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From the triangle of which the hypotenuse is twice the lesser side the three first regular solids are formed first, the equilateral pyramid or tetrahedron; secondly, the octahedron; thirdly, the icosahedron; and from the isosceles triangle is formed the cube. Let us now assign the geometrical forms to their respective elements.
He proposed to take some leading proposition of Euclid's, and show by construction that its truth was known to us, to demonstrate, for example, that the angles at the base of an isosceles triangle are equal, and that if the equal sides be produced the angles on the other side of the base are equal also, or that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the two other sides.
From the edge of a cliff just in front of this hut, there swung a long cable, which reached almost to the base of the shore beneath us; and, even as we looked, we saw what seemed a tiny bucket go swinging slowly down that strange hypotenuse. We watched it till we saw it get to the end of its journey in the valley beneath, not far from the great mill to which we were bound.
His last words were in the form of a question, addressed to Mike, on the subject of the hypotenuse and the square upon the same. 'A remarkably intelligent boy, said Psmith. 'You must let him come to tea at our flat one day. I may not be in myself I have many duties which keep me away but Comrade Jackson is sure to be there, and will be delighted to chat with him.
You will then have one side and the corner fixed, and the problem is to get another line at right angles to it. Boys who have studied geometry know that "in a right-angle triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides." It isn't necessary to understand this, but it is the principle employed in "squaring."
In this manner, by cutting carefully in parts the squares on the two sides of the right-angled triangle, and arranging these parts carefully, one makes from them the square on the hypotenuse; that is demonstrating empirically the 47th proposition of the first book of Euclid.
I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side.
In vain the aged boatman strains, His heaving sides reveal his pains; The angry water gains apace Both of his sides and half his base, Till, as he sits, he seems to lose The square of his hypotenuse. The boat advanced to X + 2, Lord Ullin reached the fixed point Q, Then the boat sank from human eye, OY, OY 2, OGY. But this is only a sample of what can be done.
The plain fact is that, after a girl has spent four years of her time and a great deal of her parents' money in equipping herself for a career that she is never going to have, the wretched creature goes and gets married, and in a few years she has forgotten which is the hypotenuse of a right-angled triangle, and she doesn't care. She has much better things to think of.
"If there is a viva voce, be sure and speak up and give your answers as though you were sure of them. They may be wrong, but on the other hand they may be right. Anyway, the one thing the examiners will not thole is a body that dithers. "Take a last keek at that Proposition they may call them Theorems, though about the Square on the Hypotenuse. It hasn't been set for four years.
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