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My brain, acted upon by two forces, was compelled to take the hypothenuse, and I think the concussion was considerably diminished thereby.
It was called the Keazletown road, from a little German village on the flank of Massanutten; and as it was the hypothenuse of the triangle, and reported good except at two points, I decided to take it. That night a pioneer party was sent forward to light fires and repair the road for artillery and trains.
Lady Deppingham is one of the angles; Mr. Browne, the American gentleman, is another; the native population is the last. Each wants to be the hypothenuse. While the interests of all three are merged in the real issue, there is, nevertheless, a decided disposition all around to make it an entirely one-sided affair." "I don't believe I grasp " muttered Deppingham blankly.
Let a man of good parts know all the maxims generally made use of in mathematics ever so perfectly, and contemplate their extent and consequences as much as he pleases, he will, by their assistance, I suppose, scarce ever come to know that the square of the hypothenuse in a right-angled triangle is equal to the squares of the two other sides.
The two theorems, that the internal angles of every right-line triangle are equal to two right angles, and that the square of the hypothenuse of every right angled triangle is equal to the sum of the squares of the other two sides, are ascribed to him.
That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe.
For the square of 3 is 9, and the square of 4 is 16, and the square of 5 is 25; but 9 added to 16 is equal to 25; and thus, out of the addition, or coming together, of the squares of the perpendicular and base, arises the square of the hypothenuse, just as, out of the coming together, in the Egyptian system, of the active and passive principles, arises, or is generated, the world.
The symbol of universal nature among the Egyptians was the right-angled triangle, of which the perpendicular side represented Osiris, or the male principle; the base, Isis, or the female principle; and the hypothenuse, their offspring, Horus, or the world emanating from the union of both principles.
Among these we may mention one, viz., "The square of the hypothenuse of a right-angled triangle equals four times the area of the triangle, plus the square of the difference of the other two sides." This principle has been known to mathematicians by means of arithmetic and algebra, but has never before, we believe, been reduced to a geometrical demonstration.
Spencer’s opinion, “disprove the validity of the test;” not only because any test whatever “is liable to yield untrue results, either from incapacity or from carelessness in those who use it,” but because the propositions in question “were complex propositions, not to be established by a test applicable to propositions no further decomposable.” “A test legitimately applicable to a simple proposition, the subject and predicate of which are in direct relation, can not be legitimately applied to a complex proposition, the subject and predicate of which are indirectly related through the many simple propositions implied.” “That things which are equal to the same thing are equal to one another, is a fact which can be known by direct comparison of actual or ideal relations.... But that the square of the hypothenuse of a right-angled triangle equals the sum of the squares of the other two sides, can not be known immediately by comparison of two states of consciousness: here the truth can be reached only mediately, through a series of simple judgments respecting the likenesses or unlikenesses of certain relations.” Moreover, even when the proposition admits of being tested by immediate consciousness, people often neglect to do it.
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