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

350 3rd point of difficulty, you say that the third sound is imperceptible: “It is believed that the third sound is not in any way perceptible on instruments with metal strings, since on these the sound is not protracted, nor can be protracted but instead ceases immediately”. This proposition is contradicted by experience, because if one didn’t adopt the device of little pieces of cloth on harpsichords and other instruments with metal strings, one would hear from the jacks an uncommon confusion of preceding sounds resounding with the successive ones, as is proved above all by the experience of bells; and none of these instruments obtain the vibrations through the use of the bow as is claimed in the aforementioned reply. Furthermore, organ pipes and other wind instruments maintain the sound with greater constancy and control, and yet from these the third sound is not heard; hence the doubt could arise that the third sound is not produced by the causes attributed. 93. Tartini to G.B. Martini I have finally had the consolation of seeing the examination begun from your letter, in which the difficulties encountered by Your Reverence and by the worthiest Signor Dottor Balbi are expressed in order. I shall reply in the same order, and I shall first quote the objection raised to me, which shall be underlined. After replying, I shall tell you my opinion with the agreed frankness. If irrational quantities can be expressed only by means of lines, these either have a common measure or they don’t. If they have it, they are rational. If they do not have it, they are irrational. And as a consequence they cannot be reduced to a proportion, etc. The first part of the dilemma is false, if they have it, they are rational. In my explanation sent there in November I believed (and with reason) to have solved forever this objection, which you raised in October. It must be said that I am not able to explain myself clearly. I discovered then the phenomenon of the equal weights adapted to the extendible sounding string. I demonstrated the sound of the second weight to the sound of the first weight in radical duple ratio; the sound of the third weight to the sound of the second in radical sesquialter ratio. But these sounds (all irrational, and all of irrational lines) have as a common measure the equal weights arithmetically numbered 1, 2, 3, 4; and the common measures in general are number, measure and weight; therefore there are irrational sounds of irrational lines, which have a common measure. Therefore the first part of the dilemma is false. As a result, one has the duty of clearing up how irrational quantities are introduced. They are introduced as necessary to the demonstration of the principal proposition. The sounds are produced by weights