Do the good people who mean so well by science imagine that sculptors will in the future chisel microscopes in marble, that painters will depict the circulation of the blood, and that poets will display in rich rhymes the principles of Euclid?The proofs and diagrams of Euclid, organised with their near-perfect artistry, have provided countless philosophers, mystics and poets with visual inspiration. In this post I discuss the response of three writers to Euclid's very first problem—the construction of an equilateral triangle on a straight line. Here's the problem, adapted for clarity from this website:
— Max Nordau, Degeneration (1892).
1. Let AB be a given finite straight line.There are several problems with this proof, known since antiquity; most of these concern necessary axioms not enumerated beforehand. But these are specialist quibbles, and need not concern us. The proof is essentially sound.
2. Describe a circle [BCD] with centre A and radius AB. Again describe a circle [ACE] with centre B and radius BA. From the point C, at which the circles cut one another, to the points A and B, join the straight lines CA and CB. [You can see this construction in the above diagram; CA and CB are indicated by dotted lines.]
3. Now, since the point A is the centre of the circle BCD, therefore AC equals AB [ie. because both lines are radii of BCD]. Again, since the point B is the center of the circle ACE, therefore BC equals BA.
4. Therefore AB = AC = BC.
5. Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB.
In the late 5th century, Proclus Diadochus, the last of the great Neoplatonists, wrote a commentary on Euclid's Elements. It is from this work that we have the famous story about Euclid and Ptolemy, from which the title of my post derives. Like his predecessors, Proclus had a mystical conception of number as the essence and foundation of the universe. It is no surprise, then, that we find him allegorising Euclid's first problem in terms of Neoplatonic metaphysics:
That an equilateral triangle is the best among triangles, and is particularly allied to a circle, having all its lines from the centre to the circumference equal, and one simple line for its external bound, is manifest to every one; but the partial comprehension of two circles in this problem, seems to exhibit in images how things which depart from principles, receive from them perfection, identity, and equality. For after this manner, things moving in a right line, roll round in a circle, on account of continual generation; and souls themselves, since they are indued with transitive intellections, resemble by restitutions and circumvolutions, the stable energy of intellect. The zoogonic or vivific fountain of souls too, is said to be contained by two intellects. If, therefore, a circle is an image of the essence of intellect, but a triangle of the first soul, on account of the equality and similitude of angles and sides; this is very properly exhibited by circles, since an equilateral triangle is included in their comprehension. But if also every soul proceeds from intellect, and to this finally returns and participates intellect in a two-fold respect; on this account also it will be proper that a triangle, since it is the symbol of the triple essence of souls, should receive its origin comprehended by two circles. But speculations of this kind, as from bright images in the mirror of phantasy, recall into our memory the nature of things.I quote the first English translation of Proclus' commentary, done by Thomas Taylor in 1789. Taylor was extremely prolific, also being the first to translate the works of Plato, Aristotle, Plotinus and Iamblichus—and you can imagine how influential these translations were on the burgeoning Romanticism of the late eighteenth century.
The passage quoted here deals with a lot of technical Neoplatonic mumbo-jumbo. I had just written a paragraph explaining this mumbo-jumbo in some depth, but then I concluded that my readers would be more interested if I instead picked out the salient points. The key is this: the human soul (psuche) proceeds at birth from the divine intellect (nous), and through philosophical contemplation seeks to return to an understanding of (and identification with) that divine intellect. This basic theme is at the core of virtually all Western speculative mysticism. Here the triangle represents the soul (which, as in Plato's Republic, has a 'triple essence'), and the two circles represent the twofold path of the soul away from, and back towards, the divine nous.
What is interesting is that Euclid's diagram is taken as a hieroglyphic to focus the contemplative mind: the general schema of Euclidean geometry is presupposed as a perfect picture of the real world, and consequently of the ideal spiritual world as well. This conception of Euclid extends all the way through the Middle Ages and Renaissance, and continues among the mystical thinkers of the eighteenth century, including Taylor himself.
Two years after the publication of Thomas Taylor's translation, S. T. Coleridge wrote a comic poem about Euclid's first problem, dated March 31, 1791. He was 18, and still at school; in October he would enrol at Jesus College, Cambrige. However, Coleridge was already devouring the Neoplatonists. In the 1817 Biographia Literaria, he refers to his 'early study of Plato and Plotinus, with the commentaries and the THEOLOGIA PLATONICA of the illustrious Florentine; of Proclus, and Gemistius Pletho'. In his 1818 Treatise on Method, Coleridge would propound a very Platonic conception of mathematics, which he classifies as a 'formal pure science', along with logic and universal grammar. He contrasts this category to the 'real pure sciences' of metaphysics, morals and theology. Mathematics, he insists, deals purely with the ideal—with essentia, not with existentia:
Now these laws are purely Ideal. It is not externally to us that the general notion of a square, or a triangle, of the number three, or the number five, exists; nor do we seek external proof of the relations of those notions; but on the contrary, by contemplating them as Ideas in the Mind, we discover truths which are applicable to external existence.By the time he attended university, Coleridge was, in his own words, an excellent Greek scholar, so it is debatable whether he would have needed to refer to Taylor in order to read the Neoplatonists. Nonetheless, it is clear that he was already fascinated with mathematics. In 1791, upon completing his poem, he wrote to his brother George,
I have often been surprized, that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. —Frequent consideration and minute scrutiny have at length unravelled the cause—viz.—that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in it's [sic] proper Paradise, Imagination is wearily travelling on a dreary desart. To assist Reason by the stimulus of Imagination is the design of the following production. In the execution of it much may be objectionable.Much of the work is, indeed, objectionable. In fact, 'A Mathematical Problem' is one of the silliest poems ever penned. But then, poetry could do with a little silliness now and then. Play it, Sam:
I.Well, I did warn you. Now it is evident that the first two parts (and the beginning of the third part) of this poem merely describe, in hilariously poor verse, Euclid's problem. The remaining lines articulate Coleridge's metaphorical reading of the construction. It is clear that Coleridge has no need of seeing, with Proclus, a mystical hieroglyph. Instead he finds in the construction an image of interpersonal 'balance' and harmony, of 'Equality's Band'. His exposition seems to reiterate itself, with an odd pointlessness, between 'Harmoniously equal for ever must stay' and 'Preserving the balance of power so true'. After that, syntax and sense become gnomic and confusing. The Autocratrix—ie. Catherine the Great—the Briton, the Prussian and the Muslim—what have their fright and fear to do with Euclid? Lucyle Werkmeister (in MLN 1959, online here), reads it as a comic response to Edmund Burke's theories about the relation of the sublime to the human imagination. She concludes, 'as a joke on Burke, it is not without some interest. Even so, one is not sorry that it turned to be unique'.
This is now — this was erst,
Proposition the first — and Problem the first.
On a given finite line
Which must no way incline;
To describe an equi —
— lateral Tri —
— A, N, G, L, E.
Now let A. B.
Be the given line
Which must no way incline;
The great Mathematician
Makes this Requisition,
That we describe an Equi —
— lateral Tri —
— angle on it:
Aid us, Reason — aid us, Wit!
From the centre A. at the distance A. B.
Describe the circle B. C. D.
At the distance B. A. from B. the centre
The round A. C. E. to describe boldly venture.
(Third postulate see.)
And from the point C.
In which the circles make a pother
Cutting and slashing one another,
Bid the straight lines a journeying go.
C. A. C. B. those lines will show.
To the points, which by A. B. are reckon’d,
And postulate the second
For Authority ye know.
A. B. C.
Triumphant shall be
An Equilateral Triangle,
Not Peter Pindar carp, nor Zoilus can wrangle.
Because the point A. is the centre
Of the circular B. C. D.
And because the point B. is the centre
Of the circular A. C. E.
A. C. to A. B. and B. C. to B. A.
Harmoniously equal for ever must stay;
Then C. A. and B. C.
Both extend the kind hand
To the basis, A. B.
Unambitiously join’d in Equality’s Band.
But to the same powers, when two powers are equal,
My mind forbodes the sequel;
My mind does some celestial impulse teach,
And equalises each to each.
Thus C. A. with B. C. strikes the same sure alliance,
That C. A. and B. C. had with A. B. before;
And in mutual affiance
None attempting to soar
The unanimous three
C. A. and B. C. and A. B.
All are equal, each to his brother,
Preserving the balance of power so true:
Ah! the like would the proud Autocratrix do!
At taxes impending not Britain would tremble,
Nor Prussia struggle her fear to dissemble;
Nor the Mah’met-sprung Wight
The great Mussulman
Would stain his Divan
With Urine the soft-flowing daughter of Fright.
But rein your stallion in, too daring Nine!
Should Empires bloat the scientific line?
Or with dishevell’d hair all madly do ye run
For transport that your task is done?
For done it is — the cause is tried!
And Proposition, gentle Maid,
Who soothly ask’d stern Demonstration’s aid,
Has proved her right, and A. B. C.
Of Angles three
Is shown to be of equal side;
And now our weary steed to rest in fine,
’Tis rais’d upon A. B. the straight, the given line.
The reference to the stallion makes one think of the proem to Parmenides' hexametric On Being, though I cannot be sure that this is what Coleridge had in mind. More generally, it is tempting, albeit wholly without foundation, to see in this ridiculous poem an anticipation of the radical, anti-imperialist politics that the young poet would concern himself with during the 1790s—the image of the harmonious triangle a precursor to his utopian Pantisocracy.
James Joyce's response to the Euclidean problem is, naturally, far more difficult to decipher. I have already mentioned this response on a number of occasions, but I have not yet offered any discussion thereof. It occurs in Finnegans Wake 2.293, in the middle of a renownedly difficult passage often referred to as the 'Night Lessons'. Here is Joyce's version of the diagram—
Here π takes the place of C, α and λ the place of A and B, and P occurs opposite π. Joyceophiles will immediately recognise here the recurring initials of ALP (Anna Livia Plurabelle)—'A is for Anna like L is for liv. Aha hahah, Ante Ann you're apt to ape aunty annalive!'. The construction itself is described in the subsequent pages, albeit heavily garbled in the usual Wake manner. For instance, when Joyce writes 'With Olaf as centrum and Olaf's lambtail for his spokesman circumscript a cyclone', we understand 'With α (aleph) as the centre, and αλ (aleph-lambda) as the radius (spoke-sman), describe a circle (Gk. kuklos)'. Similarly, 'join alfa pea and pull loose' means 'join [by straight lines] απ and πλ'. 'Loose' suggests luis (pron. loosh), the Irish L, as well as Lucia, Joyce's daughter. 'Alfa' suggests alfalfa—also called luc-erne—and along with 'pea' connotes fertility. The climax of the exposition arrives on p. 297:
Outer serpumstances being ekewilled, we carefully, if she pleats, lift by her seam hem and jabote at the spidsiest of her trickkikant (like thousands done before since fillies calpered. Ocone! Ocone!) the maidsapron of our A.L.P., fearfully! till its nether nadir is vortically where (allow me to aright to two cute winkles) its naval's napex will have to beandbe.Finally: 'Paa lickam laa lickam, apl lpa! This it is an her. You see her it'. The significance of this passage and diagram is hotly debated among Joyce scholars. It is clearly, like everything else in the book, polysemic—some have seen in it a schematic map of Dublin, others a sexual joke (the central shape seen as a vagina between two arse-cheeks), or a theological diagram, with 'Pee for Pride' (hell) at the base, and 'Pie out of Humbles' (heaven) at the top. It is all of those things, because it can be. The climax is clearly erotic—'You see her it'—recapitulating the theme of voyeurism that pervades the book.
But when it comes to Joyce—on Euclid or not—your guess is as good as mine, or anyone else's. Thomas Rice's book, Joyce, Chaos and Complexity, discusses our hero's interest in non-Euclidean geometry, an obvious sort of analogy to his own oblique methods. But Rice has nothing on the Euclid diagram. Others do, but their expositions are the usual ingenious exegeses, taking bits here and there to find a pattern. Joyce's total demolition of order and harmony, of the geometric method, is the farthest point possible from Proclus' mystical vision. Far from being an ideal figure inspiring recollection of the Neoplatonic hypostases, Euclid's diagram becomes for Joyce just another part of the dirty, playful, sexy natural world. What was once orderly and bounded, the epitome of clarity and simplicity, is now an icon of the the infinite and indefinite, something to catch the eye in a sea of words.