# Who published the law of tangents first?

The Law of Tangents is a rather obscure trigonometric identity that is sometimes used in place of its better-known counterparts, the law of sines and law of cosines, to calculate angles or sides in a triangle.

According to Wikipedia,the law of tangents for planar triangles was described in the 11th century by Ibn Muʿādh al-Jayyānī (Wikipedia cites Roshdi Rashed (ed.) Encyclopedia Of The History Of Arabic Science, p. 182. But I couldn't find any reference to the law of tangents in this book). Another source credits the Persian mathematician Abu'l-Wafa' in the 10th century CE as the first to publish it.

The following statement can be found in Merzbach and Boyer, A History of Mathematics, 3rd edition, page 278:

"Though Viete may have been the first to use this formula [the law of tangents], it was first published by the German physician and professor of mathematics Thomas Finck (1561-1656) in 1583, in Geometriae Rotundi Libri XIV."

Who was really the first to publish (or whatever that means back then) it?

I agree that the reference in Wikipedia seems at first sight to be off.

But if one reads a little further into p.184 (frame 192 of 401 in the archive.org display) of vol.2 of Roshdi Rashed (ed.) 1996, Encyclopedia Of The History Of Arabic Science -- if the archive viewer shows 'nan' instead of a page number then just click on 'vol.2' in the left panel and scroll to frame 192/page 184 --then one sees an explanation of how Ibn Muʿādh (11th-c.) derived a formula which is indeed the so-called 'law of tangents', differing only by small points of notation, not in substance, from the 'law of tangents' formulae given elsewhere:

$$\tan\left(\frac{x + y}{2}\right) = \frac{a+b}{a-b}\tan\left(\frac{\alpha}{2}\right).$$

where $$\alpha$$ has been defined as $$x - y$$ , and $$x$$ and $$y$$ (the angles) are opposite $$a$$ and $$b$$ (the sides).

The derivation in the book appears to start from the sine theorem, and indeed the tangent theorem itself can be considered as essentially a rearrangement (taking a bit of work of trig. expansion to get there!) of the sine theorem.

• Nice find! I was slightly worried that Braunmühl's might not be the last word on the subject. Fincke has also been credited with introducing the tangent function, but your source says that it goes back to to Habash al-Hasib (second half of 9th century). Commented Mar 25, 2023 at 15:49
• It appears we can use MathJax for formulae as on Math.SE e.g. $$\tan((x + y)/2) = \tan(\alpha/2) . (a + b)/(a - b)$$ Commented Mar 25, 2023 at 19:19
• Thanks for the reactions. I'm a bit lost with Mathjax but I'll come back and try it. T Commented Mar 25, 2023 at 23:13
• @njuffa : Thanks for doing that! I hope to learn from the formatting example! Commented Mar 26, 2023 at 19:30

The law of tangents is indeed attributed to the Danish polymath Thomas Fincke (1561–1656). According to his short biography by Jürgen Schönbeck (in German and behind a paywall), the law appears in Vol. 14 of Fincke's opus magnum "Geometria rotundi" (Basel, 1583) in terms of the Latin phrase:

Ut semissis summae crurum ad differentiam summae semissis alteriusque cruris, sic tangens semissis anguli crurum exterioris ad tangentem anguli, quo minor interiorum semisse dicti retiqui minor est, aut major, major.

My Latin is too rusty to provide an exact translation but with the help of Google translate the first part seems to say that the ratio of half the difference to half the sum of two adjacent sides is the same as the ratio of the tangents of half the (appropriately chosen) angular difference and sum, respectively.

Braunmühl in Vorlesungen über Geschichte der Trigonometrie states on p. 178 that the modern version of the law (i.e. its formulation as an equation) is due to Vieta (1593) and comments (my translation from German):

In this elegant form the equation appears here [Vieta, 1593] for the first time, while it has been put forward in much more complicated terms by Finck[e] ten years earlier who has always, and correctly so, been credited with its discovery.

Schönbeck (loc. cit.) also says that the underlying problem can be traced back to Ptolemy, again referring to Braunmühl's first volume (loc. cit., p. 23), where one can also find the statement that the ancient "Greeks did not know the tangent".

• Re Vol. 14 of Finck's Geometriæ rotunda: It appears on page 292 (see scan here, in the middle of the page). Commented Mar 26, 2023 at 17:24
• The Greeks may not have known the tangent, but they had at least a notion of the co-tangent, as the length of a shadow cast by a gnomon based on the angle of the sun over the horizon. Commented Mar 26, 2023 at 17:27