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Updated: June 11, 2025
The first duty of the new-comer is to teach his nether extremities to avoid instinctively the hypothenuse of the street-triangulation, and the last lesson the resident fails to learn is which of the shortcuts from point to point is the least lengthy.
We reply to you, and we show to you that we can reason by indicating that the square of the hypothenuse of a right-angled triangle is equivalent to the sum of the squares of the other two sides. Hope to hear from you further.
Take the word isosceles; a schoolboy can be shown what an isosceles triangle is, but, if he knows nothing of the derivation, he will wonder why such an apparently outlandish term should be necessary to express so simple an idea. If mathematicians had had more Greek, perhaps the misspelt form 'hypothenuse' would not have survived so long.
That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these two 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 whatever is anywhere existent in the universe.
Hence the impression of concavity; and this impression must remain, until the elevation shall bear so great a proportion to the extent of prospect, that the apparent parallelism of the base and hypothenuse disappears when the earth's real convexity must become apparent. Qui n'a plus qu'un moment a vivre N'a plus rien a dissimuler. Quinault Atys.
Her heart would hurt as tangibly as if the surface of her body were red with a wound from it, yet, sitting there at her milk and biscuit, her gaze into the monotonous repetition of wall-paper design, the thought of that Sunday dinner out there, with its invariable roast chicken, bread stuffing, candied sweet potatoes, and lemon-meringue pie; the Sunday-afternoon lethargy; the hypothenuse of her father asleep in his chair, the newspaper over his face; Albert, the celluloid toothpick moving along his lips, puttering around at favorite locks and bells; the mere visualization was such a fillip to her present that she lay back on the bed, stretching her arms and legs like a great, luxurious cat, her lips curved to a smile.
For the geometrical proposition being that the squares of the perpendicular and base are equal to the square of the hypothenuse, they may be said to produce it in the same way as Osiris and Isis are equal to, or produce, the world.
It may be jure divino that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the sides, that he is a blockhead who believes otherwise, and that a permanent apparatus should be set up in every land for teaching this mathematical faith; and yet it may be equally jure divino that no one shall be compelled to avail himself of that apparatus, or be punished for doubting or denying the proposition.
After having worked some fifty problems out of Hamilton Moore, of blessed memory, and having drawn an infinity of triangles with all possible degrees of incidence, with very neat little ships, now upon the base, now upon the hypothenuse, and now upon the perpendicular, my erudite usher pronounced me to be a perfect master of the noble science of navigation in all its branches, for the which he glorified himself exceedingly.
If we take a canvas of ordinary proportion, namely, one whose length is equal to the hypothenuse on the square of its breadth, as 28×36 or 18×24 and divide it into unequal divisions as three, five or seven, we will produce points on which good composition will result. The reason for this is that the remaining two-thirds becomes a unit as has the one-third.
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