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Forty-seventh Problem

The forty-seventh problem of Euclid's first book, which has been adopted as a symbol in the Master's Degree, is thus enunciated: "In any right-angled triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle." Thus, in a triangle whose perpendicular is three feet, the square of which is nine, and whose base is four feet, the square of which is sixteen, the hypothenuse, or subtending side, will be five feet, the square of which will be twenty-five, which is the sum of nine and sixteen. This interesting problem, on account of its great utility in making calculations and drawing plans for buildings, is sometimes called the Carpenter's Theorem.

For the demonstration of this problem the world is indebted to Pythagoras, who, it is said, was so elated after making the discovery, that he made an offering of a hecatomb, or a sacrifice of a hundred oxen, to the gods. The devotion to learning which this religious act indicated in the mind of the ancient philosopher has induced Freemasons to adopt the problem as a memento, instructing them to be lovers of the arts and sciences.

The triangle, whose base is four parts, whose perpendicular is three, and whose hypothenuse is five, and which would exactly serve for a demonstration of this problem, was, according to Plutarch, a symbol frequently employed by the Egyptian priests, and hence it is called by M. Jomard, in his Exposition du Systeme Mtrique des Amperes Egyptians, Exposition of the Ancient Egyptians System of Measurements, the Egyptian triangle. It was, with the Egyptians, the symbol of universal nature, the base representing Osiris, or the male principle; the perpendicular, Isis, or the female principle; and the hypothenuse, Horus, their son, or the produce of the two principles. They added that three was the first perfect odd number, that four was the square of two, the first even number, and that five was the result of three and two. But the Egyptians made a still more important use of this triangle. It was the standard of all their measures of extent, and was applied by them to the building of the pyramids. The researches of M. Jomard, on the Egyptian system of measures, published in the magnificent work of the French savants on Egypt, has placed us completely in possession of the uses made by the Egyptians of this forty-seventh problem of Euclid, and of the triangle which formed the diagram by which it was demonstrated.

If we inscribe within a circle a triangle, whose perpendicular shall be 300 parts, whose base shall be 400 parts, and whose hypotenuse shall be 500 parts, which, of course, bear the same proportion to each other as three, four, and five; then if we let a perpendicular fall from the angle of the perpendicular and base to the hypothenuse, and extend it through the hypothenuse to the circumference of the circle, this chord or lane will be equal to 480 parts, and the two segments of the hypothenuse, on each side of it, will be found equal, respectively, to 180 and 320. From the point where this chord intersects the hypothenuse let another lane fall perpendicularly to the shortest side of the triangle, and this line will be equal to 144 parts, while the shorter segment, formed by its junction with the perpendicular side of the triangle, will be equal to 108 parts. Hence, we may derive the following measures from the diagram: 500, 480, 400, 320, 180, 144, and 108, and all these without the slightest fraction. Supposing, then, the 500 to be cubits, we have the measure of the base of the great pyramid of Memphis. In the 400 cubits of the base of the triangle we have the exact length of the Egyptian stadium.

The 320 gives us the exact number of Egyptian cubits contained in the Hcbrew and Babylonian stadium. The stadium of Ptolemy is represented by the 480 cubits, or length of the line falling from the right angle to the circumference of the circle, through the hypothenuse. The number 180, which expresses the smaller segment of the hypothenuse being doubled, will give 360 cubits, which will be the stadum of Cleomedes. By doubling the 144, the result will be 288 cubits, or the length of the stadium of Archamedes; and by doublang the 108, we produce 216 cubits, or the precise value of the lesser Egyptian stadium.

Thus we get all the length measures used by the Egyptians; and since this triangle, whose sides are equal to three, four, and five, was the very one that most naturally would be used in demonstrating the forty-seventh problem of Euclid; and since by these three sides the Egyptians symbolized Osiris, Isis, and Horus, or the two producers and the product, the very principle, expressed in symbolic language, which constitutes the terms of the problem as enunciated by Pythagoras, that the sum of the squares of the two sides will produce the square of the third, we have no reason to doubt that the forty-seventh problem was well known to the Egyptian Priests, and by them communicated to Pythagoras.

Doctor Lardner, in his edition of Euclid, says: Whether we consider the forty-seventh proposition with reference to the peculiar and beautiful relation established in it, or to its innumerable uses in every department of mathematical science, or to its fertility in the consequences derivable from it, it must certainly be esteemed the most celebrated and important in the whole of the elements, if not in the whole range, of mathematical science. It is by the influence of this proposition, and that which establishes the similitude of equiangular triangles, in the sixth book, that geometry has been brought under the dominion of algebra, and it is upon the same principles that the whole science of trigonometry is founded. The thirty-second and forty-seventh propositions are said to have been discovered by Pythagoras, and extraordinary accounts are given of his exultation upon his first perception of their truth. It is however, supposed by some that Pythagoras acquired a knowledge of them in Egypt, and was the first to make them known in Greece.

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