CHAPTER XV.

OF THE NATURE, PROPERTIES, AND DIVERS
CONSIDERATIONS OF MOTION AND
ENDEAVOUR.

1.. Repetition of some principles of the doctrine of motion formerly set down.—2. Other principles added to them.—3. Certain theorems concerning the nature of motion.—4. Divers considerations of motion.—5. The way by which the first endeavour of bodies moved tendeth.—6. In motion which is made by concourse, one of the movents ceasing, the endeavour is made by the way by which the rest tend.—7. All endeavour is propagated in infinitum.—8. How much greater the velocity or magnitude is of a movent, so much the greater is the efficacy thereof upon any other body in its way.

1. The next things in order to be treated of are MOTION and MAGNITUDE, which are the most common accidents of all bodies. This place therefore most properly belongs to the elements of geometry. But because this part of philosophy, having been improved by the best wits of all ages, has afforded greater plenty of matter than can well be thrust together within the narrow limits of this discourse, I thought fit to admonish the reader, that before he proceed further, he take into his hands the works of Euclid, Archimedes, Apollonius, and other as well ancient as modern writers. For to what end is it, to do over again that which is already done? The little therefore that I shall say concerning geometry in some of the following chapters, shall be such only as is new, and conducing to natural philosophy.

I have already delivered some of the principles of this doctrine in the eighth and ninth chapters; which I shall briefly put together here, that the reader in going on may have their light nearer at hand.

First, therefore, in chap. VIII. art. 10, motion is defined to be the continual privation of one place, and acquisition of another.

Secondly, it is there shown, that whatsoever is moved is moved in time.

Thirdly, in the same chapter, art. 11, I have defined rest to be when a body remains for some time in one place.

Fourthly, it is there shown, that whatsoever is moved is not in any determined place; as also that the same has been moved, is still moved, and will yet be moved; so that in every part of that space, in which motion is made, we may consider three times, namely, the past, the present, and future time.

Fifthly, in art. 15 of the same chapter, I have defined velocity or swiftness to be motion considered as power, namely, that power by which a body moved may in a certain time transmit a certain length; which also may more briefly be enunciated thus, velocity is the quantity of motion determined by time and line.

Sixthly, in the same chapter, art. 16, I have shown that motion is the measure of time.

Seventhly, in the same chapter, art. 17, I have defined motions to be equally swift, when in equal times equal lengths are transmitted by them.

Eighthly, in art. 18 of the same chapter, motions are defined to be equal, when the swiftness of one moved body, computed in every part of its magnitude, is equal to the swiftness of another, computed also in every part of its magnitude. From whence it is to be noted, that motions equal to one another, and motions equally swift, do not signify the same thing; for when two horses draw abreast, the motion of both is greater than the motion of either of them singly; but the swiftness of both together is but equal to that of either.

Ninthly, in art. 19 of the same chapter, I have shown, that whatsoever is at rest will always be at rest, unless there be some other body besides it, which by getting into its place suffers it no longer to remain at rest. And that whatsoever is moved, will always be moved, unless there be some other body besides it, which hinders its motion.

Tenthly, in chap. IX. art. 7, I have demonstrated, that when any body is moved which was formerly at rest, the immediate efficient cause of that motion is in some other moved and contiguous body.

Eleventhly, I have shown in the same place, that whatsoever is moved, will always be moved in the same way, and with the same swiftness, if it be not hindered by some other moved and contiguous body.

2. To which principles I shall here add those that follow. First, I define ENDEAVOUR to be motion made in less space and time than can be given; that is, less than can be determined or assigned by exposition or number; that is, motion made through the length of a point, and in an instant or point of time. For the explaining of which definition it must be remembered, that by a point is not to be understood that which has no quantity, or which cannot by any means be divided; for there is no such thing in nature; but that, whose quantity is not at all considered, that is, whereof neither quantity nor any part is computed in demonstration; so that a point is not to be taken for an indivisible, but for an undivided thing; as also an instant is to be taken for an undivided, and not for an indivisible time.

In like manner, endeavour is to be conceived as motion; but so as that neither the quantity of the time in which, nor of the line in which it is made, may in demonstration be at all brought into comparison with the quantity of that time, or of that line of which it is a part. And yet, as a point may be compared with a point, so one endeavour may be compared with another endeavour, and one may be found to be greater or less than another. For if the vertical points of two angles be compared, they will be equal or unequal in the same proportion which the angles themselves have to one another. Or if a strait line cut many circumferences of concentric circles, the inequality of the points of intersection will be in the same proportion which the perimeters have to one another. And in the same manner, if two motions begin and end both together, their endeavours will be equal or unequal, according to the proportion of their velocities; as we see a bullet of lead descend with greater endeavour than a ball of wool.

Secondly, I define IMPETUS, or quickness of motion, to be the swiftness or velocity of the body moved, but considered in the several points of that time in which it is moved. In which sense impetus is nothing else but the quantity or velocity of endeavour. But considered with the whole time, it is the whole velocity of the body moved taken together throughout all the time, and equal to the product of a line representing the time, multiplied into a line representing the arithmetically mean impetus or quickness. Which arithmetical mean, what it is, is defined in the 29th article of chapter XIII.

And because in equal times the ways that are passed are as the velocities, and the impetus is the velocity they go withal, reckoned in all the several points of the times, it followeth that during any time whatsoever, howsoever the impetus be increased or decreased, the length of the way passed over shall be increased or decreased in the same proportion; and the same line shall represent both the way of the body moved, and the several impetus or degrees of swiftness wherewith the way is passed over.

And if the body moved be not a point, but a strait line moved so as that every point thereof make a several strait line, the plane described by its motion, whether uniform, accelerated, or retarded, shall be greater or less, the time being the same, in the same proportion with that of the impetus reckoned in one motion to the impetus reckoned in the other. For the reason is the same in parallelograms and their sides.

For the same cause also, if the body moved be a plane, the solid described shall be still greater or less in the proportions of the several impetus or quicknesses reckoned through one line, to the several impetus reckoned through another.

This understood, let A B C D, (in figure 1, chap. XVII.) be a parallelogram; in which suppose the side A B to be moved parallelly to the opposite side C D, decreasing all the way till it vanish in the point C, and so describing the figure A B E F C; the point B, as A B decreaseth, will therefore describe the line B E F C; and suppose the time of this motion designed by the line C D; and in the same time C D, suppose the side A C to be moved parallel and uniformly to B D. From the point O taken at adventure in the line C D, draw O R parallel to B D, cutting the line B E F C in E, and the side A B in R. And again, from the point Q taken also at adventure in the line C D, draw Q S parallel to B D, cutting the line B E F C in F, and the side A B in S; and draw E G and F H parallel to C D, cutting A C in G and H. Lastly, suppose the same construction done in all the points possible of the line B E F C. I say, that as the proportions of the swiftness wherewith Q F, O E, D B, and all the rest supposed to be drawn parallel to D B and terminated in the line B E F C, are to the proportions of their several times designed by the several parallels H F, G E, A B, and all the rest supposed to be drawn parallel to the line of time C D and terminated in the line B E F C, the aggregate to the aggregate, so is the area or plane D B E F C to the area or plane A C F E B. For as A B decreasing continually by the line B E F C vanisheth in the time C D into the point C, so in the same time the line D C continually decreasing vanisheth by the same line C F E B into the point B; and the point D describeth in that decreasing motion the line D B equal to the line A C described by the point A in the decreasing motion of A B; and their swiftnesses are therefore equal. Again, because in the time G E the point O describeth the line OE, and in the same time the point S describeth the line S E, the line O E shall be to the line SE, as the swiftness wherewith OE is described to the swiftness wherewith SE is described. In like manner, because in the same time H F the point Q describeth the line Q F, and the point R the line R F, it shall be as the swiftness by which Q F is described to the swiftness by which R F is described, so the line itself Q F to the line itself R F; and so in all the lines that can possibly be drawn parallel to B D in the points where they cut the line B E F C. But all the parallels to B D, as S E, R F, A C, and the rest that can possibly be drawn from the line A B to the line B E F C, make the area of the plane A B E F C; and all the parallels to the same B D, as Q F, O E, D B and the rest drawn to the points where they cut the same line B E F C, make the area of the plane B E F C D. As therefore the aggregate of the swiftnesses wherewith the plane B E F C D is described, is to the aggregate of the swiftnesses wherewith the plane A C F E B is described, so is the plane itself B E F C D to the plane itself A C F E B. But the aggregate of the times represented by the parallels A B, G E, H F and the rest, maketh also the area A C F E B. And therefore, as the aggregate of all the lines Q F, O E, D B and all the rest of the lines parallel to B D and terminated in the line B E F C, is to the aggregate of all the lines H F, G E, A B and all the rest of the lines parallel to C D and terminated in the same line B E F C; that is, as the aggregate of the lines of swiftness to the aggregate of the lines of time, or as the whole swiftness in the parallels to D B to the whole time in the parallels to C D, so is the plane B E F C D to the plane A C F E B. And the proportions of Q F to F H, and of O E to E G, and of D B to B A, and so of all the rest taken together, are the proportions of the plane D B E F C to the plane A B E F C. But the lines Q F, O E, D B and the rest are the lines that design the swiftness; and the lines H F, G E, A B and the rest are the lines that design the times of the motions; and therefore the proportion of the plane D B E F C to the plane A B E F C is the proportion of all the velocities taken together to all the times taken together. Wherefore, as the proportions of the swiftnesses, &c.; which was to be demonstrated.

The same holds also in the diminution of the circles, whereof the lines of time are the semidiameters, as may easily be conceived by imagining the whole plane A B C D turned round upon the axis B D; for the line B E F C will be everywhere in the superficies so made, and the lines H F, G E, A B, which are here parallelograms, will be there cylinders, the diameters of whose bases are the lines H F, G E, A B, &c. and the altitude a point, that is to say, a quantity less than any quantity that can possibly be named; and the lines Q F, O E, D B, &c. small solids whose lengths and breadths are less than any quantity that can be named.

But this is to be noted, that unless the proportion of the sum of the swiftnesses to the proportion of the sum of the times be determined, the proportion of the figure D B E F C to the figure A B E F C cannot be determined.

Thirdly, I define RESISTANCE to be the endeavour of one moved body either wholly or in part contrary to the endeavour of another moved body, which toucheth the same. I say, wholly contrary, when the endeavour of two bodies proceeds in the same strait line from the opposite extremes, and contrary in part, when two bodies have their endeavour in two lines, which, proceeding from the extreme points of a strait line, meet without the same.

Fourthly, that I may define what it is to PRESS, I say, that of two moved bodies one presses the other, when with its endeavour it makes either all or part of the other body to go out of its place.

Fifthly, a body, which is pressed and not wholly removed, is said to RESTORE itself, when, the pressing body being taken away, the parts which were moved do, by reason of the internal constitution of the pressed body, return every one into its own place. And this we may observe in springs, in blown bladders, and in many other bodies, whose parts yield more or less to the endeavour which the pressing body makes at the first arrival; but afterwards, when the pressing body is removed, they do, by some force within them, restore themselves, and give their whole body the same figure it had before.

Sixthly, I define FORCE to be the impetus or quickness of motion multiplied either into itself, or into the magnitude of the movent, by means whereof the said movent works more or less upon the body that resists it.

3. Having premised thus much, I shall now demonstrate, first, that if a point moved come to touch another point which is at rest, how little soever the impetus or quickness of its motion be, it shall move that other point. For if by that impetus it do not at all move it out of its place, neither shall it move it with double the same impetus. For nothing doubled is still nothing; and for the same reason it shall never move it with that impetus, how many times soever it be multiplied, because nothing, however it be multiplied, will for ever be nothing. Wherefore, when a point is at rest, if it do not yield to the least impetus, it will yield to none; and consequently it will be impossible that that, which is at rest, should ever be moved.

Secondly, that when a point moved, how little soever the impetus thereof be, falls upon a point of any body at rest, how hard soever that body be, it will at the first touch make it yield a little. For if it do not yield to the impetus which is in that point, neither will it yield to the impetus of never so many points, which have all their impetus severally equal to the impetus of that point. For seeing all those points together work equally, if any one of them have no effect, the aggregate of them all together shall have no effect as many times told as there are points in the whole body, that is, still no effect at all; and by consequent there would be some bodies so hard that it would be impossible to break them; that is, a finite hardness, or a finite force, would not yield to that which is infinite; which is absurd.

Coroll. It is therefore manifest, that rest does nothing at all, nor is of any efficacy; and that nothing but motion gives motion to such things as be at rest, and takes it from things moved.

Thirdly, that cessation in the movent does not cause cessation in that which was moved by it. For (by number 11 of art. 1 of this chapter) whatsoever is moved perseveres in the same way and with the same swiftness, as long as it is not hindered by something that is moved against it. Now it is manifest, that cessation is not contrary motion; and therefore it follows that the standing still of the movent does not make it necessary that the thing moved should also stand still.

Coroll. They are therefore deceived, that reckon the taking away of the impediment or resistance for one of the causes of motion.

4. Motion is brought into account for divers respects; first, as in a body undivided, that is, considered as a point; or, as in a divided body. In an undivided body, when we suppose the way, by which the motion is made, to be a line; and in a divided body, when we compute the motion of the several parts of that body, as of parts.

Secondly, from the diversity of the regulation of motion, it is in body, considered as undivided. sometimes uniform and sometimes multiform. Uniform is that by which equal lines are always transmitted in equal times; and multiform, when in one time more, in another time less space is transmitted. Again, of multiform motions, there are some in which the degrees of acceleration and retardation proceed in the same proportions, which the spaces transmitted have, whether duplicate, or triplicate, or by whatsoever number multiplied; and others in which it is otherwise.

Thirdly, from the number of the movents; that is, one motion is made by one movent only, and another by the concourse of many movents.

Fourthly, from the position of that line in which a body is moved, in respect of some other line; and from hence one motion is called perpendicular, another oblique, another parallel.

Fifthly, from the position of the movent in respect of the moved body; from whence one motion is pulsion or driving, another traction or drawing. Pulsion, when the movent makes the moved body go before it; and traction, when it makes it follow. Again, there are two sorts of pulsion; one, when the motions of the movent and moved body begin both together, which may be called trusion or thrusting and vection; the other, when the movent is first moved, and afterwards the moved body, which motion is called percussion or stroke.

Sixthly, motion is considered sometimes from the effect only which the movent works in the moved body, which is usually called moment. Now moment is the excess of motion which the movent has above the motion or endeavour of the resisting body.

Seventhly, it may be considered from the diversity of the medium; as one motion may be made in vacuity or empty place; another in a fluid; another in a consistent medium, that is, a medium whose parts are by some power so consistent and cohering, that no part of the same will yield to the movent, unless the whole yield also.

Eighthly, when a moved body is considered as having parts, there arises another distinction of motion into simple and compound. Simple, when all the several parts describe several equal lines; compounded, when the lines described are unequal.

5. All endeavour tends towards that part, that is to say, in that way which is determined by the motion of the movent, if the movent be but one; or, if there be many movents, in that way which their concourse determines. For example, if a moved body have direct motion, its first endeavour will be in a strait line; if it have circular motion, its first endeavour will be in the circumference of a circle.

6. And whatsoever the line be, in which a body has its motion from the concourse of two movents, as soon as in any point thereof the force of one of the movents ceases, there immediately the former endeavour of that body will be changed into an endeavour in the line of the other movent.

Wherefore, when any body is carried on by the concourse of two winds, one of those winds ceasing, the endeavour and motion of that body will be in that line, in which it would have been carried by that wind alone which blows still. And in the describing of a circle, where that which is moved has its motion determined by a movent in a tangent, and by the radius which keeps it in a certain distance from the centre, if the retention of the radius cease, that endeavour, which was in the circumference of the circle, will now be in the tangent, that is, in a strait line. For, seeing endeavour is computed in a less part of the circumference than can be given, that is, in a point, the way by which a body is moved in the circumference is compounded of innumerable strait lines, of which every one is less than can be given; which are therefore called points. Wherefore when any body, which is moved in the circumference of a circle, is freed from the retention of the radius, it will proceed in one of those strait lines, that is, in a tangent.

7. All endeavour, whether strong or weak, is propagated to infinite distance; for it is motion. If therefore the first endeavour of a body be made in space which is empty, it will always proceed with the same velocity; for it cannot be supposed that it can receive any resistance at all from empty space; and therefore, (by art. 7, chap, IX) it will always proceed in the same way and with the same swiftness. And if its endeavour be in space which is filled, yet, seeing endeavour is motion, that which stands next in its way shall be removed, and endeavour further, and again remove that which stands next, and so infinitely. Wherefore the propagation of endeavour, from one part of full space to another, proceeds infinitely. Besides, it reaches in any instant to any distance, how great soever. For in the same instant in which the first part of the full medium removes that which is next it, the second also removes that part which is next to it; and therefore all endeavour, whether it be in empty or in full space, proceeds not only to any distance, how great soever, but also in any time, how little soever, that is, in an instant. Nor makes it any matter, that endeavour, by proceeding, grows weaker and weaker, till at last it can no longer be perceived by sense; for motion may be insensible; and I do not here examine things by sense and experience, but by reason.

8. When two movents are of equal magnitude, the swifter of them works with greater force than the slower, upon a body that resists their motion. Also, if two movents have equal velocity, the greater of them works with more force than the less. For where the magnitude is equal, the movent of greater velocity makes the greater impression upon that body upon which it falls; and where the velocity is equal, the movent of greater magnitude falling upon the same point, or an equal part of another body, loses less of its velocity, because the resisting body works only upon that part of the movent which it touches, and therefore abates the impetus of that part only; whereas in the mean time the parts, which are not touched, proceed, and retain their whole force, till they also come to be touched; and their force has some effect. Wherefore, for example, in batteries a longer than a shorter piece of timber of the same thickness and velocity, and a thicker than a slenderer piece of the same length and velocity, work a greater effect upon the wall.

CHAPTER XVI.

OF MOTION ACCELERATED AND UNIFORM, AND
OF MOTION BY CONCOURSE.

1. The velocity of any body, in what time soever it be computed, is that which is made of the multiplication of the impetus, or quickness of its motion into the time.—2-5. In all motion, the lengths which are passed through are to one another, as the products made by the impetus multiplied into the time.—6. If two bodies be moved with uniform motion through two lengths, the proportion of those lengths to one another will be compounded of the proportions of time to time, and impetus to impetus, directly taken.—7. If two bodies pass through two lengths with uniform motion, the proportion of their times to one another will be compounded of the proportions of length to length, and impetus to impetus reciprocally taken; also the proportion of their impetus to one another will be compounded of the proportions of length to length, and time to time reciprocally taken.—8. If a body be carried on with uniform motion by two movents together, which meet in an angle, the line by which it passes will be a strait line, subtending the complement of that angle to two right angles.—9, &c. If a body be carried by two movents together, one of them being moved with uniform, the other with accelerated motion, and the proportion of their lengths to their times being explicable in numbers, how to find out what line that body describes.

1. The velocity of any body, in whatsoever time it be moved, has its quantity determined by the sum of all the several quicknesses or impetus, which it hath in the several points of the time of the body's motion. For seeing velocity, (by the definition of it, chap, VIII, art. 15) is that power by which a body can in a certain time pass through a certain length; and quickness of motion or impetus, (by chap. XV, art. 2, num. 2) is velocity taken in one point of time only, all the impetus, together taken in all the points of time, will be the same thing with the mean impetus multiplied into the whole time, or which is all one, will be the velocity of the whole motion.

Coroll. If the impetus be the same in every point, any strait line representing it may be taken for the measure of time: and the quicknesses or impetus applied ordinately to any strait line making an angle with it, and representing the way of the body's motion, will design a parallelogram which shall represent the velocity of the whole motion. But if the impetus or quickness of motion begin from rest and increase uniformly, that is, in the same proportion continually with the times which are passed, the whole velocity of the motion shall be represented by a triangle, one side whereof is the whole time, and the other the greatest impetus acquired in that time; or else by a parallelogram, one of whose sides is the whole time of motion, and the other, half the greatest impetus; or lastly, by a parallelogram having for one side a mean proportional between the whole time and the half of that time, and for the other side the half of the greatest impetus. For both these parallelograms are equal to one another, and severally equal to the triangle which is made of the whole line of time, and of the greatest acquired impetus; as is demonstrated in the elements of geometry.

2. In all uniform motions the lengths which are transmitted are to one another, as the product of the mean impetus multiplied into its time, to the product of the mean impetus multiplied also into its time.

For let A B (in fig. 1) be the time, and A C the impetus by which any body passes with uniform motion through the length D E; and in any part of the time A B, as in the time A F, let another body be moved with uniform motion, first, with the same impetus A C. This body, therefore, in the time A F with the impetus A C will pass through the length A F. Seeing, therefore, when bodies are moved in the same time, and with the same velocity and impetus in every part of their motion, the proportion of one length transmitted to another length transmitted, is the same with that of time to time, it followeth, that the length transmitted in the time A B with the impetus A C will be to the length transmitted in the time A F with the same impetus A C, as A B itself is to A F, that is, as the parallelogram A I is to the parallelogram A H, that is, as the product of the time A B into the mean impetus A C is to the product of the time A F into the same impetus A C. Again, let it be supposed that a body be moved in the time A F, not with the same but with some other uniform impetus, as A L. Seeing therefore, one of the bodies has in all the parts of its motion the impetus A C, and the other in like manner the impetus A L, the length transmitted by the body moved with the impetus A C will be to the length transmitted by the body moved with the impetus A L, as A C itself is to A L, that is, as the parallelogram A H is to the parallelogram F L. Wherefore, by ordinate proportion it will be, as the parallelogram A I to the parallelogram F L, that is, as the product of the mean impetus into the time is to the product of the mean impetus into the time, so the length transmitted in the time A B with the impetus A C, to the length transmitted in the time A F with the impetus A L; which was to be demonstrated.

Coroll. Seeing, therefore, in uniform motion, as has been shown, the lengths transmitted are to one another as the parallelograms which are made by the multiplication of the mean impetus into the times, that is, by reason of the equality of the impetus all the way, as the times themselves, it will also be, by permutation, as time to length, so time to length; and in general, to this place are applicable all the properties and transmutations of analogisms, which I have set down and demonstrated in chapter XIII.

3. In motion begun from rest and uniformly accelerated, that is, where the impetus increaseth continually according to the proportion of the times, it will also be, as one product made by the mean impetus multiplied into the time, to another product made likewise by the mean impetus multiplied into the time, so the length transmitted in the one time to the length transmitted in the other time.

For let A B (in fig. 1) represent a time; in the beginning of which time A, let the impetus be as the point A; but as the time goes on, so let the impetus increase uniformly, till in the last point of that time A B, namely in B, the impetus acquired be B I. Again, let A F represent another time, in whose beginning A, let the impetus be as the point itself A; but as the time proceeds, so let the impetus increase uniformly, till in the last point F of the time A F the impetus acquired be F K; and let D E be the length passed through in the time A B with impetus uniformly increased. I say, the length D E is to the length transmitted in the time A F, as the time A B multiplied into the mean of the impetus increasing through the time A B, is to the time A F multiplied into the mean of the impetus increasing through the time A F.

For seeing the triangle A B I is the whole velocity of the body moved in the time A B, till the impetus acquired be B I; and the triangle A F K the whole velocity of the body moved in the time A F with impetus increasing till there be acquired the impetus F K; the length D E to the length acquired in the time A F with impetus increasing from rest in A till there be acquired the impetus F K, will be as the triangle A B I to the triangle A F K, that is, if the triangles A B I and A F K be like, in duplicate proportion of the time A B to the time A F; but if unlike, in the proportion compounded of the proportions of A B to A F and of B I to F K. Wherefore, as A B I is to A F K, so let D E be to D P; for so, the length transmitted in the time A B with impetus increasing to B I, will be to the length transmitted in the time A F with impetus increasing to F K, as the triangle A B I is to the triangle A F K; but the triangle A B I is made by the multiplication of the time A B into the mean of the impetus increasing to B I; and the triangle A F K is made by the multiplication of the time A F into the mean of the impetus increasing to F K; and therefore the length D E which is transmitted in the time A B with impetus increasing to B I, to the length D P which is transmitted in the time A F with impetus increasing to F K, is as the product which is made of the time A B multiplied into its mean impetus, to the product of the time A F multiplied also into its mean impetus; which was to be proved.

Coroll. I. In motion uniformly accelerated, the proportion of the lengths transmitted to that of their times, is compounded of the proportions of their times to their times, and impetus to impetus.

Coroll. II. In motion uniformly accelerated, the lengths transmitted in equal times, taken in continual succession from the beginning of motion, are as the differences of square numbers beginning from unity, namely, as 3, 5, 7, &c. For if in the first time the length transmitted be as 1, in the first and second times the length transmitted will be as 4, which is the square of 2, and in the three first times it will be as 9, which is the square of 3, and in the four first times as 16, and so on. Now the differences of these squares are 3, 5, 7, &c.

Coroll. III. In motion uniformly accelerated from rest, the length transmitted is to another length transmitted uniformly in the same time, but with such impetus as was acquired by the accelerated motion in the last point of that time, as a triangle to a parallelogram, which have their altitude and base common. For seeing the length D E (in fig. 1) is passed through with velocity as the triangle A B I, it is necessary that for the passing through of a length which is double to D E, the velocity be as the parallelogram A I; for the parallelogram A I is double to the triangle A B I.

4. In motion, which beginning from rest is so aclerated, that the impetus thereof increases continually in proportion duplicate to the proportion of the times in which it is made, a length transmitted in one time will be to a length transmitted in another time, as the product made by the mean impetus multiplied into the time of one of those motions, to the product of the mean impetus multiplied into the time of the other motion.

For let A B (in fig. 2) represent a time, in whose first instant A let the impetus be as the point A; but as the time proceeds, so let the impetus increase continually in duplicate proportion to that of the times, till in the last point of time B the impetus acquired be B I; then taking the point F anywhere in the time A B, let the impetus F K acquired in the time A F be ordinately applied to that point F. Seeing therefore the proportion of F K to B I is supposed to be duplicate to that of A F to A B, the proportion of A F to A B will be subduplicate to that of F K to B I; and that of A B to A F will be (by chap. XIII. art. 16) duplicate to that of B I to F K; and consequently the point K will be in a parabolical line, whose diameter is A B and base B I; and for the same reason, to what point soever of the time A B the impetus acquired in that time be ordinately applied, the strait line designing that impetus will be in the same parabolical line A K I. Wherefore the mean impetus multiplied into the whole time A B will be the parabola A K I B, equal to the parallelogram A M, which parallelogram has for one side the line of time A B and for the other the line of the impetus A L, which is two-thirds of the impetus B I; for every parabola is equal to two-thirds of that parallelogram with which it has its altitude and base common. Wherefore the whole velocity in A B will be the parallelogram A M, as being made by the multiplication of the impetus A L into the time A B. And in like manner, if F N be taken, which is two-thirds of the impetus F K, and the parallelogram F O be completed, F O will be the whole velocity in the time A F, as being made by the uniform impetus A O or F N multiplied into the time A F. Let now the length transmitted in the time A B and with the velocity A M be the strait line D E; and lastly, let the length transmitted in the time A F with the velocity A N be D P; I say that as A M is to A N, or as the parabola A K I B to the parabola A K F, so is D E to D P. For as A M is to F L, that is, as A B is to A F, so let D E be to D G. Now the proportion of A M to A N is compounded of the proportions of A M to F L, and of F L to A N. But as A M to F L, so by construction is D E to D G; and as F L is to A N (seeing the time in both is the same, namely, A F), so is the length D G to the length D P; for lengths transmitted in the same time are to one another as their velocities are. Wherefore by ordinate proportion, as A M is to A N, that is, as the mean impetus A L multiplied into its time A B, is to the mean impetus A O multiplied into A F, so is D E to D P; which was to be proved.

Coroll. I. Lengths transmitted with motion so accelerated, that the impetus increase continually in duplicate proportion to that of their times, if the base represent the impetus, are in triplicate proportion of their impetus acquired in the last point of their times. For as the length D E is to the length D P, so is the parallelogram A M to the parallelogram A N, and so the parabola A K I B to the parabola A K F. But the proportion of the parabola A K I B to the parabola A K F is triplicate to the proportion which the base B I has to the base F K. Wherefore also the proportion of D E to D P is triplicate to that of B I to F K.

Coroll. II. Lengths transmitted in equal times succeeding one another from the beginning, by motion so accelerated, that the proportion of the impetus be duplicate to the proportion of the times, are to one another as the differences of cubic numbers beginning at unity, that is as 7, 19, 37, &c. For if in the first time the length transmitted be as 1, the length at the end of the second time will be as 8, at the end of the third time as 27, and at the end of the fourth time as 64, &c.; which are cubic numbers, whose differences are 7, 19, 37, &c.

Coroll. III. In motion so accelerated, as that the length transmitted be always to the length transmitted in duplicate proportion to their times, the length uniformly transmitted in the whole time, and with impetus all the way equal to that which is last acquired, is as a parabola to a parallelogram of the same altitude and base, that is, as 2 to 3. For the parabola A K I B is the impetus increasing in the time A B; and the parallelogram A I is the greatest uniform impetus multiplied into the same time A B. Wherefore the lengths transmitted will be as a parabola to a parallelogram, &c., that is, as 2 to 3.

5. If I should proceed to the explication of such motions as are made by impetus increasing in proportion triplicate, quadruplicate, quintuplicate, &c., to that of their times, it would be a labour infinite and unnecessary. For by the same method by which I have computed such lengths, as are transmitted with impetus increasing in single and duplicate proportion, any man may compute such as are transmitted with impetus increasing in triplicate, quadruplicate, or what other proportion he pleases.

In making which computation he shall find, that where the impetus increase in proportion triplicate to that of the times, there the whole velocity will be designed by the first parabolaster (of which see the next chapter); and the lengths transmitted will be in proportion quadruplicate to that of the times. And in like manner, where the impetus increase in quadruplicate proportion to that of the times, that there the whole velocity will be designed by the second parabolaster, and the lengths transmitted will be in quintuplicate proportion to that of the times; and so on continually.