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

374 Thirdly, I shall explain myself with regard to my proposition expressed with the precise words: that I expect to demonstrate the squaring of the circle a priori , the which proposition I confirm more than ever. I ask you, if by hypothesis it were square a priori ? If you said yes, I would say no, being sure I am telling the truth. The a priori demonstration consists substantially in providing the ratio, for which the circle must be found with the square in that precise ratio, and not in any other of those infinitely possible. This is precisely the one which I expect to demonstrate. If (by hypothesis) the diameter to the circumference were geometrically demonstrated as 7 to 22, this would be an a posteriori demonstration, because it indeed demonstrates the fact to me, but not the reason of the fact. This is still correct, and please deign to consider it well, because as well considered as it is, you shall clearly see that we have hitherto not understood each other. Lastly, in fourth and last place, I beg you to be satisfied with me that I ask you the following questions, which I will ask you to answer precisely and separately. This way we shall move forward, but certainly not in the way followed until now. The first is that you answer me with precision regarding what I said in the second part of this letter. If what I say is true, agree with me forever; if not, object. The same I ask regarding what I said in the third part, either yes or no; giving the reason if no. The second will be long, but it will reach a conclusion. Having compared the two figures, square and circle, on the basis of sines and strings reduced to series deduced from parts common to the two figures, I have demonstrated the circle to be constantly harmonic, the square variously arithmetic and counter-harmonic; please answer whether the demonstration is true. If it is true, I have concluded that the circle is intrinsically harmonic both because of this demonstration and because I have previously explained all the harmonic phenomena, causing one to experience first-hand that they are all solved in the general unity, and that among all the possible figures there is only the circle that is one in itself; and because I have pointed out that the given unit with the given indefinite term x and by demonstration incapable of another mean than of the harmonic one in the term resulting as 2; and given the unit as the compass opening, it is found to end up as 2 in the diameter, the indefinite term x in the circumference; hence a clear indication of a harmonic nature, because it is not capable if not of a harmonic mean, and the mean determines the extremes. Both because the circular figure is physically inseparable in the root of the third sound, and because the third sound, physically, is nothing but two spherical volumes of moved air, and the sphere in the solid is a circle in the plane. But in the third sound it is demonstrated as the harmonic root, and it is so physically, therefore the circle is of the same nature. Please reply whether I have concluded well or badly. So, therefore, the third sound is inseparable from the circle, and hence reciprocally. But because the circle is on a plane, and the third sound in a solid, the demonstrative method is inseparable in the examination from the physical method. Here lies the importance, so please answer me with the same precision.