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

366 the comparison I find the harmonic circle constantly: the square being either arithmetic or counter-harmonic respectively. Having established the harmonic circle, I find the infinite harmonic root by means of the third sound, which is an effect inseparable from two sounding lines played at the same time etc., etc. I then find it demonstratively by means of the four characteristic and specific signs, which converge between them identically both in the physical and in the demonstrative. Now, I ask whether one should suspect that there are two harmonic roots? The characteristic signs are identically the same; the root determined in 1/2 is the same. Therefore, etc. If the harmonic root is demonstratively and physically determined in 1/2, I ask, what is the purpose in wanting to know the physical means of the third sound? Certainly not for my proposition in general, which is the squaring; as for this need, I determine that lines, which I indeed suppose to be sounds, but only as they inseparably bring with them the effect of the third sound, by means of which one can physically detect that square and circle converge in the diameter etc., etc., which is the prime unit. But in the said case, the third sound is considered as a physically demonstrative sign of the truth I propose. Of what use is it then to know its physical means? May one add that my proposition is (in geometric rigour) that square and circle must converge in the diameter: a proposition which is known per se , and which has no need of evidence, because the figures themselves are the demonstration and I take the two figures in such a precise sense because in such a sense I have previously demonstrated the circle to be harmonic per se , the square to be arithmetic per se etc. Therefore, I cannot understand in any way how the physical explanation of the third sound, in order to adapt it to linear figures, becomes necessary. In that case a demonstration causing me to believe in such a need becomes necessary; but this shall never ever be given. On the contrary, if we entangle ourselves in physical things, goodbye forever to the completion of the examination. It takes little to see this. To sum up, my natural argument is this. There is the third sound (be it in any way); and it is the harmonic root. This being given, I demonstrate and form my treatise. It is impossible that from this argument and from this method there should arise the need to search for the physical way of producing the third sound. After all of this, which I have said in strict geometrical language (and you must allow me to speak this way when needed) tell my part to the most esteemed Signor Dottor Balbi, that having carried out the examination, which I intend to be done with geometric rigour, and in no other way, I shall promise, as the good servant that I am, to place before his eyes and into his hands also the physical way in which this third sound is produced. Moreover, it seems to me impossible that he does not see for himself in the eighth figure the physical principle in general. In the AFEDCB arc with respect to the lines or strings AB, AC, AD, etc., imagining the air being moved circularly (and this is beyond doubt) with the volumes of air intersecting in the collision (and this is also beyond doubt) encountering at that point resistance to further intersection and co-penetration, if it can be said in this way, and therefore constant at that point. It shall