Image: Eratosthenes teaching in Alexandria
“I would plead for more attention to the history of mathematics. It seems to me to be extremely desirable, if not actually indispensable, for entering into the heritage that has been bequeathed to us, and for seeking to enhance its value.”
These words were spoken by Augustus Love in 1912, in his Presidential address to the London Mathematical Society. While the quote (unfortunately) does not appear in the books under review, I believe it sets the tone for this great six-volume effort by Bloomsbury Press.
My review here will feature highlights (some good, some bad) from each volume. This will be supplemented by some comments on the remaining contents, including my favourite chapter.
Volume 1: Mathematics in Antiquity
In his Introduction, Michael Fried (Ben-Gurion University of the Negev) offers some definitional framework for the project at hand. “One should not get the impression that the division is always clear between a cultural history of mathematics and a history which is heritage, or what may be called a mathematical history of mathematics; it is sometimes very unclear…The problem is all the more challenging when it comes to ancient mathematics.”
A few pages later, he expands on this thought. “The question whether modern mathematics can be used as a legitimate tool to understand ancient mathematics is a question to be taken seriously.” The myth of the Egyptian knowledge of pi is a cautionary tale. “The importance of pi in modern mathematics is so overwhelming that it is easy to assume that anyone at another place or time must have understood it on similar terms as we do. Yet, it must not be forgotten that even thinking of pi as a number at all is part of a modern conception of number.” So much for that heralded ‘discovery’ in Egyptian mathematics! Readers may well be reassured by this Introduction by Fried who, as editor of the book, is fully attuned to the dangers of overreach. The remainder of the volume is, quite rightly, guided by Fried’s stern hand on the tiller as we navigate the sometimes unfathomable depths of ancient math.
In Volume 1, authors Francesca Rochberg and J. Lennart Berggren get the work of Eratosthenes completely wrong. As shown by C.A. Matthew (2023), he did not round the circumference of the Earth to “252,000 stades, so as to arrive at the figure divisible by 60.” Matthew arrives at the correct figure of 224,100 stades. “Using the 180 m stade identified in this research,” writes Matthew, “gives a result for Eratosthenes’ calculations of 40,338 km, which differs from the polar circumference of the Earth (40,007 km) by only 0.8 percent.” The Babylonians used a base 60, Eratosthenes did not. Having formerly reviewed a fantastic book by Rochberg (see References below), I know her knowledge of Babylonian is superior, but trying to put Eratosthenes in the Babylonian mathematical mindset doesn’t work. Coupled with using an incorrect value of the stade just compounds the misstep.
As explained in a recent issue of Quanta, “mathematicians have discovered it’s enough to look at a small fragment of an object to understand its broader properties. In the third century BC, Eratosthenes estimated the circumference of the Earth by measuring shadows cast by the sun in just two cities about 500 miles apart…The McKay conjecture is another example of this principle. It says that if you want to formulate a thorough description of a group you only need to look at a tiny piece of it.” The conjecture was named for John McKay, who posed it in the 1970s. In 2024 Marc Cabanes and Britta Spath proved the McKay conjecture is true for all groups. The discovery was an example of “beautiful, wonderful, deep mathematics,” in the words of Gabriel Navarro, who worked on the conjecture for years.
I think those words of Navarro, which I have used for my headline, best embody the sense one gets from reading this A Cultural History of Mathematics. (Two other recent advances in mathematics are given at the end of this review.)
The matter of base 60 is taken up again in the chapter by Christine Proust (Univ. Paris Cite) and Reviel Netz (Stanford Univ). It is still very much with us: 60 minutes in an hour. They adopt the analysis of Jens Hoyrups, who in 2002 concluded that Babylonian ‘algebra’ is, in fact, a ‘naïve geometry’ which “foreshadows methods developed long afterwards by Greek and Arab mathematicians for solving quadratic equations.”
Far from ascribing a particular breakthrough to the Greeks, Proust and Netz claim “The most fundamental invention of Greek mathematics is the invention of the mathematical author.” By that, they mean we can attach the name of a real person to any particular mathematical advance. In addition, they state “Greek mathematics is the most regimented form of writing in all of Greek literature. It is very distinctive, indeed, in its very reliance upon writing.”
The basis of Greek mathematics was geometrical proportion, which we can see exemplified in the Parthenon in Athens. “Mathematicians, it was understood, found the structures in the world, very often expressed through symmetries and proportions.”
But when Archimedes made several discoveries (such as the volume and surface of a sphere), he realized the stakes were high. As one of the first ‘mathematical authors’ he played a cagey game. The year was around 240 BCE. He wrote to his astronomer friend Conon where he “announced the results without providing their proofs, because it was meant as a challenge for anyone around the Greek Mediterranean.” Sometime later, Archimedes revealed that “two claims made in the letter were false. The plan was for people to claim they had proved the results – and then be shown up as empty braggarts. No one fell for this, because no one tried.” In conclusion, Proust and Netz make a very important point: by “inventing the author, the genre, the centrality of proportion and the tool of analysis, the very idea of an autonomous science” was invented.
Volume 2: The Medieval Age
The Introduction to Vol. 2 covers not just European but Arabic, Chinese and Indian mathematics. The three editors of this volume note it was Aryabhata, working around the year 500 in India, who “provides the first known systematic treatment of arithmetic using decimal place-value numerals with zero, and the first surviving procedures for constructing a table of sines.” He also asserted the Earth rotates daily, something not accepted for more a thousand years after his time.
Our words for algorithm and algebra derive from the 9th century. A scholar’s name al-Kharizmi became algorithm, and the title of his book Kitab fi hisab al-jabr, became the basis for algebra. In the Islamic world, “mathematical practice was largely dominated by administrative and career requirements,” with research into planetary theory being a major driving force.
For me, the Inventing Mathematics chapter in each book was very insightful. In this volume, Clemency Montelle (Univ. of Canterbury, NZ) and Kim Plofker (Union College, New York) provide a fine explanation of what ‘inventing mathematics’ is. “Mathematical research by advanced practitioners emerges organically from interest in particular types of problems and the sociocultural phenomena that provide incentive and impetus for understanding them. In turn, individual breakthroughs eventually reshape the standard mathematical background.”
These inventions cross cultural divides. For example, “Islamic mathematical sciences absorbed both Hellenistic chords (particularly in spherical trigonometry) and Indian sines, which seems to have been known to Arabic authors from at least the late eighth century via a translated Sanskrit astronomical work.” Indian trigonometry then made its way to China, while the biggest influence on late medieval Greek astronomy came from Persia.
To invoke the imagery of water I opened with, Montelle and Plofker state that “Hellenistic ontological and epistemological frameworks are the water that modern mathematicians swim in.” Mathematicians in the West employed an “explicitly axiomatic-deductive proof structure” that was unknown to the Chinese, who used analogy to justify results. By exploring this dichotomy, they have many fascinating insights into the two mathematical cultures.
Volume 3: The Early Modern Period
In this volume, I will confine my remarks to chapters 3, 4 and 5. In ‘Inventing Mathematics,’ Sebastian Maronne (Mathematical Institute of Toulouse) notes that “the only form of formal expression in Greek geometry was synthesis.” This was the method used by Euclid in his Elements: “rely on axioms or on previously proved propositions until it produced the statement to be proven.” They also used the reverse, analysis, which proceeded in the reverse order to reach an already known statement. Sadly, all Greek texts containing ‘analysis’ are lost. We only know of their existence due to an author named Pappus, who in 340 CE referred to a “treasury of analysis” comprised of lost works.
Pappus gave “three classical Greek problems, namely duplication of the cube, trisection of the angle, and quadrature of the circle…The demonstration of the impossibility of solving these problems by ruler and compass would only be given in the nineteenth century.” (I will return to this later in the review)
Another problem transmitted by Pappus was solved by Descartes (1596-1650, see image here): he reduced the problem by use of algebraic analysis. “There is no doubt that Descartes’ new way of using algebraic analysis had a tremendous impact,” writes Maronne. But Sir Isaac Newton “was one of Descartes’ opponents. “Newton harshly criticized the Cartesian program on the methodological level,” as Descartes asserted it was only necessary to show the nature of problem. Descartes provided no equations to solve problems, Newton did.
Maronne gives us important insights into the use of symbols in math. Today, they seem obvious and indispensable, but it was not always so. The close relationship between symbolism and algebra did not arise until after 1600, and even a math book published in 1659 used no symbols! The author covers the origins of calculus by both Leibniz and Newton. “Above all,” she writes,” it was the development of systematic algorithms and adequate notations that contributed to the birth of calculus.” For more on algorithms, see my article in SunNewsAustin
My favourite chapter in the entire set is on Mathematics and Worldview by David Rabouin (CNRS, France). This chapter provides an account of certainty as a touchstone for knowledge. Specifically, he looks at the “topic of infinity, because it offers an excellent example of how the debate about the certainty of mathematics is a game changer.”
To begin, he overturns a widely-held understanding of Aristotle vs Plato. “It is simply not true that mathematics had not been considered as a standard of truth by supporters of Aristotle and then was suddenly reappraised under a ‘Platonist’ influence in the Renaissance.” Proclus, for example, was a Platonist who “supported the uncertainty of mathematics and against the ‘Aristotelians’ who were supporting its highest certainty. The exact opposite of the picture Koyré, and so many others after him, thought as constitutive of our modernity.” The French philosopher Alexander Koyré (1892-1964) is being referred to. What Rabouin is doing in this chapter is knocking out the main pillar of our ‘comfortable’ understanding of the early modern period. Koyré asserted in 1957 that as mathematics entered into physics, it destroyed the ancient image of a ‘closed world’ and opened up to way to an infinite universe. “Yet, Koyré’s view is very general,” writes Rabouin,” and does not always concur with the many ways in which philosophers and mathematicians related to infinity.” In the 1650’s Pascal “used the pervasive presence of infinity in mathematics to refute the idea that mathematics could be based on Cartesian certainty.”
Rabouin moves on to a discussion of a “new epoch in the discussions surround foundations of mathematics.” Fames mathematicians often locked horns into very public polemics. While Newton never attacked Leibniz outright (leaving that dirty work to his proxies), others were not so circumspect. Johann Bernoulli and Leibniz engaged in a brawl over the logarithm of negative quantities. During his time in Paris (1672-76), Leibniz faced a deep problem relating to negative numbers: how to find a number the square of which is -1?
Nowhere in this set of books is the subject of negative mathematics engaged in fully. The University of Texas (Austin) professor Alberto Martinez (2005) shows that by using an algebra in which minus times minus makes minus, mathematicians can describe curves or trajectories that are not represented by traditional coordinate geometry. This very important aspect of mathematics is not mentioned in these volumes (indeed, the indexes even ignore mention of negative matters in nearly every instance: the Leibniz negative math problem on page 142 does not merit an entry). While it seems counterintuitive, it is not an obscure aspect of math that can only be reached by a professional. It is, in fact, an ever-present aspect of education. As the Princeton University Press website posits: “A student in class asks the math teacher: ‘Shouldn’t minus times minus make minus?’ Teachers soon convince most students that it does not. Yet the innocent question brings with it a germ of mathematical creativity.” Martinez revolutionized understanding of negative math 20 years ago, but its implications have yet to be incorporated by the professional community. His book is not referenced in the Bloomsbury set.
The chapter by Antoni Malet (Universitat Pompeu Fabra, Barcelona) is on ‘Describing and Understanding the World.’ He takes this title very literally. In his discussion of Pedro Nunes (1502-1578), the work of this Portuguese Jew delves into such things as map projections and his foundational work on mathematical navigation. All well and good. But since this is a ‘cultural’ history it seems strange to ignore his work on the Biblical assertion that the Sun could actually reverse course. Copernicanism triggered a “highly contentious relationship between biblical exegesis and the mathematical sciences,” according to Julia Ellinghaus and Volker Remmert (Univ. of Wuppertal). “Nunes proved (in 1537) in a purely mathematical way that the retrogradation of the Sun would in fact naturally occur twice a day.” In their 2024 book, the authors argue “that Nunes was among the first to deliver astronomers and mathematicians power over the interpretation of the Bible.” This important development, coupled with the fact that mathematics could be perverted to prove a falsity, certainly deserved mention in this volume.
Volume 4: The Eighteenth Century
In the ‘Inventing Mathematics’ chapter, Jeanne Peiffer (emerita at the Centre Alexandre Koyré, Paris) laments the legacy Newton could have had in mathematics. His “most explicit and complete treatise” was written I Latin in 1671, “presumably for Newton’s personal use.” It did not appear in English until 1736, and French in 1740. “Had this masterpiece already appeared in the 1670s, it very likely would have revolutionised mathematics. This single example illustrates how the circulation of mathematical knowledge was an integral part of a complex production process.” Such slow communication often led to the loss of important results, Pfeiffer writes. Furthermore, scholarly research is still lacking in this era. “The circulation of Newtonian and Leibnizian ideas outside Europe has received little attention and remains a relatively open problem in the history of mathematics.” The debate over which one of these great scholars invented calculus is nicely covered, and given the right degree of nuance, e.g. the two camps “had divergent understandings on what it meant to possess a valid solution of a problem.”
Divergent approaches led to debates between “three giants of eighteenth-century mathematics.” On the apparently simple matter of a vibrating string, “D’Alembert, a mathematician, was interested in the integration of partial differential equations. Daniel Bernoulli, a traditional physicist, was interested in the phenomenon itself and invited fellow mathematicians in making observations of vibrating strings. Euler, a mathematical physicist, realized that the arbitrary functions, which entered in the solution, posed completely new problems for the analysis of his time.” A fascinating chapter!
In ‘Mathematics and Worldviews,’ David Bellhouse (Univ. of Western Ontario, Canada) and Jip van Besouw (Vrije Univ. Brussel, Belgium) tell us how “the historically rather unfortunate view of Newton as having given us the cosmology of a ‘clockwork universe,’ arose.” It was Leibniz who “claimed that the God of Newton and Clarke was not reasonable and powerful enough, comparing him to a watchmaker who had to wind up his clock.” The Clarke in question was Samuel Clarke, an Anglican clergyman who conducted an “infamous correspondence” with Leibniz (pictured here). Clarke is one of those proxies I mentioned, but he went beyond what was countenanced by Newton.
Others, however, went much further. “Kant exemplifies how little of the actual Newton was left in the thought of post-Newtonians. While Newton provided empirical support for the laws of motion, Kant argued that they were synthetic a priori truths and therefore beyond empirical validation.” Deists believed God was incomprehensible, and it gave many of them a reason to dispense with an “active God. This was a move Newton would have abhorred. The most famous deist interpretation of Newton” was by Voltaire and also Maupertuis. The latter set out to prove Newton was right about the flattening of Earth at its poles, and his famous expedition to Lapland in 1737 proved it. The concept of ‘Newtonianism,’ was born and swept all competitors to the dustbin of history. Together with Einstein, Newton remains the most famous scientist/mathematician of all time.
Volume 5
In their introduction to Vol. 5, Tom Archibald (Simon Fraser Univ., Canada) and David E. Rowe (Mainz Univ., Germany) write about “shocking news” that took four decades to make mathematical headlines. In the 1820s and 30s, Lobachevsky and Bolyai conceived of “an alternative geometry, one in which some of the basic theorems of Euclidean geometry, and the parallel postulate itself, were false.” It was not until 1868 that many mathematicians and some philosophers “had begun to see the light. Once this door fell open, it became clear that the classical geometry of Euclid was merely a special case.” Eventually, “mathematics began to acquire two hallmark features of modernity: abstraction, and its companion, generality.” Strangely, Carl Gauss agreed with the new geometry, but never made his views public. This comes in a chapter by Jeremy Gray (Open University), who unfortunately ventures onto very thin ice.
In a discussion of the famous list of mathematical problems posed by Hilbert in 1900, Gray has this to say. “This selection of the 23 problems highlights the ones that proved exceptionally difficult and important. Others could never be solved (axiomatize physics), but many were.”
Luckily, the younger generation of mathematicians are not deterred by this ‘can-never-be-solved’ attitude. (Keep in mind the three problems of Greek math, which took more than 2,000 years to solve! But solved they were!)
In March 2025, Yu Deng, Zaher Hani and Xiao Ma figured out how to unify three physical theories that explain the motion of fluids. This achieves one of the subgoals established by Hilbert, on the road to solving his sixth problem. He called there for axiomatizing physics: determining the bare minimum of mathematical assumptions behind all its theories. This has been described as the loftiest of all Hilbert’s problems. (see ref below)
The chapter on Mathematics and Worldviews by Ivahn Smadja (Univ. of Nantes, France) deserves a close reading. For those who read SunNewsAustin avidly, I’m sure you will have read my review dealing with Theodor Adorno in Naples in 1925 (see reference below). In 1969, Adorno and Max Horkheimer claimed that “Enlightenment’s program was the disenchantment of the world.”
“They contended,” writes Smadja, “that the process was still ongoing and that mathematics was central to it.” He writes that Adorno- Horkheimer “held that the core of the entire historical process lay in the way the Enlightenment enthroned mathematics as its privileged self-mirroring instrument. From the outset, they argued, the Enlightenment was doomed to fail as an emancipatory project.”
Finally, I will mention the chapter on Everyday Numeracy by Adrian Rice (Randolph-Macon College, Virginia). Even though Nunes misused mathematics in 1537, it was not until three centuries later that the full implications hit. The United States census of 1840 was used by those in favour of slavery to proclaim that it gave evidence of the damaging effects of emancipation! This marked a turning point “in the history of numeracy in America, when it was realized for the first time, not only that numbers do not always present an objective truth, but that they can be manipulated and misrepresented just as easily as words.”
Volume 6
Since I have already taxed the reader of this review by some 4,000 words, I will keep my remarks on the Modern volume brief. It will suffice to quote from the Russian mathematician Andrei Kolmogorov, who contributed the ‘Mathematics’ entry to the Great Soviet Encyclopedia starting in 1938. His work was key to modern probability theory (1933), but in 1963 he was more concerned about the compatibility between pure and applied mathematics. As he mentions culture here, and appears to predict the importance of artificial intelligence, it seems a fitting end to a review of this massive and extraordinary set of books that no mathematical library can do without.
“I admire those mathematicians who become important in technology, I absolutely acknowledge the relevance of computers and cybernetics for the future of mankind, yet I feel that pure mathematics has not handed over the place of honour that it obtained among the sciences…I want to underscore the value and merit of the position of those mathematicians who understand the place and role of their science in …all of human culture, but who patiently continue to develop pure mathematics as it corresponds to the inner logic of its evolution.”
There are a few typos in the books:
In Vol. 1: pg. 15: “one the” should be “one of the”
In Vol. 4: pg. 148: “interpret” should be “interpreted”
Pg. 170: “already used” is printed twice in a single sentence
Pg. 200: “to it mass” should be “to its mass”
References
Cabanes, M. and Spath, B. (2025). The McKay Conjecture on character degrees. Arxiv.org 2410.20392
Cunningham, C. (2018). Review of Rochberg’s book Before Nature. Journal of Astronomical History & Heritage, 21, 256-257.
Cunningham, C. (2023). Review of the book The Evolution of Mathematics by G. Mitchell Reyes.
sunnewsaustin.com/2023/01/09/the-danger-of-algorithms/
Cunningham, C. (2025). Review of Martin Mittelmeier’s book Naples 1925.
sunnewsaustin.com/2025/04/08/naples-1925-a-critical-year-for-philosophy/
Deng, Y., Hani, Z. and Ma, X. (2025). Hibert’s Sixth Problem: Derivation of Fluid Equations via Boltzmann’s Kinetic Theory. Arxiv.org 2503.01800
Ellinghaus, J and Remmert, V. (2024). Manipulating the Sun. Leiden, Brill.
Martinez, A. (2005). Negative Math. Princeton, Princeton University Press.
Matthew, C.A. (2023). Eratosthenes and the Measurement of the Earth’s Circumference. Oxford, Oxford University Press.
Wang, H and Zahl, J. (2025). Volume estimates for unions of convex sets, and the Kakeya set conjectures in three dimensions. Arxiv.org 2502.17655
A Cultural History of Mathematics comprises 1,536 pages in six volumes. It is by Bloomsbury, and lists for $610. Each chapter typically has about 2 to 6 illustrations, plus diagrams, but none are in colour.
Each book contains the same chapter topics, as printed below.
The series has 2 General Editors:
David E. Rowe is Professor emeritus of History of Mathematics and Natural Sciences at Mainz University. His principal research interests concern developments in Western mathematics and physics during the period 1800 to 1950 with special focus on Göttingen University. He has written or co-edited sixteen books, most recently Emmy Noether: Mathematician Extraordinaire (Springer 2021).
Joseph W. Dauben is Distinguished Professor of History and History of Science at the City University of New York. He is the author of Georg Cantor, His Mathematics and Philosophy of the Infinite and Abraham Robinson: The Creation of Nonstandard Analysis, a Personal and Mathematical Odyssey.
Table of Contents
Volume 1:A Cultural History of Mathematics in Antiquity
Edited by Michael N. Fried, Ben-Gurion University, Israel
Introduction, Michael N. Fried
1. Everyday Numeracy, Merav Haklai
2. Practice and Profession, Cécile Michel
3. Inventing Mathematics, Christine Proust and Reviel Netz
4. Mathematics and Worldviews, G.E.R. Lloyd
5. Describing and Understanding the World, Francesca Rochberg and J. Lennart Berggren
6. Mathematics and Technological Change, Eduardo A. Escobar
7. Representing Mathematics, Liba Taub
Volume 2: A Cultural History of Mathematics in the Medieval Age
Edited by Joseph W. Dauben, CUNY, USA, Clemency Montelle, University of Canterbury, New Zealand and Kim Plofker, Union College, USA
Introduction,Joseph W. Dauben, Clemency Montelle, and Kim Plofker
1. Everyday Numeracy, Joseph W. Dauben, Clemency Montelle and Kim Plofker
2. Practice and Profession, Clemency Montelle
3. Inventing Mathematics, Kim Plofker
4. Mathematics and Worldviews, Joseph W. Dauben
5. Describing and Understanding the World, Jeff Chen
6. Mathematics and Technological Change, Joseph W. Dauben, Clemency Montelle and Kim Plofker
7. Representing Mathematics, Sonja Brentjes and Nathan Sidoli
Volume 3: A Cultural History of Mathematics in the Early Modern Period
Edited by Jeanne Peiffer, CNRS, France and Volker R. Remmert, University of Wuppertal, Germany
Introduction, Jeanne Peiffer and R. Volker Remmert
1. Everyday Numeracy, Maryvonne Spiesser
2. Practice and Profession, James Bennett
3. Inventing Mathematics, Sébastien Maronne
4. Mathematics and Worldviews, David Rabouin
5. Describing and Understanding the World, Antoni Malet
6. Mathematics and Technological Change, Thomas Morel
7. Representing Mathematics, Robert Goulding and Volker Remmert
Volume 4: A Cultural History of Mathematics in the Eighteenth Century
Edited by Maarten Bullynck, Université Paris 8, France
Introduction, Maarten Bullynck
1. Everyday Numeracy, Maarten Bullynck
2. Practice and Profession, Maarten Bullynck
3. Inventing Mathematics, Jeanne Peiffer
4. Mathematics and Worldviews, David Bellhouse and Jip van Besouw
5. Describing and Understanding the World, Helmut Pulte
6. Mathematics and Technological Change, Jane Wess
7. Representing Mathematics, Franziska Bomski
Volume 5: A Cultural History of Mathematics in the Nineteenth Century
Edited by Tom Archibald, Simon Fraser University, Canada and David E. Rowe, Mainz University, Germany
Introduction, Tom Archibald and David E. Rowe
1. Everyday Numeracy, Adrian Rice
2. Practice and Profession, Tom Archibald and David E. Rowe
3. Inventing Mathematics, Jeremy Gray
4. Mathematics and Worldviews, Ivahn Smadja
5. Describing and Understanding the World, Scott A. Walter
6. Mathematics and Technological Change, June Barrow-Green and Tony Royle
7. Representing Mathematics, Anja Sattelmacher and David E. Rowe
Volume 6: A Cultural History of Mathematics in the Modern Age
Edited by Tom Archibald, Simon Fraser University, Canada and David E. Rowe, Mainz University, Germany
Introduction, Tom Archibald and David E. Rowe
1. Everyday Numeracy, Christopher J. Phillips
2. Practice and Profession, Tom Archibald and David E. Rowe
3. Inventing Mathematics, Roger Cooke and Leo Corry
4. Mathematics and Worldviews, José Ferreirós
5. Describing and Understanding the World, Marta Jordi Taltavull and Tilman Sauer
6. Mathematics and Technological Change, Liesbeth De Mol
7. Representing Mathematics, Nina Engelhardt
Images in this review:
Lead photo: Eratosthenes teaching in Alexandria, a 1635 painting by Bernardo Strozzi. Montral Museum of Fine Arts. Courtesy of Wikimedia Commons.
Leibniz. A 1700 painting by Johann Wentzel. Courtesy of Wikimedia Commons. PD-US.
Descartes. A 1649 painting. In the Louvre. Courtesy of Wikimedia Commons