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Updated: May 11, 2025
That the square of the hypothenuse is equal to the squares of the other two sides, cannot be known, let the terms be ever so exactly defined, without a train of reasoning and enquiry. But to convince us of this proposition, that where there is no property, there can be no injustice, it is only necessary to define the terms, and explain injustice to be a violation of property.
No one, probably, ever believed that the will of a god kept parallel lines from meeting, or made two and two equal to four; or ever prayed to the gods to make the square of the hypothenuse equal to more or less than the sum of the squares of the sides. The most devout believers have recognized in propositions of this description a class of truths independent of the devine omnipotence.
"Well, my boy, I have just constructed two similar right-angled triangles; the first, the smallest, has for its sides the perpendicular pole, the distance which separates the little stick from the foot of the pole and my visual ray for hypothenuse; the second has for its sides the perpendicular cliff, the height of which we wish to measure, the distance which separates the little stick from the bottom of the cliff, and my visual ray also forms its hypothenuse, which proves to be prolongation of that of the first triangle."
The sail which they are now just hoisting is, in shape, like a right-angled triangle, with a parallelogram below its base; the hypothenuse or head of the sail is secured to a yard, like an enormous fishing-rod; the halyards are secured to it about a third of the way from the butt-end, and it is hoisted close up to the head of the mast.
But in all the three cases which I have just cited, the inconceivability seems to be apprehended directly; no train of argument was needed, as in the case of the square of the hypothenuse, to obtain the verdict of consciousness on the point. Neither is any of the three a case like that of the school-boy’s mistake, in which the mind was never really brought into contact with the proposition.
Or his idea may be that of an abstract relation, say of that between the sides and the hypothenuse of a triangle, such a relation being, of course, a reality quite as much as a glass of beer is.
It may be said that his course along this stream, on his return from the Apache country, was like the base of a triangle, while he was now following the hypothenuse. This latter route was preferable in every sense to that which he had been using for the last few days.
Nevertheless, towards evening the leaders took some wise strategic precautions; they protected the army, which rested in the rear on Sedan and the Meuse, by two battle fronts, one composed of the 7th Corps, and extending from Floing to Givonne, the other composed of the 12th Corps, extending from Givonne to Bazeilles; a triangle of which the Meuse formed the hypothenuse.
Two and two cannot make six, neither can two distinct bodies occupy the same space at the same time, nor the square of the hypothenuse be otherwise than equal to the sum of the squares of the other two sides of a right-angled triangle. Our knowledge of natural science is, however, founded on experience.
Spencer is real; but, in the case of the propositions classed by him as complex, consciousness, until the media of proof are supplied, gives no verdict at all: it neither declares the equality of the square of the hypothenuse with the sum of the squares of the sides to be inconceivable, nor their inequality to be inconceivable.
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