# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 10

*Of the motion of bodies in given superficies, and of the reciprocal motion of funependulous bodies.*

Proposition XLVI. Problem XXXII.

*Any kind of centripetal force being supposed, and the centre of force, and any plane whatsoever in which the body revolves, being given, and the quadratures of curvilinear figures being allowed; it is required to determine the motion of a body going of from a given place, with a given velocity, in the direction of a given right line in that plane.*

Let *S* (Pl. 18. *Fig.* 4.) be the centre of force, *SC* the least distance of that centre from the given plane, *P* a body issuing from the place *P* in the direction of the right line *PZ*, *Q* the same body revolving in its trajectory, and *PQR* the trajectory it self which is required to be found, described in that given plane. join *CQ*, *QS*, and if in *QS* we take *SV* proportional to the centripetal force with which the body is attracted towards the centre *S*, and draw *VT* parallel to *CQ* and meeting *SC* in *T*; then will the force *SV* be resolved into two, (by cor. 2. of the laws of motion) the force *ST*, and the force *TV*; of which *ST* attracting the body in the direction of a line perpendicular to that plane, does not at all change its motion in that plane. But the action of the other force *TV* coinciding with the position of the plane it self, attracts the body directly towards the given point *C* in that plane; and therefore causes the body to move in this plane in the same manner as if the force ST were taken away, and the body were to revolve in free space about the centre *C* by means of the Force *TV* alone. But there being given the centripetal force *TV* with which the body Q revolves in free space about the given centre *C*, there is given (by pro. 42.) the trajectory *PQR* which the body describes; the place *Q* in which the body will be found at any given time; and lastly, the velocity of the body in that place *Q*. And so *è contra*. *Q. E. I.*

Proposition XLVII. Theorem XV.

*Supposing the centripetal force to be proportional to the distance of the body from the centre; all bodies revolving in any planes whatsoever will describe ellipses, and compleat their revolutions in equal times; and those which move in right lines, running backwards and forwards alternately, will compleat their several periods of going and returning in the same times.*

For letting all things sland as in the foregoing proposition, the force *SV* with which the body *Q* revolving in any plane *PQR* is attracted towards the centre *S*, is as the distance *SQ*; and therefore because *SV* and *SQ*, *TV* and *CQ* are proportional, the force *TV* with which the body is attracted towards the given point *C* in the plane of the orbit is as the distance *C*. Therefore the forces with which bodies found in the plane *PQR* are attracted towards the point *C*, are in proportion to the distances equal to the forces with which the same bodies are attracted every way towards the centre *S*; and therefore the bodies will move in the same times, and in the same figures in any plane *PQR* about the point *C*, as they would do in free spaces about the centre *S*; and therefore (by cor. 2. prop. 10. and cor. 2.

prop. 38) they will in equal times either describe ellipses in that plane about the centre *C*, or move to and fro in right lines passing through the centre *C* in that plane; compleating the same periods of time in all cases. *Q. E. D.*

Scholium.

The ascent and descent of bodies in curve superficies has a near relation to these motions we have been speaking of. Imagine curve lines to be described on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent; their motions will be performed in planes passing through the axis, and therefore in the curve lines by whose revolution those curve superficies were generated. In those cases therefore it will be sufficient to consider the motion in those curve lenes.

Proposition XLVIII. Theorem XVI.

*If a wheel sands upon the out-side of a globe at right angles thereto, and revolving about its own axis goes forward in a great circle; the length of the curvilinear path which any point, given in the perimeter of the wheel,* *hath desribed since the time that it touched the globe, (which curvilinear path we may call the cycloid or epicycloid) will be to double the versed sine of half the arc which since that time has touched the globe in passing over it, at the sum of the diameters of the globe and the wheel, to the semidiameter of the globe.*

Proposition XLIX. Theorem XVII.

*If a wheel stand upon the inside of a concave globe at right angles thereto, and revolving about its own axis go forward in one of the great circles of the globe, the length of the curvilinear path which any point, given in the perimeter of the wheel, hath described since it touched the globe, will be to the double of the versed sine of half the arc which in all that time has touched the globe in passing over it, as the difference of the diameter of the globe and the wheel, to the semidiameter of the globe.*

Let *ABL* (PL 19. *Fig.* 1.2.) be the globe, *C* its centre. *BPV* the wheel insisting thereon, *E* the centre of the wheel, *B* the point of contact, and *P* the given point in the perimeter of the wheel. Imagine this wheel to proceed in the great circle *ABL* from *A* through *B* towards *L*. and in its progress to revolve in such a manner that the arcs *AB*, *PB* may be always equal the one to the other, and the given point *P* in the perimeter of the wheel may describe in the mean time the curvilinear path *AP*. Let *AP* be the whole curvilinear path described since the wheel touched the globe in *A*, and the length of this path *AP* will be to twice the versed line of the arc *PB*, as 2 *CE* to *CB*. For let the right line *CE* (produced if need be) meet the wheel in *V*, and join *CP*, *BP*, *EP*, *VP*; produce *CP*, and let fall thereon the perpendicular *VF*. Let *PH*, *VH*, meeting in *H*, touch the circle in *P* and *K* and let *PH* cut *VF* in *G*, and to *VP* let fall the perpendiculars *GI*, *HK*. From the centre *C* with any interval let there be described the circle *nom*, cutting the right line *CP* in *n*, the perimeter of the wheel *BP* in *o*, and the curvilinear path *AP* in *m*; and from the centre *V* with the interval *Va* let there be described a circle cutting *VP* produced in *q*.

Because the wheel in its progress always revolves about the point of contact *B*, it is manifest that the right line *BP* is perpendicular to that curve line *AP* which the point *P* of the wheel describes, and therefore that the right line *VP* will touch this curve in the point *P*. Let the radius of the circle *nom* be gradually increased or diminished so that at last it become equal to the distance *CP*; and by reason of the similitude of the evanescent figure *Pnomq*, and the figure *PFGVI*, the ultimate ratio of the evanescent lineolæ *Pm*, *Pn*, *Po*, *Pq*, that is, the ratio of the momentary mutations of the curve *AP*, the right line *CP*, the circular arc *BP*, and the right line *VP*, will be the same as of the lines *PV*, *PF*, *PG*, *PL*, respectively. But since *VF* is perpendicular to *CF*, and *VH* to *CV*, and therefore the angles *HVG*, *VCF* equal; and the angle *VHG* (because the angles of the quadrilateral figure *HVEP* are right in *V* and *P*) is equal to the angle *CEP*, the triangles *VHG*, *CEP* will be similar; and thence it will come to pass that as *EP* is to *CE* so is *HG* to *HV* or *HP*, and so *KI* to *KP*, and by composition or division as *CB* to *CE* so is *PI* to *PK*, and doubling the consequents as *CB* to *CE* so is *PI* to *PV* and so is *Pq* to *Pm*. Therefore the decrement of the line *VP*, that is the increment of the line *BV - VP* to the increment of the curve line *AP* is in a given ratio of CB to 2 CE, and therefore (by cor. lem. 4.) the lengths *BV - VP* and *AP* generated by those increments, are in the same ratio. But if *BV* be radius, *VP* is the cosine of the angle *BVP* or *BEP*, and therefore BV - VP is the versed sine of the same angle; and therefore in this wheel whose radius is *BV*, *BV - VP* will be double the versed fine of the arc *BP*. Therefore *AP* is to double the versed fine of the arc *BP* as 2 CE to *CB*. *Q. E. D.*

The line *AP* in the former of these propositions we shall name the cycloid without the globe, the other in the latter proposition the cycloid within the globe, for distinction sake.

Cor. 1. Hence if there be described the entire cycloid *ASL* and the same be bisected in S, the length of the part *PS* will be to the length *PVT* (which is the double of the line of the angle *VBP*, when *EB* is radius) as 2 CE to CB, and therefore in a given ratio.

Cor. 2. And the length of the semi-perimeter of the cycloid *AS* will be equal to a right line which is to the diameter of the wheel *BV* as 2 *CE* to *CB*.

Proposition L. Problem XXXIII.

*To cause a pendulous body to oscillate in* a given cycloid.

Let there be given within the globe *QVS*, (Pl. 19. *Fig.* 3.) described with the centre *C*, the cycloid *QRS*, bisected in *R*, and meeting the superficies of the globe with its extreme points *Q* and *S* on either hand. Let there be drawn *CR* bisecting the arc *QS* in *O*, and let it be produced to *A* in such sort that *CA* may be to *CO* as *CO* to *CR*. About the centre *C*, with the interval *CA*, let there be described an exterior globe *DAF*, and within this globe; by a wheel whose diameter is *AO*, let there be described two semi-cycloids *AQ*, *AS*, touching the interior globe in *Q* and *S*, and meeting the exterior globe in *A*. From that point *A*, with a thread *APT* in length equal to the line *AR*, let the body *T* depend, and oscillate in such manner between the two semi-cycloids *AQ*, *AS* that as often as the pendulum parts from the perpendicular *AR*, the upper part of the thread *AP* may be applied to that semi-cycloid *APS* towards which the motion tends, and fold it self round that curve line, as if it were some solid obstacle; the remaining part of the same thread *PT* which has not yet touched the semi-cycloid continuing straight. Then will the weight *T* oscillate in the given cycloid *QRS*. *Q. E. F.*

For let the thread *P* meet the cycloid *QRS* in *T* and the circle *QOS* in *V* and let *CV* be drawn; and to the rectilinear part of the thread *PT* from the extreme points *P* and *T* let there be erected the perpendiculars *BP*. *TW*, meeting the right line *CV* in *B* and *W*. It is evident from the construction and generation of the similar figures *AS*, *SR*, that those perpendiculars *PB*, *TW*, cut off from *CV* the lengths *VB*, *VW* equal to the diameters of the wheels *OA*, *OR*. Therefore *TP* is to *VP* (which is double the fine of the angle *VBP* when *BV* is radius) as *BW* to *BV*, or *AQ* + *OP* to *AO*, that is (since *CA* and *CO*, *CO* and *CR*, and by division *AO* and *OR* are proportional) as *CA* + *CO* to *CA*; or, if *BV* be bisected in *E*, as 2 CE to CB. Therefore (by cor. 1. prop. 49) the length of the rectilinear part of the thread *PT* is always equal to the arc of the cycloid *PS*, and the whole thread *APT* is always equal to the half of the cycloid *APS*, that is (by cor. 2. prop. 49.) to the length *AR*. And therefore contrary-wise, if the string remain always equal to the length *AR* the point *T* will always move in given cycloid *QRS*. *Q. E. D.*

Cor. The string *AR* is equal to the semi-cycloid *AS*, and therefore has the same ratio to *AC* the semi-diameter of the exterior globe as the like semi-cycloid *SR* has to *CO* the semi-diameter of the interior globe.

Proposition LI. Theorem XVIII.

*If a centripetal force tending on all sides to the centre* C *of a globe (Pl. 19.* Fig. *4.) be in all places as the distance of the place from the centre, and by this force alone acting upon it, the body* T *oscillate (in the manner above descrid) in the perimeter of the cycloid* QRS*; I say that all the oscillations bow unequal soever in themseves will be performed in equal times.*

For upon the tangent *TW* infinitely produced let fall the perpendicular *CX* and join *CT*. Because the centripetal force with which the body *T* is impelled towards *C* is as the distance *CT*; let this (by cor. 2. of the laws) be resolved into the parts *CX*, *TX* of which *CX* impelling the body directly from *P* stretches the thread *PT* and by the resistance the thread makes to it is totally employed, producing no other effect; but the other part *TX*, impelling the body transversely or towards *X* directly accelerates the motion in the cycloid. Then it is plain that the acceleration of the body, proportional to this accelerating force, will be every moment as the length *TX*, that is, (because *CV*, *WV*, and *TX*, *TW* proportional to them are given) a the length *TW* that is (by cor. 1. prop. 49.) as the length of the arc of the cycloid *TR*. If therefore two pendulums *APT*; *Apt* be unequally drawn aside from the perpendicular *AR*, and let fall together, their accelerations will be always as the arcs to be described *TR*, *tR*. But the parts described at the beginning of the motion are as the accelerations, that is, as the wholes that are to be described at the beginning, and therefore the parts which remain to be described and the subsequent accelerations proportional to those parts, are also as the wholes, and so on. Therefore the accelerations, and consequently the velocities generated, and the parts described with those velocities, and the parts to be described, are always as the wholes; and therefore the parts to be described preserving a given ratio to each other will vanish together, that is, the two bodies oscillating will arrive together at the perpendicular *AR*. And since on the other hand the ascent of the pendulums from the lowest place *R* through the same cycloidal arcs with a retrograde motion, is retarded in the several places they pass through by the same forces by which their descent was accelerated, 'tis plain that the velocities of their ascent and descent through the same arcs are equal, and consequently performed in equal times; and therefore since the two parts of the cycloid *RS* and *RQ* lying on either side of the perpendicular are similar and equal, the two pendulum will perform as well the wholes as the halves of their oscillations in the same time. *Q. E. D.*

Cor. The force with which the body *T* is accelerated or retarded in any place *T* of thr cycloid, is to the whole weight of the same body in the highest place *S* or *Q*, as the arc of the cycloid *TK* is to the arc *SR* or *QR*.

*To define the velocities of the pendulums in the several places, and the times in which both the entire oscillation, and the several parts of them are performed.*

About any centre *G* (Pl. 20. Fig. 1.) with the interval *GH* equal to the arc of the cycloid *RS*, describe a semi-circle *HKM* bisected by the semi-diameter *GK*. And if a centripetal force proportional to the distance of the places from the centre tend to the centre *G*, and it be in the perimeter *HIK* equal to the centripetal force in the perimeter of the globe *QOS* tending towards its centre, and at the same time that the pendulum T is let fall from the highest place *S*, a body as *L* is let fall from *H* to *G*; then because the forces which act upon the bodies are equal at the beginning, and always proportional to the spaces to be described *TR*, *LG*, and therefore if *TR* and *LG* are equal, are also equal in the places *T* and *L*, it is plain that those bodies describe at the beginning equal spaces *ST*, *HL*, and therefore are still acted upon equally, and continue 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* arrives at *L*, to the semi-periphery *HKM*, the time in which the body *H* will come to *M*. And the velocity of the pendulous body in the place *T* is to its velocity in the lowest place *R*, that is, the velocity of the body *H* in the plane *L* to its velocity in the place *G*, or the momentary increment of the line *HL* to the momentary increment of the line *HG*. (the arcs *HI*, *HK* increasing with an equable flux) as the ordinate *LI* to the radius *GK* or as, to *SR*. Hence since in unequal oscillations there are described in equal times arcs proportional to the entire arcs of the oscillations; there are obtained from the times given, both the velocities and the arcs described in all the oscillations universally. Which was first required.

Let now any pendulous bodies oscillate in different cycloids described within different globes, whose absolute forces are also different; and if the absolute force of any globe *QOS* be called *V*, the accelerative force with which the pendulum is acted on in the circumference of this globe, when it begins to move directly towards its centre, will be as the distance of the pendulous body from that centre and the absolute force of the globe conjunctly, that is, as *CO* x V. Therefore the lineola *HT* which is as this accelerative force *CO* x V will be described in a given time; and if there be erected the perpendicular *TZ* meeting the circumference in *Z*, the nascent arc *HZ* will denote that given time. But that nafcent arc *HZ* is in the subduplicate ratio of the rectangle *GHT*, and therefore as . Whence the time of an entire oscillation in the cycloid *QRS* (it being as the semi-periphery *HKM* which denotes that entire oscillation, directly; and as the arc *HZ* which in like manner denotes a given time inversely) will be as *GH* directly and, inversely, that is, because *GH* and *SR* are equal, as, , or (by cor. prop. 50.) as . Therefore the oscillations in all globes and cycloids, performed with what absolute forces soever, are in a ratio compounded of the subduplicate ratio of the length of the string directly, and the subduplicate ratio of the distance between the point of fufpension and the centre of the globe inversely, and the subduplicate ratio of the absolute force of the globe inversely also. *Q. E. I.*

Cor. 1. Hence also the times of oscillating, falling, and revolving bodies may be compared among themselves. For if the diameter of the wheel with which the cycloid is described within the globe is supposed equal to the semi-diameter of the globe, the cycloid will become a right line passing through the centre of the globe, and the oscillation will be changed into a descent and subsequent ascent in that right line. Whence there is given both the time of the descent from any place to the centre, and the time equal to it in which the body revolving uniformly about the centre of the globe at any distance describes an arc of a quadrant. For this time (by case 2.) is to the time of half the oscillation in any cycloid *QRS* as 1 to .

Cor. 2. Hence also follow what Sir *Christopher* *Wren* and M. *Huygens* have difcovered concerning the vulgar cycloid. For if the diameter of the globe be infinitely increased, its sphærical superficies will be changed into a plane, and the centripetal force will act uniformly in the direction of lines perpendicular to that plane, and this cycloid of ours will become the same with the common cycloid. But in that case the length of the arc of the cycloid between that plane and the describing point, will become equal to four times the versed sine of half the arc of the wheel between the same plane and the describing point as was discovered by Sir *Christopher Wren*. And a pendulum between two such cycloids will oscillate in a similar and equal cycloid in equal times as M. *Huygens* demonstrated. The descent of heavy bodies also in the time of one oscillation will be the same as M. *Huygens* exhibited.

The propositions here demonstrated are adapted to the true constitution of the Earth, in so far as wheels moving in an of its great circles will describe by the motions of nails fixed in their perimeters, cycloids without the globe; and pendulums in mines and deep caverns of the Earth must oscillate in cycloids without the globe, that those oscillations may be performed in equal times. For gravity (as will be shewn in the third book) decreafes in its progress from the superficies of the Earth; upwards in a duplicate ratio of the distances from the centre of the earth, downwards in a simple ratio of the same.

*Granting the quadratures of curvilinear figures, it is required to found the forces with which bodies moving in given curve lines may always perform their oscillations in equal times.*

Let the body *T* (Pl. 20. Fig. 2.) oscillate in any curve line *STRQ*, whose axis is *AR* passing through the centre of force *C*. Draw *TX* touching that curve in any place of the body *T*, and in that tangent *TX* take *TY* equal to the arc *TR*. The length of that arc is known from the common methods used for the quadratures of figures. From the point *Y* draw the right line *YZ* perpendicular to the tangent. Draw *CT* meeting that perpendicular in *Z*, and the centripetal force will be proportional to the right line TZ. *Q. E. I.*

For if the force with which the bod is attracted from *T* towards *C* be expressed by the right line *TZ* taken proportional to it, that force will be resolved into two forces *TY*, *YZ*, of which *YZ* drawing the body in the direction of the length of the thread *PT* does not at all change its motion; whereas the other force *TY* directly accelerates or retards its motion in the curve *STRQ*. Wherefore since that force is as the space to be described *TR*, the accelerations or retardations of the body in describing two proportional parts (a greater and a less) of two oscillations, will be always as those parts, and therefore will cause those parts to be described together. But bodies which continually describe together parts proportional to the wholes, will describe the wholes together also. *Q. E. D.*

Cor. 1. Hence if the body *T* (Pl. 20. Fig. 3.) hanging by a rectilinear thread *AT* from the centre *A*, describe the circular arc *STRQ*, and in the mean time be acted on by any force tending downwards with parallel directions, which is to the uniform force of gravity as the arc *TR* to its sine *TN*, the times of the several oscillations will be equal. For because *TZ*, *AR* are parallel. the triangle *ATN*, *ZTY* are similar; and therefore *TZ* will be to *AT* as *TY* to *TN*; that is, if the uniform force of gravity be expressed by the given length *AT* the force *TZ* by which the oscillations become isochronous, will be to the force of gravity *AT*; as the arc *TR* equal to *TY* is to *TN* the sine of that arc.

Cor. 2. And therefore in clocks, if forces were impressed by some machine upon the pendulum which preserves the motion, and so compounded with the force of gravity, that the whole force tending downwards should be always as a line produced by applying the rectangle under the arc *TR* and the radius *AR* to the line *TN*, all the oscillations will become isochronous.

*Granting the quadratures of curvilinear figures, it is required to find the times, in which bodies by means of any centripetal force will descend or ascend in any curve lines described in a plane passing through the centre of force.*

Let the body descend from any place *S* (Pl. 20. Fig. 4.) and move in any curve *STtR*, given in a plane passing through the centre of force *C*. Join *CS*, and let it be divided into innumerable equal parts, and let *Dd* be one of those parts. From the centre *C*, with the intervals *CD*, Cd, let the circles *DT*, *dt* be described, meeting the curve line *STtR* in *T* and *t*. And because the law of centripetal force is given, and also the altitude *CS* from which the body at firsts fell; there will be given the velocity of the body in any other altitude *CT* (by prop. 39.) But the time in which the body describes the lineola *Tt* is as the length of that lineola, that is. as the fecant of the angle *tTC* directly, and the velocity inversely. Let the ordinate *DN*, proportional to this time, be made perpendicula to the right line *CS* at the point *D*, an because *Dd* is given, the rectangle *Dd* x *DN* that is, the area *DNnd*, will be proportional to the same time. Therefore if *PNn* be a curve line in which the point *N* is perpetually found, and its asymptote be the right line *SQ* slanding upon the line *CS* at right angles, the area *SQPND* will be proportional to the time in which the body in its descent hath described the line *ST*; and therefore that area being found the time is also given. *Q. E. I.*

*If a body move in any curve superficies whose axis passes trough the centre of force, and from the body a perpendicular be let fall upon the axis; and a line parallel and equal thereto be drawn from any given point of the axis; I say, that this parallel line will describe an area proportional to the time.*

Let *BKL* (Pl. 20. Fig. 5.) be a curve superficies, *T* a body revolving in it, *STR* a trajectory which the body describes in the same, *S* the beginning of the trajectory, *OMK* the axis of the curve superficies, *TN* a right line let fall perpendicularly from the body to the axis; *OP* a line parallel and equal thereto drawn from the given point *O* in the axis; *AP* the orthographic projection of the trajectory described by the point *P* in the plane *AOP* in which the revolving line *OP* is found; A the beginning of that projection answering to the point *S*; *TC* a right line drawn from the body to the centre; *TG* a part thereof proportional to the centripetal force with which the body tends towards the centre *C*; *TM* a right line perpendicular to the curve superficies; *TI* a part thereof proportional to the force of pressure with which the body urges the superficies, and therefore with which it is again repelled by the superficies towards *M*; *PTF* a right line parallel to the axis and passing through the body, and *GF*, *IH* right lines let fall perpendicularly from the points *G* and *I* upon that 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 cor. 2. of the laws of motion) is resolved into the forces *TF*, *FG*; and the force *TI* into the forces *TH*, *HI*; but the forces *TF*, *TH* acting in the direction of the line *PF* perpendicular to the plane *AOP*, introduce no change in the motion of the body but in a direction perpendicular to that plane. Therefore its motion so far as it has the same direction with the position of the plane, that is, the motion of the point *P*, by which the projection *AP* of the trajectory is described in that plane, is the same as if the forces *TF*, *TH* were taken away, and the body were acted on by the forces *FG*, *HI* alone; that is, the same as if the body were to describe in the plane *AOP* the curve *AP* by means of a centripetal force tending to the centre *O*, and equal to the sum of the forces *FG* and *HI*. But with such a force as that (by prop. 1.) the area *AOP* will be described proportional to the time. *Q. E. D.*

Cor. By the same reasoning if a body, acted on by forces tending to two or more centres in any the same right line *CO*, should describe in a free space any curve line *ST*; the area *AOP* would always proportional to the time.

*Granting the quadratures of curvilinear figures and supposing that there are given both the law of centripetal force tending to a given centre, and the curve supesficies whos axis passes through that centre; it if required to find the trajectory which a body will describe in that superificies, when going of from a given place with a given velocity, and in a given direction in that superficies.*

The last construction remaining, let the body *T* go from the given place *S* (Pl. 20. *Fig.* 6.) in the direction of a line given by position, and turn into the trajectory sought *STR* whose orthographic projection in the plane *BLO* is *AP*. And from the given velocity of the body in the altitude *SC*, its velocity in any other altitude *TC* will be also given. With that velocity in a given moment of time let the body describe the particle *Tt* of its trajectory, and let *Pp* be the projection of that particle described in the plane *AOP*. Join *Op*, and a little circle being described upon the curve superficies about the centre *T* with the interval *Tt*, let the projection of that little circle in the plane *AOP* be the ellipsis *pQ* And because the magnitude of that little circle *Tt*, and *TN* or *PO* its distance from the axis *CO* is also given, the ellipsis *pQ* will be given both in kind and magnitude, as also its position to the right 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 thence will be given *p* the common intersection of the ellipsis and the right line *Op*, together with the angle *OPp* in which the projection *APp* of the trajectory cuts the line *OP*. But from thence (by conferring prop. 41. with its 2d cor.) the manner of determining the curve *APp* easily appears. Then from the several points *P* of that projection erecting to the plane *AOP* the perpendicular *PT* meeting the curve superficies in *T*, there will be given the several points *T* of the trajectory. *Q. E. I.*