Rheticus, and The Names of Trigonometric Ratios

 Spending lots of time lately reading old English journal articles (1825-45) sent me by Dave Renfro who tries to help me stay up on the history of math. It is kind of great reading and watching the actual history of ideas unfold as they did in the old journals.... I came across an interesting letter from Agustus De Morgan about the protege of Copernicus, George Joachim of Rhaetia, also called Rheticus. It was Rheticus who managed to convince Copernicus to publish his long withheld manuscript. In fact, the first account of the Copernican system was not published by Copernicus, but  in Rheticus’s Narratio prima in 1540.

I didn't realize for some years of teaching that in the very early days before the trigonometric functions, the early astronomers used the length of the chord of an angle.  The lost tables of Hipparchus (c. 190 BC – c. 120 BC) and Menelaus (c. 70–140 CE) and those of Ptolemy (c. AD 90 – c. 168) were all tables of chord

 lengths of central angles of a circle of specified radius.  The first step towards the common sine of an angle was by the fifth century Indian mathematician and astronomer Āryabhata, who chose to print tables of half-chords, or more specifically, a table of the first differences of the values of trigonometric sines expressed in arcminutes. were first thought conceived to be lengths of segments in a circle of a given diameter (or radius) rather than the more modern view of ratios. I did not know until I read this article, that apparently it was Rheticus who first developed the use of trig functions based on the ratios of sides of a triangle. In fact, the tables he created to include in his publication of the trigonometric sections of De Revolutionibus were the first tables to include all six functions and  (although he did not use the current names). 

" Rheticus published his first trigonometric canon, the Canon doctrinæ triangulorum, in 1551 in Leipzig . This table gave the six trigonometric functions at intervals of 10' of degree, semi-quadrantically arranged. Each function was given to 7 places, or more exactly as integers for a radius of 10^7.

Rheticus did not consider angles in circles, but considered triangles of which one of the side (the hypotenuse) was constant, and he gave the lengths of the other sides as a function of the angle at the center" *Denis Roegel

 This is a very rare table, and it was practically unknown when De Morgan happened to find a copy of it in the 1840s .

Here is the way De Morgan wrote it:
" Modern teachers (he writes in 1845) of trigonometry have pretty generally abandoned the system of independent lines, which used to be called sines, tangents, &c.; and have substituted, for the meaning of these words, the ratio of the sides of right-angled triangles. It appears that they have antiquity in their favor; indeed so completely has the idea of representing the ratios of the sides of triangles taken possession of the mind of Rheticus, that he abandons the use of the word sine. He dwells on the importance of the right-angled triangles, without any reference to the circle: his maxim expressed in the dialogue, is Triquetrum in planicie cum angulo recto, est magister Mathesos . It would also seem as if his choice of the semi-quadrantal arrangement with double descriptions was dictated merely by the convenience of heading one division with majus latus, and the other with minus latus. [Rheticus had labeled the top of his table with perpendiculum and basis, then the bottoms of these columns were reversed, much as Sine and Cosine were reversed at the top and bottom of tables used in my youth before calculators]........ The names cosine, cotangent, and cosecant are the consequence, not the cause, of this duplicate system of arrangements.......The introduction of the terms sine of the complement, complemental sine, and cosine, &c., followed after an interval of more than half a century."
De Morgan points out that one of the reasons it is so hard to find copies of much of Rheticus' work is that ", In the Index Expurgatorius, it is not Copernicus who is forbidden to be read generally; the prohibition only extends to the work De Revolutionibus, and is accompanied with a nisi corrigatur. But Rheticus is wholly forbidden to be read in any of his works. "
I think the difference in the two mens treatment in the Index may be because of the fact that Rheticus was Protestant, and in fact, was at Wittenburg, the very University where Luther had taught, and burned the Papal Bull.

(Another perhaps,  was that Rheticus was more zealous about the Copernican system than Copernicus, insisting on the physical truth of the motion of the earth.

An interesting anecdote told about Rheticus while he was, "puzzling himself about the motion of Mars, he invoked his genius or guardian angel to help him out of the difficulty: the angel accordingly lifted him up by the hair of his head to the roof and threw him down upon the pavement saying with a bitter laugh, 'That's the way Mars moves.' "

addendum James asked about the phrase "semi-quadrantal"... this just means he only went from 0 degrees to 45 degrees (1/2 of a quadrant) and then put Sin-Cos (he didn't use these words) at the top of the columns and Cos-Sin in the reverse order at the bottom... so that from 45 to 90 degrees was simply read up from the bottom....My old CRC tables were arranged the same way, and many textbooks did as well in the Fifties-sixties. (See image below)
 Giving Sin and Cosine as ratios was still pretty new when this was written by De Morgan. It appears that Peacock had initiated the practice in his lectures at Cambridge around 1830, and by 1837, according to De Morgan, it had become the accepted way to define the terms.

Another newish feature of Rheticus' tables were the use of decimal fractions, introduce  by Regiomontanus (1436–76), German astronomer and mathematician, who composed the first tables with decimal fractions.

Some Notes About the Names from MacTutor at Saint Andrews University :

The Hindu word jya for the sine was adopted by the Arabs who called the sine jiba, a meaningless word with the same sound as jya. Now jiba became jaib in later Arab writings and this word does have a meaning, namely a 'fold'. When European authors translated the Arabic mathematical works into Latin they translated jaib into the word sinus meaning fold in Latin. In particular Fibonacci's use of the term sinus rectus arcus soon encouraged the universal use of sine.

Edmund Gunter was the first to use the abbreviation sin in 1624 in a drawing. The first use of sin in a book was in 1634 by the French mathematician Hérigone while Cavalieri used Si and Oughtred S.

It is perhaps surprising that the second most important trigonometrical function during the period we have discussed was the versed sine, a function now hardly used at all. The versine is related to the sine by the formula

versin $$�=1−cos⁡�.

It is just the sine turned (versed) through 90°.

The cosine follows a similar course of development in notation as the sine. Viète used the term sinus residuae for the cosine, Gunter (1620) suggested co-sinus. The notation Si.2 was used by Cavalieri, s co arc by Oughtred 

The tangent and cotangent came via a different route from the chord approach of the sine. These developed together and were not at first associated with angles. They became important for calculating heights from the length of the shadow that the object cast. The length of shadows was also of importance in the sundial. Thales used the lengths of shadows to calculate the heights of pyramids.

The first known tables of shadows were produced by the Arabs around 860 and used two measures translated into Latin as umbra recta and umbra versa. Viète used the terms amsinus and prosinus. The name tangent was first used by Thomas Fincke in 1583. The term cotangens was first used by Edmund Gunter in 1620.

The common abbreviation used today is tan whereas the first occurrence of this abbreviation was used by Albert Girard in 1626, but tan was written over the angle.

The secant and cosecant were not used by the early astronomers or surveyors. These came into their own when navigators around the 15th Century started to prepare tables. Copernicus knew of the secant which he called the hypotenusa.

The abbreviations used by various authors were similar to the trigonometric functions already discussed. Cavalieri used Se and Se.2, Oughtred used se arc and sec co arc while Wallis used s and σ. Albert Girard used sec, written above the angle as he did for the tan.

And just for the record, the term 'trigonometry' first appears as the title of a book Trigonometria by B Pitiscus, published in 1595.