Ancient Trigonometic Tables:
Calculated using Pythagorean Triples
by Richard Allen Brown
1248 Insititue, Charleston
Summary
Between the Plimpton 322 tablet of Sumeria and first complete trigonometic table
of antiquity produced by Claudius Ptolomy in the first century AD, almost 2000
years elapsed. During the interim period many attempts were made to construct a
complete list of ratios associated with angle measures. Table 1 below is the result of methods available to the ancients before Ptolomy. It is believed by the author that such a table could have been produced during the period between 400 BC and 100 BC.
The table is not as accurate as the Ptolomy Table which was accurate to within
a few seconds per value. In ancient times the table would have been totally constructed using sexagesimal(base 60) numbers, decimal numbers were used for ease of presentation.
Table 1.
The values of the sine of 0 degrees to 45 degrees using Pythagorean triples
| Angle in Degrees | Pythagorean Triple Generators | The ratio of the even side to the hypotenuse= sine(angle)** | Error in Seconds*** |
|---|
| 0* | 0; 915 | 0/837225 = 0.0 | 0 |
|---|
|
|
| 1* | 8; 915 | 14640/837285 = 0.0174850 | +7 |
|---|
|
|
|
| 2 | 16; 915 | 29280/837481 = 0.0349619 | +13 |
|---|
| 3 | 24; 915 | 43920/837801 = 0.0524229 | +18 |
|---|
| 4 | 32; 915 | 58560/838249 = 0.0698599 | +21 |
|---|
| 5 | 40; 915 | 73200/838824 = 0.0872650 | +23 |
|---|
|
|
|
| 6 | 48; 915 | 87840/839529 = 0.1046301 | +21 |
|---|
| 7 | 56; 915 | 102480/840361 = 0.12219475 | +16 |
|---|
| 8 | 64; 915 | 117120/841321 = 0.1392096 | +8 |
|---|
| 9 | 72; 915 | 131760/842409 = 0.1564085 | -5 |
|---|
| 10 | 80; 914 | 146240/841796 = 0.1737238 | +16 |
|---|
|
|
|
| 11 | 88; 914 | 160864/843140 = 0.1907915 | -4 |
|---|
| 12 | 96; 913 | 175296/842785 = 0.2079960 | +18 | |
|---|
| 13 | 104; 913 | 189904/844385 = 0.224902 | -10 |
|---|
| 14 | 112; 912 | 204288/844288 = 0.2419648 | +9 |
|---|
|
|
|
| 15* | 120; 911 | 218640/844321 = 0.2589536 | +29 |
|---|
|
|
|
| 16 | 128; 911 | 23216/846305 = 0.2755696 | -15 |
|---|
| 17 | 136; 910 | 247520/846596 = 0.2923708 | 0 |
|---|
| 18 | 144; 909 | 261792/847017 = 0.3090752 | +12 |
|---|
| 19 | 152; 908 | 276032/847568 = 0.3256753 | +23 |
|---|
| 20 | 160; 907 | 290240/848249 = 0.3421636 | +32 |
|---|
|
|
|
| 21 | 168; 906 | 304416/849060 = 0.3585329 | +36 |
|---|
| 22 | 176; 905 | 318560/850001 = 0.3747760 | +38 |
|---|
| 23 | 184; 904 | 332672/851072 = 0.3908858 | +35 |
|---|
| 24 | 192; 903 | 346752/852273 = 0.4068555 | +27 |
|---|
| 25 | 200; 902 | 360800/853604 = 0.4226784 | +14 |
|---|
|
|
|
| 26 | 208; 901 | 374816/855065 = 0.4383479 | -5 |
|---|
| 27 | 216; 900 | 388800/856656 = 0.4538577 | -31 |
|---|
| 28 | 224; 898 | 402304/856580 = 0.4696630 | +45 |
|---|
| 29 | 232; 897 | 416208/858433 = 0.4848462 | +9 |
|---|
|
|
|
| 30* | 240; 896 | 430080/860416 = 0.4998512 | -35 |
|---|
|
|
|
| 31 | 248; 894 | 443424/860740 = 0.515166 | +31 |
|---|
| 32 | 256; 893 | 457216/862985 = 0.5298075 | -27 |
|---|
| 33 | 264; 891 | 470448/863577 = 0.5447667 | +31 |
|---|
| 34 | 272; 890 | 484160/866084 = 0.5590219 | -43 |
|---|
| 35 | 280; 888 | 497280/866944 = 0.573601 | +6 |
|---|
|
|
|
| 36 | 288; 886 | 510336/867940 = 0.5879853 | +51 |
|---|
| 37 | 296; 885 | 523920/870841 = 0.6016253 | -49 |
|---|
| 38 | 304; 883 | 536864/872105 = 0.6155956 | -17 |
|---|
| 39 | 312; 881 | 549744/873505 = 0.6293541 | +9 |
|---|
| 40 | 320; 879 | 562560/875041 = 0.6428955 | +29 |
|---|
|
|
|
| 41 | 328; 877 | 575312/876713 = 0.6562147 | +43 |
|---|
| 42 | 336; 875 | 588000/878521 = 0.6693067 | +49 |
|---|
| 43 | 344; 873 | 600624/880465 = 0.6821668 | +48 |
|---|
| 44 | 352; 871 | 613184/882545 = 0.6947906 | +38 |
|---|
|
|
|
| 45* | 360; 869 | 625680/884761 = 0.707174 | +20 |
|---|
*The values of 15,30,and 45 degrees are calculated using convergent sequences
of Pythagorean triples--see other paper. The value of the sine of 1 is calculated using the estimate 2*pi/360. The value of 0 is by definition.
**The numerator is equal to 2*generator1*generator2. The denominator is the sum of the squares of the two generators.
These are two of the sides of a Pythagorean triple. The third side is the difference between the squares of the two generators.
***The error is in seconds of a degree, the result is either X degrees 0 minutes and error seconds or X-1 degrees 59 minutes and 60 + error seconds.
All calculations were made with a Texas Instrument TI 25X solar calculator.
Methodology and Historical References
The following are the components needed to construct the table. The key addition to other work is using the a sequence of Pythagorean triples generators to interpolate the sine table given five known values.
1. Writing and Number Systems
2. Arithmetic including Addition, Multiplication, Subtraction,
Division, and Square Root
3. Knowledge of the Right Angle Rule and Pythagorean Triples including the fundamental theorem of Pythagorean Triples
4. A System of Angles and angle Measures
5. An accurate measure of the sides of a 45 degree right triangle
6. An accurate measure of the of the sides of a 30 degree right triangle
7. An accurate measure of pi or the ratio of the diameter to the
radius of a circle and it's use in estimating the sine of 1 degree
8. An accurate measure of the sides of a 15 degree right triangle
9. An interpolation method using the generators of Pythagorean Triples
In order for Pythagorean triples generators to be used in contructing a table of sines, it must be known that the there is a unique ratio of the two generators for each angle x. In fact, the ratio is (1 + cos(x))/sin(x). The ratio for 45 degrees is 1 + (2)**(1/2) and for 1 degree it is 114.58865.
In my first attempt to interpolate the sine table with triples, I used the values of (2;229) to (30;229) as reasonable close approximations for the sines of 1 through 15 degrees. In order to keep the error below 1 minute for each degree of angle I had to increase the size of the triple generators by a factor of 4, hence (8;915) to (360;869). The use of equal spacing for the smaller generators(8,16,24, etc) made the interpolation estimates based totally on the larger estimates. The smaller generators are always eight times angle measure in degrees, this makes the interpolation particularly transparent.
Using the values of (120;911) for the sine of 15 degrees and (240;896) for the sine of 30 degrees, I had a natural spacing of -4 for the first 15 degrees, -15 for the next 15 degrees and -27 for the final 15 degrees. A primitive distribtion of the changes using finite differences gave estimates for the larger generator for the values of 5,10,20,25,35, and 40 degrees.
The choice of the generators proved fortunate, From 0 degrees through 9 degrees
all used 915 as the larger generator; from 16 degrees through 31 degrees, the
larger generator decreased by one per degree change; and from 37 degrees to
45 degrees the larger generator decreased by two per degree change.
Without too much difficulty the values of 1/2 degree and 1/4 degree increments can also be calculated either with linear interpolation or by using values of the small generator between the whole degree values, ie 12 for 1.5 degrees or 14 for 1.75 degrees.
Differences between the constructed table and the Ptolomy table