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

368 and so on infinitely, without one being ever able to realise the circular portion, which one supposed to be positioned in a straight line. Therefore, the proposition remains true by which of the two straight lines necessary for the squaring, one concrete one having been assigned and determined in demonstrative precision, the other one can never be found if not by approximation. This proposition is common to all sciences; as it happens when operating by rational numbers, by geometrical lines, by algebra, and by the physical harmonic science. But the last-mentioned has a great advantage over all the others, as it does not consider immediately the material quantity of the two lines, but considers instead, immediately and mainly, the a priori ratios, from which a posteriori must proceed the quantity determined and realised by the ratios, intended and conceived as forms of the quantity determined. (I have to be allowed such ways of expressing myself, if I wish to be understood). If then one finds a ratio producing a tertium quid both physically and demonstratively, and that one has a known term of the ratio (may the other term be indeed as unknown as one wishes); if one knows in advance what the tertium quid that must be produced is, the term previously unknown shall be unfailingly known to me, when joined to the known term it produces for me the tertium quid, which I know must be produced demonstratively and physically. And it shall be known to me because of the ratio, and not because of the quantity; because although it becomes manifest to me with the effect produced, which I already know must be produced, and it becomes manifest to me because, joined to the known term, it forms the ratio sought for the production of the effect, which is the tertium quid, its precise concrete quantity is not made manifest to me, so that it is expressible either by numbers, or by a geometric line, or by algebra, or by any way. This inexpressible quantity can be assigned by approximation of an infinite progress, and nothing more. So we also agree on this, and I have not claimed in the squaring of the circle, nor do I claim, to assign the ratio a priori, so that the circle to the square must be found in the ratio thus determined. I say determined, precise, concrete, not according to the quantity of one of the two terms, but according to the tertium quid produced, which I know must be produced physically and demonstratively. This is the idea of the treatise, and it is on this that judgement must be given. May one observe then, that I say a lot more than what appears. Because given as a hypothesis the squaring of the circle in two straight lines of determined quantity, and (by hypothesis) let 14 be the line, that is to say one side of the square, 11 one quarter of the circumference, so that when the two lines are reduced to a complete figure, let the surface area of the circumscribed square be 14, the area of the inscribed circle 11, in such case, if I asked for what reason one finds the square circumscribed to the circle inscribed in the ratio of 14 to 11, what would the answer be? Nothing scientific certainly, as there is no known science that can assign such causes. The sciences of quantity, which usually deal with it, assign the effect, and not the cause. The physical harmonic science assigns the causes, and when the cause is known to me, it is impossible that the effect should be unknown to me; and to say everything