Pro Portion - 1st, terms, 2d and proportional

PRO PORTION, harmonic, is when three terms are so disposed, that as the differ ence of the first and second : the differ ence of the second and third :: first : tnird ; and they are said to be harmonical ly proportional. Thus, 10, 15, 30, are harmonically proportional. For as the difference of 10 and 15, is to the differ ence of 15 and 30, so is 10 to 30. Also, 12, 6, 4, are harmonically proportional ; for 12— 6: 6— 4 : : 12: 4. So h. 4- 3 h n -F 2 bl +2 h n, -F h n, are har monically proportional. For h n + 2 n. : h n : : h. + 3 h 2 n. : h. + h a.— Whence, if the two first terms of an bar. manic proportion be given, the third is readily found.

For if A, B, C, be harmonically propor tional. Then A—B: B—C::A: C, and A C— B C= A B—A C. There foreAB.= 2 A—B xC, andBC= 2 C— B X A. Consequently C= 2 A A —11' and A = 2 C B B C Again, when fbur terms are so disposed, that as the difference of the 1st and 2t1: the differ ence of the 3d and 4th:: 1st : 4th, they are also harmonically proportional. As 10, 16, 24,60 ; for as 10— 16 : 24— 60 : : 10 : 60. Whence, if the three first terms of such an harmonic proportional be given, the 4th . is easily found.

For if a, b, c, d, be harmonic propor tionals, then a— b:c— d::a: d; and a d—bd =a a—ad, therefore d a c b d ----- and a = 2a —b' If the terms of an harmonic proportion be continued, then it is called an harmon ic progression. Thus, supposing

h, to be the 2d term, Zd, the difference of the 1st and 2d, and that the 1st exceeds the 2d. The progression will be + h d h. h a h.+ h d h d, 11, h -1- 2 d' h + 3 d' h + 4 d' la' h d &c. Whence, if out of a rank h 5 ' of harmonic proportionals, there be taken any series of equidistant terms, that se ries will be harmonically proportional.— And this kind of proportion has several other properties common with arithmetic and geometric proportions.

When three terms are so disposed, that the difference of the 1st and 2d : differ ence of the 2d and 3d :: 3d : 1st, they are said to be in a contra-harmonic propor tion. Thus, 6, 5, 3, and 12, 10, 4, are contra-harmonics. For 6 — 5 : 5 — 3 : 3 : 6 ; and 12 — 10 : 10— 4 : : 4: 12. Or, supposing /1 greater than n, if the 2d term be greater than the 1st : Then A n + h. + h. -I- hn, are contra-harmonics, for hn— A': n—An :: h. -F h n n..

But if the 1st term exceeds the 2d, then Is' n, A' ± n', h n + a", are harmonics. For /i — a' : h. — h n it it + : 11' -F h a.