Finally arises the following simple rule for the construction of "primaries" to contain any angle—Decide upon a satellite which shall contain half the angle—say, 5, 1. Call the first figure a, the second b, then—
a² + b² = hypotenuse.
a² - b = perpendicular.
a × 2b = base.
| "Primary" | Lowest Ratio. | |||
| Thus— | 5² + 1² | = 26 | = 13 | |
| Satellite 5,1 | 5² - 1² | = 24 | = 12 | |
| 5 × 2 × 1 | = 10 | = 5 | ||
| and— | 5² + 2² | = 29 | = 29 | |
| Satellite 5,2 | 5² - 2² | = 21 | = 21 | |
| 5 × 2 × 2 | = 20 | = 20 |
Having found the lowest ratio of the three sides of a "primary" triangle, the lowest whole numbers for tangent, secant, co-secant, and co-tangent, if required, are obtained in the following manner.
Take for example the 20, 21, 29 triangle, now 20 × 21 = 420, and 29 × 420 = 12180, a new radius instead of 29 from which with the sine 20, and co-sine 21, increased in the same ratio, the whole canon of the 20, 21, 29 triangle will come out in whole numbers.
Similarly in the triangle 48, 55, 73, radius 73 × 13200 (the product of 48 × 55) makes radius in whole numbers 963600, for an even canon without fractions. This is because sine and co-sine are the two denominators in the fractional parts of the other lines when worked out at the lowest ratio of sine, co-sine, and radius.
After I found that the plan of the Gïzeh group was a system of "primary" triangles, I had to work out the rule for constructing them, for I had never met with it in any book, but I came across it afterwards in the "Penny Encyclopedia," and in Rankine's "Civil Engineering."
The practical utility of these triangles, however, does not appear to have received sufficient consideration. I certainly never met with any except the 3, 4, 5, in the practice of any surveyor of my acquaintance.
(For squaring off a line nothing could be more convenient than the 20, 21, 29 triangle; for instance, taking a base of 40 links, then using the whole chain for the two remaining sides of 42 and 58 links.)