Vive l'arithmétique! The story of 6 mathematical books from the time of Simon Stevin

In 2020 it is 400 years since Simon Stevin [pronounce: Steven] died. Today, this famous 'Bruggeling' (citizen of Bruges) is usually described as a mathematician but Stevin was so much more: he built the first sailing wagon, improved the design of the windmill, wrote about fortification and published on the calculation of interest, knowledge that until then belonged to the monopoly of the bankers.

Whereas the scientific world today uses a rather strict distinction between disciplines, early modern scientists such as Stevin were less concerned about the demarcation of their field of work. The homo universalis dealt with all kinds of challenges, from mathematics and physics over astronomy, economics and architecture to music. These were mostly part of the quadrivium, the sciences studying numbers, as they were already taught at medieval universities.

The sixteenth and seventeenth centuries were the time of the scientific revolution, in which knowledge progressed by leaps and bounds. Thanks to the printing industry, which stimulated the widespread dissemination of knowledge and international intellectual debate. At the same time, it was an era characterised at least as much by tradition as by innovation, in which magic, religion and science merged into one another.

The publications of the early modern mathematicians are wonderful testimonials to this exciting period of scientific progress.

They show how great international personalities and mathematicians from the field, among them many Flemish mathematicians, contributed to the development of mathematics as a discipline.

From elementa to 'integraal'

Today they are called 'Integraal', 'Matrix Wiskunde', 'Van basis tot limiet' or 'Uitgerekend', but the most successful textbook in the history of mathematics is without doubt the Elementa of Euclid of Alexandria, written in the third century BC. Euclid - also called 'the father of geometry' - laid the foundation for axiomatic geometry in this book. He proved, on the basis of a limited number of axioms or basic truths, numerous properties of geometric figures.

This late sixteenth-century book contains an Arabic translation of the Elementa, entitled كتاب تحرير الأصول لأقليدس. (Kitab Tahrir usul li-Uqlidis) by the thirteenth-century Persian scholar Nasir al-Din al-Tusi.

It is no coincidence that an Arab version of this key work reached our regions. After all, the influence of the Arab world played a major role in the scientific flourishing of pre-modern Europe. In the eighth and ninth centuries, Arab scholars made great progress in science and mathematics and were miles ahead of the Christian world.

Presentation of a hemisphere

As a result of a large-scale translation effort of their works into Latin, Arabic knowledge became increasingly widespread in the Western world from the eleventh century onwards. This is how the Christian world came into contact with not only classical mathematics such as Euclid's, but also with Arabic knowledge such as the Arab-Indian numerical system and the use of the number zero, the mathematical proof method of full induction and calculation with algorithms.

Browse كتاب تحرير الأصول لأقليدس.

Written in the stars

Wondering what life has in store for you? Danger and illness or happiness and wealth? In the early modern period you turned to astrological predictions for your answer. Although we now tend to laugh with horoscopes, such predictions were taken very seriously in the past. It was believed that the position of the stars at the time of your birth would greatly determine your further course of life.

M. Manili Astronomicon libri qvinqve is an example of birth astrology. Marcus Manilius composed this extensive didactic work in hexameters in the first century AD. In 1579 Joseph Justus Scaliger published a critical edition with commentary.

As a philologist and humanist, Scaliger valued classical knowledge and authors, and as a biblical scholar he attached great importance to chronology and correct time calculation. These two interests come together in Manilius' poem about the origin of the universe, the signs of the zodiac and their influence on the fate of each person.

The image shows the division of the celestial sphere into twelve houses, each representing certain characteristics, such as wealth, health, social status, business success and danger. In which part the sun, moon and planets are located at the moment of one's birth, predicts one's future.

But what do astrological predictions have in common with mathematics? Well, without mathematics, astrology would not be possible. To correctly determine the position of the celestial bodies at a given moment, a lot of calculation is needed. Until the modern period, no strict distinction was made between mathematics, astronomy and astrology.

Astrological symbols

So, too, the contemporary division between astrology as pseudoscience and astronomy as the scientific study of the universe does not hold true for this era. The more empirical approach to nature in the early modern period did provide new insights, but these did not replace traditional ideas and approaches. Astrology and astronomy, alchemy and chemistry, tradition and innovation, magical thinking, superstition and new scientific methods remained side by side for a long time.

Browse Scaliger's M. Manili Astronomicon

Looking for proof

When you say mathematics, you think of proofs. They are the foundation of mathematics, since they demonstrate the theories on which mathematical knowledge continues to build. Just think of Pythagoras' theorem, which any schoolchild knows by heart, or Fermat's theorem, which captivated the greatest mathematicians for 300 years until Andrew Wiles definitively proved it at the end of the twentieth century.

For early modern mathematicians too, proving the theorems that had puzzled countless minds for centuries was an irresistible challenge. Out of their eagerness to fully grasp the legacy of Greek mathematics, they tried to find answers to age-old questions.

Take for example the issue of squaring the circle. Ever since the ancient Greeks, mathematicians looked for a method to construct, armed only with a compass and ruler, a square whose surface area corresponded exactly to that of a given circle. The calculation of 𝜋 is key here.

In the sixteenth century mathematicians tackled this problem again, on the one hand out of admiration for their predecessors from classical antiquity, on the other hand because they wanted to surpass their great examples in the spirit of the classical aemulatio.

Jacobus Falco also contributed with his De circuli quadratura. He claimed to have finally found proof in 1589. Just like Oronce Finé about half a century earlier as well as soap merchant Jacob Marcelis about 100 years later. And Daniel Waeywel in 1712 ...

The result always turned out not to be entirely convincing but that did not stop anyone from trying again. Even though there were voices saying that the problem could not be solved. In 1586, Ludolph van Ceulen presented his 'cort klaar bewijs' (brief ready evidence) against the solution proposed by his colleague Simon Van Der Eycke. He himself was the first to succeed in calculating the number 𝜋 to 35 decimal places.

The definitive conclusion finally came in 1882: the problem is not solvable under the given parameters, since 𝜋 is what mathematicians call a purely transcendent number and these cannot be constructed. In writers such as Dante, Alexander Pope and Thomas Mann, ‘squaring the circle’ is a metaphor for a doomed endeavour, something only undertaken by foolish dreamers.

Browse Falco's De circuli quadratura

“For all the diligent practitioners”

“Als 8 El. Hollantsch zijn 5 Garten Engelsch , ende 20 guld. Hollantsch zijn 40 s. Engelsch , ende een elle Hollandtsch kost 7 guld 10 stuy. Hoeveel komt een Garte Enghelsch te staen?” [example, in Dutch, of a mathematical problem with measures of lengths and prices, using the rule of three]

Questions such as these may spontaneously evoke memories of the many arithmetic exercises that everyone had to undergo during primary school in order to practise the rule of three.

This exempel in a seventeenth-century jacket illustrates that those exercises have been around for a long time. You may read it in Arithmetica, oft Reken-konst penned by school and mathematics master Jacob Vander Schuere, originally from Menen and later from Haarlem. With his work, Vander Schuere wanted to support 'cooplieden, facteurs, cassiers, ontvanghers, etc' [merchants, sales agents, tellers, collectors, etc] in their daily usage of mathematics.

Those who practiced diligently were able to master all kinds of practical uses of elementary mathematics, such as interest calculation, accounting, measuring the contents of a wine barrel, drawing up invoices or surveying a piece of land.

The growing popularity of such practical mathematical books shows how mathematics played an essential role in the professionalisation of technology and commerce. They offer solutions to concrete issues in daily life, without too much mathematical theory.

The question discussed above trains merchants in the smooth conversion of coins and measures of length from different countries, not an unnecessary luxury at a time of a growing international economy. But the multiplication tables are also included in the work!

Browse Vander Schuere's Arithmetica

On the art of Calculating

Jacob Vander Schuere gave his exercise book the title 'Arithmetica, oft: Reken-konst' (literally translated to: Arithmetic, or: Art of calculating), for good reason. Arithmetics was one of the seven liberal arts in the medieval curriculum, as were geometry and astronomy. Besides goddesses of tragedy, lyric poetry and dance, the nine classical Muses also included Urania, Muse of astronomy, in their ranks.

One of the disciplines in which art and numbers go hand in hand is architecture. After all, no architectural masterpieces such as domes, bridges, arches or solid fortifications are possible without complicated calculations. So it was a great advantage for a master builder to have knowledge of mathematics.

This certainly applied to Hans Vredeman de Vries. As an architect, he achieved groundbreaking work in the field of fortification engineering and garden design, but unfortunately none of his creations survived. We can only see his genius in his printed drawings. Especially his studies on perspective were - and still are- impressive.

Like a rather early modern Escher, Vredeman produced almost illusionistic architectural drawings on paper, full of lines, spaces, stairs and passages. In contrast to Escher, Vredeman did not pursue a surrealistic, but rather a hyper-realistic spatial rendering.

As the title of the work La très noble perspective reveals, he explored the perfect perspective in art. The book was part of the early modern search for an exact representation of reality.

Vredeman's contemporaries were very impressed by his drawings, which translated into a large and international print run.

Nothing shows more clearly how diverse early modern mathematics was than Vredeman's drawings. Whereas today we often draw a sharp line between the sciences and the arts, in the early modern era there was in fact a complex connection.

After all, both set themselves the same goal: to discover the truth behind nature against the backdrop of the scientific revolution. Thorough observation of the world was the driving force behind scientific and artistic innovation, in which art and science stimulated each other in their progress.

Browse Vredeman's La très noble perspective

Drawings from Vredeman's study on perspective
Through turbulent waters

Besides architecture, applied mathematics also includes the design of all kinds of instruments and tools. After all, the professional construction of measuring instruments, globes and astronomical models requires a great deal of mathematical insight. The illustration shows how you can use an astrolabium catholicum to calculate the height of buildings on the basis of triangulation.

Gemma Frisius is the inventor of this instrument. This Frisian cosmographer, mathematician and physician became a professor at the University of Leuven, where he made a career at the Faculty of Medicine. However, it is mainly his achievements as a cosmographer and instrument maker and his cartographic impact that have brought him lasting fame.

Frisius' astrolabium catholicum was a big step forward compared to the previous versions of this measuring instrument. Different types of astrolabia had been circulating since the fourth century, with which one could calculate the heights and angles of objects in space. At sea, for example, sailors used it to determine the height of the sun and the stars, and thus determine the position of their ship in the wide ocean. So it can be seen as the predecessor of our modern GPS.

The problem with such a traditional astrolabium was that you needed different astrolabia for calculations at different latitudes.

Gemma Frisius created a revolutionary innovation with his astrolabium catholicum, or universal astrolabium. From now on, an observer could perform all calculations with one astrolabium, independent of the latitude at which he was located, and easily determine his location. At a time when the Northern Netherlands were emerging as a major player in overseas trade with the East, the invention represented great added value in accurate navigation at sea.

Frisius explains exactly how this innovative measuring instrument works in De astrolabio catholico. The first edition of the work appeared posthumously in 1556.

Browse Frisius' De astrolabio catholico

The unpredictable path of early modern mathematics ran along theory books and practical applications, such as measuring instruments, horoscopes and perspective drawings. This was accompanied by the necessary tension between tradition and innovation and between scholars themselves. The mathematical early printed books are physical witnesses to this intriguing mix of scientific ideas, in which magical thinking and faith had their place alongside solid evidence.

Together, all these gems contributed to the colourful field of early modern mathematics!

VISIT THE exhibition about Simon Stevin!

The exhibition you just visited is the online extension of the exhibition Simon Stevin van Brugghe. For the first time you can admire all the works and manuscripts of this renowned mathematician together in one place. The exhibition runs from 28 August to 29 November 2020 in the City Archives of Bruges.

The exhibition Simon Stevin van Brugghe is an initiative of Stadsarchief Brugge (City Archives Bruges), Openbare Bibliotheek Brugge (Public Library Bruges), Musea Brugge (Museums Bruges), Toerisme Brugge (Tourism Bruges) and Universiteit Gent (Ghent University).

About the project

This exhibition was created as part of a project of assessing the significance of mathematical early printed books from the collections of five Flemish heritage libraries, coordinated by Flanders Heritage Libraries. The approximately 250 beautiful mathematical early printed books from the years 1570 to 1620 of the project are the foundation of this exhibition.

We would like to thank Ad Meskens, Maarten Van Dyck, Geert Vanpaemel and Jean Paul Van Bendegem for their advice and support!

The six main works of the exhibition
  1. Euclid, Kitab tahòrir usòu li-Uqlidus ilakh (Rome, 1594) - Museum Plantin-Moretus (B 509)
  2. Josephus Justus Scaliger, M. Manili Astronomicon libri qvinqve (Leiden, 1600) - Public Library Bruges (560)
  3. Jacobus Falco, De circuli quadratura (Antwerp, 1591) - Museum Plantin-Moretus (8 499)
  4. Jacob Vander Schuere, Arithmetica, oft: Reken-konst: verciert met veel schoone exempelen, zeer nut voor alle vlijtige oeffenaers… (Haarlem, 1611) - Hendrik Conscience Heritage Library (G 68673 [S0-106 g])
  5. Hans Vredeman de Vries, La très noble perspective, à scavoir la theorie, practique et instruction fondamentale d'icelle (Amsterdam, 1619) - University Library Ghent (BIB.ACC.028921)
  6. Gemma Frisius, De astrolabo catholico (Antwerp, 1556) - KU Leuven Libraries Special Collections (CaaA3)
Assessing significance of mathematical early printed books at KU Leuven Libraries
Text, images and layout

An Smets (KU Leuven Libraries), Anneleen Decraene (Museum Plantin-Moretus), Hendrik Defoort (University Library Ghent), Hilde Van Parys (Public Library Bruges), Ine Calmeyn (Flanders Heritage Libraries), Marie-Charlotte Le Bailly (Hendrik Conscience Heritage Library) and Sara Moens (Flanders Heritage Libraries)

Created By
Flandrica.be Heritage Libraries online


A project of Flanders Heritage Libraries in cooperation with Hendrik Conscience Heritage Library (Antwerp), Museum Plantin-Moretus (Antwerp), Public Library Bruges, University Library Ghent and KU Leuven Libraries with the support of the Flemish Government.