Let AB be the given line to be divided in extreme and mean ratio, _i.e._, so that the whole line may be to the greater part, as the greater is to the less part. Draw BC perpendicular to AB, and equal to half AB. Join AC; and with BC as a radius from C as a centre, describe the arc DB; then with centre A, and radius AD, describe the arc DE; so shall AB be divided in E, in extreme and mean ratio, or so that AB: AE:: AE: EB. (Note that AE is equal to the side of a decagon inscribed in a circle with radius AB.) Let it be noted that since the division of a line in mean and extreme ratio is effected by means of the 2, 1 triangle, ABC, therefore, as the exponent of this ratio, another reason presents itself why it should be so important a feature in the Gïzeh pyramids in addition to its connection with the primary triangle 3, 4, 5. Fig. 66. To complete the explanation offered with figure 65, I must refer to Fig. 66, where in constructing a pentagon, the 2, 1 triangle ABC, is again made use of. The line AB is a side of the pentagon. The line BC is a perpendicular to it, and half its length. The line AC is produced to F, CF being made equal to CB; then with B as a centre, and radius BF, the arc at E is described; and with A as a centre, and the same radius, the arc at E is intersected, their intersection being the centre of the circle circumscribing the pentagon, and upon which the remaining sides are laid off. We will now refer to figure 67, in which the pentangle appears as the symbolic exponent of the division of lines in extreme and mean ratio. Thus: MC : MH :: MH : HC AF : AG :: AG : GF AB : AF :: AF : FB while MN, MH or XC: CD:: 2: 1--being the geometric template of the work. Thus every line in this beautiful symbol by its intersections with the other lines, manifests the problem. Note also that GH = GA AE = AF DH = DE I append a table showing the comparative measures of the lines in Fig. 67, taking radius of the circle as a million units. Fig 67. Table Showing the Comparative Measures of Lines. (_Fig. 67._) ME = 2000000 = diameter. AB = 1902113 = AD ÷ DB MB = 1618034 = MC + MH = MP + PB AS = 1538841·5 EP = 1453086 = AG + FB AF = 1175570 = AE = GB MC = 1000000 = radius = CD + DX = CH + CX AD = 951056·5 = DB = DS PB = 854102 QS = 812298·5 MP = 763932 = CH × 2 = base of Cheops. AG = 726543 = GH = XH = HN = PF = FB = Slant edge of Cheops. = slant edge of Pent. Pyr. DE = 690983 = DH = XD = apothem of Pentagonal Pyramid. {apothem of Cheops. MH = 618034 = MN = XC = {altitude of Pentagonal Pyramid. {side of decagon inscr'd in circle. MS = 500000 {mean proportional between MH and HC 485868 = { {altitude of Cheops. OP = 449027 = GF = GD + DF