§ 1. THE GROUND PLAN OF THE GIZEH GROUP.

I find that the Pyramid Cheops is situated on the acute angle of a right-angled triangle—sometimes called the Pythagorean, or Egyptian triangle—of which base, perpendicular, and hypotenuse are to each other as 3, 4, and 5. The Pyramid called Mycerinus, is situate on the greater angle of this triangle, and the base of the triangle, measuring three, is a line due east from Mycerinus, and joining perpendicular at a point due south of Cheops. (See Figure 1.)

Fig. 1.

I find that the Pyramid Cheops is also situate at the acute angle of a right-angled triangle more beautiful than the so-called triangle of Pythagoras, because more practically useful. I have named it the 20, 21, 29 triangle. Base, perpendicular, and hypotenuse are to each other as twenty, twenty-one, and twenty-nine.

The Pyramid Cephren is situate on the greater angle of this triangle, and base and perpendicular are as before described in the Pythagorean triangle upon which Mycerinus is built. (See Fig. 2.)


Fig. 2.	Fig. 3

Figure 3 represents the combination,—A being Cheops, F Cephren, and D Mycerinus.

Lines DC, CA, and AD are to each other as 3, 4, and 5; and lines FB, BA, and AF are to each other as 20, 21, and 29.

The line CB is to BA, as 8 to 7; the line FH is to DH, as 96 to 55; and the line FB is to BC, as 5 to 6.

The Ratios of the first triangle multiplied by forty-five, of the second multiplied by four, and the other three sets by twelve, one, and sixteen respectively, produce the following connected lengths in natural numbers for all the lines.

DC	135
CA	180
AD	225
____________
FB	80
BA	84
AF	116
____________
CB	96
BA	84
____________
FH	96
DH	55
____________
FB	80
BC	96

Figure 4 connects another pyramid of the group—it is the one to the southward and eastward of Cheops.

In this connection, A Y Z A is a 3, 4, 5 triangle, and B Y Z O B is a square.

Lines	YA to CA	are as	1 to 5
	CY to YZ	as	3 to 1
	FO to ZO	as	8 to 3
and	DA to AZ	as	15 to 4.

I may also point out on the same plan that calling the line FA radius, and the lines BA and FB sine and co-sine, then is YA equal in length to versed sine of angle AFB.

This connects the 20, 21, 29 triangle FAB with the 3, 4, 5 triangle AZY.

I have not sufficient data at my disposal to enable me to connect the remaining eleven small pyramids to my satisfaction, and I consider the four are sufficient for my purpose.

Fig. 4.

				At level of Own base	At level of Cephren's base
Of these natural numbers the bases of the pyramids are as follows.	}	=	Cheops Cephren Mycerīnus	56½ 52½ 26¼	52½ 52½ 27¼

I now establish the following list of measurements of the plan in connected natural numbers. (See Figure 4.)