and altitude of Cheops = √(MH × MC) Let us now compare the construction of the two stars:-- Fig. 69. TO CONSTRUCT THE STAR PENTALPHA FIG. 69. Describe a circle. Draw diameter MCE. Divide MC in mean and extreme ratio at H. Lay off half MH from C, to D. Draw chord ADB, at right angles to diameter ECM. Draw chord BHN, through H. Draw chord AHO, through H. Connect NE. Connect EO. Fig. 70. TO CONSTRUCT THE STAR CHEOPS, FIG. 70. Describe a circle. Draw diameter MCE. Divide MC in mean and extreme ratio, at H. Describe an inner circle with radius CH, and around it describe the square a, b, c, d. Draw diameter ACB, at right angles to diameter ECM. Draw Aa, aE, Eb, bB, Bd, dM, Mc, and cA. The question now arises, does this pyramid Cheops set forth by the relations of its altitude to perimeter of base the ratio of diameter to circumference; or, does it set forth mean proportional, and extreme and mean ratio, by the proportions of its apothem, altitude, and half-base? The answer is--from the practical impossibility of such extreme accuracy in such a mass of masonry, that it points alike to all, and may as fairly be considered the exponent of the one as of the others. Piazzi Smyth makes Cheops 761·65 feet base, and 484·91 feet altitude, which is very nearly what he calls a [Pi] pyramid, for which I reckon the altitude would be about 484·87 feet with the same base: and for a pyramid of extreme and mean ratio the altitude would be 484·34 feet. The whole difference, therefore, is only about six inches in a height of nearly five hundred feet. This difference, evidently beyond the power of man to discover, now that the pyramid is a ruin, would even in its perfect state have been inappreciable. It appears most probable that the star Pentalpha led to the star Cheops, and that the star Cheops (_Fig_. 70) was the plan used by the ancient architect, and the ratio of 34 to 21, hypotenuse to base, the template used by the ancient builders. Suppose some king said to his architect, "Make me a plan of a pyramid, of which the base shall be 420 cubits square, and altitude shall be to the perimeter of the base as the radius of a circle to the circumference."--Then might the architect prepare an elaborate plan in which the relative dimensions would be about-- R. B. CUBITS {Base 420 Base angle 51° 51′ 14·3″ {Altitude 267·380304 &c. {Apothem 339·988573 &c. The king then orders another pyramid, of the same base, of which altitude is to be a mean proportional between apothem and half-base--and apothem and half-base taken as one line are to be in mean and extreme ratio. The architect's plan of this pyramid will be the simple figure illustrated by me (_Fig_. 70), and the dimensions about--