *SP*. And is as , and therefore (by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as the cube distance *SP*. *Q. E. I.*

The perpendicular *ST* let fall upon the tangent and the chord *PV* of the circle concentrically cutting the spiral are in given ratio's to the height *SP*; and therefore is as , that is (by corol. 3. and 5. prop. 6.) reciprocally as the centripetal force.

Lemma 12.

*All parallelograms circmscribed about any* conjugate diameters of a given ellipsis or hyperbola are equal among themselves.

This is demonstrated by the writers on the conic sections.

Proposition X. Problem V.

*If a body revolve: in an ellipsis: it is proposed to find the law of the centripetal force tending to the centre of the ellipsus*. Pl. 4. Fig. 1.

Suppose *CA*, *CB* to be semi-axes of the ellippse; *GP*, *DK* conjugate diameters; *PF*, *Qf* perpendiculars to those diameters; *Qv* an ordinate to the diameter *GP*; and if the parallelogram *QvPR* be compleated; then (by the properties of the conic sections) the rectangle *PvG* will be to as to to , and (because of the similar triangles *QvT*, *PCF) to as to ; and by composition,* the ration of PvG to is compounded of the ratio of to and of the ratio of to , that is, *vG* to as to . Put *QR* for *Pv*, and (by lem. 11.) *BC* x *CA* for *CD* x *PF* PM-C, also (the points *P* and *Q* coinciding,) 2*PC* for *vG*; and multiplying the extremes and means together, we shall have equal to . Therefore (by cor. 5. prop. 6.) the centripetal force us reciprocally as ; that is (because is given) reciprocally as ; that is, directly as the disŧance *PC*. *Q. E. I.*

*The same otherwise.*

In the right line PG on the other side of the point *T*; take the point *u* so that *Tu* may be eqal to *Tv*; then take *uV*, such as shall be to *vG* as to . And because is to *PvG* as to (by the conic sections) we shall have . Add the rectangle *uPv* to both sides, and the square of the chord of the arc *PQ* will be equal to the rectangle *VPv*; and therefore a circle, which touches the conic section in *P*, and passes thro' the point *Q* will pals also thro' the point *V*. Now let the points *P* and *Q* meet, and the ratio of *uV* to *vG*, which is the same with the ratio of to , will become the ratio of *PV* to *PG* or *PV* to 2*PC*; and therefore *PV* will be equal to . And therefore the force, by which the body *P* revolves in the ellipses, will be reciprocally as

(by cor. 3. prop. 6.) that is, (because is given) directly as *PC*. *Q. E. I.*

Cor. 1. And therefore the force is as the disŧance of the body from the centre of the ellipsis; and *vice versa* if the force is as the distance, the body will move in an ellipsis whose centre coincides with the centre of force, or perhaps in a circle into which the ellipsis may degenerate.

Cor. 2. And the periodic times of the revolutions made in all ellipses whatsoever about the same centre will be equal. For those times in similar ellipses will be equal (by corol. 3 and 8. prop. 4.) but in ellipses that have their greater axe common, they are one to another as the whole areas of the ellipses directly, and the parts of the areas described in the same time inversly; that is, as the lesser axes directly. and the velocities of the bodies in their principal vertices inversely; that is. as those lesser axes directly, and the ordinates to the same point of the common axis inversely; and therefore (because of the equality of the direct and inverse ratio's) in the ratio of equality.

Scholium.

If the ellipsis by having its centre removed to an infinite distance degenerates into a parabola, the body will move in this parabola; and the force, now tending to a centre infinitely remote, will become equable. which is *Ga1ileo'*s theorem. And if the parabolic section of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in like manner as in the circle, or in the ellipsis, if the forces are directed to the centre of the figure placed in the abscissa, those forces by increasing or diminishing the ordinates in any given ratio, or even by changing the angle of the inclination of the ordinates to the abscissa, are always augmented or diminished in the ratio of the distances from the centre; provided the periodic times remain equal; so also in all figures whatsoever, if the ordinates are augmented or diminished in any given ratio, or their inclination is any way changed, the periodic time remaining the same); the forces directed to any centre placed in the abscissa, are in the several ordinates augmented or diminished in the ratio of the distances from the centre.