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

by Thomas Hobbes

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.

6. If two bodies with uniform motion transmit two lengths, each with its own impetus and time, the proportion of the lengths transmitted will be compounded of the proportions of time to time, and impetus to impetus, directly taken.

Let two bodies be moved uniformly (as in [fig. 3]), one in the time A B with the impetus A C, the other in the time A D with the impetus A E. I say the lengths transmitted have their proportion to one another compounded of the proportions of A B to A D, and of A C to A E. For let any length whatsoever, as Z, be transmitted by one of the bodies in the time A B with the impetus A C; and any other length, as X, be transmitted by the other body in the time A D with the impetus A E; and let the parallelograms A F and A G be completed. Seeing now Z is to X (by [art. 2]) as the impetus A C multiplied into the time A B is to the impetus A E multiplied into the time A D, that is, as A F to A G; the proportion of Z to X will be compounded of the same proportions, of which the proportion of A F to A G is compounded; but the proportion of A F to A G is compounded of the proportions of the side A B to the side A D, and of the side A C to the side A E (as is evident by the Elements of Euclid), that is, of the proportions of the time A B to the time A D, and of the impetus A C to the impetus A E. Wherefore also the proportion of Z to X is compounded of the same proportions of the time A B to the time A D, and of the impetus A C to the impetus A E; which was to be demonstrated.

Coroll. I. When two bodies are moved with uniform motion, if the times and impetus be in reciprocal proportion, the lengths transmitted shall be equal. For if it were as A B to A D (in the same [fig. 3]) so reciprocally A E to A C, the proportion of A F to A G would be compounded of the proportions of A B to A D, and of A C to A E, that is, of the proportions of A B to A D, and of A D to A B. Wherefore, A F would be to A G as A B to A B, that is, equal; and so the two products made by the multiplication of impetus into time would be equal; and by consequent, Z would be equal to X.

Coroll. II. If two bodies be moved in the same time, but with different impetus, the lengths transmitted will be as impetus to impetus. For if the time of both of them be A D, and their different impetus be A E and A C, the proportion of A G to D C will be compounded of the proportions of A E to A C and of A D to A D, that is, of the proportions of A E to A C and of A C to A C; and so the proportion of A G to D C, that is, the proportion of length to length, will be as A E to A C, that is, as that of impetus to impetus. In like manner, if two bodies be moved uniformly, and both of them with the same impetus, but in different times, the proportion of the lengths transmitted by them will be as that of their times. For if they have both the same impetus A C, and their different times be A B and A D, the proportion of A F to D C will be compounded of the proportions of A B to A D and of A C to A C; that is, of the proportions of A B to A D and of A D to A D; and therefore the proportion of A F to D C, that is, of length to length, will be the same with that of A B to A D, which is the proportion of time to time.

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.

7. If two bodies pass through two lengths with uniform motion, the proportion of the times in which they are moved will be compounded of the proportions of length to length and impetus to impetus reciprocally taken.

For let any two lengths be given, as (in the same [fig. 3]) Z and X, and let one of them be transmitted with the impetus A C, the other with the impetus A E. I say the proportion of the times in which they are transmitted, will be compounded of the proportions of Z to X, and of A E, which is the impetus with which X is transmitted, to A C, the impetus with which Z is transmitted. For seeing A F is the product of the impetus A C multiplied into the time A B, the time of motion through Z will be a line, which is made by the application of the parallelogram A F to the strait line A C, which line is A B; and therefore A B is the time of motion through Z. In like manner, seeing A G is the product of the impetus A E multiplied into the time A D, the time of motion through X will be a line which is made by the application of A G to the strait line A D; but A D is the time of motion through X. Now the proportion of A B to A D is compounded of the proportions of the parallelogram A F to the parallelogram A G, and of the impetus A E to the impetus A C; which may be demonstrated thus. Put the parallelograms in order A F, A G, D C, and it will be manifest that the proportion of A F to D C is compounded of the proportions of A F to A G and of A G to D C; but A F is to D C as A B to A D; wherefore also the proportion of A B to A D is compounded of the proportions of A F to A G and of A G to D C. And because the length Z is to the length X as A F is to A G, and the impetus A E to the impetus A C as A G to D C, therefore the proportion of A B to A D will be compounded of the proportions of the length Z to the length X, and of the impetus A E to the impetus A C; which was to be demonstrated.

In the same manner it may be proved, that in two uniform motions the proportion of the impetus is compounded of the proportions of length to length and of time to time reciprocally taken.

For if we suppose A C (in the same [fig. 3]) to be the time, and A B the impetus with which the length Z is passed through; and A E to be the time, and A D the impetus with which the length X is passed through, the demonstration will proceed as in the last article.