# History of trigonometric tables including Lambert's Algebraische Formeln für die Sinus von drei zu drei Graden

I'm looking for references about trigonometric tables, especially to those with exact values, like Lambert's Algebraische Formeln für die Sinus von drei zu drei Graden. More precisely:

1. Is there an English/French/Spanish translation of Lambert's work?

2. Why was Lambert interested in making such tables?

3. Are there others examples of exact trigonometric tables construction?

Thanks!

• My answer to For which angles we know the $\sin$ value algebraically (exact)? gives some information. Incidentally, all trig values of all rational-degree measure angles are algebraic numbers of various types. For information/details about which rational-degree measure angles give rise to ruler-and-compass constructible trig values and real radical expressible trig values and explicit algebraic expressible trig values, see this 26 September 2005 sci.math post. Jul 15, 2022 at 13:51
• Regarding incentives for obtaining such tables, it was partly of theoretical interest and partly of practical interest, the latter being a way of independently verifying correctness of trig tables as well as a way of obtaining very good approximations of larger angles (those not near zero) from which to then successively approximate nearby values by various numerical methods. Many of the 1800s trig texts have chapters on how tables of trig values are calculated (example), including the use of the exact expressions in calculating these values. Jul 15, 2022 at 13:58
• Two additional references: [1] An 1853 French publication of Lambert's table (Numdam and google-books); [2] On the Construction of these Tables in later editions of Charles Hutton's Mathematical Tables. For example, pp. xxxvii-xl in the 1834 7th edition, (continued) Jul 15, 2022 at 14:27
• where in the middle of p. xxxix Hutton writes "The following values of the natural sine for every third degree in the [first] quadrant, have greatly contributed to facilitate the computations". By the way, "natural sine" refers to the values of sine, as opposed to the then much more useful logarithms of these values -- more useful because much of the computations needed in practice were carried out by the use of logarithms, the same way one often did in mathematics classes (higher level high school and beyond) before roughly 1975 or so, when calculators began appearing in classrooms. Jul 15, 2022 at 14:33
• @J. W. Tanner I don't think it make sense to modernize the spelling of a publication published 250 years ago, rather than using the original title. A modern reader could not make sense of "drei zu drei Graden" either, as the idiom used here has fallen out of use. Jul 16, 2022 at 1:26

This is a partial answer that focuses on the second part of the question. The author of

J. H. Lambert, "Algebraische Formeln für die Sinus von drey zu drey Graden", In J. H. Lambert, Beyträge zum Gebrauche der Mathematik und deren Anwendung, Part 2. Berlin: Verlag der Buchhandlung der Realschule 1770, pp. 133-139 (scan online)

discusses his motivation for publishing his table in some detail, in particular in the first two sections. He desires to use neither classical formulas (going back to antiquity) for sectioning the circle, nor use modern methods that employ infinitesimal approaches or make use of imaginary operands, seeking instead to provide straightforward algebraic formulas:

§. 1. In den Anfangsgründen der Trigonometrie wird die Verfertigung der trigonometrischen Tafeln gewöhnlich so angegeben, wie sie sich aus der Elementargeometrie begreiflich machen läßt, und wie diese Tafeln in der That anfangs verfertigt worden. Man kann nemlich vermittelst des Circuls und Lineales den Circul in 2, 3, 4, 5, gleiche Theile theilen, und jeden Bogen, so vielmal man will, halbiren. Daraus läßt sich sodann herleiten, wie sich der Circul von 3 zu 3 Graden, oder von 45 zu 45 Min., oder von 675 zu 675 Secunden eintheilen, und die Sinus, Tangenten und Secanten von allen diesen Bögen berechnen lassen. Diese Berechnungsart ist ungemein weitläuftig. Man hat daher seit der Erfindung der Infinitesimalrechnung auf Mittel gedacht, sie abzukürzen, und jede Sinus, Tangenten und Secanten für sich zu finden, ohne daß sie erst aus einander hergeleitet werden müßten. Dazu müßten nun allerdings unendliche Reihen gebraucht werden, dafern man nicht bey den imaginairen Formeln, die Herr Joh. Bernoulli zuerst gefunden, stehen bleiben wollte, als welche, wenn man sie nicht in unendliche Reihen verwandelt, fürnehmlich nur zu Erfindung von Lehrsätzen dienen.

§. 2. Will man aber für die Sinus und Tangenten algebraische Formeln haben, die weder imaginär sind, noch aus unendlich vielen Gliedern bestehen, so bleibt man eben so zurücke, wie die ersten Berechner der trigonometrischen Tafeln, weil man sie ebenfalls nur von 3 zu 3 Graden berechnen kann, und die dazwischen fallenden, durch fortgesetztes Halbiren, finden muß. Da mir solche Formeln noch nicht vorgekommen, so unterzog ich mich der Arbeit sie zu finden, um ausführlich zu sehen, auf welche Art die Sinus von 3 zu 3 Graden mehr oder minder irrational sind. Folgende Tabelle stellt sie mit einem Anblicke vor Augen.

My rather loose translation, with additions in square brackets to enhance readability:

"§. 1. In fundamental [texts] on trigonometry the computation of trigonometric tables is typically presented in a manner that makes it understandable based on elementary geometry, and tables were actually initially constructed in this manner. It is in fact possible to section a circle into 2, 3, 4, 5 equal parts with a compass and a straightedge, and halve each [resulting] arc as often as desired. From this one can derive how the circle can be partitioned into multiples of 3 degrees, or 45 minutes [of arc], or 675 seconds [of arc], and how the sines, tangents, and secants of all these arcs can be calculated. This manner of computation is exceedingly common. Since the invention of infinitesimal computation one has therefore tried to devise ways of shortening [such computation], and find each sine, tangent, or secant by itself, without the need to derive them from each other. However, doing so would require the use of infinite series, if one does not want to stay with the imaginary formulas first found by Mr. Joh. Bernoulli, which are mostly only useful for the formulation of theorems, if one does not convert them into infinite series.

§. 2. If, however, one desires algebraic formulas for the sines and tangents that are neither imaginary nor comprise an infinite number of terms, one is back [in the same place] as the first computers of trigonometric tables, since one can also only compute them to multiples of 3 degrees, and has to compute intermediate [values] by repeated halving. Since I had not yet come across such formulas, I undertook the work to create them, to examine in detail in which way the sines of multiples of 3 degrees are more or less irrational. The following table presents them at a single view."

The author proceeds to present the trigonometric identities used in the construction of his table, and emphasizes their application in a systematic order. He summarizes his insights in section 4, as follows:

§. 4. In eben dieser Ordnung werden auch die Formeln selbst zusammengesetzter. Vergleicht man sie miteinander, in Absicht auf die Irrationalität, so findet sich, daß sie aus 15 verschiedenen Arten von Wurzelgrössen zusammengesetzt sind, und daß sie sich, wenn man einmal diese gefunden, durch bloßes Addiren und Subtrahiren berechnen lassen. Die Wurzelgrössen sind [...]

My translation:

"§. 4. In just this order the formulas themselves become increasingly more composite. If one compares them with a view towards their irrationality one finds that they are composed of 15 different kinds of radical terms, and they can computed by simply adding and subtracting [these terms] once they have been found. The radical terms are [...]"

I was intrigued by the reference to Jacob Bernoulli's "imaginary formulas". On MacTutor, J J O'Connor and E F Robertson write:

The 18th Century saw trigonometric functions of a complex variable being studied. Johann Bernoulli found the relation between $$\sin^{-1}z$$ and $$\log z$$ in 1702 [...]

and in "Trigonometric Delights", p. 58, Eli Maor writes:

[...] we mention that the general formulas expressing $$\cos n \alpha$$ and $$\sin n \alpha$$ in terms of $$\cos \alpha$$ and $$\sin \alpha$$ were found by Jakob Bernoulli in 1702, more than a hundred years after Viète's work.

Neither source provides a reference to the relevant work by Jacob Bernoulli. The closest match I could find in his collected works is:

"Section Indefinie des Arcs Circulaires, [...]" (July 13, 1702). In Jacob Bernoulli, Opera, tomus II. Geneva: Cramer & Philibert, 1744, pp. 921-929 (scan online)

However, from cursory reading, it is not immediately apparent how this publication relates to the two remarks I quoted above. Likewise, I cannot find an exact match in the collected works of Jacob's brother Johann Bernoulli. The publication closest in time and topic I found is

"Multisectio anguli vel arcus [...]" (April 1701) In Johann Bernoulli, Opera Omnia, Tomus I. Lausanne and Geneva: Marc Michel Bousquet, 1742, pp. 386-392 (scan online)