Mathematical Collections (Commandino 1588)

...that is, BE to EA, he not only attempts to seek what cannot be found, but he thinks we do as well. Thus, having posited the proportion which KH has to HR (that is, BE to EA) and given HK, the smaller straight line of the third proportion is given. But point H is given. Therefore, the other end of the smallest straight line is also given. It is manifest that this falls either between HR or between RT. But we shall demonstrate that point T also falls between RS. And the first point Φ sometimes falls between HR and sometimes between RT, according to the position of the proportion which the given KH has to HR. For let the given proportion of KH to HR—that is, BE to EA—be double that of BA to AC. Therefore, the proportion of KH to HR is that which two have to one, namely four to two. And the proportion of KH to HS is that which four have to three. And as KH is to HS (namely, four to three), so is SH to HT, that is, three to 2 1/4. And as 3 is to 2 1/4, so is 2 1/4 to another certain value. Therefore, if it is done thus, it will be to a value smaller than 2 (which is HR), such that the smaller straight line of the third proportion—and indeed the smallest of all—is smaller than HR; and the point of section, such as Φ, falls between HR. But let the given proportion be quadruple. Therefore, the proportion of KH to HR is that which 8 has to 2; and the proportion of KH to HS is that which 8 has to 5. And as 8 is to 5, so is 5 to 3 1/8; and as 5 is to 3 1/8, so is 3 1/8 to a value smaller than 2. Wherefore, again, the section of the third proportion falls between HR. Let the proportion of KH to HR be then quintuple. Therefore, the proportion of KH to HR is that which 10 has to 2. And the proportion of KH to HS is that which 10 has to 6. But as 10 is to 6, so is 6 to 3 3/5, and as 6 is to 3 3/5, so is 3 3/5 to a value greater than 2. Thus, the point of section of the third proportion falls between RT. And it is manifest that all proportions which are less than quadruple cause such a section to fall between RH, while those which are greater than quintuple cause the point of section to fall between RT, just as we have set forth a lemma useful for a proportion of this kind. Therefore, since we have shown that the point of section such as Φ sometimes falls between HR and sometimes between RT (a fact not observed by him for the reason we have stated), he himself says the proposal is demonstrated whether point Φ is between HR or between RT. One must consider this before all else: wherever one takes the point Φ, whether below R or above it, it is not the case that as SH is to HT (that is, as KH is to HS), so is TH to HR. If, therefore, he said, "Let it be done such that KH is to HS, as SH is to HT, and TH is to HR," he refutes himself, assuming the sought-after conclusion as a premise. For with XK extended, and Kπ made equal to XK, and πH joined, and with lines drawn through points S, T, and R parallel to Kπ, that which is sought will be accomplished. And it is clear how this follows. For it was as Kπ to Sρ, so Sρ to Tσ, and Tσ to RY. But Kπ is equal to BD, and KR is equal to BA, and BE is equal to KH, so that AC is also equal to Rυ, and of the two BD and AC (that is, of the two Kπ and Rυ), two mean proportionals Sρ and Tσ have been found, which cannot be done—manifestly, with the straight line HK existing, and point R upon it. For it is not possible for two points like T and S to be taken between RK by the contemplation that occurs in the plane, such that it is as KH is to HS, so is SH to HT, and TH to HR. And although he takes F for S, the problem still cannot be performed, because it is solid by nature. Wherefore, he himself, knowing that the sought-after result is taken as a premise, did not dare to say that the other end of the smallest straight line is point R. He completes the rest of the construction above, that is, taking it between RH, as he wishes. And nonetheless, imprudently, he falls into the difficulty proposed from the beginning, not because he wished to write false things while acting with many, so as to lead anyone into error, but because he brings forward non-concluding reasons in the ratios themselves. I shall show this first in the corrupt and then in the sound way, running through that which is proposed, and then criticizing his position as not correctly taken. Since, therefore, a proportion is given...