...belongs to one of a subspecies or class will be common to every individual of the class. For example : " the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides," is a proposition equally true of every right-angled...

...implied in the hypothesis of the problem or theorem in question. Thus, in the proof of the theorem, " the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the aides," the formation of the right-angled triangle and of the squares is not technically...

...fit of enthusiasm, without any compunction one hundred oxen in commemoration of his discovery, that a square on the hypothenuse of a rightangled triangle, is equal to the sum of the two squares on the base and the perpendicular. Indeed such a cruel result of a scientific discovery...

...a fallacy in that scientific process by which we are conducted to the conclusion that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of its two sides, as to point out the want of connection between the theory of knowledge...

...determining the sum of the interior angles of any rectangular figure. 15. The semicircle described upon the hypothenuse of a right-angled triangle is equal to the sum of the semicircles described upon the sides. 16. If AB, CD be the opposite sides of a quadrilateral figure...

...APQB = QjRSC + CTUA. It is obvious that a special case of this theorem is the theorem proved before, that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the sides. (2) If through any point O within or without a circle, chords are drawn,...

...is really a process of generalization presenting a striking illustration of our principle. To prove that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the sides containing the right angle, Euclid takes only a single example of such a triangle,...

...Classes. Suppose a student should fail in his attempt to demonstrate the proposition that, The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other sides, would his failure invalidate the proposition? Aberrancies -of this...

...the perpendicular. 411. By Geometry there may be readily demonstrated the following 1. The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. Thus, Let h denote the hypothenuse, 6 the base, and p the perpendicular...

...comparatively simple case of geometric deduction, the 47th of the First Book of Euclid, ' the square described on the hypothenuse of a right-angled triangle is equal to the sum of the squares described on the two sides,' we shall find that the proof can be accomplished by two main...