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and geological sign for the operations of surveying operations indispensable for the regulation of the annual alluviums of the Nile in its bed and in its valley."

37. TO ILLUSTRATE GEOMETRIC TRUTH.

The square and the triangle, with their several properties, together with the relation of diameter and circumference, as well as the correspondence of the radius of a circle and the side of a square, are all well brought before the intelligent eye in the pyramid.

Mr. H. C. Agnew, in a work published 1838, says, "The pyramids of Egypt appear in general to have been emblems of the sacred sphere and its great circle exhibited in the most convenient architectural form. The chief objects of these buildings being to serve for sepulchral monuments, the Egyptians sought in the appropriate figure of the pyramid to perpetuate at the same time a portion of their geometrical science."

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This gentleman was, perhaps, the first to point out an interesting mathematical discovery. "The Third Pyramid," said he, was the spirit of this holy circle, since it defined the square equal to it in perimeter and in area by showing the difference between their sides and the diameter of the circle."

38. TO SHOW THE PROPORTION OF DIAMETER AND

CIRCUMFERENCE.

Mr. John Taylor, in The Great Pyramid: why was it built? lays down the proposition that the vertical height is to the double of its base as the diameter is to the circumference of a circle. This question is dependent on the angle made by the face of the pyramid with its base.

The slope angle has been more accurately determined since the discovery of the casing-stones by Col. Vyse in 1837. Mr. Taylor speaks of 51° 49′ 46′′, and Mr. Smyth of 51° 51' 14". That slope with a certain base establishes the so-called Rho theory (7), the proportion of diameter to circumference, 3·14159. Archimedes made it 3-14286. The Hindoos' Vija Ganita, of 3927 to 1250, brings out 3·1416. On this Mr. Taylor remarks, "The Hindoo proportion is identical with that, so far as its numbers go, which was expressed in English inches when the pyramids were founded." While the real amount is 3·1415927, he obtains 3.141792 from the pyramid. He thinks no other pyramid has so true a relation.

The Queen's Chamber has a niche. This, 185 inches, multiplied by 10 and then by 3.14159, yields 5812, the vertical height of the pyramid. The wall of the chamber, 182-62 pyramid inches, multiplied by 100 and divided by 2, will show 9131 for the side of the pyramid in pyramid inches.

Prof. Piazzi Smyth utilises the Rho theory in connection with the King's Chamber, to get the length of a cubit, saying, "On being simply computed according to the modern determination of the value of π, and length of the year, and comes out from the local measure of 412.545 British inches to be 25.0250 + British inches."

Although an ingenious discovery, and admitted by Sir John Herschel, this latter distinguished man adds, "We are not entitled to conclude that they (the Egyptians) were aware of this coincidence (3.14159), and intended to embody both results in their building."

Sir Edmund Beckett calls attention to the assumed 11 to 7

theory; that, with the slope of 51° 51' 14", the width is to the height as the length of a quadrant is to its radius. He does not think with Mr. Smyth and others that this was a primary motive of construction, "though," says he, "they did use it for fixing the size, probably taking it approximately from the slopes."

He shows that with the angle at 51° 50' the height is a mean proportional between the length down the middle of each slope and half the width of the base. The 51° he esteems "about the slope at which mounds of earth will stand naturally." He points out another singular coincidence. The diagonal angle at the top, 96°, or four times 24°, would equal that of the four sectors of a quindecagon. (Euclid, iv. 10, 11, 16.)

The parallelism is exhibited in the coffer, whose height is to the two adjacent sides as the diameter is to the circumference. Captain Tracey, taking for the radius of a circle the height of the pyramid, 232.52 cubits, pyramid measure, finds the diameter bear the same proportion to the periphery of a square whose side is 365 243, the length of the base in cubits, or days in a year, as 1 is to 3 1416. With 412-132, the length of the King's Chamber in inches, as the diameter, the circle would equal a square whose side, 365 242, in cubits, measures the base of the pyramid.

"Never have any monuments," says M. Dufeu, "exercised the sagacity of the learned as the pyramids of Gizeh."

39. TO MASONIFY THE QUADRATURE OF THE CIRCLE. Mr. H. C. Agnew conceives that one purpose of the erection was to masonify approximately the relation of square and circle.

"Here we find," writes he, "the quadrature of the circle exemplified in a curious manner, with all practicable approach and correctness, by the Egyptians." He, however, admits that "its arithmetical solution is known now to be impossible; the geometrical solution, in all probability, is so likewise; but whether the Egyptian priests were of this opinion I cannot venture to say."

Only a few quotations can be given from his publication, just sufficient to indicate his object :—

"If a square described about a circle be conceived to be drawn up from the centre in the form of a pyramid, having the perpendicular equal to the radius of a circle, and the superficies of the square be supposed to adjust itself equally among the planes of the four isosceles triangles of the faces of the pyramid, each face of such pyramid will, of course, be equal in area to one quarter of the square, or equal to the square of the radius; and the new square formed by the four lines of the bases of the triangles of the faces of the pyramid will be equal in perimeter to the circumference of the circle, with an error in excess of about one part in fourteen hundred."

"In the original diagram we find the proportion of five to four very dominant; the diameter of the circle is five, and that of the great square four, and thence, of course, the perpendicular of the pyramid is to half its base as five to four."

"If the tangent be to the radius as five to four the angle is 51° 20′ 25′′, and this being so very near the result of my observations, I am justified in concluding that the perpendicular of the Great Pyramid was to half its base as five to four, or to its base as five to eight."

"Two perpendiculars, being radii of circles, are together equal to the sum of the perimeters of the bases."

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He was particularly emphatic in his observations forty years ago upon the superiority of the Third Pyramid. Its true angle is affirmed to be 51° 51′ 14′′, and this, adds he, would be a perfection which neither of the two great pyramids separately possessed; namely, that its perpendicular was the radius of a circle, the circumference of which was equal to the square of its base." He concludes with this statement :"The Third Pyramid appears to be an emanation (if I may so say) from the first great principle of the system, the circle of origin, of which it is the spirit or essence.

Hekekyan Bey of Constantinople holds a similar high conception of the Third Pyramid, saying, "Of the Siriadic monuments erected in the land of Egypt, hers was considered to be the richest in scientific records, and the most perfect; it was, also, the most beautiful from its high ornamentation, being of a ruddy complexion, from its exterior casing of polished granite."

Sir Henry James brings out a similar result to that shown by Mr. Taylor first. He speaks of a pyramid rising at the corners nine to ten as a π pyramid, and its height being equal to the radius of a circle whose circumference is very approximately equal to the length of the four sides of the base." The height 486 × 2 x 3.1416 3053.6. But four times the length, 764, 3056.

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A curious thing is noted respecting the floor of the Antechamber. The granite part is, according to Mr. Casey, 103-03 pyramid inches, and the limestone 116.26 inches. Taking the first as the side of a square, and the last as the diameter of a circle, the areas of the two figures will be about equal. The side of

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