The Mathematical Principles of Natural Philosophy

Definitions

THE Mathematical PRINCIPLES OF Natural Philosophy.

Definitions

Def. I

1The Quantity of Matter is the measure of the same, arising from its density and bulk conjunctly.

THUS AIR of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes 2whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of Body or Mass. And the same is known by the weight of each body: For it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

Definition II

The Quantity of Motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the Sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

Definition III

The Vis Insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.

This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the Mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without 3difficulty put out of its state of rest or motion. Upon which account, this Vis insita, may, by a most significant name, be called Vis Inertiæ, or Force of Inactivity. But a body exerts this force only, when another force, impress'd upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse: It is resistance in so far as the body, for maintaining its present state withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impress'd force of another, endeavours to change the state of that other. Resistance is usually ascrib'd to bodies at rest, and impulse to those in motion: But motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.

Definition IV

An impress'd force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its Vis Inertiæ only. Impress'd forces are of different origines; as from percussion, from pressure, from centripetal force.

Definition V

4A Centripetal force is that by which bodies are drawn or impell'd, or any way tend, towards a point as to a centre.

Of this sort is Gravity by which bodies tend to the centre of the Earth; Magnetism, by which iron tends to the loadstone; and that force, whatever it is, by which the Planets are perpetually drawn aside from the rectilinear motions, which otherwise they wou'd pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolv'd with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes it self to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because 'tis directed to the hand as the centre of the orbit, I call the Centripetal force. And the same thing is to be understood of all bodies, revolv'd in any orbits. They all endeavour to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call Centripetal, would fly off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the Earth, but would go off from it in a right line and that with an uniform motion, if the resistance of the air was taken away. 'Tis by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the Earth, more or less, according 5to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball projected from the top of a mountain by the force of gun-powder, with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground; the same, if the resistance of the Air was took away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole Earth before it falls; or lastly, so that it might never fall to the Earth, but go forwards into the Celestial Spaces, and proceed in its motion in infinitum. And after the same manner, that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole Earth, the Moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impells it towards the Earth, may be perpetually drawn aside towards the Earth, out of the rectilinear way, which by its innate force it would pursue; and be made to revolve in the orbit which it now describes: nor could the Moon without some such force, be retain'd in its orbit. If this force was too smal', it would not sufficiently turn the Moon out of a rectilinear course: if it was too great, it would turn it too much, and draw down the Moon from its orbit towards the Earth. It is necessary, that the force be of a just quantity, and it belongs to the Mathematicians to find the force, 6that may serve exactly to retain a body in a given orbit with a given velocity; and vice versa, to determine the curvilinear way into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.

The quantity of any Centripetal Force may be considered as of three kinds, Absolute, Accelerative, and Motive.

Definition VI

The absolute quantity of a centripetal force is the measure of the same, proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.

Thus the magnetic force is greater in one load-stone and less in another, according to their sizes and strength.

Definition VII

The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.

Thus the force of the same load-stone is greater at a less distance, and less at a greater: also the force of gravity is greater in valleys, less on tops of exceeding high mountains; and yet less (as shall hereafter be shewn) at greater distances from the body of the 7Earth; but at equal distances, it is the same every where; because (taking away, or allowing for, the resistance of the Air), it equally accelerates all falling bodies, whether heavy or light, great or small.

Definition VIII

The motive quantity of a centripetal force, is the measure of the same, proportional to the motion which it generates in a given time.

Thus the Weight is greater in a greater body, less in a less body; it is greater near to the Earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or as I may say, its Weight; and it is ever known by the quantity of a force equal and contrary to it, that is just sufficient to hinder the descent of the body.

These quantities of Forces, we may for brevity's sake, call by the names of Motive, Accelerative and Absolute forces; and for distinction sake consider them, with respect to the Bodies that tend to the centre; to the Places of those bodies; and to the Centre of force towards which they tend: That is to say, I refer the Motive force to the Body, as an endeavour and propensity of the whole towards a centre, arising from the propensities of the several parts taken together; the Accelerative force to the Place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them; and the Absolute force to the Centre, as indued with some cause, without which those motive forces would not be propagated through the spaces round about; whether that cause is some central body, (such as is the Load-stone, in 8the centre of the force of Magnetism, or the Earth in the centre of the gravitating force) or any thing else that does not yet appear. For I here design only to give a Mathematical notion of those forces, without considering their Physical causes and seats.

Wherefore the Accelerative force will stand in the same relation to the Motive, as celerity does to motion. For the quantity of motion arises from the celerity, drawn into the quantity of matter; and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the Accelerative force, upon the several particles of the body, is the Motive force of the whole. Hence it is, that near the surface of the Earth, where the accelerative gravity, or force productive of gravity in all bodies is the same, the motive gravity or the Weight is as the Body: but if we should ascend to higher regions, where the accelerative gravity is less, the Weight would be likewise diminished, and would always be as the product of the Body, by the Accelerative gravity. So in those regions, where the accelerative gravity is diminished into one half, the Weight of a body two or three times less, will be four or six times less.

I likewise call Attractions and Impulses, in the same sense, Accelerative, and Motive; and use the words Attraction, Impulse or Propensity of any sort towards a centre, promiscuously, and indifferently, one for another; considering those forces not Physically but Mathematically: Wherefore, the reader is not to imagine, that by those words, I any where take upon me to define the kind, or the manner of any Action, the causes or the physical reason thereof, or that I attribute Forces, in a true and Physical sense, to certain centres (which are only Mathematical points); when 9at any time I happen to speak of centres as attracting, or as endued with attractive powers.

Scholium

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define Time, Space, Place and Motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into Absolute and Relative, True and Apparent, Mathematical and Common.

I. Absolute, True, and Mathematical Time, of it self, and from its own nature, flows equably without regard to any thing external, and by another name is called Duration: Relative, Apparent, and Common Time is some sensible and external (whether accurate or unequable) measure of Duration by the means of motion, which is commonly used instead of True time; such as an Hour, a Day, a Month, a Year.

II Absolute Space, in its own nature, without regard to any thing external, remains always similar and immoveable. Relative Space is some moveable dimension or measure of the absolute spaces; which our senses determine, by its position to bodies; and which is vulgarly taken for immoveable space; Such is the dimension of a subterraneous, an aereal, or celestial space, determin'd by its position in respect of the Earth. Absolute and Relative space, are the same in figure and magnitude; but they do not remain always numerically the same. For if the Earth, for instance, moves; 10a space of our Air, which relatively and in respect of the Earth, remains always the same, will at one time be one part of the absolute space into which the Air passes; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable.

III. Place is a part of space which a body takes up, and, is according to the space, either absolute or relative. I say, a Part of Space; not the situation, nor the external surface of the body. For the places of equal Solids, are always equal; but their superficies, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts, that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and therefore the Place of the whole, is the same thing with the Sum of the places of the parts, and for that reason, it is internal, and in the whole body.

IV. Absolute motion, is the translation of a body from one absolute place into another; and Relative motion, the translation from one relative place into another. Thus in a Ship under sail, the relative place of a body is that part of the Ship, which the Body possesses; or that part of its cavity which the body fills, and which therefore moves together with the Ship: And Relative rest, is the continuance of the Body in the same part of the Ship, or of its cavity. But Real, absolute rest, is the continuance of the Body in the same part of that Immoveable space, in which the Ship it self, its cavity, and all that it contains, is moved. Wherefore, if the Earth is really at rest, the Body 11which relatively rests in the Ship, will really and absolutely move with the same velocity which the Ship has on the Earth. But if the Earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the Earth, in immoveable space; partly from the relative motion of the Ship on the Earth: and if the body moves also relatively in the Ship; its true motion will arise, partly from the true motion of the Earth, in immoveable space, and partly from the relative motions as well of the Ship on the Earth, as of the Body in the Ship; and from these relative motions, will arise the relative motion of the Body on the Earth. As if that part of the Earth where the Ship is, was truly mov'd toward the East, with a velocity of 10010 parts; while the Ship it self with a fresh gale, and full sails, is carry'd towards the West, with a velocity express'd by 10 of those parts; but a Sailor walks in the Ship towards the East, with 1 part of the said velocity: then the Sailor will be moved truly and absolutely in immoveable space towards the East with a velocity of 10001 parts, and relatively on the Earth towards the West, with a velocity of 9 of those parts.

Absolute time, in Astronomy, is distinguish'd from Relative, by the Equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly consider'd as equal, and used for a measure of time: Astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the True, or equable progress, of Absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions 12are swift or slow, or none at all: and therefore it ought to be distinguish'd from what are only sensible measures thereof; and out of which we collect it, by means of the Astronomical equation. The necessity of which Equation, for determining the Times of a phænomenon, is evinc'd as well from the experiments of the pendulum clock, as by eclipses of the Satellites of Jupiter.

As the order of the parts of Time is immutable, so also is the order of the parts of Space. Suppose those parts to be mov'd out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the Places as well of themselves as of all other things. All things are placed in Time as to order of Succession; and in Space as to order of Situation. It is from their essence or nature that they are Places; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places; and translations out of those places, are the only Absolute Motions.

But because the parts of Space cannot be seen, or distinguished from one another by our Senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body consider'd as immoveable, we define all places: and then with respect to such places, we estimate all motions, considering bodies as transfer'd from some of those places into others. And so instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs: but in Philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referr'd.

13But we may distinguish Rest and Motion, absolute and relative, one from the other by their Properties, Causes and Effects. It is a property of Rest, that bodies really at rest do rest in respect of one another. And therefore as it is possible, that in the remote regions of the fixed Stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions.

It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axe of motion; and the impetus of bodies moving forwards, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are mov'd, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determin'd by the translation of it from those which only seem to rest: For the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions; and tho' that translation was not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded, as the exteriour part of a whole does to the interior, or as the shell does to the kernel; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

14A property near a kin to the preceding, is this, that if a place is mov'd, whatever is placed therein moves along with it; and therefore a body, which is mov'd from a place in motion, partakes also of the motion of its place. Upon which account all motions from places in motion, are no other than parts of entire and absolute motions; and every entire motion is composed out of the motion of the body out of its first place, and the motion of this place out of its place, and so on; until we come to some immoveable place, as in the before mention'd example of the Sailor. Wherefore entire and absolute motions can be no otherwise determin'd than by immoveable places; and for that reason I did before refer those absolute motions to immoveable places, but relative ones to moveable places. Now no other places are immoveable, but those that, from infinity to infinity, do all retain the same given positions one to another; and upon this account, must ever remain unmov'd; and do thereby constitute, what I call, immoveable space.

The Causes by which true and relative motions are distinguished, one from the other, are the forces impress'd upon bodies to generate motion. True motion is neither generated nor alter'd, but by some force impress'd upon the body moved: but relative motion may be generated or alter'd without any force impress'd upon the body. For it is sufficient only to impress some force on other bodies with which the former is compar'd, that by their giving way, that relation may be chang'd, in which the relative rest or motion of this other body did consist. Again, True motion suffers always some change from any force impress'd upon the moving body; but Relative motion does not necessarily undergo any change, by such forces. For if the same forces are likewise impress'd on those other 15bodies, with which the comparison is made, that the relative position may be preserved, then that condition will be preserv'd, in which the relative motion consists. And therefore, any relative motion may be changed, when the true motion remains unalter'd, and the relative may be preserv'd, when the true suffers some change. Upon which accounts, true motion does by no means consist in such relations.

The Effects which distinguish absolute from relative motion are, the forces of receding from the axe of circular motion. For there are no such forces in circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then fill'd with water, and held at rest together with the water; after by the sudden action of another force, it is whirl'd about the contrary way, and while the cord is untwisting it self, the vessel continues for some time in this motion; the surface of the water will at first be plain, as before the vessel began to move: but the vessel, by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming it self into a concave figure, (as I have experienced) and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shews its endeavour to recede from the axe of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers it self, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was 16greatest it produc'd no endeavour to recede from the axe: the water shew'd no tendency to the circumference, nor any ascent towards the sides of the vessel, but remain'd of a plain surface, and therefore its True circular motion had not yet begun. But afterwards, when the relative motion of the water had decreas'd, the ascent thereof towards the sides of the vessel, prov'd its endeavour to recede from the axe; and this endeavour shew'd the real circular motion of the water perpetually increasing, till it had acquir'd its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defin'd by such translations. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axe of motion, as its proper and adequate effect: but relative motions in one and the same body are innumerable, according to the various relations it bears to external bodies, and like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps participate of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixt Stars, carry the Planets along with them; the several parts of those heavens, and the Planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest) and being carried together with their heavens, participate of their motions, and as parts of revolving wholes, endeavour to recede from the axe of their motions.

Wherefore relative quantities, are not the quantities themselves, whose names they bear, but those sensible 17measures of them (either accurate or inaccurate) which are commonly used instead of the measur'd quantities themselves. And if the meaning of words is to be determin'd by their use; then by the names Time, Space, Place and Motion, their measures are properly to be understood; and the expression will be unusual, and purely Mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the Sacred Writings, who there interpret those words for the measur'd quantities. Nor do those less defile the purity of Mathematical and Philosophical truths, who confound real quantities themselves with their relations and vulgar measures.

It is indeed a matter of great difficulty to discover, and effectually to distinguish, the True motions of particular bodies from the Apparent: because the parts of that immoveable space in which those motions are perform'd, do by no means come under the observation of our senses. Yet the thing is not altogether desperate; for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes kept at a given distance one from the other, by means of a cord that connects them, were revolv'd about their common centre of gravity; we might, from the tension of the cord, discover the endeavour of the globes to recede from the axe of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impress'd at once on the alternate faces of the globes to augment or diminish their circular motions; from the encrease or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found, on what faces 18those forces ought to be impress'd, that the motions of the globes might be most augmented, that is, we might discover their hindermost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently, the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, ev'n in an immense vacuum, where there was nothing external or sensible with which the globes could be compar'd. But now if in that space some remote bodies were plac'd that kept always a given position one to another, as the Fixt Stars do in our regions; we cou'd not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observ'd the cord, and found that its tension was that very tension which the motions of the globes requir'd, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions. But how we are to collect the true motions from their causes, effects, and apparent differences; and vice versa, how from the motions, either true or apparent, we may come to the knowledge of their causes and effects, shall be explain'd more at large in the following Tract. For to this end it was that I compos'd it.

Axioms, or Laws of Motion

Axioms or Laws of Motion.

Law I

19Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impress'd thereon.

PROJECTILES persevere in their motions, so far as they are not retarded by the resistance of the air, or impell'd downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the Planets and Comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

Law II

The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impress'd altogether and 20at once, or gradually and successively. And this motion (being always directed the same way with the generating force) if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joyned, when they are oblique, so as to produce a new motion compounded from the determination of both.

Law III

To every Action there is always opposed an equal Reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tyed to a rope, the horse (if I may so say) will be equally drawn back towards the stone: For the distended rope, by the same endeavour to relax or unbend it self, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other; that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities, but in the motions of bodies; that is to say, if the bodies are not hinder'd by any other impediments. For, because the motions 21are equally changed, the changes of the velocities made towards contrary parts, are reciprocally proportional to the bodies. This Law takes place also in Attractions, as will be proved in the next Scholium.

Corollary I

A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides, by those forces apart. (Pl. 1. Fig. 1.)

If a body in a given time, by the force M impress'd apart in the place A, should with an uniform motion be carried from A to B; and by the force N impress'd apart in the same place, should be carried from A to C: compleat the parallelogram ABCD, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the force N be impress'd or not; and therefore at the end of that time, it will be found somewhere in the line BD. By the same argument, at the end of the same time it will be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I.

Corollary II

22And hence is explained the composition of any one direct force AD, out of any two oblique forces AB and BD; and, on the contrary the resolution of any one direct force AD into two oblique forces AB and BD: which composition and resolution are abundantly confirmed from Mechanics. (Fig. 2.)

As if the unequal Radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP; and the forces of those weights to move the wheel were required. Through the centre O draw the right line KOL, meeting the cords perpendicularly in K and L; and from the centre O, with OL the greater of the distances OK and OL, describe a circle, meeting the cord MA in D: and drawing OD, make AC parallel and DC perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be fixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the Line AD, and let it be resolved into the forces AC and CD; of which the force AC, drawing the radius OD directly from the centre, will have no effect to move the wheel: but the other force DC, drawing the radius DO perpendicularly, will have the same effect as if it drew perpendicularly the radius OL equal to OD; that is, it will have the 23same effect as the weight P, if that weight is to the weight A, as the force DC is to the force DA; that is (because of the similar triangles ADC, DOK,) as OK to OD or OL. Therefore the weights A and P, which are reciprocally as the radii OK and OL that lye in the same right line, will be equipollent, and so remain in equilibrio: which is the well known property of the Ballance, the Lever, and the Wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater.

If the weight p, equal to the weight P, is partly suspended by the cord Np, partly sustained by the oblique plane pG; draw pH, NH, the former perpendicular to the horizon, the latter to the plane pG; and if the force of the weight p tending downwards is represented by the line pH, it may be resolved into the forces pN, HN. If there was any plane perpendicular to the cord pN, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those planes pQ, pG, it would press those planes perpendicularly with the forces pN, HN; to wit, the plane pQ with the force pN, and the plane pG with the force HN. And therefore if the plane pQ was taken away, so that the weight might stretch the cord, because the cord, now sustaining the weight, supplies the place of the plane that was removed, it will be strained by the same force pN which press'd upon the plane before. Therefore the tension of this oblique cord pN will be to that of the other perpendicular cord PN as pN to pH. And therefore if the weight p is to the weight A in a ratio compounded of the reciprocal ratio of the least distances of the cords PN, AM, from the centre of the wheel, 24and of the direct ratio of pH to pN, the weights will have the same effect towards moving the wheel and will therefore sustain each other, as any one may find by experiment.

But the weight p pressing upon those two oblique planes, may be consider'd as a wedge between the two internal surfaces of a body split by it; and hence the forces of the Wedge and the Mallet may be determin'd; for because the force with which the weight p presses the plane pQ, is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line pH towards both the planes, as pN to pH; and to the force with which it presses the other plane pG, as pN to NH. And thus the force of the Screw may be deduced from a like resolution of forces; it being no other than a Wedge impelled with the force of a Lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther confirmed. For on what has been said depends the whole doctrine of Mechanics variously demonstrated by different authors. For from hence are easily deduced the forces of Machines, which are compounded of Wheels, Pulleys, Leavers, Cords and Weights, ascending directly or obliquely, and other Mechanical Powers; as also the force of the Tendons to move the Bones of Animals.

Corollary III

25The Quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

For Action and its opposite Re-action are equal, by Law 3, and therefore, by Law 2, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same.

Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity; the motion of A will be to that of B, as 6 to 10. Suppose then their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4 or 5 parts of motion, B will lose as many; and therefore after reflexion A will proceed with 9, 10 or 11 parts, and B with 7, 6 or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11 or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17 or 18 parts; the body 26B, losing so many parts as A has got, will either proceed with one part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts: or it will go back with one part, having not only lost its whole motion, but (if I may so say) one part more; or it will go back with 2 parts, because a progressive motion of 12 parts is took off. And so the Sums of the conspiring motions $15+1$, or $16+0$, and the Differences of the contrary motions $17-1$ and $18-2$, will always be equal to 16 parts, as they were before the meeting and reflexion of the bodies. But, the motions being known with which the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of the body A was of 6 parts before reflexion and of 18 parts after, and the velocity was of 2 parts before reflexion, the velocity thereof after reflexion will be found to be of 6 parts; by saying, as the 6 parts of motion, before to 18 parts after, so are 2 parts of velocity before reflexion to 6 parts after.

But if the bodies are either not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their motions after reflexion are required: in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse; then the motion of each body (by Corol. 2) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, because the bodies act upon each other in the direction of a line perpendicular to this plane, the parallel motions are to be retained the same after reflexion as before; and to the perpendicular motions we are to assign equal changes towards the contrary parts; in 27such manner that the sum of the conspiring, and the difference of the contrary motions, may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I don't consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.

Corollary IV

The common centre of gravity of two or more bodies, does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line.

For if two points proceed with an uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated hereafter in Lem. 23. and its Corol. when the points are moved in the same plane; and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of bodies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or proceeds uniformly in a right line; because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those 28two and that of a third body will be either at rest or moving uniformly in a right line; because at that centre, the distance between the common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line; because the distance between the common centre of the three bodies, and the centre of the fourth is there also divided in a given ratio, and so on in infinitum. Therefore in a system of bodies where there is neither any mutual action among themselves, nor any foreign force impress'd upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest or moves uniformly forwards in a right line.

Moreover, in a system of two bodies mutually acting upon each other, since the distances between their centres and the common centre of gravity of both, are reciprocally as the bodies; the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor retarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that action; and much less the common centre of gravity of the others with which that action does not intervene; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts reciprocally proportional to the total sums of those bodies 29whose centres they are; and therefore while those two centres retain their state of motion or rest, the common centre of all does also retain its state: It is manifest, that the common centre of all never suffers any change in the state of its motion or rest from the actions of any two bodies between themselves. But in such a system all the actions of the bodies among themselves, either happen between two bodies, or are composed of actions interchanged between some two bodies; and therefore they do never produce any alteration in the common centre of all as to its state of motion or rest. Wherefore since that centre when the bodies do not act mutually one upon another, either is at rest or moves uniformly forward in some right line; it will, notwithstanding the mutual actions of the bodies among themselves, always persevere in its state, either of rest, or of proceeding uniformly in a right line, unless it is forc'd out of this state by the action of some power impress'd from without upon the whole system. And therefore the same law takes place in a system, consisting of many bodies, as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion whether of one single body or of a whole system of bodies, is always to be estimated, from the motion of the centre of gravity.

Corollary V

30The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.

For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law 2.) the effects of those collisions will be equal in both cases; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship: where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line.

Corollary VI

31If bodies, any how moved among themselves are urged in the direction of parallel lines by equal accelerative forces; they will all continue to move among themselves, after the same manner as if they had been urged by no such forces.

For these forces acting equally (with respect to the quantities of the bodies to be moved) and in the direction of parallel lines, will (by Law 2.) move all the bodies equally (as to velocity) and therefore will never produce any change in the positions or motions of the bodies among themselves.

Scholium

Hitherto I have laid down such principles as have been receiv'd by Mathematicians, and are confirm'd by abundance of experiments. By the two first Laws and the first two Corollaries, Galileo discover'd that the descent of bodies observ'd the duplicate ratio of the time, and that the motion of projectiles was in the curve of a Parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resistance of the Air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities: and in the whole time impresses a whole force and generates a whole velocity, proportional to the time. And the spaces described in 32proportional times are as the velocities and the times conjunctly; that is, in a duplicate ratio of the times. And when a body is thrown upwards, its uniform gravity impresses forces and takes off velocities proportional to the times; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunctly, or in the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection is compounded with the motion arising from its gravity. As if the Body A by its motion of projection alone (Fig. 3.) could describe in a given time the right line AB, and with its motion of falling alone could describe in the same time the altitude AC; compleat the parallelogram ABDC, and the body by that compounded motion will at the end of the time be found in the place D; and the curve line AED, which that body describes, will be a Parabola, to which the right line AB will be a tangent in A; and whose ordinate BD will be as the square of the line AB. On the same laws and corollaries depend those things which have been demonstrated concerning the times of the vibration of Pendulums, and are confirm'd by the daily experiments of Pendulum clocks. By the same together with the third Law Sir Christ. Wren, Dr. Wallis and Mr. Huygens, the greatest Geometers of our times, did severally determine the rules of the Congress and Reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves, as to those rules. Dr. Wallis indeed was something more early in the publication; then followed Sir Christopher Wren, and lastly, Mr. Huygens. But Sir Christopher Wren confirmed the 33truth of the thing before the Royal Society by the experiment of pendulums, which Mr Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air, as to the elastic force of the concurring bodies. Let the spherical bodies AB, be suspended by the parallel and equal strings, AC, BD, Fig. 4. from the centres C, D. About these centres, with those intervals, describe the semicircles EAF, GBH, bisected by the radii CA, DB. Bring the body A to any point R of the arc EAF, and (withdrawing the body B) let it go from thence, and after one oscillation suppose it to return to the point V: then RV will be the retardation arising from the resistance of the air. Of this RV let ST be a fourth part situated in the middle, to wit, so as RS and TV may be equal, and RS may be to ST as 3 to 2: then will ST represent very nearly the retardation during the descent from S to A. Restore the body B to its place: and supposing the body A to be let fall from the point S, the velocity thereof in the place of reflexion A, without sensible error, will be the same as if it had descended in vacuo from the point T. Upon which account this velocity may be represented by the chord of the arc TA. For it is a proposition well known to Geometers, that the velocity of a pendulous body in the lowest point is as the chord of the arc which it has described in its descent. After reflexion, suppose the body A comes to the place s, and the body B to the place k. Withdraw the body B, and find the place v, from which if the body A, being let go, should after one oscillation return to the place r, st may be a fourth part of rv, so placed in the middle thereof as to leave rs equal to tv, and 34let the chord of the arc tA represent the velocity which the body A had in the place A immediately after reflexion. For t will be the true and correct place to which the Body A should have ascended, if the resistance of the Air had been taken off. In the same way we are to correct the place k to which the body B ascends, by finding the place l to which it should have ascended in vacuo. And thus every thing may be subjected to experiment, in the same manner as if we were really placed in vacuo. These things being done we are to take the product (if I may so say) of the body A, by the chord of the arc TA (which represents its velocity), that we may have its motion in the place A immediately before reflexion; and then by the chord of the arc tA, that we may have its motion in the place A immediately after reflexion. And so we are to take the product of the body B by the chord of the arc Bl, that we may have the motion of the same immediately after reflexion. And in like manner, when two bodies are let go together from different places, we are to find the motion of each, as well before as after reflexion; and then we may compare the motions between themselves, and collect the effects of the reflexion. Thus trying the thing with pendulums of ten feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an errour of 3 inches, that when the bodies concurr'd together directly, equal changes toward the contrary parts were produced in their motions, and, of consequence, that the action and reaction were always equal. As if the body A imping'd upon the body B at rest with 9 parts of motion, and losing 7, proceeded after reflexion with 2; the body B was carried backwards with those 7 parts. If the bodies 35concurr'd with contrary motions, A with twelve parts of motion, and B with six, then if A receded with 2, B receded with 8, to wit; with a deduction of 14 parts of motion on each side. For from the motion of A subducting 12 parts; nothing will remain: but subducting 2 parts more, a motion will be generated of 2 parts towards the contrary way; and so, from the motion of the body B of 6 parts, subducting 14 parts, a motion is generated of 8 parts towards the contrary way. But if the bodies were made both to move towards the same way; A, the swifter, with 14 parts of motion, B, the slower, with 5, and after reflexion A went on with 5, B likewise went on with 14 parts; 9 parts being transferr'd from A to B. And so in other cases. By the congress and collision of bodies, the quantity of motion, collected from the sum of the motions directed towards the same way, or from the difference of those that were directed towards contrary ways, was never changed. For the error of an inch or two in measures may be easily ascrib'd to the difficulty of executing every thing with accuracy. It was not easy to let go the two pendulums so exactly together that the bodies should impinge one upon the other in the lowermost place AB; nor to mark the places s, and k, to which the bodies ascended after congress. Nay, and some errors too might have happen'd from the unequal density of the parts of the pendulous bodies themselves, and from the irregularity of the texture proceeding from other causes.

But to prevent an objection that may perhaps be alledged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic; whereas no such bodies are to be found in nature 36; I must add that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to diminish the reflexion in such a certain proportion, as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies absolutely hard return one from another with the same velocity with which they meet. But this may be affirm'd with more certainty of bodies perfectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminish'd together with the elastic force; because that force (except when the parts of bodies are bruised by their congress, or suffer some such extension as happens under the strokes of a hammer,) is (as far as I can perceive) certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met. This I tried in balls of wool, made up tightly and strongly compres'd. For first, by letting go the pendulous bodies and measuring their reflexion, I determined the quantity of their elastic force; and then, according to this force, estimated the reflexions that ought to happen in other cases of congress. And with this computation other experiments made afterwards did accordingly agree; the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met, as about 5 to 9. Balls of steel return'd with almost the same velocity: those of cork with a velocity something less: but in balls of glass the proportion was as about 15 to 16. And thus the third law, so far as it regards percussions and reflexions, is prov'd by a theory, exactly agreeing with experience.

37In attractions, I briefly demonstrate the thing after this manner. Suppose an obstacle is interpos'd to hinder the congress of any two bodies A, B, mutually attracting one the other: then if either body as A, is more attracted towards the other body B, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B, and therefore will not remain in æquilibrio: but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly towards the parts on which B lies; and in free spaces, to go forward in infinitum with a motion perpetually accelerated. Which is absurd, and contrary to the first law. For by the first law, the system ought to persevere in it's state of rest, or of moving uniformly forward in a right line; and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. I made the experiment on the loadstone and iron. If these plac'd apart in proper vessels, are made to float by one another in standing water; neither of them will propel the other, but by being equally attracted, they will sustain each others pressure, and rest at last in an equilibrium.

So the gravitation betwixt the Earth and its parts, is mutual. Let the Earth FI (Fig. 5.) be cut by any plane EG into two parts EGF and EGI: and their weights one towards the other will be mutually equal. For if by another plane HK, parallel to the former EG, the greater part EGI is cut into two parts EGKH and HKI, whereof HKI is equal to the part EFG first cut off: it is evident that the middle part EGKH will have no propension by its proper weight towards either side, but will hang as it were and rest in an equilibrium betwixt both. But the one extreme part HKI 38will with its whole weight bear upon and press the middle part toward the other extreme part EGF; and therefore the force, with which EGI, the sum of the parts HKI and EGKH, tends towards the third part EGF, is equal to the weight of the part HKI, that is, to the weight of the third part EGF. And therefore the weights of the two parts EGI and EGF, one towards the other, are equal, as I was to prove. And indeed if those weights were not equal, the whole Earth floating in the non-resisting æther, would give way to the greater weight, and, retiring from it, wou'd be carried off in infinitum.

And as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate forces: so in the use of mechanic instruments, those agents are equipollent and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces.

So those weights are of equal force to move the arms of a Ballance, which during the play of the ballance are reciprocally as their velocities upwards and downwards: that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axe of the ballance; but if they are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will be equipollent, which are reciprocally as the heights of their ascent and descent taken according to the perpendicular; and that on account of the determination of gravity downwards.

And in like manner in the Pully, or in a combination of Pullies, the force of a hand drawing the rope directly, that is to the weight, whether ascending directly or 39obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight.

In Clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impress'd, will mutually sustain the one the other.

The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand, is to the progressive velocity of the Screw towards the press'd body.

The forces by which the Wedge presses or drives the two parts of the wood it cleaves, are to the force of the mallet upon the wedge, as the progress of the wedge in the direction of the force impress'd upon it by the mallet, is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all Machines.

The power and use of Machines consists only in this, that by diminishing the velocity we may augment the force, and the contrary: From whence in all sorts of proper Machines, we have the solution of this problem; To move a given weight with a given power, or with a given force to overcome any other given resistance. For if Machines are so contriv'd, that the velocities of the agent and resistant are reciprocally as their forces, the agent will just sustain the resistant: but with a greater disparity of velocity will overcome it. So that if the disparity of velocities is so great, as to overcome all that resistance, which commonly 40arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised; the excess of the force remaining, after all those resistances are overcome, will produce an acceleration of motion proportional thereto, as well in the parts of the Machine, as in the resisting body. But to treat of Mechanics is not my present business. I was only willing to shew by those examples, the great extent and certainty of the third law of motion. For if we estimate the action of the agent from its force and velocity conjunctly; and likewise the re-action of the impediment conjunctly from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts; the action and re-action in the use of all sorts of Machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impress'd upon the resisting body, the ultimate determination of the action will be always contrary to the determination of the re-action.

Section I. Of the method of first and last ratio's of quantities, by the help whereof we demonstrate the propositions that follow.

OF THE MOTION OF BODIES.

Book I

Section I

41Of the method of first and last ratio's of quantities, by the help whereof we demonstrate the propositions that follow.

Lemma I

Quantities, and the ratio's of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the 42other than by any given difference, become ultimately equal.

If you deny it; suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is against the supposition.

Lemma II

If in any figure AacE (Pl.1.Fig.6.) terminated by the right lines Aa, AE, and the curve acE, there be inscrib'd any number of parallelograms Ab, Bc, Cd, &c. comprehended under equal bases AB, BC, CD, &c. and the sides Bb, Cc, Dd, &c. parallel to one side Aa of the figure; and the parallelograms aKbl, bLcm, cMdn, &c. are compleated. Then if the breadth of those parallelograms be suppos'd to be diminished, and their number to be augmented in infinitum: I say that the ultimate ratio's which the inscrib'd figure AKbLcMdD, the circumscribed figure AalbmcndoE, and curvilinear figure AabcdE, will have to one another, are ratio's of equality.

For the difference of the inscrib'd and circumscrib'd figures is the sum of the parallelograms Kl, Lm, Mn, Do, that is, (from the equality of all their bases) the rectangle under one of their bases Kb and the sum of their altitudes Aa, that is, the rectangle ABla, 43But this rectangle, because its breadth AB is suppos'd diminished in infinitum, becomes less than any given space. And therefore (by Lem. 1.) the figures inscribed and circumscribed become ultimately equal one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either.

Q. E. D.

Lemma III

The same ultimate ratio's are also ratio's of equality, when the breadths AB, BC, DC, &c. of the parallelograms are unequal, and are all diminished in infinitum.

For suppose AF equal to the greatest breadth, and compleat the parallelogram FAaf. This parallelogram will be greater than the difference of the inscrib'd and circumscribed figures; but, because its breadth AF is diminished in infinitum, it will become less than any given rectangle. Q. E. D.

Cor. 1. Hence the ultimate sum of those evanescent parallelograms will in all parts coincide with the curvilinear figure.

Cor. 2. Much more will the rectilinear figure, comprehended under the chords of the evanescent arcs ab, bc, cd, &c. ultimately coincide with the curvilinear figure.

Cor. 3. And also the circumscrib'd rectilinear figure comprehended under the tangents of the same arcs.

Cor. 4. And therefore these ultimate figures (as to their perimeters acE,) are not rectilinear, but curvilinear limits of rectilinear figures.

Lemma IV

44If in two figures AacE, PprT, (Pl.i.Fig.7.) you inscribe (as before) two ranks of parallelograms, an equal number in each rank, and when their breadths are diminished in infinitum, the ultimate ratio's of the parallelograms in one figure to those in the other each to each respectively, are the same; I say that those two figures AacE, PprT, are to one another in that same ratio.

For as the parallelograms in the one are severally to the parallelograms in the other, so (by composition) is the sum of all in the one to the sum of all in the other; and so is the one figure to the other, because (by Lem. 3.) the former figure to the former sum, and the latter figure to the latter sum are both in the ratio of equality. Q. E. D.

Cor. Hence if two quantities of any kind are any how divided into an equal number of parts: and those parts, when their number is augmented and their magnitude diminished in infinitum, have a given ratio one to the other, the first to the first, the second to the second, and so on in order; the whole quantities will be one to the other in that same given ratio. For if, in the figures of this lemma, the parallelograms are taken one to the other in the ratio of the parts, the sum of the parts will always be as the sum of the parallelograms; and therefore supposing the 45number of the parallelograms and parts to be augmented, and their magnitudes diminished in infinitum, those sums will be in the ultimate ratio of the parallelogram in the one figure to the correspondent parallelogram in the other; that is, (by the supposition) in the ultimate ratio of any part of the one quantity to the correspondent part of the other.

Lemma V

In similar figures, all sorts of homologous sides, whether curvilinear or rectilinear, are proportional; and the area's are in the duplicate ratio of the homologous sides.

Lemma VI

If any arc ACB (Pl.2.Fig.1.) given in position is subtended by its chord AB, and in any point A in the middle of the continued curvature, is touch'd by a right line AD, produced both ways; then if the points A and B approach one another and meet, I say the angle BAD, contained between the chord and the tangent, will be diminished in infinitum, and ultimately will vanish.

For if that angle does not vanish, the arc ACB will contain with the tangent AD an angle equal to a rectilinear angle; and therefore the curvature at the point A will not be continued, which is against the supposition.

Lemma VII

46The same things being supposed; I say, that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality. Pl. 2. Fig. 1.

For while the point B approaches towards the point A, consider always AB and AD as produc'd to the remote points b and d, and parallel to the secant BD draw bd: and let the arc Acb be always similar to the arc ACB. Then, supposing the points A and B to coincide, the angle dAb will vanish, by the preceding lemma; and therefore the right lines Ab, Ad (which are always finite) and the intermediate arc Acb will coincide, and become equal among themselves. Wherefore the right lines AB, AD, and the intermediate arc ACB (which are always proportional to the former) will vanish; and ultimately acquire the ratio of equality. Q.E.D.

Cor. 1. Whence if through B (Pl.2. Fig.2.) we draw BF parallel to the tangent, always cutting any right line AF passing through A in F, this line BF will ultimately be in the ratio of equality with the evanescent arc ACB; because, compleating the parallelogram AFBD, it is always in a ratio of equality with AD.

Cor. 2. And if through B and A more right lines are drawn as BE, BD, AF, AG cutting the tangent AD and its parallel BF; the ultimate ratio of all the abscissa's AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the ratio of equality.

47Cor. 3. And therefore in all our reasoning about ultimate ratio's, we may freely use any one of those lines for any other.

Lemma VIII

If the right lines AR, BR, (Pl.2.Fig.1.) with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB, RACB, RAD, and the points A and B approach and meet: I say that the ultimate form of these evanescent triangles is that of similitude, and their ultimate ratio that of equality.

For while the point B approaches towards the point A consider always AB, AD, AR, as produced to the remote points b, d, and r, and rbd as drawn parallel to RD, and let the arc Acb be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bAd will vanish; and therefore the three triangles rAb, rAcb, rAd, (which are always finite) will coincide, and on that account become both similar and equal. And therefore the triangles RAB, RACB, RAD, which are always similar and proportional to these, will ultimately become both similar and equal among themselves. Q.E.D.

Cor. And hence in all our reasonings about ultimate ratio's, we may indifferently use any one of those triangles for any other.

Lemma IX

48If a right line AE, (Pl.2.Fig.3.) and a curve line ABC, both given by position, cut each other in a given angle A; and to that right line, in another given angle, BD, CE are ordinately applied, meeting the curve in B, C; and the points B and C together, approach towards, and meet in, the point A: I say that the area's of the triangles ABD, ACE, will ultimately be one to the other in the duplicate ratio of the sides.

For while the points B, C approach towards the point A, suppose always AD to be produced to the remote points d and e, so as Ad, Ae may be proportional to AD, AE; and the ordinates db, ec, to be drawn parallel to the ordinates DB and EC, and meeting AB and AC produced in b and c. Let the curve Abc be similar to the curve ABC, and draw the right line Ag so as to touch both curves in A, and cut the ordinates DB, EC, db, ec, in F, G, f, g. Then, supposing the length Ae to remain the same, let the points B and C meet in the point A; and the angle cAg vanishing, the curvilinear areas Abd, Ace will coincide with the rectilinear areas Afd, Age; and therefore (by Lem 5) will be one to the other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas; and so the sides AD, AE are to these sides. And therefore the areas ABD, ACE are ultimately 49one to the other in the duplicate ratio of the sides AD, AE. Q.E.D.

Lemma X

The spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are in the very beginning of the motion one to the other in the duplicate ratio of the times.

Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC. The spaces described with these velocities will be as the areas ABD, ACE, described by those ordinates, that is, at the very beginning of the motion (by Lem. 9.) in the duplicate ratio of the times AD, AE. Q.E.D.

Cor. 1. And hence one may easily infer, that the errors of bodies describing similar parts of similar figures in proportional times, are nearly in the duplicate ratio of the times in which they are generated; if so be these errors are generated by any equal forces similarly applied to the bodies, and measur'd by the distances of the bodies from those places of the similar figures, at which, without the action of those forces, the bodies would have arrived in those proportional times.

50Cor. 2. But the errors that are generated by the proportional forces, similarly applied to the bodies at similar parts of the similar figures, are as the forces and the squares of the times conjunctly.

Cor. 3. The same thing is to be understood of any spaces whatsoever describ'd by bodies urged with different forces. All which, in the very beginning of the motion, are as the forces and the squares of the times conjunctly.

Cor. 4. And therefore the forces are as the spaces described in the very beginning of the motion directly, and the squares of the times inversly.

Cor. 5. And the squares of the times are as the spaces describ'd directly, and the forces inversly.

Scholium

If in comparing indetermined quantities of different sorts with another, any one is said to be as any other directly or inversly: the meaning is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. And if any one is said to be as any other two or more directly or inversly: the meaning is, that the first is augmented or diminished in the ratio compounded of the ratio's in which the others, or the reciprocals of the others, are augmented or diminished. As if A is said to be as B directly and C directly and D inversly: the meaning is, that A is augmented or diminished in the same ratio with $B\times C\times \frac{1}{D}$, that is to say, that A and $\frac{BC}{D}$ are one to the other in a given ratio.

Lemma XI

51The evanescent subtense of the angle of contact, in all curves which at the point of contact have a finite curvature, is ultimately in the duplicate ratio of the subtense of the conterminate arc. Pl. 2. Fig 4.

Case 1. Let AB be that arc, AD its tangent, BD the subtense of the angle of contact perpendicular on the tangent, AB the subtense of the arc. Draw BG perpendicular to the subtense AB, and AG to the tangent AD, meeting in G; then let the points D, B, and G, approach to the points d, b, and g, and suppose J to be the ultimate intersection of the lines BG, AG, when the points D, B have come to A. It is evident that the distance GJ may be less than any assignable. But (from the nature of the circles passing through the points A, B, G; A, b, g) $A{B}^2=AG\times BD$, and $A{b}^2=Ag\times bd$; and therefore the ratio $A{B}^2$ to $A{b}^2$ is compounded of the ratio's of AG to Ag, and of BD to bd. But because GJ may be assum'd of less length than any assignable, the ratio of AG to Ag may be such as to differ from the ratio of equality by less than any assignable difference; and therefore the ratio $A{B}^2$ to $A{b}^2$ may be such as to differ from the ratio of BD to bd by less than any assignable difference. Therefore, by Lem. 1. the ultimate ratio of $A{B}^2$ to $A{b}^2$ is the same with the ultimate ratio of BD to bd. Q. E. D.

Case 2. Now let BD be inclined to AD in any given angle, and the ultimate ratio of BD to bd will 52always be the same as before, and therefore the same with the ratio of $A{B}^2$ to $A{b}^2$. Q.E.D.

Case 3. And if we suppose the angle D not to be given, but that the right line BD converges to a given point, or is determined by any other condition whatever; nevertheless the angles D, d, being determined by the same law, will always draw nearer to equality, and approach nearer to each other than by any assigned difference, and therefore, by Lem. 1, will at last be equal; and therefore the lines BD, bd are in the same ratio to each other as before.

Cor. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, and their sines, BC, bc, become ultimately equal to the chords AB, Ab; their squares will ultimately become as the subtenses BD, bd.

Cor. 2. Their squares are also ultimately as the versed sines of the arcs, bisecting the chords, and converging to a given point. For those versed sines are as the subtenses BD, bd.

Cor. 3. And therefore the versed sine is in the duplicate ratio of the time in which a body will describe the arc with a given velocity.

Cor. 4. The rectilinear triangles ADB, Adb are ultimately in the triplicate ratio of the sides AD, Ad, and in a sesquiplicate ratio of the sides DB, db; as being in the ratio compounded of the sides AD to DB, and of Ad to db. So also the triangles ABC, Abc are ultimately in the triplicate ratio of the sides BC, bc. What I call the sesquiplicate ratio is the subduplicate of the triplicate, as being compounded of the simple and subduplicate ratio.

Cor. 5. And because DB, db are ultimately parallel and in the duplicate ratio of the lines AD, Ad: the ultimate curvilinear areas ADB, Adb will be 53(by the nature of the parabola) two thirds of the rectilinear triangles ADB, Adb and the segments AB, Ab will be one third of the same triangles. And thence those areas and those segments will be in the triplicate ratio as well of the tangents AD, Ad; as of the chords and arcs AB, Ab.

Scholium

But we have all along supposed the angle of contact to be neither infinitely greater nor infinitely less, than the angles of contact made by circles and their tangents; that is, that the curvature at the point A is neither infinitely small nor infinitely great, or that the interval AJ is of a finite magnitude. For DB may be taken as AD${}^3$: in which case no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. And by a like reasoning if DB be made successively as AD${}^4$, AD${}^5$, AD${}^6$, AD${}^7$, &c., we shall have a series of angles of contact, proceeding in infinitum, wherein every succeeding term is infinitely less than the preceding. And if DB be made successively as AD${}^2$, AD${}^{3/2}$, AD${}^{4/3}$, AD${}^{5/4}$, AD${}^{6/5}$, AD${}^{7/6}$, &c., we shall have another infinite series of angles of contact, the first of which is of the same sort with those of circles, the second infinitely greater, and every succeeding one infinitely greater than the preceding. But between any two of these angles another series of intermediate angles of contact may be interposed proceeding both ways in infinitum, wherein every succeeding angle shall be infinitely greater, or infinitely less than the preceding. As if between the 54terms AD${}^2$ and AD${}^3$ there were interposed the series AD${}^{13/6}$, AD${}^{11/5}$, AD${}^{9/4}$, AD${}^{7/3}$, AD${}^{5/2}$, AD${}^{8/3}$, AD${}^{11/4}$, AD${}^{14/5}$, AD${}^{17/6}$, &c. And again, between any two angles of this series, a new series of intermediate angles may be interposed, differing from one another by infinite intervals. Nor is nature confin'd to any bounds.

Those things which have been demonstrated of curve lines and the superficies which they comprehend, may be easily applied to the curve superficies and contents of solids. These lemmas are premised, to avoid the tediousness of deducing perplexed demonstrations ad absurdum, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles: But because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratio's of nascent and evanescent quantities, that is, to the limits of those sums and ratio's; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is perform'd as by the method of indivisibles; and now those principles being demonstrated, we may use them with more safety. Therefore if hereafter, I should happen to consider quantities as made up of particles, or should use little curve lines for right ones; I would not be understood to mean indivisibles, but evanescent divisible quantities; not the sums and ratio's of determinate parts, but always the limits of sums and ratio's: and that the force of such demonstrations always depends on the method lay'd down in the foregoing lemma's.

Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities; because the 55proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument it may be alledged, that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished.) There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical.

It may also be objected, that if the ultimate ratio's of evanescent quantities are given, their ultimate magnitudes will be also given: and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th book of his Elements. But this objection is 56founded on a false supposition. For those ultimate ratio's with which quantities vanish, are not truly the ratio's of ultimate quantities, but limits towards which the ratio's of quantities decreasing without limit, do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum. This thing will appear more evident in quantities infinitely great. If two quantities, whose difference is given, be augmented in infinitum, the ultimate ratio of these quantities will be given, to wit, the ratio of equality; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate; you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceiv'd to be always diminished without end.

Section II. Of the invention of centripetal forces.

Section II

Of the invention of centripetal forces.

Proposition I. Theorem I

57The areas, which revolving bodies describe by radii drawn to an immoveable centre of force, do lie in the same immoveable planes, and are proportional to the times in which they are described. Pl. 2. Fig. 5.

FOR 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 58parallel, 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 SEF to SDE. And therefore in equal times, equal areas are describ'd in one immoveable 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. 3.) 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 immoveable 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 ABCV and the diagonal BV of this parallelogram, in the position which it ultimately acquires when those arcs are diminished in infinitum, is produced 59both 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 ABCV, 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

60Every 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, SBC, SCD, &c. about the immoveable 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 immoveable 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.

61Cor. 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 consequentia, 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 of 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

62Every body, that, by a radius drawn to the centre of another body howsoever 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 urgeed, 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 63simple, 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 is 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; 64why 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 equable 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 of 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 any 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.

65Cor. 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 $R^n$ of the radius R, and therefore the velocity reciprocally as the power $R^{n-1}$ of the radius; the centripetal force will be reciprocally as the power $R^{2n-1}$ 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.

66Cor. 9. From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force 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, and 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 Horologio Oscillatorio, has compared the force of gravity with the centrifugal 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 reflected from the circle at the several angular points; the force, with which at 67every reflexion 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 radius: and therefore if the polygon, by having its sides diminished in infinitum, 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 describes 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 erect 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, 68meeting 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 construction, 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 resistance, 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 supposed 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. 11.) 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 69will 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 $\frac{S{P}^2\times Q{T}^2}{QR}$, 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 $SP\times QT$ 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 $\frac{S{T}^2\times Q{P}^2}{QR}$; if ST is a perpendicular from the centre of force on PR the tangent of the orbit. For the rectangles $ST\times QP$ and $SP\times QT$ 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 PV be a chord of this circle, drawn from the body through the centre of force; the centripetal force will be reciprocally as the solid $S{T}^2\times PV$. For PV is $\frac{Q{P}^2}{QR}$,

70Cor. 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 rectilinear 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 $\frac{S{P}^2\times Q{T}^2}{QR}$ or the solid $S{T}^2\times PV$, 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 71PR 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 similar triangles ZQR, ZTP, VPA, we shall have $R{P}^2$, that is, QRL, to $Q{T}^2$, as $A{V}^2$ to $P{V}^2$. And therefore $\frac{QRL\times P{V}^2}{A{V}^2}$ is equal to $Q{T}^2$. Multiply those equals by $\frac{S{P}^2}{QR}$, and the points P and Q coinciding, for RL write PV; then we shall have $\frac{S{P}^2\times P{V}^3}{A{V}^2}=\frac{S{P}^2\times Q{T}^2}{QR}$. And therefore (by cor. 1. and 5. prop. 6.) the centripetal force is reciprocally as $\frac{S{P}^2\times P{V}^3}{A{V}^2}$, that is (because $A{V}^2$ is given) reciprocally as the square of the distance or altitude SP, and the cube of the chord PV conjunctly. Q. E. I.

The same otherwise.

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 $\frac{SP\times PV}{AV}=ST$, and $\frac{S{P}^2\times P{V}^3}{A{V}^2}=S{T}^2\times PV$. And therefore (by corol. 3 and 5. prop. 6.) the centripetal force is reciprocally as $\frac{S{P}^2\times P{V}^3}{A{V}^2}$; that is, (because AV is given) reciprocally as $S{P}^2\times P{V}^3$. Q. E. I.

72Cor. 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 centripetal 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 $R{P}^2\times SP$ 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 $R{P}^2\times P{T}^3$ to $S{P}^2\times P{V}^3$; that is, as $SP\times R{P}^2$ to $\frac{S{P}^3\times P{V}^3}{P{T}^3}$ or, (because of the similar triangles PSG, TPV) to $S{G}^3$.

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 $SP\times R{P}^2$, 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

73If 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 N, and join CP. Because of the similar triangles CPM, PZT and RZQ we shall have $C{P}^2$ to $P{M}^2$ as $P{R}^2$ to $Q{T}^2$; and from the nature of the circle, $P{R}^2$ is equal to the rectangle $QR\times \overline{RN+QN}$, or the points P, Q coinciding, to the rectangle $QR\times 2PM$. Therefore $C{P}^2$ is to $P{M}^2$ as $QR\times 2PM$ to $Q{T}^2$; and $\frac{Q{T}^2}{QR}=\frac{2P{M}^3}{C{P}^2}$, and $\frac{Q{T}^2\times S{P}^2}{QR}=\frac{2P{M}^3\times S{P}^2}{C{P}^2}$. And therefore (by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as $\frac{2P{M}^3\times S{P}^2}{C{P}^2}$; that is, (neglecting the given ratio $\frac{2S{P}^2}{C{P}^2}$) reciprocally as $P{M}^3$. Q. E. I.

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

Scholium

74And by a like reasoning, a body will be moved in an ellipsis, or even in an hyperbole, or parabola, by a centripetal 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 spiral PQS, cutting all the radii SP, SQ, &c. in a given angle: it is proposed to find the law of the centripetal force tending to the centre of that spiral. 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 $\frac{QT}{QR}$ is also given, and $\frac{Q{T}^2}{QR}$ 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 $\frac{Q{T}^2}{QR}$ remains the same as before, that is as SP. And $\frac{Q{T}^2\times S{P}^2}{QR}$ is as $S{P}^3$, and therefore (by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as the cube of the distance SP. Q. E. I.

The same otherwise.

75The 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 $S{P}^3$ is as $S{T}^2\times PV$, that is (by corol. 3. and 5. prop. 6.) reciprocally as the centripetal force.

Lemma XII

All parallelograms circumscribed 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 revolves in an ellipsis: it is proposed to find the law of the centripetal force tending to the centre of the ellipsis. Pl. 4. Fig. 1.

Suppose CA, CB to be semi-axes of the ellippse; GP, DK conjugate diameters; PF, QT 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 $Q{v}^2$ as $P{C}^2$ to $C{D}^2$, and (because of the similar triangles QvT, PCF) $Q{v}^2$ to $Q{T}^2$ as $P{C}^2$ to $P{F}^2$; and by composition, the ratio of PvG to $Q{T}^2$ is compounded of 76the ratio of $P{C}^2$ to $C{D}^2$ and of the ratio of $P{C}^2$ to $P{F}^2$, that is, vG to $\frac{Q{T}^2}{Pv}$ as $P{C}^2$ to $\frac{C{D}^2\times P{F}^2}{P{C}^2}$. Put QR for Pv, and (by lem. 12.) $BC\times CA$ for $CD\times PF$, also (the points P and Q coinciding,) 2PC for vG; and multiplying the extremes and means together, we shall have $\frac{Q{T}^2\times P{C}^2}{QR}$ equal to $\frac{2B{C}^2\times C{A}^2}{PC}$. Therefore (by cor. 5. prop. 6.) the centripetal force is reciprocally as $\frac{2B{C}^2\times C{A}^2}{PC}$; that is (because $2B{C}^2\times C{A}^2$ is given) reciprocally as $\frac{1}{PC}$; that is, directly as the distance 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 equal to Tv; then take uV, such as shall be to vG as $D{C}^2$ to $P{C}^2$. And because $Q{v}^2$ is to PvG as $D{C}^2$ to $P{C}^2$, (by the conic sections) we shall have $Q{v}^2=Pv\times uV$. 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 pass 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 $D{C}^2$ to $P{C}^2$, will become the ratio of PV to PG or PV to 2PC; and therefore PV will be equal to $\frac{2D{C}^2}{PC}$. And therefore the force, by which the body P revolves in the ellipsis, will be reciprocally as 77$\frac{2D{C}^2}{PC}\times P{F}^2$ (by cor. 3. prop. 6.) that is, (because $2D{C}^2\times P{F}^2$ is given) directly as PC. Q. E. I.

Cor. 1. And therefore the force is as the distance 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 inversely; 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 Galileo'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 78circle, 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.

Section III. Of the motion of bodies in eccentric Conic sections.

Section III

Of the motion of bodies in eccentric Conic sections.

Proposition XI. Problem VI

79If a body revolves in an ellipsis: it is required to find the law of the centripetal force tending to the focus of the ellipsis. Pl. 4. Fig. 2.

Let S be the focus of the ellipsis. Draw SP cutting the diameter DK of the ellipsis in E, and the ordinate Qv in x; and compleat the parallelogram QxPR. It is evident that EP is equal to the greater semi-axis AC: for drawing HI from the other focus H of the ellipsis parallel to EC, because CS, CH are equal ES, EI will be also equal, so that EP is the half sum of PS, PI that is, (because of the parallels HI, PR, and the equal angles IPR, HPZ) of PS, PH, which taken together are equal to the whole axis 2AC. Draw QT perpendicular to SP, and putting L for the principal latus rectum of the ellipsis (or for $\frac{2B{C}^2}{AC}$,) we shall have $L\times QR$ to 80$L\times Pv$ as QR to Pv, that is, as PE or AC to PC; and $L\times Pv$ to GvP as L to Gv; and GvP to $Q{v}^2$ as $P{C}^2$ to $C{D}^2$; and (by corol. 2. lem. 7.) the points Q and P coinciding, $Q{v}^2$ is to $Q{x}^2$ in the ratio of equality; and $Q{x}^2$ or $Q{v}^2$ is to $Q{T}^2$ as $E{P}^2$ to $P{F}^2$, that is, as $C{A}^2$ to $P{F}^2$ or (by lem. 12.) as $C{D}^2$ to $C{B}^2$. And compounding all those ratio's together, we shall have $L\times QR$ to $Q{T}^2$ as $AC\times L\times P{C}^2\times C{D}^2$ or $2C{B}^2\times P{C}^2\times C{D}^2$ to $PC\times Gv\times C{D}^2\times C{B}^2$, or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gv are equal. And therefore the quantities $L\times QR$ and $Q{T}^2$, proportional to these, will be also equal. Let those equals be drawn into $\frac{S{P}^2}{QR}$, and $L\times S{P}^2$ will become equal to $\frac{S{P}^2\times Q{T}^2}{QR}$. And therefore (by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as $L\times S{P}^2$, that is, reciprocally in the duplicate ratio of the distance SP. Q. E. I.

The same otherwise.

Seeing the force tending to the centre of the ellipsis, by which the body P may revolve in that ellipsis, is (by corol. 1. prop. 10.) as the distance CP of the body from the centre C of the ellipsis; let CE be drawn parallel to the tangent PR of the ellipsis; and the force, by which the same body P may revolve about any other point S of the ellipsis, if CE and PS intersect in E, will be as $\frac{P{E}^3}{S{P}^2}$ (by cor. 3. prop. 7.) that is, if the point S is the focus of the ellipsis, and therefore PE be given, as $S{P}^2$ reciprocally. Q. E. I.

81With the same brevity with which we reduced the fifth problem to the parabola and hyperbola, we might do the like here: But because of the dignity of the problem and its use in what follows, I shall confirm the other cases by particular demonstrations.

Proposition XII. Problem VII

Suppose a body to move in an hyperbola: it is required to find the law of centripetal force tending to the focus of that figure. Pl. 5. Fig. 1.

Let CA, CB be the semi-axes of the hyperbola; PG, KD other conjugate diameters; PF a perpendicular to the diameter KD; and Qv an ordinate to the diameter GP. Draw SP cutting the diameter DK in E, and the ordinate Qv in x, and compleat the parallelogram QRPx. It is evident that EP is equal to the semi-transverse axe AC; for, drawing HI, from the other focus H of the hyperbola, parallel to EC, because CS, CH are equal, ES, EI will be also equal; so that EP is the half difference of PS, PI; that is, (because of the parallels IH, PR, and the equal angles IPR, HPZ) of PS, PH, the difference of which is equal to the whole axis 2AC. Draw QT perpendicular to SP. And putting L for the principal latus rectum of the hyperbola, (that is, for $\frac{2B{C}^2}{AC}$,) we shall have $L\times QR$ to $L\times Iv$ as QR to Pv, or Px to Pv, that is, (because of the similar triangles Pxv, PEC) as PE to PC, or AC to PC. And 82$L\times Pv$ will be to $Gv\times Pv$ as L to Gv; and (by the properties of the conic sections) the rectangle GvP is to $Q{v}^2$ as $P{C}^2$ to $C{D}^2$; and (by cor. 2. lem. 7.) $Q{v}^2$ to $Q{x}^2$, the points Q and P coinciding, becomes a ratio of equality; and $Q{x}^2$ or $Q{v}^2$ is to $Q{T}^2$ as $E{P}^2$ to $P{F}^2$, that is, as $C{A}^2$ to $P{F}^2$, or (by lem. 12.) as $C{D}^2$ to $C{B}^2$: and, compounding all those ratio's together, we shall have $L\times QR$ to $Q{T}^2$ as $AC\times L\times P{C}^2\times C{D}^2$ or $2C{B}^2\times P{C}^2\times C{D}^2$ to $PC\times Gv\times C{D}^2\times C{B}^2$, or as 2PC to Gv. But the points P and Q coinciding, 2PC and Gv are equal. And therefore the quantities $L\times QR$ and $Q{T}^2$, proportional to them, will be also equal. Let those equals be drawn into $\frac{S{P}^2}{QR}$, and we shall have $L\times S{P}^2$ equal to $\frac{S{P}^2\times Q{T}^2}{QR}$. And therefore (by cor. 1 & 5. prop. 6.) the centripetal force is reciprocally as $L\times S{P}^2$, that is, reciprocally in the duplicate ratio of the distance SP. Q. E. I.

The same otherwise.

Find out the force tending from the centre C of the hyperbola. This will be proportional to the distance CP. But from thence (by cor. 3. prop. 7.) the force tending to the focus S will be as $\frac{P{E}^3}{S{P}^2}$, that is, because PE is given, reciprocally as $S{P}^2$. Q. E. I.

And the same way it may be demonstrated, that the body having its centripetal changed into a centrifugal force, will move in the conjugate hyperbola.

Lemma XIII

83The latus rectum of a parabola belonging to any vertex is quadruple the distance of that vertex from the focus of the figure.

This is demonstrated by the writers on the conic sections.

Lemma XIV

The perpendicular let fall from the focus of a parabola on its tangent, is a mean proportional between the distances of the focus from the point of contact, and from the principal vertex of the figure. Pl. 5. Fig. 2.

For, let AP be the parabola, S its focus, A its principal vertex, P the point of contact, PO an ordinate to the principal diameter, PM the tangent meeting the principal diameter in M, and SN the perpendicular from the focus on the tangent. Join AN, and because of the equal lines MS and SP, MN and NP, MA and AO; the right lines AN, OP, will be parallel; and thence the triangle SAN will be right angled at A, and similar to the equal triangles SNM, SNP: therefore PS is to SN as SN to SA. Q. E. D.

Cor. 1. $P{S}^2$ is to $S{N}^2$ as PS to SA.

84Cor. 2. And because SA is given, $S{N}^2$ will be as PS.

Cor. 3. And the concourse of any tangent PM with the right line SN, drawn from the focus perpendicular on the tangent, falls in the right line AN, that touches the parabola in the principal vertex.

Proposition XIII. Problem VIII

If a body moves in the perimeter of a parabola: it is required to find the law of the centripetal force tending to the focus of that figure. Pl. 5. Fig. 3.

Retaining the construction of the preceding lemma, let P be the body in the perimeter of the parabola; and from the place Q, into which it is next to succeed draw QR parallel and QT perpendicular to SP, as also Qv parallel to the tangent, and meeting the diameter PG in v, and the distance SP in x. Now, because of the similar triangles Pxv, SPM, and of the equal sides SP, SM of the one, the sides Px or QR and Pv of the other will be also equal. But (by the conic sections) the square of the ordinate Qv is equal to the rectangle under the latus rectum and the segment Pv of the diameter, that is, (by lem. 13.) to the rectangle $4PS\times Pv$, or $4PS\times QR$; and the points P and Q coinciding, the ratio of Qv to Qx (by cor. 2. lem. 7.) becomes a ratio of equality. And therefore $Q{x}^2$, in this case, becomes equal to the rectangle $4PS\times QR$. But (because of the similar triangles QxT, SPN) $Q{x}^2$ is to $Q{T}^2$ as $P{S}^2$ to $S{N}^2$, that is (by cor. 1. lem. 14.) as PS to SA; that is, as $4PS\times QR$ to $4SA\times QR$, and therefore 85(by prop. 9. lib. 5. elem.) $Q{T}^2$ and $4SA\times QR$ are equal. Multiply these equals by $\frac{S{P}^2}{QR}$, and $\frac{S{P}^2\times Q{T}^2}{QR}$ will become equal to $S{P}^2\times 4SA$: and therefore (by cor. 1. and 5. prop. 6.) the centripetal force is reciprocally as $S{P}^2\times 4SA$; that is, because 4SA is given, reciprocally in the duplicate ratio of the distance SP. Q. E. I.

Cor. 1. From the three last propositions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the same time is urged by the action of a centripetal force, that is reciprocally proportional to the square of the distance of the places from the centre; the body will move in one of the conic sections, having its focus in the centre of force; and the contrary. For the focus, the point of contact, and the position of the tangent being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and the bodies velocity given: and two orbits mutually touching one the other, cannot be described by the same centripetal force and the same velocity.

Cor. 2. If the velocity, with which the body goes from its place P, is such, that in any infinitely small moment of time the lineola PR may be thereby described; and the centripetal force such as in the same time to move that body through the space QR; the body will move in one of the conic sections, whose principal latus rectum is the quantity $\frac{Q{T}^2}{QR}$ in its ultimate state, when the lineolæ PR, QR are 86diminished in infinitum. In these corollaries, I consider the circle as an ellipsis; and I except the case, where the body descends to the centre in a right line.

Proposition XIV. Theorem VI

If several bodies revolve about one common centre, and the centripetal force is reciprocally in the duplicate ratio of the distance of places from the centre; I say, that the principal latera recta of their orbits are in the duplicate ratio of the area's, which the bodies by radii drawn to the centre describe in the same time. Pl. 6. Fig. 1.

For (by cor. 2. prop. 13.) the latus rectum L is equal to the quantity $\frac{Q{T}^2}{QR}$ in its ultimate state when the points P and Q coincide. But the lineola QR in a given time is as the generating centripetal force; that is (by supposition) reciprocally as $S{P}^2$. And therefore $\frac{Q{T}^2}{QR}$ is as $Q{T}^2\times S{P}^2$ that is, the latus rectum L is in the duplicate ratio of the area $QT\times SP$. Q. E. D.

Cor. Hence the whole area of the ellipsis, and the rectangle under the axes, which is proportional to it, is in the ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. For the whole area is as the area $QT\times SP$ described in a given time, multiplied by the periodic time.

Proposition XV. Theorem VII

87The same things being supposed, I say that the periodic times in ellipses are in the sesquiplicate ratio of their greater axes.

For the lesser axe is a mean proportional between the greater axe and the latus rectum; and therefore the rectangle under the axes is in the ratio compounded of the subduplicate ratio of the latus rectum and the sesquiplicate ratio of the greater axe. But this rectangle (by cor. prop. 14.) is in a ratio compounded of the subduplicate ratio of the latus rectum and the ratio of the periodic time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the sesquiplicate ratio of the greater axe, equal to the ratio of the periodic time. Q. E. D.

Cor. Therefore the periodic times in ellipses are the same as in circles whose diameters are equal to the greater axes of the ellipses.

Proposition XVI. Theorem VIII

The same things being supposed, and right lines being drawn to the bodies that shall touch the orbits, and perpendiculars being let fall on the tangents from the common focus: I say that the velocities of the bodies are in a ratio compounded of the ratio of the perpendiculars inversely, and the subduplicate 88ratio of the principal latera recta directly. Pl. 6. Fig; 2.

From the focus S, draw SY perpendicular to the tangent PR, and the velocity of the body P will be reciprocally in the subduplicate ratio of the quantity $\frac{S{Y}^2}{L}$. For that velocity is as the infinitely small arc PQ described in a given moment of time, that is, (by lem. 7.) as the tangent PR; that is, (because of the proportionals PR to QT and SP to SY) as $\frac{SP\times QT}{SY}$, or as SY reciprocally and $SP\times QT$ directly; but $SP\times QT$ is as the area described in the given time, that is (by prop. 14.) in the subduplicate ratio of the latus rectum. Q. E. D.

Cor. 1. The principal latera recta are in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the velocities.

Cor. 2. The velocities of bodies, in their greatest and least distances from the common focus, are in the ratio compounded of the ratio of the distance inversely, and the subduplicate ratio of the principal latera recta directly. For those perpendiculars are now the distances.

Cor. 3. And therefore the velocity in a conic section, at its greatest or least distance from the focus, is to the velocity in a circle at the same distance from the centre, in the subduplicate ratio of the principal latus rectum to the double of that distance.

Cor. 4. The velocities of the bodies revolving in ellipses, at their mean distances from the common focus, are the same as those of bodies revolving in circles, at the same distances; that is (by cor. 6. prop. 4.) reciprocally in the subduplicate ratio of the distances.

89For the perpendiculars are now the lesser semi-axes, and these are as mean proportionals between the distances and the latera recta. Let this ratio inversely be compounded with the subduplicate ratio of the latera recta directly, and we shall have the subduplicate ratio of the distances inversely.

Cor. 5. In the same figure, or even in different figures, whose principal latera recta are equal, the velocity of a body is reciprocally as the perpendicular let fall from the focus on the tangent.

Cor. 6. In a parabola, the velocity is reciprocally in the subduplicate ratio of the distance of the body from the focus of the figure; it is more variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by cor. 2. lem. 14.) the perpendicular let fall from the focus on the tangent of a parabola is in the subduplicate ratio of the distance. In the hyperbola the perpendicular is less variable, in the ellipsis more.

Cor. 7. In a parabola, the velocity of a body at any distance from the focus, is to the velocity of a body revolving in a circle at the same distance from the centre, in the subduplicate ratio of the number 2 to 1; in the ellipsis it is less, and in the hyperbola greater, than according to this ratio. For (by cor. 2. of this prop.) the velocity at the vertex of a parabola is in this ratio, and (by cor. 6. of this prop. and prop. 4.) the same proportion holds in all distances. And hence also in a parabola, the velocity is every where equal to the velocity of a body revolving in a circle at half the distance; in the ellipsis it is less, and in the hyperbola greater.

90Cor. 8. The velocity of a body revolving in any conic section is to the velocity of a body revolving in a circle at the distance of half the principal latus rectum of the section, as that distance to the perpendicular let fall from the focus on the tangent of the section. This appears from cor. 5.

Cor. 9. Wherefore since (by cor. 6. prop. 4.) the velocity of a body revolving in this circle is to the velocity of another body revolving in any other circle, reciprocally in the subduplicate ratio of the distances; therefore ex æquo the velocity of a body revolving in a conic section will be to the velocity of a body revolving in a circle at the same distance, as a mean proportional between that common distance and half the principal latus rectum of the section, to the perpendicular let fall from the common focus upon the tangent of the section.

Proposition XVII. Problem IX

Supposing the centripetal force to be reciprocally proportional to the squares of the distances of places from the centre, and that the absolute quantity of that force is known; it is required to determine the line, which a body will describe that is let go from a given place with a given velocity in the direction of a given right line.

Let the centripetal force tending to the point S (Pl. 6. Fig. 3.) be such, as will make the body p revolve in any given orbit pq; and suppose the velocity of this body in the place p is known. Then from the place P, suppose the body P to be let go with a given velocity 91in the direction of the line PR; but by virtue of a centripetal force to be immediately turned aside from that right line into the conic section PQ. This the right line PR will therefore touch in P. Suppose likewise that the right line pr touches the orbit pq in p; and if from S you suppose perpendiculars let fall on those tangents, the principal latus rectum of the conic section (by cor. 1. prop. 16.) will be to the principal latus rectum of that orbit, in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the velocities; and is therefore given. Let this latus rectum be L. The focus S of the conic section is also given. Let the angle RPH be the complement of the angle RPS to two right; and the line PH, in which the other focus H is placed, is given by position. Let fall SK perpendicular on PH, and erect the conjugate semi-axe BC; this done, we shall have $S{P}^2-2KPH+P{H}^2=S{H}^2$ = $4C{H}^2$ = $4B{H}^2-4B{C}^2$ = $\overline{SP+P{H}^2}-L\times \overline{SP+PH}$ = $S{P}^2+2SPH+P{H}^2-L\times \overline{SP+PH}$. Add on both sides $2KPH-S{P}^2-P{H}^2+L\times \overline{SP+PH}$, and we shall have $L\times \overline{SP+PH}$ = $2SPH+2KPH$, or $SP+PH$ to PH as $2SP+2KP$ to L. Whence PH is given both in length and position. That is, if the velocity of the body in P is such that the latus rectum L is less than $2SP+2KP$, PH will lie on the same side of the tangent PR with the line SP; and therefore the figure will be an ellipsis, which from the given foci S, H and the principal axe $SP+PH$, is given also. But if the velocity of the body is so great, that the latus rectum L becomes equal to $2PS+2KP$, the length PH will be infinite; and therefore the figure will be a parabola, which has its axe SH parallel to 92the line PK, and is thence given. But if the body goes from its place P with a yet greater velocity, the length PH is to be taken on the other side the tangent; and so the tangent passing between the foci, the figure will be an hyperbola having its principal axe equal to the difference of the lines SP and PH, and thence is given. For if the body, in these cases, revolves in a conic section so found, it is demonstrated in prop. 11, 12, and 13. that the centripetal force will be reciprocally as the square of the distance of the body from the centre of force S; and therefore we have rightly determined the line PQ, which a body let go from a given place P with a given velocity, and in the direction of the right line PR given by position, would describe with such a force. Q. E. F.

Cor. 1. Hence in every conic section, from the principal vertex D, the latus rectum L, and the focus S given, the other focus H is given, by taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. For the proportion, $SP+PH$ to PH as $2PS+2KP$ to L, becomes, in the case of this corollary, $DS+DH$ to DH as 4DS to L, and by division DS to DH as $4DS-L$ to L.

Cor. 2. Whence if the velocity of a body in the principal vertex D is given, the orbit may be readily found; to wit, by taking its latus rectum to twice the distance DS, in the duplicate ratio of this given velocity to the velocity of a body revolving in a circle at the distance DS (by cor. 3. prop. 16.) and then taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS.

Cor. 3. Hence also if a body move in any conic section, and is forced out of its orbit by any impulse; you may discover the orbit in 93which it will afterwards pursue its course. For by compounding the proper motion of the body with that motion, which the impulse alone would generate, you'll have the motion with which the body will go off from a given place of impulse, in the direction of a right line given in position.

Cor. 4. And if that body is continually disturbed by the action of some foreign force, we may nearly know its course, by collecting the changes which that force introduces in some points, and estimating the continual changes it will undergo in the intermediate places, from the analogy that appears in the progress of the series.

Scholium

If a body P (Pl. 6. Fig. 4.) by means of a centripetal force tending to any given point R move in the perimeter of any given conic section, whose centre is C; and the law of the centripetal force is required: Draw CG parallel to the radius RP, and meeting the tangent PG of the orbit in G; and the force required (by cor. 1. & schol. prop. 10. & cor. 3. prop. 7.) will be as $\frac{C{G}^3}{R{P}^2}$.

Section IV. Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given.

Section IV

94Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given.

Lemma XV

If from the two foci S, H, (Pl. 7. Fig. 1.) of any ellipsis or hyperbola, we draw to any third point V the right lines SV, HV, whereof one HV is equal to the principal axis of the figure, that is, to the axis in which the foci are situated, the other SV is bisected in T by the perpendicular TR let fall upon it; that perpendicular TR will somewhere touch the conic section: and vice versa, if it does touch it, HV will be equal to the principal axis of the figure.

For, let the perpendicular TR cut the right line HV, produced if need be, in R; and join SR. Because 95TS, TV are equal, therefore the right lines SR, VR, as well as the angles TRS, TRV, will be also equal. Whence the point R will be in the conic section, and the perpendicular TR will touch the same: and the contrary. Q. E. D.

Proposition XVIII. Problem X

From a focus and the principal axes given, to describe elliptic and hyperbolic trajectories, which shall pass through given points, and touch right lines given by position. Pl. 7. Fig. 2.

Let S be the common focus of the figures; AB the length of the principal axis of any trajectory; P a point through which the trajectory should pass; and TR a right line which it should touch. About the centre P, with the interval $AB-SP$, if the orbit is an ellipsis, or $AB+SP$ if the orbit is an hyperbola, describe the circle HG. On the tangent TR let fall the perpendicular ST and produce the same to V, so that TV may be equal to ST; and about V as a centre with the interval AB describe the circle FH. In this manner whether two points P, p, are given, or two tangents TR, tr, or a point P and a tangent TR, we are to describe two circles. Let H be their common intersection, and from the foci S, H with the given axis describe the trajectory. I say the thing is done. For (because $PH+SP$ in the ellipsis, and $PH-SP$ in the hyperbola is equal to the axis) the described trajectory will pass through the point P, and (by the preceding lemma) will touch the right line TR. And by the same argument it 96will either pass through the two points P, p, or touch the two right lines TR, tr. Q. E. F.

Proposition XIX. Problem XI

About a given focus, to describe a parabolic trajectory, which shall pass through given points, and touch right lines given by position. Pl. 7. Fig. 3.

Let S be the focus, P a point, and TR a tangent of the trajectory to be described. About P as a centre, with the interval PS, describe the circle FG. From the focus let fall ST perpendicular on the tangent, and produce the same to V, so as TV may be equal to ST. After the same manner another circle fg is to be described, if another point p is given; or another point v is to be found, if another tangent tr is given; then draw the right line IF, which shall touch the two circles FG, fg, if two points P, p are given, or pass through the two points V, v, if two tangents TR, tr are given, or touch the circle FG and pass through the point V, if the point P and the tangent TR are given. On FI let fall the perpendicular SI and bisect the same in K; and with the axis SK, and principal vertex K describe a parabola. I say the thing is done. For this parabola (because SK is equal to IK, and SP to FP) will pass through the point P; and (by cor. 3. lem. 14.) because ST is equal to TV, and STR a right angle, it will touch the right line TR. Q. E. F.

Proposition XX. Problem XII

97About a given focus to describe any trajectory given in specie, which shall pass thro' given points and touch right lines given by position.

Case 1. About the focus S (Pl. 7. Fig. 4.) it is required to describe a trajectory ABC, passing thro' two points B, C. Because the trajectory is given in specie, the ratio of the principal axe to the distance of the foci will be given. In that ratio take KB to BS and LC to CS. About the centres B, C, with the intervals BK, CL describe two circles, and on the right line KL, that touches the same in K and L, let fall the perpendicular SG; which cut in A and a, so that GA may be to AS, and Ga to aS, as KB to BS; and with the axe Aa, and vertices A, a, describe a trajectory. I say the thing is done. For let H be the other focus of the described figure, and seeing GA is to AS as Ga to aS, then by division we shall have $Ga-GA$ or Aa to $aS-AS$ or SH in the same ratio, and therefore in the ratio which the principal axe of the figure to be described has to the distance of its foci; and therefore the described figure is of the same species with the figure which was to be described. And since KB to BS, and LC to CS are in the same ratio, this figure will pass thro' the points B, C, as is manifest from the conic sections

Case 2. About the focus S (Pl. 7. Fig. 5.) it is required to describe a trajectory, which shall somewhere touch two right lines TR, tr. From the focus on those tangents let fall the perpendiculars ST, St, 98which produce to V, v, so that TV, tv may be equal to TS, tS. Bisect Vv in O, and erect the indefinite perpendicular OH, and cut the right Line VS infinitely produced in K and k, so that VK be to KS, and Vk to kS as the principal axe of the trajectory to be described is to the distance of it's foci. On the diameter Kk describe a circle cutting OH in H; and with the foci S, H, and principal axe equal to VH, describe a trajectory. I say the thing is done. For, bisecting Kk in X, and joining HX, HS, HV, Hv, because VK is to KS, as Vk to kS; and by composition, as $VK+Vk$ to $KS+kS$; and by division as $Vk-VK$ to $kS-KS$, that is, as 2VX to 2KX and 2KX to 2SX, and therefore as VX to HX and HX to SX the triangles VXH, HXS will be similar; Therefore VH will be to SH, as VX to XH; and therefore as VK to KS. Wherefore VH the principal axe of the described trajectory has the same ratio to SH the distance of the foci, as the principal axe of the trajectory which was to be described has to the distance of its foci; and is therefore of the same species. And seeing VH, vH, are equal to the principal axe, and VS, vS are perpendicularly bisected by the right lines TR, tr; 'tis evident (by lem. 15.) that those right lines touch the described trajectory. Q. E. F.

Case 3. About the focus S (Pl. 7. Fig. 6.) it is required to describe a trajectory, which shall touch a right line TR in a given point R. On the right line TR let fall the perpendicular ST; which produce to V, so that TV may equal to ST, join VR, and cut the right line VS indefinitely produced in K and k, so that VK may be to SK, and Vk to Sk as the principal axe of the ellipsis to be described, to the distance of its foci; and on the diameter Kk describing 99a circle, cut the right line VR produced in H, then with the foci S, H and principal axe equal to VH, describe a trajectory. I say the thing is done. For VH is to SH as VK to SK, and therefore as the principal axe of the trajectory which was to be described to the distance of its foci, (as appears from what we have demonstrated in Case 2.) and therefore the described trajectory is of the same species with that which was to be described; but that right line TR, by which the angle VRS is bisected, touches the trajectory in the point R, is certain from the properties of the conic sections. Q. E. F.

Case 4. About the focus S (Pl. 7. Fig. 7.) it is required to describe a trajectory APB that shall touch a right line TR, and pass thro' any given point P without the tangent, and shall be similar to the figure apb, described with the principal axe ab, and foci s, h. On the tangent TR let fall the perpendicular ST; which produce to V, so that TV may be equal to ST. And making the angles hsq, shq equal to the angles VSP, SVP; about q as a centre, and with an interval which shall be to ab as SP to VS describe a circle cutting the figure apb in p: join sp, and draw SH, such that it may be to sh, as SP is to sp, and may make the angle PSH equal to the angle psh; and the angle VSH equal to the angle psq. Then with the foci S, H, and principal axe AB equal to the distance VH, describe a conic section. I say the thing is done. For if sv is drawn so that it shall be to sp as sh is to sq, and shall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be similar, and therefore vh will be to pq, as sh is to sq, that is, (because of the similar triangles VSP, hsq) as VS is to SP or as ab to pq. Wherefore 100vh and ab are equal. But because of the similar triangles VSH, vsh, VH is to SH as vh to sh; that is, the axe of the conic section now described is to the distance of its foci, as the axe ab to the distance of the foci s, h; and therefore the figure now described is similar to the figure apb. But, because the triangle PSH is similar to the triangle psh, this figure passes through the point P, and because VH is equal to its axis, and VS is perpendicularly bisected by the right line TR, the said figure touches the right line TR. Q. E. F.

Lemma XVI

From three given points to draw to a fourth point that is not given three right lines whose differences shall be either given or none at all.

Case 1. Let the given points be A, B, C (Pl. 8. Fig. 1.) and Z the fourth point which we are to find; because of the given difference of the lines AZ, BZ, the locus of the point Z will be an hyperbola, whose foci are A and B, and whose principal axe is the given difference. Let that axe be MN. Taking PM to MA, as MN is to AB, erect PR perpendicular to AB, and let fall ZR perpendicular to PR; then, from the nature of the hyperbola, ZR will be to AZ as MN is to AB. And by the like argument, the locus of the point Z will be another hyperbola, whose foci are A, C, and whose principal axe is the difference between AZ and CZ; and QS a perpendicular on AC may be drawn to which (QS) if from any point Z of this hyperbola 101a perpendicular ZS is let fall, this (ZS) shall be to AZ as the difference between AZ and CZ is to AC. Wherefore the ratio's of ZR and ZS to AZ are given, and consequently the ratio of ZR to ZS one to the other; and therefore if the right lines RP, SQ meet in T, and TZ and TA are drawn, the figure TRZS will be given in specie, and the right line TZ, in which the point Z is somewhere placed, will be given in position. There will be given also the right line TA, and the angle ATZ; and because the ratio's of AZ and TZ to ZS are given, their ratio to each other is given also; and thence will be given likewise the triangle ATZ whose vertex is the point Z. Q. E. I.

Case 2. If two of the three lines, for example AZ and BZ, are equal, draw the right line TZ so as to bisect the right line AB; then find the triangle ATZ as above. Q. E. I.

Case 3. If all the three are equal, the point Z will be placed in the centre of a circle that passes thro' the points A, B, C. Q. E. I.

This problematic lemma is likewise solved in Apollonius's Book of Tactions restored by Vieta.

Proposition XXI. Problem XIII

About a given focus to describe a trajectory, that shall pass through given points and touch right lines given by position.

Let the focus S, (Pl. 8. Fig. 2.) the point P, and the tangent TR be given, suppose that the other focus H is to be found. On the tangent let fall the perpendicular ST, which produce to T, so that TT 102may be equal to ST; and TH will be equal to the principal axe. Join SP, HP, and SP will be the difference between HP and the principal axe. After this manner if more tangents TR are given, or more points P, we shall always determine as many lines TH or PH, drawn from the said points T or P, to the focus H, which either shall be equal to the axes, or differ from the axes by given lengths SP; and therefore which shall either be equal among themselves, or shall have given differences; from whence (by the preceding lemma) that other focus H is given. But having the foci and the length of the axe (which is either TH; or, if the trajectory be an ellipsis, $PH+SP$, or $PH-SP$ if it be an hyperbola) the trajectory is given. Q. E. I.

Scholium

When the trajectory is an hyperbola, I do not comprehend its conjugate hyperbola under the name of this trajectory. For a body going on with a continued motion can never pass out of one hyperbola into its conjugate hyperbola.

The case when three points are given is more readily solved thus. Let B, C, D (Pl. 8. Fig. 3.) be the given points. Join BC, CD, and produce them to E, F; so as EB may be to EC, as SB to SC; and FC to FD, as SC to SD. On EF drawn and produced let fall the perpendiculars SG, BH and in GS produced indefinitely take GA to AS, and Ga to aS, as HB is to BS; then A will be the vertex, and Aa the principal axe of the trajectory: Which, according as GA is greater than, equal to, or less than AS, will be either an ellipsis, a parabola 103or an hyperbola; the point a in the first case falling on the same side of the line GF as the point A; in the second, going off to an infinite distance; in the third, falling on the other side of the line GF. For if on GF, the perpendiculars CI, DK are let fall, IC will be to HB as EC to EB; that is, as SC to SB; and by permutation IC to SC as HB to SB, or as GA to SA. And, by the like argument, we may prove that KD is to SD in the same ratio. Wherefore the points B, C, D lie in a conic section described about the focus S, in such manner that all the right lines drawn from the focus S to the several points of the section, and the perpendiculars let fall from the same points on the right line GF are in that given ratio.

That excellent geometer M. De la Hire has solved this problem much after the same way in his conics, prop. 25. lib. 8.

Section V. How the orbits are to be found when neither focus is given.

Section V

How the orbits are to be found when neither focus is given.

Lemma XVII

104If from any point P of a given conic section, to the four produced sides AB, CD, AC, DB of any trapezium ABDC inscribed in that section, as many right lines PQ, PR, PS, PT are drawn in given angles, each line to each side; the rectangle $PQ\times PR$ of those on the opposite sides AB, CD, will be to the rectangle $PS\times PT$ of those on the other two opposite sides AC, BD, in a given ratio.

Case 1. Let us suppose first that the lines drawn to one pair of opposite sides are parallel to either of the other sides; as PQ and PR (Pl. 8. Fig. 4) to the side AC, and PS and PT to the side AB. And farther, that one pair of the opposite sides, as AC and BD, are parallel betwixt themselves; then 105the right line which bisects those parallel sides will be one of the diameters of the conic section, and will likewise bisect RQ. Let O be the point in which RQ is bisected, and PO will be an ordinate to that diameter. Produce PO to K so that OK may be equal to PO, and OK will be an ordinate on the other side of that diameter. Since therefore the points A, B, P, and K are placed in the conic section, and PK cuts AB in a given angle, the rectangle PQK (by prop. 17. 19. 21. & 23. book 3. of Apollonius's conics) will be to the rectangle AQB in a given ratio. But QK and PR are equal, as being the differences of the equal lines OK, OP, and OQ, OR; whence the rectangles PQK and $PQ\times PR$ are equal; and therefore the rectangle $PQ\times PR$ is to the rectangle AQB, that is, to the rectangle $PS\times PT$ in a given ratio. Q. E. D.

Case 2. Let us next suppose that the opposite sides AC and BD (Pl. 8. Fig. 5.) of the trapezium, are not parallel. Draw Bd parallel to AC and meeting as well the right line ST in t, as the conic section in d. Join Cd cutting PQ in r, and draw DM parallel to PQ, cutting Cd in M and AB in N. Then (because of the similar triangles BTt, DBN,) Bt or PQ is to Tt as DN to NB. And so Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antecedents by the antecedents and the consequents by the consequents, as the rectangle $PQ\times Rr$ is to the rectangle $PS\times Tt$, so will the rectangle NDM be to the rectangle ANB, and (by case 1.) so is the rectangle $PQ\times Pr$ to the rectangle $PS\times Pt$, and by division, so is the rectangle $PQ\times PR$ to the rectangle $PS\times PT$. Q. E. D.

Case 3. Let us suppose lastly the four lines PQ, PR, PS, PT (Pl. 8. Fig. 6.) not to be parallel 106to the sides AC, AB, but any way inclined to them. In their place draw Pq, Pr parallel to AC; and Ps, Pt parallel to AB; and because the angles of the triangles PQq, PRr, PSs, PTt are given, the ratio's of PQ to Pq, PR to Pr, PS to Ps, PT to Pt will be also given; and therefore the compounded ratio's $PQ\times PR$ to $Pq\times Pr$, and $PS\times PT$ to $Ps\times Pt$ are given. But from what we have demonstrated before, the ratio of $Pq\times Pr$ to $Ps\times Pt$ is given; and therefore also the ratio of $PQ\times PR$ to $PS\times PT$. Q. E. D.

Lemma XVIII

The same things supposed, if the rectangle $PQ\times PR$ of the lines drawn to the two opposite sides of the trapezium is to the rectangle $PS\times PT$ of those drawn to the other two sides, in a given ratio; the point P, from whence those lines are drawn, will be placed in a conic section described about the trapezium. (Pl. 8. Fig. 7.)

Conceive a conic section to be described passing through the points A, B, C, D, and any one of the infinite number of points P, as for example p; I say the point P will be always placed in this section. If you deny the thing, join AP cutting this conic section somewhere else if possible than in P, as in b. Therefore if from those points p and b, in the given angles to the sides of the trapezium, we draw the right lines pq, pr, ps, pt, and bk, bn, bf, bd, we shall have 107as $bk\times bn$ to $bf\times bd$, so (by Lem. 17.) $pq\times pr$ to $ps\times pt$; and so (by supposition) $PQ\times PR$ to $PS\times PT$. And because of the similar trapezia bkAf, PQAS, as bk to bf; so PQ to PS. Wherefore by dividing the terms of the preceding proportion by the correspondent terms of this, we shall have bn to bd as PR to PT. And therefore the equiangular trapezia Dnbd, DRPT are similar, and consequently their diagonals Db, DP do coincide. Wherefore b falls in the intersection of the right lines AP, DP, and consequently coincides with the point P. And therefore the point P where-ever it is taken falls to be in the assigned conic section. Q. E. D.

Cor. Hence if three right lines PQ, PR, PS, are drawn from a common point P to as many other right lines given in position AB, CD, AC, each to each, in as many angles respectively given, and the rectangle $PQ\times PR$ under any two of the lines drawn be to the square $P{S}^2$ of the third in a given ratio: The point P, from which the right lines are drawn, will be placed in a conic section that touches the lines AB, CD in A and C; and the contrary. For the position of the three right lines AB, CD, AC remain the same, let the line BD approach to and coincide with the line AC; then let the line PT come likewise to coincide with the line PS; and the rectangle $PS\times PT$ will become $P{S}^2$, and the right lines AB, CD, which before did cut the curve in the points A and B, C, and D, can no longer cut, but only touch, the curve in those co-inciding points.

Scholium

108In this lemma, the name of conic section is to be understood in a large sense, comprehending as well the rectilinear section thro' the vertex of the cone, as the circular one parallel to the base. For if the point p happens to be in a right line, by which the points A and D or C and B are joined, the conic section will be changed into two right lines, one of which is that right line upon which the point p falls, and the other is a right line that joins other two of the four points. If the two opposite angles of the trapezium taken together are equal to two right angles, and if the four lines PQ, PR, PS, PT are drawn to the sides thereof at right angles, or any other equal angles, and the rectangle $PQ\times PR$ under two of the lines drawn PQ and PR, is equal to the rectangle $PS\times PT$ under the other two PS and PT, the conic section will become a circle. And the same thing will happen, if the four lines are drawn in any angles, and the rectangle $PQ\times PR$ under one pair of the lines drawn, is to the rectangle $PS\times PT$ under the other pair, as the rectangle under the sines of the angles S, T in which the two last lines PS, PT are drawn, to the rectangle under the sines of the angles Q, R, in which the two first PQ, PR are drawn. In all other cases the locus of the point P will be one of the three figures, which pass commonly by the name of the conic sections. But in room of the trapezium ABCD, we may substitute a quadrilateral figure whose two opposite sides cross one another like diagonals. And one or two of the four points A, B, C, D may be supposed to be removed 109to an infinite distance, by which means the sides of the figure which converge to those points, will become parallel: And in this case the conic section will pass through the other points, and will go the same way as the parallels in infinitum.

Lemma XIX

To find a point P (Pl. 8. Fig. 8.) from which if four right lines PQ, PR, PS, PT are drawn to as many other right lines AB, CD, AC, BD given by position, each to each, at given angles, the rectangle $PQ\times PR$, under any two of the lines drawn, shall be to the rectangle $PS\times PT$, under the other two, in a given ratio.

Suppose the lines AB, CD, to which the two right lines PQ, PR, containing one of the rectangles, are drawn to meet two other lines, given by position, in the points A, B, C, D. From one of those as A, draw any right line AH, in which you would find the point P. Let this cut the opposite lines BD, CD, in H and I; and, because all the angles of the figure are given, the ratio of PQ to PA, and PA to PS, and therefore of PQ to PS will be also given. Subducting this ratio from the given ratio of $PQ\times PR$ to $PS\times PT$, the ratio of PR to PT will be given; and adding the given ratio's of PI to PR, and PT to PH the ratio of PI to PH, and therefore the point P will be given. Q. E. I.

Cor. 1. Hence also a tangent may be drawn to any point D of the locus of all the points P. For 110the chord PD, where the points P and D meet, that is, where AH is drawn thro' the point D, becomes a tangent. In which case the ultimate ratio of the evanescent lines IP and PH will be found as above. Therefore draw CF parallel to AD, meeting BD in F, and cut it in E in the same ultimate ratio, then DE will be the tangent; because CF, and the evanescent IH are parallel, and similarly cut in E and P.

Cor. 2. Hence also the locus of all the points P may be determined. Through any of the points A, B, C, D, as A, (Pl. 9. Fig. 1.) draw AE touching the locus, and through any other point B parallel to the tangent, draw BF meeting the locus in F: And find the point F by this lemma. Bisect BF in G, and drawing the indefinite line AG, this will be the position of the diameter to which BG, and FG are ordinates. Let this AG meet the locus in H, and AH will be its diameter or latus transversum, to which the latus rectum will be as $B{G}^2$ to $AG\times GH$. If AG no where meets the locus, the line AH being infinite the locus will be a parabola; and its latus rectum corresponding to the diameter AG will be $\frac{B{G}^2}{AG}$. But if it does meet it any where, the locus will be an hyperbola, when the points A and H are placed on the same side the point G; and an ellipsis, if the point G falls between the points A and H; unless perhaps the angle AGB is a right angle, and at the same time $B{G}^2$ equal to the rectangle AGH, in which case the locus will be a circle.

And so we have given in this corollary a solution of that famous problem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius; and this not an analytical calculus, but a geometrical composition, such as the ancients required.

Lemma XX

111If the two opposite angular points A and P (Pl. 9. Fig. 2.) of any parallelogram ASPQ touch any conic section in the points A and P; and the sides AQ, AS of one of those angles, indefinitely produced, meet the same conic section in B and C; and from the points of concourse B and C to any fifth point D of the conic section, two right lines BD, CD are drawn meeting the two other sides PS, PQ of the parallelogram, indefinitely produced, in T and R; the parts PR and PT, cut off from the sides, will always be one to the other in a given ratio. And vice versa, if those parts cut off are one to the other in a given ratio, the locus of the point D will be a conic section, passing through the four points A, B, C, P.

Case 1. Join BP, CP, and from the point D draw the two right lines DG, DE, of which the first DG shall be parallel to AB, and meet PB, PQ, CA in H, I, G; and the other DE shall be parallel to AC, and meet PC, PS, AB, in F, K, E; and (by Lem. 17.) the rectangle $DE\times DF$ will be to the rectangle $DG\times DH$, in a given ratio. But PQ is to DE (or IQ) as PB to HB, and consequently as PT to DH; and by permutation, PQ 112is to PT, as DE to DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS to DG; and, by permutation, PR is to PS as DF to DG; and, by compounding those ratio's, the rectangle $PQ\times PR$ will be to the rectangle $PS\times PT$ as the rectangle $DE\times DF$ is to the rectangle $DG\times DH$, and consequently in a given ratio. But PQ and PS are given, and therefore the ratio of PR to PT is given. Q. E. D.

Case 2. But if PR and PT are supposed to be in a given ratio one to the other, then by going back again by a like reasoning, it will follow that the rectangle $DE\times DF$ is to the rectangle $DG\times DH$ in a given ratio; and so the point D (by lem. 18.) will lie in a conic section passing thro' the points A, B, C, P, as its locus. Q. E. D.

Cor. 1. Hence if we draw BC cutting PQ in r, and in PT take Pt to Pr in the same ratio which PT has to PR: Then Bt will touch the conic section in the point B. For suppose the point D to coalesce with the point B, so that the chord BD vanishing, BT shall become a tangent, and CD and BT will coincide with CB and Bt.

Cor. 2. And vice versa, if Bt is a tangent, and the lines BD, CD meet in any point D of a conic section; PR will be to PT as Pr to Pt. And on the contrary, if PR is to PT as Pr to Pt, then BD, and CD will meet in some point D of a conic section.

Cor. 3. One conic section cannot cut another conic section in more than four points. For, if it is possible, let two conic sections pass thro' the five points A, B, C, P, O; and let the right line BD cut them in the points D, d, and the right line Cd cut the right line PQ in q. Therefore PR is to PT as Pq to PT:

113Whence PR and Pq are equal one to the other, against the supposition.

Lemma XXI

If two moveable and indefinite right lines BM, CM drawn through given points B, C, at poles, do by their point of concourse M describe a third right line MN given by position; and other two indefinite right lines BD, CD are drawn, making with the former two at those given points B, C, given angles, MBD, MCD: I say that those two right lines BD, CD will by their point of concourse D describe a conic section passing through the points B, C. And vice versa, if the right lines BD, CD do by their point of concourse D describe a conic section passing through the given points B, C, A, and the angle DBM is always equal to the given angle ABC, as well as the angle DCM always equal to the given angle ACB: the point M will lie in a right line given ly position, as its locus. Pl. 9. Fig. 3.

For in the right line MN let a point N be given, and when the moveable point M falls on the immoveable point N, let the moveable point D fall on an immoveable point P. Join CN, BN, CP

114BP, and from the point P draw the right lines PT, PR meeting BD, CD in T and R, and making the angle BPT equal to the given angle BNM and the angle CPR equal to the given angle CNM. Wherefore since (by supposition) the angles MBD, NBP are equal, as also the angles MCD, NCP; take away the angles NBD and NCD that are common, and there will remain the angles NBM and PBT, NCM and PCR equal; and therefore the triangles NBM, PBT are similar, as also the triangles NCM, PCR. Wherefore PT is to NM as PB to NB; and PR to NM, as PC to NC. But the points B, C, N, P are immoveable: Wherefore PT and PR have a given ratio to NM, and consequently a given ratio between themselves; and therefore, (by lem. 20.) the point D wherein the moveable right lines BT and CR perpetually concur, will be placed in a conic section passing through the points B, C, P. Q. E. D.

And vice versa, if the moveable point D (Pl. 9. Fig. 4.) lies in a conic section passing through the given points B, C, A; and the angle DBM is always equal to the given angle ABC, and the angle DCM always equal to the given angle ACB, and when the point D falls successively on any two immoveable points p, P, of the conic section, the moveable point M falls successively on two immoveable points n, N: through these points n, N, draw the right line nN, this line nN will be the perpetual locus of that moveable point M. For if possible, let the point M be placed in any curve line. Therefore the point D will be placed in a conic section passing through the five points B, C, A, p, P, when the point M is perpetually placed in a curve line. But from what was demonstrated before, the point D 115will be also placed in a conic section, passing through the same five points B, C, A, p, P, when the point M is perpetually placed in a right line. Wherefore the two conic sections will both pass through the same given points, against corol. 3. lem. 20. It is therefore absurd to suppose that the point M is placed in a curve line. Q. E. D.

Proposition XXII. Problem XIV

To describe a trajectory that shall pass through five given points. Pl. 9 Fig. 5.

Let the five given points be A, B, C, P, D. From any one of them as A, to any other two as B, C, which may be called the poles, draw the right lines AB, AC, and parallel to those the lines TPS, PRQ through the fourth point P. Then from the two poles B, C, draw through the fifth point D two indefinite lines BDT, CRD, meeting with the last drawn lines TPS, PRQ (the former with the former, and the latter with the latter) in T and R. Then drawing the right line tr parallel to TR, cutting off from the right lines PT, PR, any segments Pt, Pr, proportional to PT, PR; and if through their extremities t, r, and the poles B, C, the right lines Bt, Cr are drawn, meeting in d, that point d will be placed in the trajectory required. For (by lem. 20.) that point d is placed in a conic section passing through the four points A, B, C, P; and the lines Rr, Tt vanishing, the point d comes to coincide with the point D. Wherefore the conic section passes through the five points A, B, C, P, D. Q. E. D.

The same otherwise. Pl. 9. Fig. 6.

116Of the given points join any three as A, B, C; and about two of them B, C, as poles, making the angles ABC, ACB of a given magnitude to revolve, apply the legs BA, CA, first to the point D, then to the point P, and mark the points M, N, in which the other legs BL, CL intersect each the other in both cases. Draw the indefinite right line MN and let those moveable angles revolve about their poles B, C, in such manner that the intersection, which is now supposed to be m, of the legs BL, CL, or BM, CM, may always fall in that indefinite right line MN; and the intersection which is now supposed to be d, of the legs BA, CA, or BD, CD, will describe the trajectory required PADdB. For, (by lem. 21.) the point d will be placed in a conic section passing through the points B, C; and when the point m comes to coincide with the points L, M, N, the point d will (by construction) come to coincide with the points A, D, P. Wherefore a conic section will be described that shall pass through the five points A, B, C, P, D. Q. E. F.

Cor. 1. Hence a right line may be readily drawn which shall be a tangent to the trajectory in any given point B. Let the point d come to coincide with the point B, and the right line Bd will become the tangent required.

Cor. 2. Hence also may be found the centres, diameters, and latera recta of the trajectories, as in cor. 2. lem. 19.

Scholium

117The former of these constructions (Fig. 5.) will become something more simple by joining BP, and in that line, produced if need be, taking Bp to BP as PR is to PT; and through p drawing the indefinite right line pe parallel to SPT; and in that line pe taking always pe equal to Pr; and drawing the right lines Be, Cr to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in the same ratio, pe and Pr will be always equal. After this manner the points of the trajectory are most readily found, unless you would rather describe the curve mechanically as in the second construction.

Proposition XXIII. Problem XV

To describe a trajectory that shall pass through four given points, and touch a right line given by position. Pl. 10. Fig. 1.

Case 1. Suppose that HB is the given tangent, B the point of contact, and C, D, P, the three other given points. Join BC, and draw PS parallel to BH, and PQ parallel to BC, compleat the parallelogram BSPQ. Draw BD cutting SP in T and CD cutting PQ in R. Lastly, drawing any line tr parallel to TR, cutting off from PQ, PS, the segments Pr, Pt proportional to PR, PT respectively; and drawing Cr, Bt, their point of concourse d will (by lem. 20.) always fall on the trajectory to be described.

The same otherwise. Pl. 10. Fig. 2.

118Let the angle CBH of a given magnitude revolve about the pole B, as also the rectilinear radius DC both ways produced, about the pole C. Mark the points M, N, on which the leg BC of the angle cuts that radius when BH the other leg thereof meets the same radius in the points P and D. Then drawing the indefinite line MN, let that radius CP or CD and the leg BC of the angle perpetually meet in this line; and the point of concourse of the other leg BH with the radius will delineate the trajectory required.

For if in the constructions of the preceding problem the point A comes to a coincidence with the point B the lines CA and CB will coincide, and the line AB, in its last situation, will become the tangent BH; and therefore the constructions there set down will become the same with the constructions here described. Wherefore the concourse of the leg BH with the radius will describe a conic section passing through the points C, D, P, and touching the line BH in the point B. Q. E. F.

Case 2. Suppose the four points B, C, D, P, (Pl. 10. Fig. 3.) given, being situated without the tangent HI join each two by the lines BD, CP, meeting in G, and cutting the tangent in H and I. Cut the tangent in A in such manner that HA may be to IA, as the rectangle under a mean proportional between CG and GP, and a mean proportional between BH and HD, is to a rectangle under a mean proportional between GD and GB, and a mean proportional between PI and IC; and A will be the point of contact. For if HX, a parallel to the right 119line PI, cuts the trajectory in any points X and Y; the point A (by the properties of the conic sections) will come to be so placed, that $H{A}^2$ will become to $A{I}^2$ in a ratio that is compounded out of the ratio of the rectangle XHY to the rectangle BHD, or of the rectangle CGP to the rectangle DGB; and the ratio of the rectangle BHD to the rectangle PIC. But after the point of contact A is found, the trajectory will be described as in the first case. Q. E. F. But the point A may be taken either between or without the points H and I; upon which account a twofold trajectory may be described.

Proposition XXIV. Problem XVI

To describe a trajectory that shall pass through three given points, and touch two right lines given by position. Pl. 10. Fig. 4.

Suppose HI, KL to be the given tangents, and B, C, D, the given points. Through any two of those points as B, D, draw the indefinite right line BD meeting the tangents in the points H, K. Then likewise through any other two of these points as C, D, draw the indefinite right line CD, meeting the tangents in the points I, L. Cut the lines drawn in R and S, so that HR may be to KR, as the mean proportional between BH and HD is to the mean proportional between BK and KD; and IS to LS, as the mean proportional between CI and ID is to the mean proportional between CL and LD. But you may cut, at pleasure, either within 120or between the points K and H, I and L, or without them; then draw RS cutting the tangents in A and P, and A and P will be the points of contact. For if A and P are supposed to be the points of contact situated any where else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI a right line IT is drawn, parallel to the other tangent KL, and meeting the curve in X and Y, and in that right line there be taken IZ equal to a mean proportional between IX and IY; the rectangle XIY or $I{Z}^2$, will (by the properties of the conic sections) be to $L{P}^2$, as the rectangle CID is to the rectangle CLD, that is (by the construction) as $S{I}^2$ is to $S{L}^2$, and therefore IZ is to LP, as SI to SL. Wherefore the points S, P, Z, are in one right line. Moreover, since the tangents meet in G, the rectangle XIY or $I{Z}^2$ will (by the properties of the conic sections) be to $I{A}^2$ as $G{P}^2$ is to $G{A}^2$, and consequently IZ will be to IA, as GP to GA. Wherefore the points P, Z, A, lie in one right line, and therefore the points S, P, and A are in one right line. And the same argument will prove that the points R, P, and A are in one right line. Wherefore the points of contact A and P lie in the right line RS. But after these points are found the trajectory may be described as in the first case of the preceding problem. Q. E. F.

In this proposition, and case 2. of the foregoing, the constructions are the same, whether the right line XY cut the trajectory in X and Y or not; neither do they depend upon that section. But the constructions being demonstrated where that right line does cut the trajectory, the constructions, where it does not, 121are also known; and therefore, for brevity's sake, I omit any farther demonstration of them.

Lemma XXII

To transform figures into other figures of the same kind. Pl. 10. Fig. 5.

Suppose that any figure HGI is to be transformed. Draw, at pleasure, two parallel lines AO, BL, cutting any third line AB given by position, in A and B, and from any point G of the figure, draw out any right line GD, parallel to OA, till it meet the right line AB. Then from any given point O in the line OA, draw to the point D the right line OD, meeting BL in d, and from the point of concourse raise the right line dg containing any given angle with the right line BL, and having such ratio to Od, as DG has to OD; and g will be the point in the new figure hgi, corresponding to the point G. And in like manner the several points of the first figure will give as many correspondent points of the new figure. If we therefore conceive the point G to be carried along by a continual motion through all the points of the first figure, the point g will be likewise carried along by a continual motion through all the points of the new figure, and describe the same. For distinction's sake, let us call DG the first ordinate, dg the new ordinate, AD the first abscissa, ad the new abscissa; O the pole, OD the abscinding radius, OA the first ordinate radius, and Oa (by which the parallelogram 122OABa is compleated) the new ordinate radius.

I say, then, that if the point G is placed in a right line given by position, the point g will be also placed in a right line given by position. If the point G is placed in a conic section, the point g will be likewise placed in a conic section. And here I understand the circle as one of the conic sections. But farther, if the point G is placed in a line of the third analytical order, the point g will also be placed in a line of the third order, and so on in curve lines of higher orders. The two lines in which the points G, g, are placed, will be always of the same analytical order. For as ad is to OA, so are od to OD, dg to DG, and AB to AD; and therefore AD is equal to $\frac{OA\times AB}{ad}$ and DG equal to $\frac{OA\times dg}{ad}$. Now if the point G is placed in a right line, and therefore, in any equation by which the relation between the abscissa AD and the ordinate DG is expressed, those indetermined lines AD and DG rise no higher than to one dimension, by writing this equation $\frac{OA\times AB}{ad}$ in place of AD, and $\frac{OA\times dg}{ad}$ in place of DG, a new equation will be produced, in which the new abscissa ad and new ordinate dg rise only to one dimension; and which therefore must denote a right line, But if AD and DG (or either of them) had risen to two dimensions in the first equation, ad and dg would likewise have risen to two dimensions in the second equation. And so on in three or more dimensions. The indetermined 123lines ad, dg in the second equation, and AD, DG, in the first will always rise to the same number of dimensions; and therefore the lines in which the points G, g, are placed are of the same analytical order.

I say farther, that if any right line touches the curve line in the first figure, the same right line transferred the same way with the curve into the new figure, will touch that curve line in the new figure, and vice versa. For if any two points of the curve in the first figure are supposed to approach one the other till they come to coincide; the same points transferred will approach one the other till they come to coincide in the new figure; and therefore the right lines with which those points are joined will come together tangents of the curves in both figures. I might have given demonstrations of these assertions in a more geometrical form; but I study to be brief.

Wherefore if one rectilinear figure is to be transformed into another we need only transfer the intersections of the right lines of which the first figure consists, and through the transferred intersections to draw right lines in the new figure. But if a curvilinear figure is to be transformed we must transfer the points, the tangents, and other right lines, by means of which the curve line is defined. This lemma is of use in the solution of the more difficult problems. For thereby we may transform the proposed figures if they are intricate into others that are more simple, Thus any right lines converging to a point are transformed into parallels; by taking for the first ordinate radius any right line that passes through the point of concourse of the converging lines, and that, because their point of concourse is by this means made 124to go off in infinitum, and parallel lines are such as tend to a point infinitely remote. And after the problem is solved in the new figure, if by the inverse operations we transform the new into the first figure we shall have the solution required.

This lemma is also of use in the solution of solid problems. For as often as two conic sections occur by the intersection of which a problem may be solved; any one of them may be transformed if it is an hyperbola or a parabola, into an ellipsis, and then this ellipsis may be easily changed into a circle. So also a right line and a conic section, in the construction of plane problems, may be transformed into a right line and a circle.

Proposition XXV. Problem XVII

To describe a trajectory that shall pass through two given points, and touch three right lines given by position. Pl. 10. Fig. 6.

Through the concourse of any two of the tangents one with the other, and the concourse of the third tangent with the right line which passes through the two given points, draw an indefinite right line; and taking this line for the first ordinate radius transform the figure by the preceding lemma into a new figure. In this figure those two tangents will become parallel to each other, and the third tangent will be parallel to the right line that passes through the two given points. Suppose hi, kl to be those two parallel tangents, ik the third tangent, and hl a right line parallel thereto, passing through those points a, b, 125through which the conic section ought to pass in this new figure; and compleating the parallelogram hikl, let the right lines hi, ik, kl be so cut in c, d, e, that hc may be to the square root of the rectangle ahb, ic to id, and ke to kd, as the sum of the right lines hi and kl is to the sum of the three lines, the first whereof is the right line ik, and the other two are the square roots of the rectangles ahb and alb; and c, d, e, will be the points of contact. For by the properties of the conic sections $h{c}^2$ to the rectangle ahb, and $i{c}^2$ to $i{d}^2$, and $k{e}^2$ to $k{d}^2$, and $e{l}^2$ to the rectangle alb, are all in the same ratio; and therefore hc to the square root of ahb, ic to id, ke to kd, and el to the square root of alb, are in the subduplicate of that ratio; and by composition in the given ratio of the sum of all the antecedents $hi+kl$, to the sum of all the consequents $\sqrt{ahb}+ik+\sqrt{alb}$. Wherefore from that given ratio we have the points of contact c, d, e, in the new figure. By the inverted operations of the last lemma, let those points be transferred into the first figure, and the trajectory will be there described by prob. 14. Q. E. F. But according as the points a, b, fall between the points h, l, or without them, the points c, d, e, must be taken either between the points h, i, k, l, or without them. If one of the points a, b, falls between the points h, l, and the other without the points h, l, the problem is impossible.

Proposition XXVI. Problem. XVIII

126To describe a trajectory that shall pass through a given point, and touch four right lines given by position. Pl. 11. Fig. 1.

From the common intersections of any two of the tangents to the common intersection of the other two draw an indefinite right line; and taking this line for the first ordinate radius transform the figure (by lem. 22.) into a new figure, and the two pairs of tangents each of which before concurred in the first ordinate radius will now become parallel. Let hi and kl, ik and hl, be those pairs of parallels compleating the parallelogram hikl. And let p be the point in this new figure corresponding to the given point in the first figure. Through O the centre of the figure draw pq; and Oq being equal to Op, q will be the other point through which the conic section must pass in this new figure. Let this point be transferred by the inverse operation of lem. 22. into the first figure, and there we shall have the two points, through which the trajectory is to be described. But through those points that trajectory may be described by prob. 17. Q. E. F.

Lemma XXIII

127If two right lines as AC, BD given by position, and terminating in given points A, B, are in a given ratio one to the other, and the right line CD, by which the indetermined points C, D are joined, is cut in K in a given ratio; I say that the point K will be placed in a right line given by position. Pl. 11. Fig. 2.

For let the right lines AC, BD meet in E, and in BE take BG to AE, as BD is to AC, and let FD be always equal to the given line EG; and by construction, EC will be to GD, that is, to EF, as AC to BD, and therefore in a given ratio; and therefore the triangle EFC will be given in kind. Let CF be cut in L so as CL may be to CF in the ratio of CK to CD; and because that is a given ratio, the triangle EFL will be given in kind, and therefore the point L will be placed in the right line EL given by position. Join LK and the triangles CLK, CFD will be similar; and because FD is a given line, and LK is to FD in a given ratio, LK will be also given. To this let EH be taken equal, and ELKH will be always a parallelogram. And therefore the point K is always placed in the side HK (given by position) of that parallelogram. Q. E. D.

128Cor. Because the figure EFLC is given in kind, the three right lines EF, EL and EC, that is GD, HK and EC will have given ratio's to each other.

Lemma XXIV

If three right lines, two whereof are parallel, and given by position, touch any conic section; I say, that the semidiameter of the section which is parallel to those two is a mean proportional between the segments of those two, that are intercepted between the points of contact and the third tangent. Pl. 11. Fig. 3.

Let AF, GB be the two parallels touching the conic section ADB in A and B; EF the third right line touching the conic section in I, and meeting the two former tangents in F and G, and let CD be the semi-diameter of the figure parallel to those tangents; I say, that AF, CD, BG are continually proportional.

For if the conjugate diameters AB, DM meet the tangent FG in E and H, and cut one the other in C, and the parallelogram IKCL be compleated; from the nature of the conic sections, EC will be to CA as CA to CL, and so by division, $EC-CA$ to $CA-CL$ or EA to AL; and by composition, EA to $EA+AL$ or EL, as EC to $EC+CA$ or EB; and therefore (because of 129the similitude of the triangles EAF, ELI, ECH, EBG) AF is to LI as CH to BG. Likewise from the nature of the conic sections, LI (or CK) is to CD as CD to CH; and therefore (ex æquo perturbate) AF is to CD, as CD to BG. Q. E. D.

Cor. 1. Hence if two tangents FG, PQ meet two parallel tangents AF, BG in F and G, P and Q, and cut one the other in O; AF (ex æquo perturbate) will be to BQ as AP to BG, and by division, as FP to GQ, and therefore as FO to OG.

Cor. 2. Whence also the two right lines PG, FQ drawn through the points P and G, F and Q, will meet in the right line ACB, passing through the centre of the figure and the points of contact A, B.

Lemma XXV

If four sides of a parallelogram indefinitely produced touch any conic section, and are cut by a fifth tangent; I say, that taking those segments of any two conterminous sides that terminate in opposite angles of the parallelogram, either segment is to the side from which it is cut off, as that part of the other conterminous side which is intercepted between the point of contact and the third side, is to the other segment. Pl. 11. Fig. 4.

Let the four sides ML, IK, KL, MI of the parallelogram MLIK touch the conic section in 130A, B, C, D; and let the fifth tangent FQ cut those sides in F, Q, H and E, and taking the segments ME, KQ of the sides MI, KI; or the segments KH, MF of the sides KL, ML; I say, that ME is to MI as BK to KQ; and KH to KL, as AM to MF. For, by cor. 1. of the preceding lemma, ME is to EL as (AM or) BK to BQ; and, by composition, ME is to MI as BK to KQ. Q. E. D. Also KH is to HL as (BK or) AM to AF and by division KH to KL, as AM to MF. Q. E. D.

Cor. 1. Hence if a parallelogram IKLM described about a given conic section is given, the rectangle $KQ\times ME$, as also the rectangle $KH\times MF$ equal thereto, will be given. For, by reason of the similar triangles KQH, MFE, those rectangles are equal.

Cor. 2. And if a sixth tangent eq is drawn meeting the tangents KI, MI in q and e; the rectangle $KQ\times ME$ will be equal to the rectangle $Kq\times Me$, and KQ will be to Me, as Kq to ME, and by division as Qq to Ee.

Cor. 3. Hence also if Eq, eQ, are joined and bisected, and a right line is drawn through the points of bisection, this right line will pass through the centre of the conic section. For since Qq is to Ee, as KQ to Me; the same right line will pass through the middle of all the lines Eq, eQ, MK (by lem. 22.) and the middle point of the right line MK is the centre of the section.

Proposition XXVII. Problem XIX

131To describe a trajectory that may touch five right lines given by position. Pl. 11. Fig. 5.

Supposing ABG, BCF, GCD, FDE, EA to be the tangents given by position. Bisect in M and N, AF, BE the diagonals of the quadrilateral figure ABFE contained under any four of them; and (by cor. 3. lem. 25) the right line MN drawn through the points of bisection will pass through the centre of the trajectory. Again, bisect in P and Q the diagonals (if I may so call them) BD, GF of the quadrilateral figure BGDF contained under any other four tangents, and the right line PQ drawn through the points of bisection will pass through the centre of the trajectory. And therefore the centre will be given in the concourse of the bisecting lines. Suppose it to be O. Parallel to any tangent BC draw KL, at such distance that the centre O may be placed in the middle between the parallels; this KL will touch the trajectory to be described. Let this cut any other two tangents GCD, FDE, in L and K. Through the points C and K, F and L, where the tangents not parallel CL, FK meet the parallel tangents CF, KL, draw CK, FL meeting in R; and the right line OR drawn and produced, will cut the parallel tangents CF, KL, in the points of contact. This appears from cor. 3. lem. 24. And by the same method the other points of contact may be found, and then the trajectory may be described by prob. 14. Q. E. F.

Scholium

132Under the preceding propositions are comprehended those problems wherein either the centres or asymptotes of the trajectories are given. For when points and tangents and the centre are given, as many other points and as many other tangents are given at an equal distance on the other side of the centre. And an asymptote is to be considered as a tangent, and its infinitely remote extremity (if we may say so) is a point of contact. Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an asymptote, and the constructions of the preceding problems will be changed into the constructions of those problems wherein the asymptote is given.

After the trajectory is described, we may find its axes and foci in this manner. In the construction and figure of lem. 21. (Pl. 12. Fig. 1.) let those legs BP, CP, of the moveable angles PBN, PCN, by the concourse of which the trajectory was described, be made parallel one to the other; and retaining that position, let them revolve about their poles B, C, in that figure. In the mean while let the other legs CN, BN, of those angles, by their concourse K or k, describe the circle BKGC. Let O be the centre of this circle; and from this centre upon the ruler MN, wherein those legs CN, BN did concur while the trajectory was described, let fall the perpendicular OH meeting the circle in K and L. And when those other legs CK, BK meet in the point K that is nearest to the ruler, the first legs CP, BP will be parallel to the greater axis 133and perpendicular on the lesser; and the contrary will happen if those legs meet in the remotest point L. Whence if the centre of the trajectory is given, the axes will be given; and those being given, the foci will be readily found.

But the squares of the axes are one to the other as KH to LH, and thence it is easy to describe a trajectory given in kind through four given points. For if two of the given points are made the poles C, B, the third will give the moveable angles PCK, PBK; but those being given, the circle BGKC may be described. Then, because the trajectory is given in kind, the ratio of OH to OK, and therefore OH it self, will be given. About the centre O, with the interval OH, describe another circle, and the right line that touches this circle and passes through the concourse of the legs CK, BK, when the first legs CP, BP, meet in the fourth given point, will be the ruler MN, by means of which the trajectory may be described. Whence also on the other hand a trapezium given in kind (excepting a few cases that are impossible) may be inscribed in a given conic section.

There are also other lemma's by the help of which trajectories given in kind may be described through given points, and touching given lines. Of such a sort is this, that if a right line is drawn through any point given by position, that may cut a given conic section in two points, and the distance of the intersections is bisected, the point of bisection will touch another conic section of the same kind with the former, and having its axes parallel to the axes of the former. But I hasten to things of greater use.

Lemma XXVI

134To place the three angles of a triangle, given both in kind and magnitude, in respect of as many right lines given by position, provided they are not all parallel among themselves, in such manner that the several angles may touch the several lines. Pl. 12. Fig. 2.

Three indefinite right lines AB, AC, BC, are given by position, and it is required so to place the triangle DEF that its angle D may touch the line AB, its angle E the line AC, and its angle F the line BC. Upon DE, DF and EF, describe three segments of circles DRE, DGF, EMF, capable of angles equal to the angles BAC, ABC, ACB respectively. But those segments are to be described towards such sides of the lines DE, DF, EF, that the letters DRED may turn round about in the same order with the letters BACB; the letters DGFD in the same order with the letters ABCA; and the letters EMFE in the same order with the letters ACBA; then compleating those segments into entire circles, let the two former circles cut one the other in G, and suppose P and Q to be their centres. Then joining GP, PQ, take Ga to AB, as GP is to PQ; and about the centre G, with the interval Ga describe a circle that may cut the first circle DGE in a. Join aD cutting the second circle DFG in b, as 135well as aE cutting the third circle EMF in c. Compleat the figure ABCdef similar and equal to the figure abcDEF. I say the thing is done.

For drawing Fc meeting aD in n, and joining aG, bG, QG, QD, PD; by construction the angle EaD is equal to the angle CAB, and the angle acF equal to the angle ACB; and therefore the triangle anc equiangular to the triangle ABC. Wherefore the angle anc or FnD is equal to the angle ABC, and consequently to the angle FbD; and therefore the point n falls on the point b. Moreover the angle GPQ which is half the angle GPD at the centre is equal to the angle GaD at the circumference; and the angle GQP, which is half the angle GQD at the centre, is equal to the complement to two right angles of the angle GbD at the circumference, and therefore equal to the angle Gab. Upon which account the triangles GPQ, Gab, are similar, and Ga is to ab as GP to PQ; that is (by construction) as Ga to AB. Wherefore ab and AB are equal; and consequently the triangles abc, ABC, which we have now proved to be similar, are also equal. And therefore since the angles D, E, F, of the triangle DEF do respectively touch the sides ab, ac, bc of the triangle abc, the figure ABCdef may be compleated similar and equal to the figure abcDEF, and by compleating it the problem will be solved. Q. E. F.

Cor. Hence a right line may be drawn whose parts given in length may be intercepted between three right lines given by position. Suppose the triangle DEF, by the access of its point D to the side EF, and by having the sides DE, DF placed in directum to be changed into a right line 136whose given part DE is to be interposed between the right lines AB, AC given by position; and its given part DF is to be interposed between the right lines AB, BC, given by position; then by applying the preceding construction to this case the problem will be solved.

Proposition XXVIII. Problem XX

To describe a trajectory given both in kind and magnitude, given parts of which shall be interposed between three right lines given by position. Pl. 12. Fig. 3.

Suppose a trajectory is to be described that may be similar and equal to the curve line DEF, and may be cut by three right lines AB, AC, BC given by position, into parts DE and EF, similar and equal to the given parts of this curve line.

Draw the right lines DE, EF, DF; and place the angles D, E, F, of this triangle DEF, so as to touch those right lines given by position (by lem. 26). Then about the triangle describe the trajectory, similar and equal to the curve DEF. Q. E. F.

Lemma XXVII

137To describe a trapezium given in kind, the angles whereof may be so placed in respect of four right lines given by position, that are neither all parallel among themselves nor converge to one common point, that the several angles may touch the several lines. Pl. 13. Fig. 1.

Let the four right lines ABC, AD, BD, CE, be given by position; the first cutting the second in A, the third in B, and the fourth in C; and suppose a trapezium fghi is to be described, that may be similar to the trapezium FGHI; and whose angle f, equal to the given angle F, may touch the right line ABC; and the other angles g, h, i, equal to the other given angles G, H, I, may touch the other lines AD, BD, CE, respectively. Join FH, and upon FG, FH, FI describe as many segments of circles FSG, FTH, FVI; the first of which FSG may be capable of an angle equal to the angle BAD; the second FTH capable of an angle equal to the angle CBD; and the third FVI of an angle equal to the angle ACE. But the segments are to be described towards those sides of the lines FG, FH, FI, that the circular order of the letters FSGF, may be the same as of the letters BADB, and that the letters FTHF may turn about in the same order as the letters CBDC, and the letters FVIF in the same order 138as the letters ACEA. Compleat the segments into entire circles, and let P be the centre of the first circle FSG, Q the centre of the second FTH. Join and produce both ways the line PQ, and in it take QR in the same ratio to PQ, as BC has to AB. But QR is to be taken towards that side of the point Q, that the order of the letters P, Q, R may be the same, as of the letters A, B, C; and about the centre R with the interval RF describe a fourth circle FNc cutting the third circle FVI in c. Join Fc cutting the first circle in a, and the second in b. Draw aG, bH, cI, and let the figure ABCfghi be made similar to the figure abcFGHI; and the trapezium fghi will be that which was required to be described.

For let the two first circles FSG, FTH cut one the other in K; join PK, QK, RK, aK, bK, cK, and produce QP to L. The angles FaK, FbK, FcK at the circumferences are the halves of the angles FPK, FQK, FRK, at the centres, and therefore equal to LPK, LQK, LRK the halves of those angles. Wherefore the figure PQRK is equiangular and similar to the figure abcK, and consequently ab is to bc as PQ to QR, that is, as AB to BC. But by construction the angles fAg, fBh, fCi are equal to the angles FaG, FbH, FcI. And therefore the figure ABCfghi may be compleated similar to the figure abcFGHI. Which done a trapezium fghi will be constructed similar to the trapezium FGHI, and which by its angles f, g, h, i will touch the right lines ABC, AD, BD, CE. Q. E. F.

Cor. Hence a right line may be drawn whose parts intercepted in a given order, between four right lines given by position, shall have a given 139proportion among themselves. Let the angles FGH, GHI, be so far increased that the right lines FG, GH, HI, may lie in directum, and by constructing the problem in this case, a right line fghi will be drawn, whose parts fg, gh, hi, intercepted between the four right lines given by position, AB and AD, AD and BD, BD and CE, will be one to another as the lines FG, GH, HI, and will observe the same order among themselves. But the same thing may be more readily done in this manner.

Produce AB to K (Pl. 13. Fig. 2.) and BD to L, so as BK may be to AB, as HI to GH; and DL to BD as GI to FG; and join KL meeting the right line CE in i. Produce iL to M, so as LM may be to iL as GH to HI; then draw MQ parallel to LB and meeting the right line AD in g, and join gi cutting AB, BD in f, h. I say the thing is done.

For let Mg cut the right line AB in Q, and AD the right line KL in S, and draw AP parallel to BD, and meeting iL in P, and gM to Lh (gi to hi, Mi to Li, GI to HI, AK to BK) and AP to BL will be in the same ratio. Cut DL in R, so as DL to RL may be in that same ratio; and because gS to gM, AS to AP, and DS to DL are proportional; therefore (ex æquo) as gS to Lh, so will AS be to BL, and DS to RL; and mixtly $BL-RL$ to $Lh-BL$, as $AS-DS$ to $gS-AS$. That is, BR is to Bh, as AD is to Ag, and therefore as BD to gQ. And alternately BR is to BD, as Bh to gQ, or as fh to fg. But by construction the line BL was cut in D and R, in the same ratio as the line FI in G and H; and therefore 140BR is to BD as FH to FG. Wherefore fh is to fg, as FH to FG. Since therefore gi to hi likewise is as Mi to Li, that is, as GI to HI, it is manifest that the lines FI, fi, are similarly cut in G and H, g and h. Q. E. F.

In the construction of this corollary, after the line LK is drawn cutting CE in i, we may produce iE to V, so as EV may be to Ei as FH to HI, and then draw Vf parallel to BD. It will come to the same, if about the centre i, with an interval IH, we describe a circle cutting BD in X, and produce iX to Y so as iY may be equal to IF, and then draw Yf parallel to BD.

Sir Christopher Wren, and Dr. Wallis have long ago given other solutions of this problem.

Proposition XXIX. Problem XXI

To describe a trajectory given in kind, that may be cut by four right lines given by position, into parts given in order, kind, and proportion.

Suppose a trajectory is to be described that may be similar to the curve line FGHI (Pl. 13. Fig. 3.) and whose parts, similar and proportional to the parts FG, GH, HI of the other, may be intercepted between the right lines AB and AD, AD and BD, BD and CE given by position, viz. the first between the first pair of those lines, the second between the second, and the third between the third. Draw the right lines FG, GH, HI, FI; and (by lem. 27.) describe a trapezium 141fghi that may be similar to the trapezium FGHI, and whose angles f, g, h, i, may touch the right lines given by position, AB, AD, BD, CE, severally according to their order. And then about this trapezium describe a trajectory, that trajectory will be similar to the curve line FGHI.

Scholium

This problem may be likewise constructed in the following manner. Joining FG, GH, HI, FI, (Pl. 13. Fig. 4.), produce GF to V, and join FH, IG, and make the angles CAK, DAL equal to the angles FGH, VFH. Let AK, AL meet the right line BD in K and L, and thence draw KM, LN, of which let KM make the angle AKM equal to the angle GHI, and be it self to AK, as HI is to GH; and let LN make the angle ALN equal to the angle FHI, and be it self to AL, as HI to FH. But AK, KM, AL, LN are to be drawn towards those sides of the lines AD, AK, AL, that the letters CAKMC, ALKA, DALND may be carried round in the same order as the letters FGHIF; and draw MN meeting the right line CE in i. Make the angle iEP equal to the angle IGF, and let PE be to Ei, as FG to GI; and through P draw PQf that may with the right line ADE contain an angle PQE equal to the angle FIG, and may meet the right line AB in f, and join fi. But PE and PQ are to be drawn towards those sides of the lines CE, PE, that the circular order of the letters PEiP and PEQP may be the same, as of the letters FGHIF, and if upon the line fi, in 142the same order of letters, and similar to the trapezium FGHI, a trapezium fghi is constructed, and a trajectory given in kind is circumscribed about it, the problem will be solved.

So far concerning the finding of the orbits. It remains that we determine the motions of bodies in the orbits so found.

Section VI. How the motions are to be found in given orbits.

Section VI

143How the motions are to be found in given orbits.

Proposition XXX. Problem XXII

To find at any assigned time the place of a body moving in a given parabolic trajectory.

Let S (Pl. 14. Fig. 1.) be the focus, and A the principal vertex of the parabola; and suppose $4AS\times M$ equal to the parabolic area to be cut off APS, which either was described by the radius SP, since the bodies departure from the vertex, or is to be described thereby before its arrival there. Now the quantity of that area to be cut off is known from the time which is proportional to it. Bisect AS in G, and erect the perpendicular GH equal to 3M, and a circle described about the centre H, with the interval HS, will cut the parabola in the place P required. For letting fall PO 144perpendicular on the axis, and drawing PH, there will be $A{G}^2+G{H}^2$ $(=H{P}^2=\overline{AO-AG}{|}^2+\overline{PO-GH}{|}^2)$ $=A{O}^2+P{O}^2-2GAO-2GH+PO+A{G}^2+G{H}^2$. Whence $2GH\times PO$$(=A{O}^2+P{O}^2-2GAO)$$=A{O}^2+\frac{3}{4}P{O}^2$. For $A{O}^2$ write $AO\times \frac{P{O}^2}{4AS}$; then dividing all the terms by 3PO and multiplying them by 2AS, we shall have $\frac{4}{3}GH\times AS$ $(=\frac{1}{6}AO\times PO+\frac{1}{2}AS\times PO=$$\frac{AO+3AS}{6}\times PO=$$\frac{4AO-3SO}{6}\times PO=$ to the area $\overline{APO-SPO}|=$ to the area APS. But GH was 3M, and therefore $\frac{4}{3}GH\times AS$ is $4AS\times M$. Wherefore the area cut off APS is equal to the area that was to be cut off $4AS\times M$. Q. E. D.

Cor. 1. Hence GH is to AS, as the time in which the body described the arc AP, to the time in which the body described the arc between the vertex A and the perpendicular erected from the focus S upon the axis.

Cor. 2. And supposing a circle ASP, perpetually to pass through the moving body P, the velocity of the point H, is to the velocity which the body had in the vertex A, as 3 to 8; and therefore in the same ratio is the line GH to the right line which the body, in the time of its moving from A to P, would describe with that velocity which it had in the vertex A.

Cor. 3. Hence also on the other hand, the time may be found, in which the body has described any assigned arc AP. Join AP, and on its middle point erect a perpendicular meeting the right line GH in H.

Lemma XXVIII

145There is no oval figure whose area, cut off by right lines at pleasure, can be universally found by means of equations of any number of finite terms and dimensions.

Suppose that within the oval any point is given, about which as a pole a right line is perpetually revolving, with an uniform motion, while in that right line a moveable point going out from the pole, moves always forward with a velocity proportional to the square of that right line within the oval. By this motion that point will describe a spiral with infinite circumgyrations. Now if a portion of the area of the oval cut off by that right line could be found by a finite equation, the distance of the point from the pole, which is proportional to this area, might be found by the same equation, and therefore all the points of the spiral might be found by a finite equation also; and therefore the intersection of a right line given in position with the spiral might also be found by a finite equation. But every right line infinitely produced cuts a spiral in an infinite number of points; and the equation by which any one intersection of two lines is found, at the same time exhibits all their intersections by as many roots, and therefore rises to as many dimensions as there are intersections. Because two circles mutually cut one 146another in two points, one of those intersections is not to be found but by an equation of two dimensions, by which the other intersection may be also found. Because there may be four intersections of two conic sections, any one of them is not to be found universally but by an equation of four dimensions, by which they may be all found together. For if those intersections are severally sought, because the law and condition of all is the same, the calculus will be the same in every case, and therefore the conclusion always the same, which must therefore comprehend all those intersections at once within it self, and exhibit them all indifferently. Hence it is that the intersections of the conic sections with the curves of the third order, because they may amount to six, come out together by equations of six dimensions; and the intersections of two curves of the third order, because they may amount to nine come out together by equations of nine dimensions. If this did not necessarily happen, we might reduce all solid to plane problems, and those higher than solid to solid problems. But here I speak of curves irreducible in power. For if the equation by which the curve is defined may be reduced to a lower power, the curve will not be one single curve, but composed of two or more; whose intersections may be severally found by different calculus's. After the same manner the two intersections of right lines with the conic sections, come out always by equations of two dimensions; the three intersections of right lines with the irreducible curves of the third order by equations of three dimensions; the four intersections of right lines with the irreducible curves of the fourth order, 147by equations of four dimensions, and so on in infinitum. Wherefore the innumerable intersections of a right line with a spiral, since this is but one simple curve, and not reducible to more curves, require equations infinite in number of dimensions and roots, by which they may be all exhibited together. For the law and calculus of all is the same. For if a perpendicular is let fall from the pole upon that intersecting right line, and that perpendicular together with the intersecting line revolves about the pole, the intersections of the spiral will mutually pass the one into the other; and that which was first or nearest, after one revolution, will be the second, after two, the third, and so on; nor will the equation in the mean time be changed, but as the magnitudes of those quantities are changed, by which the position of the intersecting line is determined. Wherefore since those quantities after every revolution return to their first magnitudes, the equation will return to its first form, and consequently one and the same equation will exhibit all the intersections, and will therefore have an infinite number of roots, by which they may be all exhibited. And therefore the intersection of a right line with a spiral cannot be universally found by any finite equation; and of consequence there is no oval figure whose area, cut off by right lines at pleasure, can be universally exhibited by any such equation.

By the same argument, if the interval of the pole and point by which the spiral is described, is taken proportional to that part of the perimeter of the oval which is cut off; it may be proved that the length of the perimeter cannot be universally exhibited by any finite equation. But here I speak 148of ovals that are not touched by conjugate figures running out in infinitum.

Cor. Hence the area of an ellipsis, described by a radius drawn from the focus to the moving body, is not to be found from the time given, by a finite equation; and therefore cannot be determined by the description of curves geometrically rational. Those curves I call geometrically rational, all the points whereof may be determined by lengths that are defineable by equations, that is, by the complicated ratio's of lengths. Other curves (such as spirals, quadratrixes, and cycloids) I call geometrically irrational. For the lengths which are or are not as number to number (according to the tenth book of elements) are arithmetically rational or irrational. And therefore I cut off an area of an ellipsis proportional to the time in which it is described by a curve geometrically irrational, in the following manner.

Proposition XXXI. Problem XXIII

To find the place of a body moving in a given elliptic trajectory at any assigned time.

Suppose A (Pl. 14. Fig. 2.) to be the principal vertex, S the focus, and O the centre of the ellipsis APB; and let P be the place of the body to be found. Produce OA to G, so as OG may be to OA as OA to OS. Erect the perpendicular GH; and about the centre O, with the interval OG, describe the circle GEF; and on the ruler GH 149as a base, suppose the wheel GEF to move forward, revolving about its axis, and in the mean time by its point A describing the cycloid ALI. Which done, take GK to the perimeter GEFG of the wheel, in the ratio of the time in which the body, proceeding from A, described the arc AP, to the time of a whole revolution in the ellipsis. Erect the perpendicular KL meeting the cycloid in L, then LP drawn parallel to KG will meet this ellipsis in P the required place of the body.

For about the centre O with the interval OA describe the semi-circle AQB, and let LP, produced, if need be, meet the arc AQ in Q, and join SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the perpendicular SR. The area APS is as the area AQS, that is, as the difference between the sector OQA and the triangle OQS, or as the difference of the rectangles $\frac{1}{2}OQ\times AQ$, and $\frac{1}{2}OQ\times SR$, that is, because $\frac{1}{2}OQ$ is given, as the difference between the arc AQ and the right line SR; and therefore (because of the equality of the given ratios SR to the sine of the arc AQ, OS to OA, OA to OG, AQ to GF, and by division, $AQ-SR$ to $GF-$ sine of the arc AQ) as GK the difference between the arc GF and the sine of the arc AQ. Q. E. D.

Scholium

But since the description of this curve is difficult, a solution by approximation will be preferable. First then let there be found a certain angle B which may be to an angle of 57,29578 degrees, 150which an arc equal to the radius subtends, as SH (Pl. 14. Fig. 3.) the distance of the foci, to AB the diameter of the ellipsis. Secondly, a certain length L, which may be to the radius in the same ratio inversely. And these being found, the problem may be solved by the following analysis. By any construction (or even by conjecture) suppose we know P the place of the body near its true place p. Then letting fall on the axis of the ellipsis the ordinate PR, from the proportion of the diameters of the ellipsis, the ordinate RQ of the circumscribed circle AQB will be given; which ordinate is the sine of the angle AOQ supposing AO to be the radius, and also cuts the ellipsis in P. It will be sufficient if that angle is found by a rude calculus in numbers near the truth. Suppose we also know the angle proportional to the time, that is, which is to four right angles, as the time in which the body described the arc Ap, to the time of one revolution in the ellipsis. Let this angle be N. Then take an angle D, which may be to the angle B as the sine of the angle AOQ to the radius; and an angle E which may be to the angle $N-AOQ+D$, as the length L to the same length L diminished by the co-sine of the angle AOQ, when that angle is less than a right angle, or increased thereby when greater. In the next place take an angle F that may be to the angle B, as the sine of the angle $AOQ+E$ to the radius, and an angle G, that may be to the angle $N-AOQ-E+F$, as the length L to the same length L diminished by the co-sine of the angle $AOQ+E$, when that angle is less than a right angle, or increased thereby when greater. For the third time take an angle H, that may be to 151the angle B as the sine of the angle $AOQ+E+G$ to the radius; and an angle I to the angle $N-AOQ-E-G+H$, as the length L is to the same length L diminished by the co-sine of the angle $AOQ+E+G$ when that angle is less than a right angle, or increased thereby when greater. And so we may proceed in infinitum. Lastly, take the angle AOq equal to the angle $AOQ+E+G+I+$ &c. and from its co-sine Or and the ordinate pr, which is to its sine qr as the lesser axis of the ellipsis to the greater, we shall have p the correct place of the body. When the angle $N-AOQ+D$ happens to be negative, the sign + of the angle E must be every where changed into -, and the sign - into +. And the same thing is to be understood of the signs of the angles G and I, when the angles $N-AOQ-E+F$, and $N-AOQ-E-G+H$ come out negative. But the infinite series $AOQ+E+G+I$ &c. converges so very fast, that it will be scarcely ever needful to proceed beyond the second term E. And the calculus is founded upon this theorem, that the area APS is as the difference between the arc AQ and the right line let fall from the focus S perpendicularly upon the radius OQ.

And by a calculus not unlike, the problem is solved in the hyperbola. Let its centre be O, (Pl. 14. Fig. 4.) its vertex A, its focus S, and asymptote OK. And suppose the quantity of the area to be cut off is known, as being proportional to the time. Let that be A, and by conjecture suppose we know the position of a right line SP, that cuts off an area APS near the truth. Join OP, and from A and P to the asymptote draw AI, PK parallel to the other asymptote: 152and by the table of logarithms the area AIKP will be given, and equal thereto the area OPA, which subducted from the triangle OPS will leave the area cut off APS. And by applying $2APS-2A$, or $2A-2APS$, the double difference of the area A that was to be cut off, and the area APS that is cut off, to the line SN that is let fall from the focus S, perpendicular upon the tangent TI, we shall have the length of the chord PQ. Which chord PQ is to be inscribed between A and P, if the area APS that is cut off be greater than the area A that was to be cut off, but towards the contrary side of the point P, if otherwise: and the point Q will be the place of the body more accurately. And by repeating the computation the place may be found perpetually to greater and greater accuracy.

And by such computations we have a general analytical resolution of the problem. But the particular calculus that follows, is better fitted for astronomical purposes. Supposing AO, OB, OD (Pl. 14. Fig. 5.) to be the semi-axes of the ellipsis, and L its latus rectum, and D the difference betwixt the lesser semi-axis OD, and $\frac{1}{2}L$ the half of the latus rectum: let an angle Y be found, whose sine may be to the radius, as the rectangle under that difference D and $AO+OD$ the half sum of the axes, to the square of the greater axis AB. Find also an angle Z whose sine may be to the radius, as the double rectangle under the distance of the foci SH and that difference D, to triple the square of half the greater semi-axis AO. Those angles being once found, the place of the body may be thus determined. Take the angle T proportional to the time in which the arc BP 153was described, or equal to what is called the mean motion; and an angle V, the first equation of the mean motion to the angle Y, the greatest first equation, as the sine of double the angle T is to the radius; and an angle X, the second equation, to the angle Z, the second greatest equation, as the cube of the sine of the angle T is to the cube of the radius. Then take the angle BHP the mean motion equated equal to $T+X+V$ the sum of the angles T, V, X, if the angle T is less than a right angle; or equal to $T+X-V$ the difference of the same, if that angle T is greater than one and less than two right angles; and if HP meets the ellipsis in P, draw SP, and it will cut off the area BSP nearly proportional to the time.

This practice seems to be expeditious enough, because the angles V and X, taken in second minutes if you please, being very small, it will be sufficient to find two or three of their first figures. But it is likewise sufficiently accurate to answer to the theory of the planets motions. For even in the orbit of Mars, where the greatest equation of the centre amounts to ten degrees, the error will scarcely exceed one second. But when the angle of the mean motion equated BHP is found, the angle of the true motion BSP, and the distance SP are readily had by the known methods.

And so far concerning the motion of bodies in curve lines. But it may also come to pass that a moving body shall ascend or descend in a right line; and I shall now go on to explain what belongs to such kind of motions.

Section VII. Concerning the rectilinear ascent and descent of bodies.

Section VII

Concerning the rectilinear ascent and descent of bodies.

Proposition XXXII. Problem XXIV

154Supposing 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, 155making 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 to 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 same; 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 therefore the area BDEB will be proportional to the time in which the body C, by its 156perpendicular 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 parabolic 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 falling body in any place C, is to the velocity 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 $\frac{1}{2}AB$ 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, 157and 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 $\frac{1}{2}L\times SP$ to $S{Y}^2$. For by the properties of the conic sections ACB is to $C{P}^2$ as $2AO$ to L, and therefore $\frac{2C{P}^2\times AO}{ACB}$ is equal to L. Therefore those velocities are to each other in the subduplicate ratio of $\frac{C{P}^2\times AO\times SP}{ACB}$ to $S{Y}^2$. 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) BO - or + CO will be to BO, as CT to BT, that is, AC will be to AO as CP to BQ; and therefore $\frac{C{P}^2\times AO\times SP}{ACB}$ is equal to $\frac{B{Q}^2\times AC\times SP}{AO\times BC}$. 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 $\frac{B{Q}^2\times AC\times SP}{AO\times BC}$ to $S{Y}^2$, that is (neglecting the ratio's of equality of 158SP to BC, and $B{Q}^2$ to $S{Y}^2$) in the subduplicate ratio of AC to AO or $\frac{1}{2}AB$. Q. E. D.