Joost Burgi"s Arithmetical Method of Producing complete Sine Tables

From the time of Menelaus and Ptolemy, the tables of chords (or sines) were found by dividing the circle by inscribing regular polygons to find one chord, and then using methods to subdivide that chord into smaller (usualy half) chords. But this method did not provide some exact sines geometrically for all angles, only those that could be expressed as \(\frac{3m}{2^n}\) degrees. One degree, for example could not be found by this method. Regiomontanus knew that 3/4 degree was already established to significant accuracy, so he found upper and lower limits on the value of one degree by approximation. his upper limit for \( sin 1^o\) he used the fact that the sine rate of growth grew smaller by each equal increase of arc, allowing him to conclude that \( sin 1^o \) must be less than 4/3 of the value of the sine of 3/4 of a degree.

Others attacked the problem with creative algebra. Around 1400 the Persian mathematician astrologer, Al-Kashi, produced a very accurate approximation for \(Sin(1^o) \) by equating the trisection of an angle sine to a cubic equation ( \(sin(3x(=3sin x -4 Sin^3 x\) ) and using an approach similar to Newton's Method. Both Viete and Henry Briggs would do a similar approach in the 1th and 16th century, but probably without knowing of al-Kashi, since little of his work was translated in the west.

But all of these methods began from the geometric approach and worked to produce a single chord length or sine value. Imagine if someone could come up with a direct method using only simple arithmetic to produce a complete table of sines from zero to 90^o.

 On 22 July of 1588, such a method was delivered to the emperor Rudolph II by Joost Burgi. Later I will provide sources (one book, one paper) to give more about the explanation of his method, but since the method was lost until the 21st Century, and is still, I believe, little known; I want to first startle you with its simplicity and accuracy.

My exploration began when the author of the book I first learned of this method in commented that Burgi had began with a column on numbers that were rough estimates of the sine values of the sines from 0^o t0 90^o in ten degree increments, the whispered the challenge in print, "although any column of numbers whatever would do here". OH, Ho, Challenge Accepted.

Burgi's method was to divide 90^o into any number, n, of equal arcs and write n+1 numbers in a row. Being Lazy I chose 0, 30, 60, and 90, degrees, and since his method has a divide by two step, I started with the even numbers from 0 to 4. Burgi's method worked in columns from left to right but I will use rows from top to bottom with the sequential approximations improving with each pass. 0^o --------------- 30^o------------- 60^o-----------90 >br>
0 ----------------- 2---------------- 4-------------6

This Last value in each column serves as the sinus totus, or radius (hypotenuse) of the ratio
For the next step we walk across writing between the previous columns and beginning with 1/2 the right most number and adding the column above at each step as we walk back to the left.

0^o --------------- 30^o------------- 60^o-----------90

0 ----------------- 2---------------- 4-------------6

---------9-------------------7-----------------3

Now we go back from left to right beginning with a zero and adding as we go along.

0^o --------------- 30^o------------- 60^o-----------90 >br>
0 ----------------- 2---------------- 4-------------6

---------9-------------------7-----------------3

0--------------------9-----------------16-------------19

Notice after one cycle the ratio of the 30^o value, 9, over the radius, 19, already gives .473.. for sin 30^o


0^o --------------- 30^o------------- 60^o-----------90 >br>
0 ----------------- 2---------------- 4-------------6

---------9-------------------7-----------------3

0--------------------9-----------------16-------------19

---------35---------------------26-------------10--------

0--------------------35----------------61-------------71---

Now the Sin(30) is .492, and the sin(60) is .859. let's try one more pass

0^o --------------- 30^o------------- 60^o-----------90 >br>
0 ----------------- 2---------------- 4-------------6

---------9-------------------7-----------------3

0--------------------9-----------------16-------------19

---------35---------------------26-------------10--------

0--------------------35----------------61-------------71---

----------132-----------------97---------------36----------

0------------------132-----------------229------------265--

At this point the Sin(30) is accurate to less than .002, and sin(60) is about the same. Take it out two more steps on your own and be amazed at the precision.
Next, I pushed the "any number we are whatever," to another level. I used 0, 30, 60, and 90 Sin(60^o) = 229,26 = again, but I started with a zero and three ones. after five passes my column was 0, 571, 989, and 1142. thus produces an estimate of sin(30^o) = 571/1142 or exactly .5, and the Sin(60^o)=989/1142 =.866024518, which seems incredible for so little effort and so casual an initial choice of values. From the ARXIV link below, here is Burgi's illustration of the method, which seems to be in sexagesimal numbers. In The Doctrine of Triangles by Glenn Van Brummelem, the same chart appears on page 70 in decimals.

REF: The Doctrine of Triangles, Glenn Van Brummelen Pgs 19-20, and 69-71.

https://arxiv.org/pdf/110.03180
Jamshid al-Kashi Wikipedia