If any three magnitudes be propounded, or three things whatsoever that have any proportion one to another, as three numbers, three times, three degrees, &c.; the proportions of the first to the second, and of the second to the third, together taken, are equal to the proportion of the first to the third. Let there be three lines, for any proportion may be reduced to the proportion of lines, A B, A C, A D; and in the first place, let the proportion as well of the first A B to the second A C, as of the second A C to the third AD, be the proportion of defect, or of less to greater; I say the proportions together taken of A B to A C, and of A C to A D, are equal to the proportion of A B to A D. Suppose the point A to be moved over the whole line A D with uniform motion; then the proportions as well of A B to A C, as of A C to A D, are determined by the difference of the times in which they are described; that is, A B has to A C such proportion as is determined by the different times of their description; and A C to AD such proportion as is determined by their times. But the proportion of A B to A D is such as is determined by the difference of the times in which A B and A D are described; and the difference of the times in which A B and A C are described, together with the difference of the times in which A C and A D are described, is the same with the difference of the times in which A B and A D are described. And therefore, the same cause which determines the two proportions of A B to A C and of A C to A D, determines also the proportion of A B to A D. Wherefore, by the definition of the same proportion, delivered above in the 6th article, the proportion of A B to A C together with the proportion of A C to A D, is the same with the proportion of A B to A D. A B C D ------------- In the second place, let A D be the first, A C the second, and A B the third, and let their proportion be the proportion of excess, or the greater to less; then, as before, the proportions of A D to A C, and of A C to A B, and of A D to A B, will be determined by the difference of their times; which in the description of A D and A C, and of A C and A B together taken, is the same with the difference of the times in the description of A D and A B. Wherefore the proportion of A D to A B is equal to the two proportions of A D to A C and of A C to A B.