The English works of Thomas Hobbes of Malmesbury, Volume 01 (of 11)

by Thomas Hobbes

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.

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.

In all motion, the lengths which are passed through are to one another, as the products made by the impetus multiplied into time.

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.