MH = AH - AM = AC - BC.
Four times area of triangle ABC = 4 × (1/2) AC × BC.
Hence (AB)2 = 4 × (1/2) AC × BC + (AC - BC) (AC - BC) = 2 AC × BC + (AC)2 - 2 AC × BC + (BC)2. Or, (AB)2 = (AC)2 + (BC)2. Q. E. D.
“Now, admiral,” continued the delighted professor, “I’m going to send Mr. Osborn’s demonstration to some of the colleges and mathematical societies. Although it is original with Mr. Osborn, at least I imagine that it is, I really cannot believe that it is possible that he is the first discoverer of this method. If it should prove that Mr. Osborn is the original person who has ever used this method it will go down in the books as ‘The Osborn Demonstration.’”
“How did you happen to fall on that method, Mr. Osborn?” asked the admiral.
“Why, sir, I failed miserably on this question in the December examination and afterward I was determined to get it without referring to the book. One time when I was working at it, wondering why I couldn’t do it, I happened to erect the square on the hypotenuse and somehow drew in the triangles. Then when I looked at the figure I started to add up the areas of the different triangles and the square in the center and it all worked out naturally.”
“It’s an algebraic rather than a geometrical proof, or rather a combination of both,” remarked Professor Scott, “and Mr. Osborn deserves much credit. And as for the statement that Mr. Osborn cheated by carrying books in the examination with him, why that is as ridiculous as it is false and contemptible.”
“That is just my notion,” assented the superintendent. “Now, Mr. Osborn, don’t worry about this letter and don’t talk with anybody about it. There is undoubtedly somebody determined to do you terrible injury, but I think we can take care of you. Keep your eyes wide open, say nothing, not even to your closest friend, and if you learn anything whatever come to me immediately.”
Ralph left the superintendent’s office in a very happy and comfortable state of mind. He was indeed perplexed at the persistent hidden enmity that had been displayed against him, and for which he could imagine no cause, but he felt that he had a powerful friend who would protect him.
The next month, February, Ralph was in the fourth section in mathematics. The regular instructor assigned to this section was sick so that the head of the department, Professor Scott, took the section. He displayed much interest in Ralph’s work; this was apparent to all, and inspired by this Ralph devoted himself to preparing his recitations with a zest he had never felt before. He worked enough on his rhetoric and French to get satisfactory marks in these subjects, but in most of his study hours and much of his leisure after drill and on Saturdays and Sundays he devoted himself to his algebra. Before going to recitations he had always studied the principles carefully and worked out most, if not all of the problems.