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.