18. If there be never so many continual proportionals, as the first, second, third, &c. their differences will be proportional to them. For the second, third, &c. are severally consequents of the preceding, and antecedents of the following proportion. But (by art. 10) the difference of the first antecedent and consequent, to difference of the second antecedent and consequent, is as the first antecedent to the second antecedent, that is, as the first term to the second, or as the second to the third, &c. in continual proportionals. 19. If there be three continual proportionals, the sum of the extremes, together with the mean twice taken, the sum of the mean and either of the extremes, and the same extreme, are continual proportionals. For let A. B. C be continual proportionals. Seeing, therefore, A. B :: B. C are proportionals, by composition also A + B. B :: B + C. C will be proportionals; and by permutation A + B. B + C :: B. C will also be proportionals; and again, by composition A + 2B + C. B + C :: B + C. C; which was to be proved. 20. In four continual proportionals, the greatest and the least put together is a greater quantity than the other two put together. Let A. B :: C. D be continual proportionals; whereof let the greatest be A, and the least be D; I say A + D is greater than B + C. For by art. 10, A - B. C - D :: A. C are proportionals; and therefore A - B is, by art. 11, greater than C - D. Add B on both sides, and A will be greater than C + B - D. And again, add D on both sides, and A + D will be greater than B + C; which was to be proved. [Sidenote: The definition and properties of continual proportion.] 21. If there be four proportionals, the extremes multiplied into one another, and the means multiplied into one another, will make equal products. Let A. B :: C. D be proportionals; I say A D is equal to B C. For the proportion of A D to B C is compounded, by art. 13, of the proportions of A to B, and D to C, that is, its inverse B to A; and therefore, by art. 14, this compounded proportion is the proportion of equality; and therefore also, the proportion of A D to B C is the proportion of equality. Wherefore they are equal.