It is generally acknowledged that the mathematisation of science was a central factor in the so-called scientific revolution. When I first came to the history of science there was widespread agreement that this mathematisation took place because of a change in the underlaying philosophy of science from Aristotelian to Platonic philosophy. However, as we saw in the last episode of this series, the renaissance in Platonic philosophy was largely of the Neoplatonic mystical philosophy rather than the Pythagorean, mathematical Platonic philosophy, the Plato of “Let no one ignorant of geometry enter here” inscribed over the entrance to The Academy. This is not to say that Plato’s favouring of mathematics did not have an influence during the Renaissance, but that influence was rather minor and not crucial or pivotal, as earlier propagated.

It shouldn’t need emphasising, as I’ve said it many times in the past, but Galileo’s infamous, *Philosophy is written in this grand book, which stands continually open before our eyes (I say the ‘Universe’), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth*, is not the origin of the mathematisation, as is falsely claimed by far too many, who should know better. One can already find the same sentiment in the Middle Ages, for example in Islam, in the work of Ibn al-Haytham (c. 965–c. 1040) or in Europe in the writings of both Robert Grosseteste (c. 1168–1253) and Roger Bacon, (c. 1219–c. 1292) although in the Middle Ages, outside of optics and astronomy, it remained more hypothetical than actually practiced. We find the same mathematical gospel preached in the sixteenth century by several scholars, most notably Christoph Clavius (1538–1612).

As almost always in history, it is simply wrong to look for a simple mono-casual explanation for any development. There were multiple driving forces behind the mathematisation. As we have already seen in various earlier episodes, the growing use and dominance of mathematics was driving by various of the practical mathematical disciplines during the Renaissance.

The developments in cartography, surveying, and navigation (which I haven’t dealt with yet) all drove an increased role for both geometry and trigonometry. The renaissance of astrology also served the same function. The commercial revolution, the introduction of banking, and the introduction of double entry bookkeeping all drove the introduction and development of the Hindu-Arabic number system and algebra, which in turn would lead to the development of analytical mathematics in the seventeenth century. The development of astro-medicine or iatromathematics led to a change in the status of mathematic on the universities and the demand for commercial arithmetic led to the establishment of the abbacus or reckoning schools. The Renaissance artist-engineers with their development of linear perspective and their cult of machine design, together with the new developments in architecture were all driving forces in the development of geometry. All of these developments both separately and together led to a major increase in the status of the mathematical sciences and their dissemination throughout Europe.

This didn’t all happen overnight but was a gradual process spread over a couple of centuries. However, by the early seventeenth century and what is generally regarded as the start of the scientific revolution the status and spread of mathematics was considerably different, in a positive sense, to what it had been at the end of the fourteenth century. Mathematics was now very much an established part of the scholarly spectrum.

There was, however, another force driving the development and spread of mathematics and that was surprisingly the, on literature focused, original Renaissance humanists in Northern Italy. In and of itself, the original Renaissance humanists did not measure mathematics an especially important role in their intellectual cosmos. So how did the humanists become a driving force for the development of mathematics? The answer lies in their obsession with all and any Greek or Latin manuscripts from antiquity and also with the attitude to mathematics of their ancient role models.

Cicero admired Archimedes, so Petrarch admired Archimedes and other humanists followed his example. In his *Institutio Oratoria* Quintilian was quite enthusiastic about mathematics in the training of the orator. However, both Cicero and Quintilian had reservations about how too intense an involvement with mathematics distracts one from the active life. This meant that the Renaissance humanists were, on the whole, rather ambivalent towards mathematics. They considered it was part of the education of a scholar, so that they could converse reasonably intelligently about mathematics in general, but anything approaching a deep knowledge of the subject was by and large frowned upon. After all, socially, mathematici were viewed as craftsmen and not scholars.

This attitude stood in contradiction to their manuscript collecting habits. On their journeys to the cloister libraries and to Byzantium, the humanists swept up everything they could find in Latin and/or Greek that was from antiquity. This meant that the manuscript collections in the newly founded humanist libraries also contained manuscripts from the mathematical disciplines. A good example is the manuscript of Ptolemaeus’ *Geographia* found in Constantinople and translated into Latin by Jacobus Angelus for the first time in 1406. The manuscripts were now there, and scholars began to engage with them leading to a true mathematical renaissance of the leading Greek mathematicians.

We have already seen, in earlier episodes, the impact that the works of Ptolemaeus, Hero of Alexander, and Vitruvius had in the Renaissance, now I’m going to concentrate on three mathematicians Euclid, Archimedes, and Apollonius of Perga, starting with Archimedes.

He also translated texts by Hero. Although, he was an excellent translator, he was not a mathematician, so his translations were somewhat difficult to comprehend. Archimedes was to a large extent ignored by the universities in the Middle Ages. In 1530, Jacobus Cremonensis (c. 1400–c. 1454) (birth name Jacopo da San Cassiano), a humanist and mathematician, translated, probably at request of the Pope, Nicholas V (1397–1455), a Greek manuscript of the works of Archimedes into Latin. He was also commissioned to correct George of Trebizond’s defective translation of Ptolemaeus’ *Mathēmatikē Syntaxis*. It is thought that the original Greek manuscript was lent or given to Basilios Bessarion (1403–1472) and has subsequently disappeared.

Bessarion had not only the largest humanist library but also the library with the highest number of mathematical manuscripts. Many of Bessarion’s manuscripts were collected by Regiomontanus (1436–1476) during the four to five years (1461–c. 1465) that he was part of Bessarion’s household.

When Regiomontanus moved to Nürnberg in 1471 he brought a large collection of mathematical, astronomical, and astrological manuscripts with him, including the Cremonenius Latin Archimedes and several manuscripts of Euclid’s *Elements*, that he intended to print and publish in the printing office that he set up there. Unfortunately, he died before he really got going and had only published nine texts including his catalogue of future intended publications that also listed the Cremonenius Latin Archimedes.

The invention of moving type book printing was, of course, a major game changer. In 1482, Erhard Ratdolt (1447–1522) published the first printed edition of *The Elements of Euclid* from one of Regiomontanus’ manuscripts of the Latin translation from Arabic by Campanus of Novara (c. 1220–1296).

In 1505, a Latin translation from the Greek by Bartolomeo Zamberti (c. 1473–after 1543) was published in Venice in 1505, because Zamberti regarded the Campanus translation as defective. The first Greek edition, edited by Simon Grynaeus (1493–1541) was published by Jacob Herwegens in Basel in 1533.

Numerous editions followed in Greek and/or Latin. The first modern language edition, in Italian, translated by the mathematician Niccolò Fontana Tartaglia (1499/1500–1557) was published in 1543.

Other editions in German, French and Dutch appeared over the years and the first English edition, translated by Henry Billingsley (died 1606) with a preface by John Dee (1527–c. 1608) was published in 1570.

In 1574, Christoph Clavius (1538–1612) published the first of many editions of his revised and modernised *Elements*, to be used in his newly inaugurated mathematics programme in Catholic schools, colleges, and universities. It became the standard version of Euclid throughout Europe in the seventeenth century. In 1607, Matteo Ricci (1552–1610) and Xu Guanqui (1562–1633) published their Chinese translation of the first six books of Clavius’ *Elements*.

From being a medieval university textbook of which only the first six of the thirteen books were studied if at all, *The Elements* was now a major mathematical text.

Unlike his Euclid manuscript, Regiomontanus’ Latin Archimedes manuscript had to wait until the middle of the sixteenth century to find an editor and publisher. In 1544, Ioannes Heruagius (Johannes Herwagen) (1497–1558) published a bilingual, Latin and Greek, edition of the works of Archimedes, edited by the Nürnberger scholar Thomas Venatorius (Geschauf) (1488–1551).

The Latin was the Cremonenius manuscript that Regiomontanus had brought to Nürnberg, and the Greek was a manuscript that Willibald Pirckheimer (1470–1530) had acquired in Rome.

Around the same time Tartaglia published partial editions of the works of Archimedes both in Italian and Latin translation. We will follow the publication history of Archimedes shortly, but first we need to go back to see what happened to *The Conics* of Apollonius, which became a highly influential text in the seventeenth century.

Although, *The Conics* was known to the Arabs, no translation of it appears to have been made into Latin during the twelfth-century scientific Renaissance. Giovanni-Battista Memmo (c. 1466–1536) produced a Latin translation of the first four of the six books of *The Conics*, which was published posthumously in Venice in 1537. Although regarded as defective this remained the only edition until the latter part of the century.

We now enter the high point of the Renaissance reception of both Archimedes and Apollonius in the work of the mathematician and astronomer Francesco Maurolico (1494–1575) and the physician Federico Commandino (1509-1575). Maurolico spent a large part of his life improving the editions of a wide range of Greek mathematical works.

Unfortunately, he had problems finding sponsors and/or publishers for his work. His heavily edited and corrected volume of the works of Archimedes first appeared posthumously in Palermo in 1585. His definitive Latin edition of *The Conics*, with reconstructions of the fifth and sixth books, completed in 1547, was first published in 1654.

Maurolico corresponded with Christoph Clavius, who had visited him in Sicily in 1574, when the observed an annular solar eclipse together, and with Federico Commandino, although the two never met.

Federico Commandino produced and published a whole series of Greek mathematical works, which became something like standard editions.

His edition of the works of Archimedes appeared in 1565 and his Apollonius translation in 1566.

Two of Commandino’s disciples were Guidobaldo del Monte (1545–1607) and Bernardino Baldi (1553–1617).

Baldi wrote a history of mathematics the *Cronica dei Matematici*, which was published in Urbino in 1707. This was a brief summary of his much bigger *Vite de’ mathematici*, a two-thousand-page manuscript that was never published.

Guidobaldo del Monte, an aristocrat, mathematician, philosopher, and astronomer

became a strong promoter of Commandino’s work and in particular the works of Archimedes, which informed his own work in mechanics.

In the midst of that darkness Federico Commandino shone like the sun, for his learning he not only restored the lost heritage of mathematics but actually increased and enhanced it … In him seem to have lived again Archytas, Diophantus, Theodosius, Ptolemy, Apollonius, Serenus, Pappus and even Archimedes himself.

Guidobaldo.Liber Mechanicorum, Pesaro 1577, Preface

When the young Galileo wrote his first essay on the hydrostatic balance, his theory how Archimedes actually detected the substitution of silver for gold in the crown made for King Hiero of Syracuse, he sent it to Guidobaldo to try and win his support and patronage. Guidobaldo was very impressed and got his brother Cardinal Francesco Maria del Monte (1549–1627), the de’ Medici family cardinal, to recommend Galileo to Ferinando I de’ Medici, Grand Duke of Tuscany, (1549–1609) for the position of professor of mathematics at Pisa University. Galileo worked together with Guidobaldo on various projects and for Galileo, who rejected Aristotle, Archimedes became his philosophical role model, who he often praised in his works.

Galileo was by no means the only seventeenth century scientist to take Archimedes as his role model in pursuing a mathematical physics, for example Kepler used a modified form of Archimedes’ method of exhaustion to determine the volume of barrels, a first step to the development of integral calculus. The all pervasiveness of Archimedes in the seventeenth century is wonderfully illustrated at the end of the century by Sir William Temple, Jonathan Swift’s employer, during the so-called battle of the Ancients and Moderns. In one of his essays, Temple an ardent supporter of the superiority of the ancients over the moderns, asked if John Wilkins was the seventeenth century Archimedes, a rhetorical question with a definitively negative answer.

During the Middle Ages Euclid was the only major Greek mathematician taught at the European universities and that only at a very low level. By the seventeenth century Euclid had been fully restored as a serious mathematical text and the works of both Archimedes and Apollonius had entered the intellectual mainstream and all three texts along with other restored Greek texts such as the *Mathematical Collection* of Pappus, also published by Commandino and the *Arithmetica* of Diophantus, another manuscript brought to Nürnberg by Regiomontanus and worked on by numerous mathematicians, became influential in development of the new sciences.

“the de’ Medici family cardinal, to recommend Galileo to Ferinando (Ferdinando I think) I de’ Medici, Grand Duke of Tuscany, (1549–1609) for the position of professor of mathematics at Pizza (Pisa maybe?) University.”

Oh dear, yes 🙄

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