# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 7

Section VII.

*Concerning the rectilinear ascent and descent of bodies.*

Proposition XXXII. Problem XXIV.

*Supposing that the centripetal force is reciprocally proportional to the square* of the distance of the places from the centre; it is required to define the spaces which a body, falling directly, describes in given times.

Case 1. If the body does not fall perpendicularly it will (by cor. 1. prop. 13.) describe some conic section whose focus is placed in the centre of force. Suppose that conic section to be *ARPB* (*Pl.* 15. *Fig.* 1.) and its focus S. And first, if the figure be an ellipsis; upon the greater axe thereof *AB* describe the semi-circle *ADB*, and let the right line *DPC* pass through the falling body,

making right angles with the axis; and drawing *DS*, *PS*, the area *ASD* will be proportional to the area *ASP*, and therefore also to the time. The axis *AB* still remaining the same, let the breadth of the ellipsis be perpetually diminished, and the area *ASD* will always remain proportional the time. Suppose that breadth to be diminished *in infinitum*; and the orbit *APB* in that case coinciding with the axis *AB*, and the focus *S* with the extreme point of the axis *B*, the body will descend in the right line *AC*, and the area *ABD* will become proportional to the time. Wherefore the space *AC* will be given which the body describes in a given time by its perpendicular fall from the place *A*, if the area *ABD* is taken proportional to the time, and from the point *D*, the right line *DC* is let fall perpendicularly on the right line *AB*. *Q. E. I.*

Case 2. If the figure *RPB* is an hyperbola, (*Fig.* 2.) on the same principal diameter AB describe the rectangular hyperbola *BED*; and because the areas *CSP*, *CBfP*, *SPfB*, are severally to the several areas *CSD*, *CBED*, *SDEB* in the given ratio of the heights *CP*, *CD*; and the area *SPfB* is proportional to the time in which the body *P* will move through the arc *PfB*, the area *SDEB* will be also proportional to that time. Let the latus rectum of the hyperbola *RPB* be diminished in infinitum, the latus transversum remaining the seme; and the arc *PB* will come to coincide with the right line *CB*, and the focus *S* with the vertex *B*, and the right line *SD* with the right line *BD*. And there fire the area *BDEB* will be proportional to the time in which the body *C*, by its perpendicular descent, describes the line *CB*. *Q. E. I.*

Case 3. And by the like argument if the figure *RPB* is a parabola, (Fig. 3.) and to the same principal vertex *B* another parabola *BED* is described, that may always remain given while the former parabola in whose perimeter the body *P* moves, by having its latus rectum diminished and reduced to nothing, comes to coincide with the line *CB*; the segment *BDEB* will be proportional to the time in which that body *P* or *C* will descend to the centre *S* or *B*. *Q. E. I.*

Proposition XXXIII. Theorem IX.

*The things above found being supposed, I say, that the velocity of a body in any place* C *is to the veolocity of a body, describing a circle about the centre* B *at the distance* BC*, in the subduplicate ratio of* AC , *the distance of the body from the remoter vertex* A *of the circle or rectangular hyperbola, to , the principal semi-diameter of the figure.* Pl. 15. Fig. 4.

Let AB the common diameter of both figures *RPB*, *DEB* be bisected in *O*; and draw the right line *PT* that may touch the figure *RPB* in P, and likewise cut that common diameter *AB* (produced, if need be) in *T*; and let *SY* be perpendicular to this line. and *BQ* to this diameter, and suppose the latus rectum of the figure *RPB* to be *L*. From cor. 9. prop. 16. it is manifest that the velocity of a body, moving in the line *RPB* about the centre *S*, in any place *P*, is to the velocity of a body describing a circle about the same centre, at the distance *SP*, in the subduplicate ratio of the rectangle to .. For by the properties of the conic sections *ACB* is to as 2*AO* to *L* and therefore is equal to *L*. Therefore those velocities are to each other in the subduplicate to . Moreover by the properties of the conic sections, *CO* is to *BO* as *BO* to *TO*, and (by composition or division) as *CB* to *BT*. Whence (by division or composition) as *CB* to *BT*. Whence (by division or composition) *BO* - or + *CO* will be to *BO* as *CT* to *BT*, that is *AC* will be to *AO* as *CP* to *BQ*; and therefore is equal to . Now suppose *CP*, the breadth of the figure *RPB*, to be diminished *in infinitum*, so as the point *P* may come to coincide with the point *C*, and the point *S* with the point *B*, and the line *SP* with the line *BC*, and the line *ST* with the line *BQ*; and the velocity of the body now descending perpendicularly in the line *CB* will be to the velocity of a body describing a circle about the centre *B* at the distance *BC*, in the subduplicate ratio of to , that is (neglecting the ratio's of equality of *SP* to *BC*, and to ) in the subduplicate ratio of *AC* to *AO* or . *Q. E. D.*

Cor. 1. When the points *B* and *S* come to coincide, *TC* will become to *TS*, as *AC* to *AO*.

Cor. 2. A body revolving in any circle at a given distance from the centre, by its motion converted upwards will ascend to double its distance from the centre.

Proposition XXXUV. Theorem X.

*If the figure* BED *is a parabola, I say that the velocity of a falling body in any place* C *is equal to the velocity by which a body may uniformly describe a circle about the centre* B *at half the interval* BC. Pl. 15. fig. 5.

For (by cor. 7. prop. 16.) the velocity of a body describing a parabola *RPB* about the centre *S*, in any place *P*, is equal to the velocity of a body uniformly describing a circle about the same centre *S* at half the interval *SP*. Let the breadth *CP* of the parabola be diminished in infinitum, so as the parabolic arc *PfB* may come to coincide with the right line *CB*, the centre *S* with the vertex *B*, and the interval *SP* with the interval *BC*, and the proposition will be manifest. *Q. E. D.*

Proposition XXXV. Theorem XI.

*The same things supposed, I say that the area of the figure* DES*, described by the indefinite radius* SD, *is equal to the area which a body with a radius equal to half the latus rectum of the figure* DES, *by uniformly revolving about the centre* S, *may be described in the same time.*. Pl. 16. Fig. 1.

For suppose a body *C* in the smallest moment of time describes in falling the infinitely little line *Cc*, while another body *K* uniformly revolving about the centre *S* in the circle *OKk*, describes the arc *Kk*. Erect the perpendiculars *CD*, *cd*, meeting the figure *DES* in *D*. Join *SD*, *Sd*, *SK*, *Sk*, and draw *Dd* meeting the axis *AS* in *T*, and thereon let fall the perpendicular *SY*.

Case 1. If the figure *DES* is a circle or a rectangular hyperbola, bisect its transverse diameter *AS* in *O*, and *SO* will be half the latus rectum. And because *TC* is to *TD* as *Cc* to *Dd*, and *TD* to *TS* as *CD* to *ST*; *ex æquo* *TC* will be to *TS*, as *CD* x *Cc* to *ST* x *Dd*. But (by cor. 1. prop. 33) *TC* is to *TS* as *AC* to *AO*, to wit, if in the coalescence of the points *D*, *d*, the ultimate ratio's of the lines are taken. Wherefore *AC* is to *AO* or *SK* as *CD* x *Cc* to *ST* x *Dd*. Farther, the velocity of the descending body in *C* is to the velocity of a body describing a circle about the centre *S*, at the interval *SC*, in the subduplicate ratio of *AC* to *AO* or *SK* (by prop. 33.) and this velocity is to the velocity of a body describing the circle *OKk* in the subduplicate ratio of *SK* to *SC* (by cor. 6. prop. 4.) and *ex* *æquo*, the first velocity to the last, that is the little line *Cc* to the arc *Kk*, in the subduplicate ratio of *AC* to *SC*, that is in the ratio of *AC* to *CD*. Wherefore *CD* x *Cc* is equal to *AC* x *Kk*, and consequently *AC* to *SK* as *AC* x *Kk* to *ST* x *Dd*, and thence *SK* x *KL* equal to *ST* x *Dd*. and equal to , that is, the area *KSk* equal to the area *SDd*. Therefore in every moment of time two equal particles, *KSk* and *SDd*, of areas are generated which, if their magnitude is diminished and their number increased in infinitum, obtain the ratio of equality, and consenquently (by cor. lem. 4.) the whole areas together generated are always equal. *Q. E. D.*

Case 2. But if the figure *DES* (Fig. 2.) is a parabola, we shall find as above *CD* x *Cc* to *ST* x *Dd* as *TC* to *TS*, that is, as 2 to 1; and that therefore is equal to . But the velocity of the falling body in *C* is equal to the velocity with which a circle may be uniformly described at the interval , (by prop. 34.) And this velocity to the velocity with which a circle may be described with the radius SK, that is, the little line *Cc* to the arc *Kk* is (by cor. 5. prop. 4.) in the subduplicate ratio of *SK* to ; that is, in the ratio of *SK* to . Wherefore is equal to , and therefore equal to ; that is, the area *KSk* is equal to the area *SDd* as above. *Q. E. D.*

Proposition XXXVI. Problem XXV.

*To determine the times of the descent of a body falling from a given place* A. Pl. 16. Fig. 3.

Upon the diameter AS, the distance of the body from the centre at the beginning. describe the semi-circle *ADS*, as likewise the semi-circle *OKH* equal thereto, about the centre *S*. From any place *C* of the body, erect the ordinate *CD*, join *SD*, and make the sector *OSI* equal to the area *ASD*. It is evident by prop. 35. that the body in falling will describe the space *AC* in the same time in which another body, uniformly revolving about the centre *S*, may describe the arc *OK*. *Q. E. F.*

Proposition XXXVII. Problem XXVI.

*To describe the times of the ascent or descent* of a body projected upwards or downwards from a given place. *Pl. 16. Fig. 4.*

Suppose the body to go off from the given place *G*, in the direction of the line *GS*, with any velocity. In the duplicate ratio of this velocity to the uniform velocity in a circle, with which the body may revolve about the centre centre *S* at the given interval *SG*, take *GA* to . If that ratio is the same as of the number 2 to 1 the point *A* is infinitely remote; in which case a parabola is to be described with any latus rectum to the vertex *S*, and axis *SG*; as appears by prop. 34. But if that ratio is less or greater than ratio of 2 to 1, in the former case a circle, in the latter a rectangular hyperbola, is to be described on the diameter *SA*; as appears by prop. 33. Then about the centre *S*. with an interval equal to half the latus rectum, describe the circle *HkK*, and at the place *G* of the ascending or descending body, and at any other place *C*, erect the perpendiculars *GI*, *CD*; meeting the conic section or circle in *I* and *D*. Then joining *SI,* *SD*, let the sectors *HSK*, *HSk* be made equal to the segments *SEIS*, *SEDS*, and by prop. 35. the body *G* will describe the space *GC* in the same time in which the body *K* may describe the arc *Kk*. *Q. E. F.*

Proposition XXXVIII. Theorem XII.

*Supposing that the centripetal force is proportional to the altitude or distance of places from the centre, I say, that the times and velocities of falling bodies, and the spaces which they describe, are respectively proportional to the arcs, and the right and versed sines of the arcs.* Pl. 17. Fig. 1.

Suppose the body to fall from any place *A* in the right line *AS*; and about the centre of force *S*

with the interval *AS*, describe the quadrant of a circle *AE*; and let *CD* be the right line of any arc *AD*; and the body *A* will in the time *AD* in falling describe the space *AC*, and in the place *C* will acquire the velocity *CD*.

This is demonstrated the same way from prop. 10. as prop. 32. was demonstrated from prop. 11.

cor. 1. Hence the times are equal in which one body falling from the place *A* arrives at the centre *S*, and another body revolving describes the quadrantal arc *ADE*.

cor. 2. Wherefore all the times are equal in which bodies falling from whatsoever places arrive at the centre. For all the periodic times of revolving bodies are equal, by cor. 3. prop. 4.

Proposition XXXIX. Problem XXVII.

Supossing a centripetal force of any kind, and granting the quadratures of curvilinear figures; it it required to find the velocity of a body, ascending or descending in a right line, in the several places through which it passes; as also the time in which it will arrive at any place; And vice versa.

Suppose the body E (Pl. 17. Fig. 1.) to fall from any place *A* in the right line *ADEC*; and from its place *E* imagine a perpendicular *EG* always erected, proportional to the centripetal force in that place tending to the centre *C*; and let *BFG* be a curve line, the locus of the point *G*. And in the beginning of the motion suppose *EG* to coincide with the perpendicular *AB*; and the velocity of the body in any place *E* will be as a right line whose power is the curvilinear area . *Q. E. I.*

In *EG* take *EM* reciprocally proportional to a right line whose power is the area , and let *VLM* be a curve line wherein the point *M* is always placed, and to which the right line *AB* produced is an asymptote, and the time in which the body is falling describes the line *AE*, will be as the curvilinear area *ABTVME*. *Q. E. I.*

For in the right line *AE* let there be taken the very small line *DE* of a given length, and let *DLF* be the place of the line *EMG*, when the body was in *D*; and if the centripetal force be such, that a right line whose power is the area , is as the velocity of the descending body, the area it self will be as the square of that velocity; that is, if for the velocities in *D* and *E* we write *V* and *V* + *I*, the area *ABFD* will be as VV, and the area as VV + 2VI + II; and by division the area *DFGE* as 2*VI* + *II* and therefore will be as , that is, if we take the first ratio's of those quantities when just nascent, the length *DF* is as the quantity and therefore also as half that quantity . But the time, in which the body in falling describes the ver small line *DE* is as that line directly and the velocity *V* inversely, and the force will be as the increment *I* of the velocity directly and the time inversely, and therefore if we take the first ratio's when those quantities are just nascent as , that is as the length *DF*. Therefore a force proportional to *DF* or *EG* will cause the body to descend with a velocity that is as the right line whose power is the area *ABGE*. *Q. E. D.*

Moreover since the time, in which a very small line *DE* of a given length may be described, is as the velocity inversely, and therefore also inversely as a right line whose square is equal to the area *ABFD*; and since the line *DL*, and by consequence the nascent area *DLME*, will be as the same right line inversely: the time will be as the area *DLME*, and the sum of all the times will be as the sum of all the area's; that is (by cor. lem. 4.) the whole time in which the line *AE* is describe be will be as the whole area *ATVME*. *Q. E. D,*

Cor. 1. Let *P* be the place from whence a body ought to fall, so as that when urged by any known uniform centripetal force (such as gravity is vulgarly supposed to be) it may acquire in the place *D* a velocity, equal to the velocity which another body, falling by any force whatever, hath acquired in that place *D*. In the perpendicular *DF* let there be taken *DR*, which may be to *DF* as that uniform force to the other force in the place *D*. Compleat the rectangle *PDRQ*, and cut off the area *ABFD* equal to that rectangle. Then *A* will be the place from whence the other body fell. For compleating the rectangle *DRSE*, since the area *AbFD* is to the area *DFGE* as *VV* to 2*VI*, and therefore as , that is, as half the whose velocity to the increment of the velocity of the body falling by the unequable force; and in like manner the area *PQRD* to the area *DRSE*, as half the whole velocity to the increment of the velocity of the body falling by the uniform force; and since those increments (by reason of the equality of the nascent times) are as the generating forces, that is, as the ordinates *DF*, *DR*, and consequently as the nascent area's *DFGE*, *DRSE*; therefore *ex æquo* the whole areas *ABFD*, *PQRD* will be to one another as the halves of the whole velocities, and therefore, because the velocities are equal, they become equal also.

Cor. 2. Whence if any body be projected either upwards or downwards with a given velocity from any place *D*, and there be given the law of centripetal force acting on it, its velocity will be found in any other place as *e*, by erecting the ordinate *eg*, and taking that velocity to the velocity in the place *D*, as a right line whose power is the rectangle *PQRD*, either increased by the curvilinear area *Dfge*, if the place *e* is below the place *D*, or diminished by the same area *DFge* if it be higher, is to the right line whole power is the rectangle *PQRD* alone.

Cor. 3. The time is also known by erecting the ordinate *em* reciprocally proportional to the square root of *PQRD* + or - *DFge*, and taking the time in which the body has described the line *De*, to the time in which another body has fallen with an uniform force from *P*, and in falling arrived at *D*, in the proportion of the curvilinear area *DLme* to the rectangle 2*PD* x *DL*. For the time in which a body falling with an uniform force hath described the line *PD*, is to the time in which the same body has described the line *PE*, in the subduplicate ratio of *PD* to *PE*; that is (the very small line *DE* being just nascent) in the ratio of PD to to 2*PD* + *DE*, and by division to the time in which the body hath described the small line *DE*, as 2*PD* to *DE*, and therefore as the rectangle 2*PD* + *DL* to the area *DLME*; and the time in which both the bodies described the very small line *DE* is to the time in which the body moving unequably hath described the line *De*, as thearea *DLME* to the area *DLme*; and *ex æquo* the first mentioned of these times is to the last as the rectangle 2*PD* x 'DL *to the* area *DLme*.