german-historians-of-mathematics

"[…] whatever is closely connected with the ancient Indian religion must be considered as having sprung up among the Indian themselves, unless positive evidence of the strongest kind point to a contrary conclusion."

"I was amazed at how this outstanding connoisseur of Indian knowledge in the field of exact sciences showed himself to be so captive to Cantor’s authority."

"I cannot help thinking that it shows battle weariness rather than a considered opinion."

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it... If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’?"

"For Egypt, in my opinion, the one fact of experience was sufficient that there was a right-angled triangle 4, 3, 5. Then Pythagoras took the matter geometrically in hand and proved the theorem named after him by ascending in ancient Greek manner from individual case to individual case, finally to the quite general theorem. (Cantor 1905: 70)."

"[…] should it […] boil down to the fact that individual parts of the Sulbasutras are relatively modern interpolations?"

"Therefore, we do not doubt for a moment that Pythagoras’ stay in Egypt, that the lessons he received from the priests there, were among the things that were common truth."

"So much appears certain to us, that Pythagoras could have been in Babylon."

"Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)"

"Pythagoras remarked, we think, that 9+16=25. When he made this under all circumstances interesting remark, he already knew, no matter from which source, the experienced fact that a right angle is constructed by assuming the measure numbers 3, 4, 5 for the lengths of the two legs and for the distance of the end points of the same. We have pointed out that the Egyptians and the Babylonians perhaps possessed the same knowledge that the Chinese certainly had (Cantor 1880: 153)"

"Pythagoras, so we tried to prove, certainly acquired mathematical knowledge in Egypt, perhaps in the Euphrates countries. The former can be seen from the explicit traditions, as well as from the Egyptian character of some geometrical developments, the latter from the Babylonian seeming number differences of the Pythagoreans. The sum of geometrical knowledge, which was made accessible by Pythagoras and his school to the Greeks before the year 400, is not quite a minor one [...] It included the Pythagorean theorem […]. (Cantor 1880: 158)"

"It is hardly conceivable that these writers [Aryabhata, Brahmagupta, Bhaskaracharya etc.], should have been unacquainted with the Greek sources, which were available to Varahamihira [the earliest of these writers]; it is even less conceivable that they [the Indian writers], acquainted with the same, did not let themselves be influenced by them [the Greek sources]. Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works."

"But instead of Cantor admitting the independence or at least the relative independence of the Indians, i.e. the independence of their geometry from the Greeks, he expresses himself extremely tortuous, even he did not give up his Heron hypothesis by hiding it behind the doubtful question at the end, whether there are not at the end in the Sulbasutras relatively modern insertions."

"Cantor concludes that Indian geometry and Greek geometry, especially of Heron, are related; and the only question is, Who borrowed from whom? He expresses the opinion that the Indians were, in geometry, the pupils of the Greeks."

"The first writer to attribute this proposition to Pythagoras is Vitruvius, hardly a reliable witness. From then on, this account became more widespread, but always in connection with the famous hecatomb that Pythagoras is said to have offered in celebration of discovering the proposition—an anecdote that severely undermines the credibility of the entire story. This sacrifice is incompatible with the strict prohibition of all bloody sacrifices, which writers of the same period, indeed often the very same ones who elsewhere recount the hecatomb, have handed down to us from the Pythagorean ritual laws. Cicero himself took offense at this anecdote, and in the latest Neopythagorean tradition, the bloody sacrifice is replaced by that of an ox formed from flour. For this reason, the hecatomb is not only incongruous with the Pythagoreans, but also with the Pythagoreans themselves. Proclus, an insightful writer, expresses himself remarkably vaguely: ‘When we listen to those who want to tell old stories, we find that they trace this theorem back to Pythagoras.’ As this shows, he too was unaware of any reliable source."