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 angle 51° 51′ 14·3″ | } | Base Altitude Apothem | 420 267·380304 &c. 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—
| R. B. CUBITS | ||||
| Base angle 51° 49′ 37 | 42″ 471 | } | Base Altitude Apothem | 420 267·1239849 &c. 339·7875153 &c. |
But the builder practically carries out both plans when he builds to my templates of 34 to 21 with—
| R. B. CUBITS | |||
| Base angle 51° 51′ 20″ | } | Base Altitude Apothem | 420 267·394839 &c. 340 |
and neither king nor architect could detect error in the work.