Giuseppe Tartini - Lettere e documenti / Pisma in dokumenti / Letters and Documents - Volume / Knjiga / Volume II

362 The harmonic series until the sextuple was there demonstrated to be 60, 30, 20, 15, 12, 10; and 40, 45, 48, 50 to be the sum of the harmonic differences, that is to say, harmonic series 60 30 20 15 12 10 harmonic differences 30 10 5 3 2 30 40 45 48 40 45 48 50 therefore, the two equal series proceed from the harmonic series as intrinsically as the harmonic series is intrinsic to the harmonic system and its differences; and this means that it is identically the harmonic series in its first principle. Equal then to such series can be found the respective means of the sines 3200, 2700, 2304, 2000, and of the extended sine 3600 (always equal), all centres of the above-mentioned harmonic series, and they were demonstrated as such in the right-angled triangles ABC first, ABC second, etc. , in the same proposition. It was likewise demonstrated that all converge in AB (eighth figure) as the common term. This term AB is found in its square with these two demonstrative characters; namely, the first 1/2 is the infinite constant; and the second 1/2 is relative to the harmonic series. For if the extremes of the harmonic series are altered, or the degrees forming the harmonic series are changed, AB is likewise altered, and is changed into another term; and therefore, like 1/2, it is absolutely relative to the specific harmonic series. In the above-given harmonic series one finds that in relation to the middle C 1/3, which is the sesquialter sine, and 1/3 C, which is the extended sesquialter sine, the above-mentioned series begins from the term 2400, with 4800 as the relative extreme. But when the same is exposed to mathematical rigour, it starts from the term 3600. Because the first sine, in which square and circle converge, is B 1/2; and so the other sine will be 1/2 B; the two extremes will be A 1/2, 1/2 M; and all the series shall start from the unit with four equal terms; namely A 1/2, extreme 3600, B 1/2 sine 3600, 1/2 B sine 3600, 1/2 M extreme 3600; always being aware that the sines are the respective means etc. This is as true as that, when finding the demonstrated series of extremes 2400, 4800 like 1 to 2, 1800, 5400 like 1 to 3, 1440, 5760 like 1 to 4, 1200, 6000 like 1 to 5 and therefore equal to the multiple series, it is known per se that the multiple series starts from 1 to 1 to continue from 1 to 2, 1 to 3 etc. Therefore, it is known per se that the given series will have to start from the two extremes, 3600, 3600; that the above- mentioned series in mathematical rigour begins from four equal terms in 3600, and as a consequence remains excluded from the harmonic series 3600, 2400, 1800, 1440, 1200, the duple 7200 formed by the diameter AM, which should strictly be the first term of the series. Equal then to this series starting from 3600, one finds that AB, hypotenuse of the right-angled triangle A 1/2 B, begins in its square from 7200, and begins from