§ 12. PRIMARY TRIANGLES AND THEIR SATELLITES;—OR THE ANCIENT SYSTEM OF RIGHT-ANGLED TRIGONOMETRY UNFOLDED BY A STUDY OF THE PLAN OF THE PYRAMIDS OF GIZEH.
TABLE TO EXPLAIN FIGURE 60.
Main Triangular Dimensions of Plan are Represented by the Following Eight Right-angled Triangles.
| AB BJ JA | 28 45 53 | } | × 3 | { | 84 135 159 | } | × 8 | { | 672 1080 1272 | DG GE ED | 3 4 5 | } | × 24 | { | 72 96 120 | } | × 8 | { | 576 768 960 |
| DC CA AD | 3 4 5 | } | × 45 | { | 135 180 225 | } | × 8 | { | 1080 1440 1800 | FW WV VF | 48 55 73 | } | × 1 | { | 48 55 73 | } | × 8 | { | 384 440 584 |
| EB BA AE | 3 4 5 | } | × 21 | { | 63 84 105 | } | × 8 | { | 504 672 840 | FB BA AF | 20 21 29 | } | × 1 | { | 80 84 116 | } | × 8 | { | 640 672 928 |
| FH HN NF | 3 4 5 | } | × 32 | { | 96 128 160 | } | × 8 | { | 768 1024 1280 | Note.—In the above table the firstcolumn is the Ratio, the second the third column represents the length each line in R.B. cubits. | |||||||||
| AY YZ ZA | 3 4 5 | } | × 12 | { | 36 48 60 | } | × 8 | { | 288 384 480 | ||||||||||
Fig. 60.
Reference to Fig. 60 and the preceding table, will show that the main triangular dimensions of this plan (imperfect as it is from the lack of eleven pyramids) are represented by four main triangles, viz:—
| Ratio. | |||
| C A D C .. .. | 3, | 4, | 5 |
| F B A F .. .. | 20, | 21, | 29 |
| A B J A .. .. | 28, | 45, | 53 |
| F W V F .. .. | 48, | 55, | 73 |
Figures 30 to 36 illustrate the two former, and Figures 61 and 62 illustrate the two latter. I will call triangles of this class "primary triangles," as the most suitable term, although it is applied to the main triangles of geodetic surveys.
We have only to select a number of such triangles and a system of trigonometry ensues, in which base, perpendicular, and hypotenuse of every triangle is a whole measure without fractions, and in which the nomenclature for every angle is clear and simple.
An angle of 43° 36′ 10·15″ will be called a 20, 21 angle, and an angle of 36° 52′ 11·65″ will be called a 3, 4 angle, and so on.
In the existing system whole angles, such as 40, 45, or 50 degrees, are surrounded by lines, most of which can only be described in numbers by interminable fractions.
In the ancient system, lines are only dealt with, and every angle in the table is surrounded by lines measuring whole units, and described by the use of a couple of simple numbers.
Connecting this with our present system of trigonometry would effect a saving in calculation, and general use of certain peculiar angles by means of which all the simplicity and beauty of the work of the ancients would be combined with the excellences of our modern instrumental appliances. Surveyors should appreciate the advantages to be derived from laying out traverses on the hypotenuses of "primary" triangles, by the saving of calculation and facility of plotting to be obtained from the practice.
The key to these old tables is the fact, that in "primary" triangles the right-angled triangle formed by the sine and versed sine, also by the co-sine and co-versed-sine, is one in which base and perpendicular are measured by numbers without fractions. These I will call "satellite" triangles.
Thus, to the "primary" triangle 20, 21, 29, the ratios of the co-sinal and sinal satellites are respectively 7 to 3, and 2 to 5. (See Figure 35.) To the 48, 55, 73 triangle the satellites are 11, 5 and 8, 3 (Fig. 62); to the 3, 4, 5 triangle they are 2, 1 and 3, 1 (Fig. 30); and to the 28, 45, 53 triangle, they are 9, 5 and 7, 2 (Fig. 61). The primary triangle, 7, 24, 25, possesses as satellites the "primary" triangle, 3, 4, 5, and the ordinary triangle, 4, 1; and the primary triangle 41, 840, 841, is attended by the 20, 21, 29 triangle, as a satellite with the ordinary triangle 41, 1, and so on.
Fig. 61.
The 28-45-53 Triangle.
Fig. 62.
The 48-55-73 Triangle.
Since any ratio, however, whose terms, one or both, are represented by fractions, can be transformed into whole numbers, it evidently follows that every conceivable relative measure of two lines which we may decide to call co-sine and co-versed-sine, becomes a satellite to a corresponding "primary" triangle.
Now, since the angle of the satellite on the circumference must be half the angle of the adjacent primary triangle at the centre, it follows that in constructing a list of satellites and their angles, the angles of the corresponding primary triangles can be found. For instance—
Satellite 8, 3, contains 20° 33′ 21·76″
Satellite 2, 7, contains 15° 56′ 43·425″
Each of these angles doubled, gives the angle of a "primary" triangle as follows, viz.:—
The 48, 55, 73 triangle = 41° 6′ 43·52″
The 28, 45, 53 triangle = 31° 53′ 26·85″
The angles of the satellites together must always be 45°, because the angle at the circumference of a quadrant must always be 135°.
From the Gïzeh plan, as far as I have developed it, the following order of satellites begins to appear, which may be a guide to the complete Gïzeh plan ratio, and to those "primary" triangles in use by the pyramid surveyors in their ordinary work.
| 1, | 2 | 2, | 3 | 3, | 4 | 4, | 5 | 5, | 6 | 6, | 7 | 7, | 8 | 8, | 9 |
| 1, | 3 | 2, | 5 | 3, | 5 | 4, | 7 | 5, | 7 | 7, | 9 | ||||
| 1, | 4 | 2, | 7 | 3, | 7 | 4, | 9 | 5, | 8 | ||||||
| 1, | 5 | 2, | 9 | 3, | 8 | 5, | 9 | 7, | 1 | ||||||
| 1, | 6 | 5, | 11 | ||||||||||||
| 1, | 7 | 3, | 11 | 5, | 13 | ||||||||||
| 1, | 8 | 3, | 13 | ||||||||||||
| 1, | 9 | ||||||||||||||
| 1, | 11 | ||||||||||||||
| 1, | 13 | ||||||||||||||
| 1, | 15 | ||||||||||||||
| 1, | 17 |
Primary triangles may be found from the angle of the satellite, but it is an exceedingly round-about way. I will, however, give an example.
Let us construct a primary triangle from the satellite 4, 9.
| Rad. × 4 | = ·4444444 = Tangt. < 23° 57′ 45·041″ |
| 9 | |
| ∠ 23° 57′ 45·041″ × 2 = 47° 55′ 30·083″. | |
| therefore the angles of the "primary" are 47° 55′ 30·083″. | |
| and 42° 4′ 29·917″. | |
| The natural sine of 42° 4′ 29·917″ = ·6701025. | |
| The natural co-sine 42° 4′ 29·917″ = ·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—
| sine² | + 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. |
| (2) | a² + b² | = radius. |
| 2b | ||
| (3) | a² - b² | = 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—
| From Satellite 5, 1. | ||
| ||
| 5 | = sine. | = 5 |
| 5² + 1² | = radius. | = 13 |
| 2 × 1 | ||
| 5² - 1² | = co-sine. | = 12 |
| 2 × 1 |
| From Satellite 5, 2. |
|
|
| × 4 |
|
|
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.)