The Trigonometric Tables of Georg Joachim Rheticus' Opus Palatinum: The Greatest Computational Effort in the History of Humankind
If one were to ask who put the greatest effort to calculate
mathematical tables in the history of mankind, the best answer would be Georg Joachim
Rheticus. His highly accurate trigonometric tables contained in “Opus
Palatinum” were used until the wake of 20th century in various
fields such as astronomy, geography, surveying , navigation, to name a few. His
orginal trigonometric tables, particularly the table of sines, were accurate
upto 10 decimal places. But it contained few errors in cotangents, cosecants etc which were later corrected by a
mathematician named Bartholomous Pitiscus, who then republished the sine table
of Rheticus to 15 decimal places (which he had already calculated, but did not
publish). But nothing that was comparable to his trigonometric tables ever
existed before and it remained the standard mathematical table for centuries to
come. The aim of this presentation is to shed some light on how Rheticus was
able to compute such a magnificent table.
The geometric methods employed by Rheticus does not differ
in any way from what Ptolemy did more than a millennia ago to calculate his
“table of chords”, the oldest extant trigonometric table. (As we have seen, a
chord table is equivalent to a sine table if one chooses the correct constant
of proportionality). Rheticus produced more accurate values at much finer
intervals (the chord table of Ptolemy gave values at intervals of ½°) which
means he had put an effort orders of magnitudes greater than that of Ptolemy. In
fact, it took Rheticus 12 years assisted by calculators to finish his Magnum
Opus, part of which was undertaken by his disciple Valentine otho after
Rheticus' death in 1574.
Rheticus had also computed a smaller trigonometric table in
1551, in his “Canon Doctrinae Triangulorum” which gave values at intervals of 10’
and with an accuracy of 7 decimal places.
The final work was published in 1596 as “Opus Palatinum” , a monumental treatise on trigonometry, of which his great trigonometric tables formed a part.
Rheticus first expounds a lot of geometric theorems relating to circles, which are similar to the ones given by Ptolemy in his Mathematikae Syntaxis also known as the Almagest, which he used to calculate his chord table.
With this, he calculates the sines and cosines of basic
angles such as 36°, 18°, 30°, 15° and so on, and uses the half angle formula
to calculate values for their sub-multiples. One should keep in mind
that calculating trigonometric ratios of base values require nothing more than
a grasp of simple geometry and arithmetic . An excerpt is shown below:
These methods are fundamentally not different from the ones
used by Regiomontanus almost a century ago to calculate his 7 digit sine table
given at intervals of 1’.
He then produces an intermediate set of values for base and
perpendicular (cosine and sine in modern language) which he uses as a basis for
his upcoming computations. All values are given to 15 decimal places.
Now using the value of 1° 30’ and other angles, Rheticus
uses the sum and difference formulae in trigonometry to calculate sines and
cosines of angles upto 90° at intervals of 1.5°. A few excerpts of numerical
calculations from the text are given below:
Now
Rheticus finishes the work and produces the intermediate sine table, accurate
to 15 decimal places, given at intervals of 1°30’
Rheticus now goes one step further and uses the value of
sine(45’) and produces a finer table at intervals of 45’ (0.75°), using the
same method of sum and difference formulae.
The table is given below:
Rheticus now starts with the first steps of getting to his
gigantic table. He applies the half angle formula successively to 45’ and
computes the values to 15 decimal places until it vanishes. Then he uses an
ingenious method to calculate the sine of an angle very close to 30”(30
seconds) by bisecting 55 ½° 14 times (The mathematics of the Heavens and the
Earth: The Early History of Trigonometry by Glen Van Brummelen).
Rheticus then approximates the value of sin(30”) by a method
similar to the one used by Ptolemy more than a millennia ago( Ptolemy was faced
with the problem of computing the value of crd (1°)).
Source: Van Brummelen
As Brummelen notes, there was a slight error in the
calculation of sin(30”) which crept into his table of cosecants and cotangents,
but not his sine table.
Now equipped with the value of sin(30”), Rheticus is able to
produce an intermediate table containing the values for 1’, 2’, 4’ etc.
Excerpts from the calculations are shown below:
He now
has all the data to compute the sine values at intervals of 1’(1 minute). For
this, he produces another intermediate table for every minute of angle.
Excerpts from the calculations and the table are given below:
Using similar computations, Rheticus produces an
intermediate table of secants and tangents, and from that a complete
trigonometric table given to 15 decimal places, at intervals of 45’.
By a method similar to the one he used for calculating
sin(30”), Rheticus produces the values of sin(5”), sin(10”), sin (20”), sin(30”),
sin(40”) and sin(50”).
Rheticus is now equipped with everything he needs to finish
his table at intervals of 10” : Complete trigonometric tables given at 1°30’
and 45’, sine and cosine values of every arc minutes, which enables him to
calculate ratios at 1’ apart, and finally sine values for 10” and its
multiples, leading him to the final form of the table, at intervals of 10” (10
seconds or (1/360)° ).
Rheticus then gives a
few numerical examples showing how the final form of the table can be
calculated, using sum and difference formula with the sine values of 10” and
its multiples.
Now, as Brummelen puts it, it was a downhill task for
Rheticus to finish the rest of his table, requiring an enormous amount of
computational effort.
But Rheticus along with his disciple Valentine Otho,
assisted by calculators over a period of 12 years or more, did finish the table
and it saw the light of the day in his “Opus Palatinum”, published posthumously
in 1596. As it was known, computing trigonometric tables using geometry was a
herculian task, and it becomes orders of magnitude more difficult when one
intends to improve accuracy ( number of correct decimal places) and resolution (
length of interval).
Rheticus braved all these hurdles, and was fortunate enough
to have a disciple like Valentine Otho to carry on with his work despite his
untimely death, and get his Magnum Opus published. The work “Opus Palatinum”
was more than 1400 folios long, with his tables alone taking up to 700 pages.
The tables has 16,200 entries for sine values alone. A few pages will help us
grasp the monstrous amount of numerical data contained in them.
His magnificent tables stand as beacons of human intellect
and hardwork. Rheticus is akin to a hero who conquered heights only a handful
have imagined of, only to bewilder even them of the efforts he had put in. Kepler
for instance remarked that sine values were calculated for sufficient accuracy
once and for all and there’s no need for further improved. François Viete even
amused at the need for such a huge table and the gigantic computational effort
behind it. But his tables transcended every last question.
