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

381 LETTERS be the first to acknowledge the truth that I know is there, and which you will in fact understand better than me. If you do not understand me due to my inadequacy, it is entirely superfluous to proceed, because I shall never be able to tell you more than what I say in the present letter. But as in this letter I shall speak more clearly, so (whatever the judgement may be) I encourage both of you to reveal nothing to anyone of what you shall detect in peculiar and in detail. In conclusion, the usual secrecy both in the whole and in the parts, because in this letter there will be something precise both for the Most Illustrious Signor Dottor Balbi and for Your Reverence. The demonstration of the circle as harmonic by its own intrinsic nature, and indeed as a harmonic root, either wasn’t understood by the Most Illustrious Signor Dottor Balbi, or he didn’t wait enough or I didn’t expound it in its clarity and force. It must be the latter, so I beg him to reflect on the fact that the aforementioned demonstration concluded that the circle is not a result of infinite polygons, as the geometricians defined it, but the result of the infinite harmonic means of discrete geometric proportions. And this by force of legitimate geometric demonstration. Please reflect on the fact that given, for example, a square infinitely full of straight lines in a plane, the circular line does no more than cut, that is to say subtract, from all the infinite aforementioned straight lines, such a portion of line which, after subtraction, always remains the harmonic mean of that ratio in which the same line has divided the diameter. So if it were possible without the circle to assign the harmonic means respectively to the same whole line, as arithmetic means always relative to the ratio in which the diameter is divided by the same line, a circular line would be constructed through such harmonic means. But that’s exactly it, since it is not possible if not through the circular line, therefore the circle is the harmonic root, in which all the infinite harmonic means are contained. One must therefore consider this figure carefully, and fully understand the strength of the demonstration, by which the circle intrinsically changes the appearance in which it has hitherto been seen and considered. Otherwise, it is impossible to understand my proposition, and my system. Signor Dottor Balbi, to whom this pertains in particular, should then trouble himself to delve into it; and then, having intrinsically understood the demonstration may he deny, if he can, that the circle is none other than the harmonic root, demonstrated in the opening of the compass in the first sine AB, that through BO passes to C, and forms the quadrant ABOC. And when one says compass opening in AB, one says 1/2 compared to the diameter EC. Having said that, I shall come to the phenomenon of the third sound, and I find it as the root of the infinite harmonic progression in 1/2. The infinite harmonic progression is the position of the infinite harmonic means, as, given 1, 1/2, 1/3, the harmonic mean 1/2; given 1/2 1/3 1/4 the harmonic mean 1/3; given 1/3 1/4 1/5 the harmonic mean 1/4, and so on infinitely. Hence, etc. So it starts from 1/2, and proceeds infinitely. But this is identically what happens in the circle, because, identically, the root 1/2 of the infinite harmonic progression. Therefore, the same identically is the