Hitherto Unpublished Works, Fascicle 4

A full investigation of any such matter regarding the seventh [proposition] of the tenth book of Euclid original: "septimam decimi Euclidis"; Euclid’s Elements, Book X, Proposition 7 states that commensurable magnitudes have the ratio of a number to a number. is required; nevertheless, so that the pure natural philosopher may see that the mathematician can lead him wherever he wishes, into both false and true matters, I will touch upon a certain contradiction of this. For 94 a 2. from the second corollary of the eighth proposition of the sixth book of the Elements, it is clear that a line drawn perpendicularly from a right angle to the base of a triangle O. 5 b. is a mean proportional between the whole base and the segment of the base adjacent to the said line. But this base is the diameter original: "dyameter"; in this context, the diagonal of a square., and the side of the triangle is the side of the square original: "costa quadrati".; therefore, the ratio of the diameter to the side is as the side is to the segment of the diameter adjacent to it. It follows from this, first, that there is some ratio and consequently a commensurability commensuracio: the property of two lengths being measurable by the same unit corresponding to that ratio; and second, that the ratio and commensurability are known original: "note"; meaning evident or mathematically defined..

This is because the tenth principle that is, the definition of the fifth book of the Elements teaches how to investigate the ratio of the first to the last in three continuously proportional quantities through the square of the ratio of the first to the second. But the ratio of the first to the last is known here because it is double, by the ninth [proposition] of the first book and the twenty-sixth of the same. Therefore, that which leads us to its knowledge, and which is the means for knowing it, will be even better known; this is, namely, the ratio here between the first and the second, which are the diameter and the side. Furthermore, if a composite whole made of several parts is known, then the simple parts from which it is composed will also be known. But the squared ratio original: "duplicata"; literally a duplicate or squared ratio. here is composed of the ratio of the first to the second, and the second to the third, as I demonstrated in The Common Principles of Mathematics, and all competent mathematicians know this. Therefore, if the composite ratio is known, those two parts from which it is composed will also be known.

Likewise, if the relationship of the middle term to the extremes is known, then the relationship of the extremes is also known; similarly, if one is known, the other will be. Likewise, that ratio which is between the middle and the extremes is the definition of the one between the extremes. But a definition makes the thing defined known, and is more known in an absolute sense and according to nature. Likewise, that ratio which is between the diameter and the semi-diameter...