§ 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.
The 3, 4, 5 triangle contains 36° 52′ 11·65″ and the complement or greater angle 53° 7′ 48·35″
| Radius | 5 | = | 60 | whole numbers.[6] |
| Co-sine | 4 | = | 48 | " |
| Sine | 3 | = | 36 | " |
| Versed sine | 1 | = | 12 | " |
| Co-versed sine | 2 | = | 24 | " |
| Tangent | 3¾ | = | 45 | " |
| Secant | 6¼ | = | 75 | " |
| Co-tangent | 6⅔ | = | 80 | " |
| Co-secant | 8⅓ | = | 100 | " |
Tangent + Secant = Diameter or 2 Radius
Co-tan + Co-sec = 3 Radius
Sine : Versed-sine :: 3 : 1
Co-sine : Co-versed sine :: 2 : 1
Figure 30 illustrates the preceding description. Figure 31 shows the 3·1 triangle, and the 2·1 triangle built up on the sine and co-sine of the 3, 4, 5 triangle.
The 3·1 triangle contains 18° 26′ 5·82″ and the 2·1 triangle 26° 33′ 54·19″; the latter has been frequently noticed as a pyramid angle in the gallery inclinations.
Figure 32 shows these two triangles combined with the 3, 4, 5 triangle, on the circumference of a circle.
[6] 60 = 3 × 4 × 5
The 20, 21, 29 triangle contains 43° 36′ 10·15″ and the complement, 46° 23′ 49·85″.
Expressed in whole numbers—
| Radius | 29 | = | [7]12180 |
| Sine | 20 | = | 8400 |
| Co-sine | 21 | = | 8820 |
| Versed sine | 8 | = | 3360 |
| Co-versed sine | 9 | = | 3780 |
| Tangent | = | 11600 | |
| Co-tangent | = | 12789 | |
| Secant | = | 16820 | |
| Co-sec | = | 17661 |
Tangent + Secant = 2⅓ radius
Co-tan + Co-sec = 2½ radius
Sine : Versed sine :: 5 : 2
Co-sine : Co-versed sine :: 7 : 3
[7] 12180 = 20 × 21 × 29
It is noticeable that while the multiplier required to bring radius 5 and the rest into whole numbers, for the 3, 4, 5 triangle is twelve, in the 20, 21, 29 triangle it is 420, the key measure for the bases of the two main pyramids in R.B. cubits.[8]
[8] 12 = 3 × 4, and 420 = 20 × 21
I am led to believe from study of the plan, and consideration of the whole numbers in this 20, 21, 29 triangle, that the R.B. cubit, like the Memphis cubit, was divided into 280 parts.
The whole numbers of radius, sine, and co-sine divided by 280, give a very pretty measure and series in R.B. cubits, viz., 43½, 30, and 31½, or 87, 60, and 63, or 174, 120 and 126;—all exceedingly useful in right-angled measurements. Notice that the right-angled triangle 174, 120, 126, in the sum of its sides amounts to 420.
Figure 33 illustrates the 20, 21, 29 triangle. Figure 34 shows the 5·2 and 7·3 triangles built up on the sine and co-sine of the 20, 21, 29 triangle.
The 5·2 triangle contains 21° 48′ 5·08″ and the 7·3 triangle 23° 11′ 54·98″.
Figure 35 shows how these two triangles are combined with the 20, 21, 29 triangle on the circumference, and Figure 36 gives a general view and identification of these six triangles which occupied an important position in the trigonometry of a people who did all their work by right angles and proportional lines.
Fig. 36. Ratios of Leading Triangles.