In the last two episodes we have looked at developments in printing and art that, as we will see later played an important role in the evolving Renaissance sciences. Today, we begin to look at another set of developments that were also important to various areas of the newly emerging practical sciences, those in mathematics. It is a standard cliché that mathematisation played a central roll in the scientific revolution but contrary to popular opinion the massive increase in the use of mathematics in the sciences didn’t begin in the seventeenth century and certainly not as the myth has it, with Galileo.

Medieval science was by no means completely devoid of mathematics despite the fact that it was predominantly Aristotelian, and Aristotle thought that mathematics was not scientia, that is, it did not deliver knowledge of the natural world. However, the mathematical sciences, most prominent astronomy and optics, had a fairly low status within medieval university culture.

One mathematical discipline that only really became re-established in Europe during the Renaissance was trigonometry. This might at first seem strange, as trigonometry had its birth in Greek spherical astronomy, a subject that was taught in the medieval university from the beginning as part of the quadrivium. However, the astronomy taught at the university was purely descriptive if not in fact even prescriptive. It consisted of very low-level descriptions of the geocentric cosmos based largely on John of Sacrobosco’s (c. 1195–c. 1256) *Tractatus de Sphera* (c. 1230). Sacrobosco taught at the university of Paris and also wrote a widely used Algorismus, *De Arte Numerandi*. Because Sacrobosco’s *Sphera* was very basic it was complimented with a *Theorica planetarum*, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos. Mathematical astronomy requiring trigonometry was not hardy taught and rarely practiced.

Both within and outside of the universities practical astronomy and astrology was largely conducted with the astrolabe, which is itself an analogue computing device and require no knowledge of trigonometry to operate.

Before we turn to the re-emergence of trigonometry in the medieval period and its re-establishment during the Renaissance, it pays to briefly retrace its path from its origins in ancient Greek astronomy to medieval Europe.

The earliest known use of trigonometry was in the astronomical work of Hipparchus, who reputedly had a table of chords in his astronomical work. This was spherical trigonometry, which uses the chords defining the arcs of circles to measure angles. Hipparchus’ work was lost and the earliest actual table of trigonometrical chords that we know of is in Ptolemaeus’ *Mathēmatikē Syntaxis* or *Almagest*, as it is usually called today.

When Greek astronomy was appropriated in India, the Indian astronomers replaced Ptolemaeus’ chords with half chords thus creating the trigonometrical ratios now known to us, as the sine and the cosine.

It should be noted that the tangent and cotangent were also known in various ancient cultures. Because they were most often associated with the shadow cast by a gnomon (an upright pole or post used to track the course of the sun) they were most often known as the shadow functions but were not considered part of trigonometry, an astronomical discipline. So-called shadow boxes consisting of the tangent and cotangent used for determine heights and depths are often found on the backs of astrolabes.

Islamic astronomers inherited their astronomy from both ancient Greece and India and chose to use the Indian trigonometrical half chord ratios rather than the Ptolemaic full cords. Various mathematicians and astronomers made improvements in the discipline both in better ways of calculating trigonometrical tables and producing new trigonometrical theorems. An important development was the integration of the tangent, cotangent, secant and cosecant into a unified trigonometry. This was first achieved by al-Battãnī (c.858–929) in his*Exhaustive Treatise on Shadows*, which as its title implies was a book on gnonomics (sundials) and not astronomy. The first to do so for astronomy was Abū al-Wafā (940–998) in his

*Almagest*.

It was this improved, advanced Arabic trigonometry that began to seep slowly into medieval Europe in the twelfth century during the translation movement, mostly through Spain. It’s reception in Europe was very slow.

The first medieval astronomers to seriously tackle trigonometry were the French Jewish astronomer, Levi ben Gershon (1288–1344), the English Abbot of St Albans, Richard of Wallingford (1292–1336) and the French monk, John of Murs (c. 1290–c. 1355) and a few others.

However, although these works had some impact it was not particularly widespread or deep and it would have to wait for the Renaissance and the first Viennese School of mathematics, Johannes von Gmunden (c. 1380–1442), Georg von Peuerbach (1423–1461) and, all of whom were Renaissance humanist scholars, for trigonometry to truly establish itself in medieval Europe and even then, with some delay.

Johannes von Gmunden was instrumental in establishing the study of mathematics and astronomy at the University of Vienna, including trigonometry. His work in trigonometry was not especially original but displayed a working knowledge of the work of Levi ben Gershon, Richard of Wallingford, John of Murs as well as John of Lignères (died c. 1350) and Campanus of Novara (c. 200–1296). His *Tractatus de sinibus, chordis et arcubus* is most important for its probable influence on his successor Georg von Peuerbach.

Peuerbach produced an abridgement of Gmunden’s *Tractatus *and he also calculated a new sine table. This was not yet comparable with the sine table produced by Ulugh Beg (1394–1449) in Samarkand around the same time but set new standards for Europe at the time. It was Peuerbach’s student Johannes Regiomontanus, who made the biggest breakthrough in trigonometry in Europe with his *De triangulis omnimodis* (*On triangles of every kind*) in 1464. However, both Peuerbach’s sine table and Regiomontanus’ *De triangulis omnimodis* would have to wait until the next century before they were published. Regiomontanus’ *On triangles* did not include tangents, but he rectified this omission in his *Tabulae Directionum*. This is a guide to calculating Directions, a form of astrological prediction, which he wrote at the request for his patron, Archbishop Vitéz. This still exist in many manuscript copies, indicating its popularity. It was published posthumously in 1490 by Erhard Ratdolt and went through numerous editions, the last of which appeared in the early seventeenth century.

Peuerbach and Regiomontanus also produced their abridgement of Ptolemaeus’ *Almagest*, the *Epitoma in Almagestum Ptolemae,* published in 1496 in Venice by Johannes Hamman. This was an updated, modernised version of Ptolemaeus’ magnum opus and they also replaced his chord tables with modern sine tables. A typical Renaissance humanist project, initialled by Cardinal Basilios Bessarion (1403–1472), who was a major driving force in the Humanist Renaissance, who we will meet again later. The *Epitoma* became a standard astronomy textbook for the next century and was used extensively by Copernicus amongst others.

Regiomontanus’ *De triangulis omnimodis* was edited by Johannes Schöner and finally published in Nürnberg in 1533 by Johannes Petreius, together with Peuerbach’s sine table, becoming a standard reference work for much of the next century. This was the first work published, in the European context, that treated trigonometry as an independent mathematical discipline and not just an aide to astronomy.

Copernicus (1473–1543,) naturally included modern trigonometrical tables in his *De revolutionibus. *When Georg Joachim Rheticus (1514–1574) travelled to Frombork in 1539 to visit Copernicus, one of the books he took with him as a present for Copernicus was Petreius’ edition of *De triangulis omnimodis*. Together they used the Regiomontanus text to improve the tables in *De revolutionibus*. When Rheticus took Copernicus’ manuscript to Nürnberg to be published, he took the trigonometrical section to Wittenberg and published it separately as *De lateribus et angulis triangulorum* (*On the Sides and Angles of Triangles*) in 1542, a year before *De revolutionibus* was published.

Rheticus’ action was the start of a career in trigonometry. Nine years later he published his *Canon doctrinae triangvlorvm*in in Leipzig. This was the first European publication to include all of the six standard trigonometrical ratios six hundred years after Islamic mathematics reached the same stage of development. Rheticus now dedicated his life to producing what would become the definitive work on trigonometrical tables his *Opus palatinum de triangulis*, however he died before he could complete and publish this work. It was finally completed by his student Valentin Otto (c. 1548–1603) and published in Neustadt and der Haardt in 1596.

In the meantime, Bartholomäus Piticus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, *Trigonometria*: sive de solution triangulorum tractatus brevis et perspicuous, one year earlier, in 1595.

This work was republished in expanded editions in 1600, 1608 and 1612. The tables contained in Pitiscus’ *Trigonometria* were calculated to five or six places, whereas those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. In comparison Peuerbach’s sine tables from the middle of the fifteenth century were only accurate to three places of decimals. However, on inspection, despite the years of effort that Rheticus and Otho had invested in the work, some of the calculations were found to be defective. Pitiscus recalculated them and republished the work as *Magnus canon doctrinae triangulorum* in 1607.

He published a second further improved version under the title *Thesaurus mathematicus *in 1613. These tables remained the definitive trigonometrical tables for three centuries only being replaced by Henri Andoyer’s tables in 1915–18.

In the seventeenth century a major change in trigonometry took place. Whereas throughout the Renaissance it had been handled as a branch of practical mathematics, used to solve spherical and plane triangles in astronomy, cartography, surveying and navigation, the various trigonometrical ratios now became mathematical functions in their own right, a branch of purely theoretical mathematics. This transition mirroring the general development in the sciences that occurred between the Renaissance and the scientific revolution, from practical to theoretical science.

I maintain – without being able to justify it, because I’m too lazy to do the work – that another reason for the revolution in mathematics in the sixteenth and seventeenth centuries is that standardized symbols and notation began to be introduced at that time.

https://en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date

It cannot be easy to do advanced mathematics when you have to write everything as a word problem, or decipher someone else’s personal notation.

It does indeed play a role and will be part of a later episode

I presume the reason for the table of chords in Greek astronomy is because they needed to be able to calibrate cross-staffs for navigation.

As the cross-staff was first described by the French-Jewish mathematician Levi ben Gershon (1288–1344) of Provence, in his “Book of the Wars of the Lord” (published in Latin and Hebrew), I very much doubt it.

Ptolemaeus made his observational measurements with an armillary sphere.

Thony, knowledge of the principle of the cross-staff is far older than 14th Century. 400 BCE might be a better estimate: https://exploration.marinersmuseum.org/object/cross-staff/

The question you really need to be asking is why did Hipparchos create a table of chords if he wasn’t measuring angles with something like a cross-staff?

If I see anything giving Cynthia Clark Northrup, as its source I project it in the direction of the waste paper basket. The cross-staff in by no means the only angle measuring astronomical instrument.

The four editors of the “Encyclopedia of World Trade: From Ancient Times to the Present: From Ancient Times to the Present, Volumes 1-4” are all economic historians and every quote on history of science or scientific instruments that I have seen from this work was flat out wrong.

All the experts on the history of scientific instruments state quite clearly that the earliest reference to a cross-staff is Levi ben Gershon.

We have no surviving texts from Hipparchus. However, Ptolemaeus, in his Mathēmatikē Syntaxis, says that Hipparchus made his observations/measurements with a dioptra. There is one passage that can be interpreted such that Hipparchus used an armillary sphere like Ptolemaeus himself but some historians dispute this interpretation.