The natural co-sine 42 4' 29917? = 7422684.

The greatest common measure of these numbers is about 102717, therefore--

Radius 10000000 102717 = 97 Co-sine 7422684 102717 = 72 Sine 6701025 102717 = 65

and 65, 72, 97 is the primary triangle to which the satellites are 4, 9, and 5, 13. (_See Fig_. 63.) The figures in the calculation do not balance exactly, in consequence of the insufficient delicacy of the tables or calculations.

Fig. 63.

The connection between primaries and satellites is shown by figure 64.

Fig. 64.

Let the triangle ADB be a satellite, 5, 2, which we will call BD 20, and AD 8. Let C be centre of semi-circle ABE.

AD : DB :: DB : DE = 50 (_Euc. VI_. 8)

AD + DE = AE = 58 = diameter

AE 2 = AC = BC = 29 = radius

AC-AD = DC = 21 = co-sine

and DB = 20 = sine

From the preceding it is manifest that--

sine ----- + ver-s = dia.

ver-s

The formula to find the "primary triangle" to any satellite is as follows:--

Let the long ratio line of the satellite or sine be called _a_, and the short ratio line or versed-sine be called _b_. Then--

(1) a = sine.

a + b (2) ------- = radius.

2b

a - b (3) ------- = co-sine.

2b

Therefore various primary triangles can be constructed on a side DB (_Fig_. 64) as sine, by taking different measures for AD as versed-sine.

For example--

} 5 = sine = 5 } } 5 + 1 From } ------- = radius = 13 Satellite } 2 1 5, 1. } } 5 - 1 } ------- = co-s. = 12 } 2 1

} 5 = sine = 5 } {20 } } { } 5 + 2 } { From } ------- = radius = 7 } 4 {29 Satellite} 2 2 } { 5, 2. } } { } 5 + 2 } { } ------- = co-s. = 5 } {21 } 2 2 } {

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--

a + b = hypotenuse.

a - b = perpendicular.

a 2b = base.

"PRIMARY" LOWEST RATIO.

Thus--

5 + 1 = 26 = 13 Satellite 5,1

5 - 1 = 24 = 12

5 2 1 = 10 = 5 ---------------

---------------------------- and--

5 + 2 = 29 = 29 Satellite 5,2

5 - 2 = 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 Gzeh 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.)

Table of Some Primary Triangles and their Satellites.

ANGLE OF PRIMARY PRIMARY SATELLITE. ANGLE OF SATELLITE DEG. MIN. SEC. RAD. CO.-S. SINE. DEG. MIN. SEC.

2 47 3970 841 840 41 41 1 1 23 4985

6 43 5862 145 144 17 17 1 3 21 5931

8 47 5069 85 84 13 13 1 4 23 5534

10 23 1989 61 60 11 11 1 5 11 3994

12 40 4937 41 40 9 9 1 6 20 2468

14 14 5910 65 63 16 8 1 7 7 2955

16 15 3673 25 24 7 7 1 8 7 4836

18 55 2871 37 35 12 6 1 9 27 4435