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Updated: May 18, 2025
To his surprise, although a fire smouldered on the hearth, there was no other sign of human occupancy. Then the young man searched in vain for some hit of writing, such as had guided him to this point. "I declare!" he exclaimed at length; "the corollary is worse than the theorem, and things are becoming so decidedly mixed that we must begin to go slow.
The mind has succeeded in discovering an attribute common to two dissimilar figures; and it is this discovery which may lead to a series of conclusions by means of which the theorem of Pythagoras will be finally demonstrated.
"While you're talking about peculiar things, I'll tell you one. In class I came right up against Oka Sayye on the solution of a theorem in trigonometry. We both had the answer, the correct answer, but we had arrived at it by widely different routes, and it was up to me to prove that my line of reasoning was more lucid, more natural, the inevitable one by which the solution should be reached.
Just now, he felt as dogged as any ambitious school-boy ever did over an obstinate theorem in Euclid here was a problem there were all the rules for its clear solution, yet the answer never would come right. Perhaps he was preparing for another attempt, as he drew his chair closer to her and looked into her face, while they sat in the spot of all spots, the most flattering to his designs.
I've been grinding theology till I'm fairly sick of the word, and I've learned just one thing, Ardea, and that is that you can't prove a single theorem in it." "But there are some things that don't appear to need any proof; one seems to have been born knowing them. Don't you feel that way?" He shook his head slowly. "I used to think I did; but now I'm afraid I don't.
And it is reported that Pythagoras, upon the discovery of this problem, offered a sacrifice to the gods; for this is a much more exquisite theorem than that which lays down, that the square of the hypothenuse in a right-angled triangle is equal to the squares of the two sides. Right, said Diogenianus, but what is this to the present question?
An example of the way in which this theorem is practically applied is the well-known case of a rower who sets out from P in order to cross at right angles a river indicated by the parallel lines. He has to overcome the velocity a of the water of the river flowing to the right by steering obliquely left towards B in order to arrive finally at C.
The latter immediately accepted the result, and published a paper on the subject, in which the theorem was made public for the first time. It is very singular that a man of such acuteness never achieved anything else of significance. He was at my station on one occasion when a total eclipse of the sun was to be observed, and made a report on what he saw.
Dissenters will of course be found, sporadically, who do not readily fall in with the prevailing animus; but as a general proposition it will still hold true that any such quarrel forthwith becomes a just quarrel in the eyes of those who have so been committed to it. A corollary following from this general theorem may be worth noting in the same connection.
Only, with singular lack of logic, I have sometimes allowed regret to overtake me, and I have looked at conduct founded upon exceptional principles with the eyes of the ordinary man. I should have been ascetic to the end; contemplation ought to have been enough for me, especially now, when the hair begins to whiten. But, after all, I am a man, and not a theorem.
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