**By Clare Moriarty**

The topic of mathematical beauty is not a new one. Aristotle, in Book 13 of the *Metaphysics*, ruminated on the beauty of various mathematical ideas and theorems, chastising those who sought to deny mathematics its aesthetic merit:

‘[T]hose who assert that the mathematical sciences say nothing of the beautiful or the good are in error … The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.’

My own favourite mathematician, Leonhard Euler, has the niche honour of having his eponymous identity equation (𝑒^{𝑖𝜋} + 1 = 0) voted ‘most beautiful’ in a 1990 poll by the *Mathematical Intelligencer* journal. Why is it beautiful? As with any aesthetic judgement, motivations for admiring a subject will vary, but my own appreciation stems from the fact that it is a simple statement linking some of mathematics’ most fundamental constants via an uncomplicated set of operations—it is parsimonious yet *powerful.*

One might think that these ascriptions of mathematical beauty are mostly metaphorical, that there is no real beauty to be found in proofs. Recently, however, philosopher of mathematics Adam Rieger has argued persuasively that this aesthetic vocabulary in mathematics should often be understood literally, and that mathematics contains genuinely important aesthetic cases. It is easy to wonder what Aristotle and Euler might have made of the beautiful geometrical productions of Irish mathematician Oliver Byrne.

**OLIVER BYRNE**

Byrne was born in Avoca, Co. Wicklow, in 1810. Not much is known about his early life. He describes himself as a graduate of Trinity College, Dublin, but this claim is not borne out in the relevant matriculation records. He married Eleanor Rugg (1822–97) in 1845. In the 1860s she published work on novel balloon designs and the impact of the moon on weather systems—impressive achievements for a woman in that period of scientific publishing. Byrne spent much of his adult life between England and the United States, combining projects in engineering and mathematics with writing on a broad range of topics. The later years of the couple’s life together were marred by prolonged and recurring illness, as well as by constant financial hardship. Oliver died of pneumonia in 1880 while Eleanor lived until 1897.

**VIBRANTLY COLOURFUL AND METHODOLOGICALLY UNIQUE**

Byrne published the book for which he is chiefly remembered—*The first six books of the Elements of Euclid—*in 1847. Euclid’s *Elements *is arguably the most important and influential book in mathematical history and represents a gold standard of rigorous mathematical proof. Over a thousand editions have been published. One might think that such a flooded market might make it difficult to stand out, and yet, though many editions of Euclid have featured magnificent artwork, Byrne’s is a true original. It is vibrantly colourful and methodologically unique.

Despite the centrality of colour illustrations to Byrne’s intentions in producing the book, aesthetic plaudits were expressly not what he was after:

‘This work has a greater aim than mere illustration: we do not introduce colours for the purpose of entertainment, or to amuse by certain combinations of tint and form, but to assist the mind in its researches after truth, to increase the facilities of instruction, and to diffuse permanent knowledge’.

Byrne’s strategy in his edition of Euclid was to replace the two-tiered system of traditional geometrical exposition. Historically, illustrated mathematics featured pairings of textual proof statements with corresponding black and white diagrams. Instead, in Byrne’s text we find a complex of proof and diagram, with pictorial elements incorporated directly into the framing text. Colour plays a crucial role in his process and the results are stunning. The idea is that the reference or target of the parts of the proof are more immediate, and the learners are doing less work than usual in their computations because they are not constantly grappling with cognitive translations between the proof and the pictures—scanning eyes back and forth between words and images to see which part of the diagram is intended by the term *‘ABC’ *in the written section. In Byrne’s scheme, diagrammatic elements are embedded in the grammar of the proof.

Whatever Byrne’s intentions, the book has garnered more attention as a work of beauty than as a pedagogical instrument. That is unsurprising: it is visually unlike anything else. There is a bold modernity to the illustrations; the bright colours and geometric forms have a very twentieth-century feel, and yet the text has elements that hark back to a much earlier era of printing. The publisher, William Pickering, was something of a maverick. He was an early pioneer of cloth binding and miniature books. His ‘Diamond Classics’ range offered affordable access to great works of literature (e.g. Homer, Virgil, Shakespeare, Milton) that sat comfortably in the pocket (they measure 8.8cm x 5cm x 1.5cm). In Byrne’s mathematical text, the modern, brightly coloured diagrams are shown alongside two printing features of old: (i) pages begin with a large decorative initial letter (by famed engraver and illustrator Mary Byfield); and (ii) Pickering reintroduced ‘old-type’ Caslon font, which features the ‘long’ lower-case ‘s’, or ‘ſ’, more familiar from medieval and early modern print. These elements combine to produce pages with a truly individual appearance.

**PHILOSOPHY OF COLOUR**

Byrne believed that the use of colours helped learners to understand aspects of mathematical objects that black-and-white illustrations could not. The use of colour makes possible the accurate depiction of certain conceptually demanding aspects of the landscape of geometry. Thus there is a kind of philosophy underlying Byrne’s presentation—he is concerned with depicting the precise, abstract nature of mathematical entities. For example, take a mathematical point, a fundamental component of Euclidean geometry. Technically speaking*, *a point is *zero-dimensional*. It has no width, length or breadth. Its point (forgive the pun) is to convey position. But how should one draw a point? How can we accurately represent this thing that is simultaneously a thing and a nothing on paper? Similarly, a line is defined as ‘length without breadth’. Again, how could you draw this faithfully? Length that truly has no breadth won’t show up on the page.

Instead of grappling with the conceptual complexities implied by the formal definition of a line, Byrne believed that a student should begin with an image of two blocks of colour beside one another—one red, one blue. The boundary between them is neither red nor blue; it is simply the boundary itself, a ‘junction [which] occupies no part of the plane, and therefore cannot have breadth but only length: from which we can readily form an idea of what is meant by a mathematical line’. Thus colour facilitates a more accurate rendering of important properties of cornerstone mathematical ideas.

In introducing his mathematical drawings, Byrne thought of more than just the stereotypical mathematical trainee. Women—rarely a focus of general mathematical instructions—are explicitly mentioned in his introduction. In this case, he believed that his visual teaching system might appeal more to young women because of how their education encouraged them to appreciate and respond to beauty. He thus recommended those involved in educating girls to attend to ‘this very attractive mode of communicating knowledge’. He also considered the benefits of his system for students with different learning abilities: ‘It may also be worthy of remark, as tangible diagrams afford the only medium through which geometry and other linear arts and sciences can be taught to the blind, this visible system is no less adapted to the exigencies of the deaf and dumb’.

Byrne wrote on a broad range of subjects beyond mathematics. Among various other projects, he co-authored an anti-phrenology text with his brother, describing the head-measuring pseudoscience as a great fallacy of the age. He also wrote practical ‘reckoning’ texts aimed at railroad engineers, shipbuilders and lumber dealers, and oversaw the compilation of engineering dictionaries.

**KANDINSKY-LIKE**

Returning to Byrne’s edition of Euclid’s *Elements*, certain artistic comparisons are hard to resist. The bright primary colours and geometric focus easily bring to mind the work of Piet Mondrian, Sophie Taeuber-Arp, Kazimir Malevich and Wassily Kandinsky. And, though Byrne had few artistic aspirations, it might be fair to say that he was united, at least in his general principles, with artists like those of the De Stijl movement. The De Stijl artists and architects gravitated towards more abstract presentations of form and colour because of their universality. This restrained artistic vocabulary—in most instances limited to primary colours and linear elements—provides a certain accessibility. You do not need to have travelled to other continents or be able to read classical languages to be familiar with the key notions. Describing Taeuber-Arp’s work, Kandinsky said that she

‘… expressed herself by means of the “colored relief” … using almost exclusively the simplest forms, geometric forms. The forms, by their sobriety, their silence, their way of being sufficient unto themselves, invite the hand, if it is skilful, to use the language that is suitable to it and which is often only a whisper; but often too the whisper is more expressive, more convincing, more persuasive, than the “loud voice” that here and there lets itself burst out.’

These sentiments directed at democratising fine art sit happily alongside Byrne’s mathematical pedagogy, which appears more interested in a general audience than many mathematical introductions. Byrne expanded his system of mathematical illustration in *The young geometrician *in 1865. In that work—aimed at trainee engineers and draughtsmen—Byrne again appears concerned with accessibility. In his system the tool kit is simplified to decrease the cost to the learner, and pictorial elements are aimed at giving the reader a new way of connecting with the material. This time colour represents movement. Byrne uses blue to convey that certain shapes should remain fixed in position in the course of a geometric construction (as opposed to red ones, which are to be moved).

This emphasis on practical demonstration and physical activity in learning puts him in the company of acclaimed mathematician and educator Mary Boole, working in a similar period. Boole published texts on the *Philosophy and fun of algebra *(1909) and on *The preparation of the child for science *(1904)*. *She suggested that the best way to teach a baby the general principles of algebra would be to present a ten-month-old with two copies of the same vessel (a teapot, for instance), one carrying cold water, the other hot, to introduce the central idea of generality—that seemingly identical containers can have importantly different contents (she warns us to take care that the baby not be seriously hurt!). Oliver Byrne and Mary Boole are also conspicuous for having been in scientific marriages; Byrne’s wife, Eleanor Rugg Byrne, was a published meteorologist, and Boole’s husband, George Boole, was a mathematician, philosopher and key figure in the development of mathematical logic and computing.

*Clare Moriarty lectures in philosophy at University College Dublin.*

**Further reading**

S.M. Hawes & S. Kolpas, ‘Oliver Byrne: the Matisse of mathematics—biography 1810–1829’, *Convergence* (August 2015). Mathematical Association of America.

A. Rieger, ‘The beautiful art of mathematics’, *Philosophia Mathematica* **26** (2) (2018), 234–50.

D. Wells, ‘Which is the most beautiful?’, *The Mathematical Intelligencer* **10** (4) (1988), 30–1.