EXCURSUS II — POPULAR NAMES FOR EUCLIDEAN PROPOSITIONS.¶
Although some of these time-honoured names are familiar to most educated people, it seems to be impossible to trace them to their original sources, or to say who applied them for the first time respectively. It may be that they were handed down by oral tradition for long periods in each case before they found their way into written documents.
We begin with
I. 5.¶
This proposition is in this country universally known as the Pons Asinorum, “Asses’ Bridge.” Even in this case opinion is not unanimous as to the exact implication of the term. Perhaps the more general view is that taken in the Stanford Dictionary of Anglicised Words and Phrases (by C. A. M. Fennell) where the description is: “Name of the fifth proposition of the first Book of Euclid, suggested by the figure and the difficulties which poor geometricians find in mastering it.” This is certainly the equivalent of what I gathered, in my early days at school, from a former Fellow of St John’s, the Reverend Anthony Bower, who was a high Wrangler in 1846 and a friend of Todhunter’s. The “ass” on this interpretation is a synonym for “fool.” But there is another view (as I have learnt lately) which is more complimentary to the ass. It is that, the figure of the proposition being like that of a trestle-bridge, with a ramp at each end which is the more practicable the flatter the figure is drawn, the bridge is such that, while a horse could not surmount the ramp, an ass could; in other words, the term is meant to refer to the surefootedness of the ass rather than to any want of intelligence on his part. (I may perhaps mention that Sir George Greenhill is a strong supporter of this view.)
An epigram of 1780 is the earliest reference to the term in Murray’s English Dictionary:
The writer’s own view is not too clear. He seems to imply that, while the inventor of the name meant that only the fool finds the bridge difficult to pass, the more proper view would be that, since the ass can get over, and “ass” is synonymous with “fool,” therefore it must be the fool who can get over; in other words, he seems to object to the phrase as being a contradiction in terms.
But we have also to take account of the fact that the French apply the term to I. 47. Now in Euclid’s figure for I. 47 there is no suggestion of a bridge, and the reference can only be to the nature of the theorem, its difficulty or otherwise. It is curious that the French dictionaries give two different explanations of Pont aux ânes. Littré makes it “ce que personne ne doit ni ne peut ignorer; ce qui est si facile que tout le monde doit y réussir.” Now no intelligent person could have applied the name to Eucl. I. 47 for this reason, namely that it was so easy that even a fool could not help knowing it. Larousse is better informed; there we find “Pont aux ânes, certaine difficulté, certaine question qui n’arrête que les ignorants, et qui sert de critérium pour juger l’intelligence de quelqu’un, et particulièrement d’un écolier. C’est ainsi que, dans les classes de mathématiques, on ne manque jamais de dire que le carré de l’hypoténuse est le pont aux ânes de la géométrie. La plupart des dictionnaires entendent par ce mot une chose si simple, si facile, que personne ne doit l’ignorer: c’est une erreur évidente.” Larousse is clearly right. But it will be observed that, so far as it goes, Larousse’s interpretation rather supports the first of the two alternative explanations of the meaning of “Asses’ Bridge” as applied to I. 5, namely that it is difficult for the fool (=”ass”) to master.
In the Stanford Dictionary it is added that “in logic the term was in the 16 c. applied to the conversion of propositions by the aid of a difficult diagram for finding middle terms”; and if the mathematicians borrowed the term from logic, this again would be rather in favour of the first explanation of its use for I. 5.
If it is permitted desipere in loco, I would add for the benefit of future generations (in the hope that they will still be able to appreciate the joke or, in the alternative, will be tempted to discuss learnedly what could possibly have been meant) a very topical allusion in a recent Punch (14 Oct. 1925): “When they film Euclid, as is suggested, we shall no doubt see a very thrilling rescue over the burning Pons Asinorum.”—And yet it is safe to prophesy that the “Asses’ Bridge” will outlive the “film”!
Elefuga.
This name for Eucl. I. 5 is mentioned by Roger Bacon (about 1250), who also gives an explanation of it (Opus Tertium, c. vi). He observes that in his day people in general, finding no utility in any science such as geometry, for example, recoiled from the idea of studying it unless they were boys forced to it by the rod, so that they would hardly learn so much as three or four propositions. Hence it is, he says, that the fifth proposition is called “Elefuga, id est, fuga miserorum; elegia enim Graece dicitur, Latine miseria; et elegi sunt miseri.” That is, according to Roger Bacon, Elefuga is “flight of the miserable.” This explanation no doubt accounts for the verses about Dulcarnon in Chaucer’s Troilus and Criseyde, III, ll. 933–5:
since “fleminge of wrecches,” “banishment of the miserable,” is a translation of “fuga miserorum.” Only Dulcarnon is there wrongly taken to be the same proposition as Elefuga, i.e. I. 5, whereas, as we shall see, Dulcarnon was really the name for the Pythagorean theorem I. 47.
Etymologically, Roger Bacon’s explanation leaves something to be desired. The word would really seem to be an attempt to compound the two Greek words ἔλεος, pity (or the object of pity), and φυγή, flight (cf. note ad loc. in Skeat’s edition of Chaucer). Notwithstanding the confusion of tongues, the object seems to be a play upon the two words Elementa and ἔλεος, which both begin with the same three letters, and the implication is that “escape from the Elements” (which normally came when Prop. 5 was reached) was equivalent to “escape from misery” or “trouble.” A better form for the word would perhaps be Eleufuga; and this form actually occurs in Alanus’ Anticlaudianus, III, c. 6 (cited by Du Cange, Glossarium, s.v.). The word also occurs, according to Skeat’s note, in Richard of Bury’s Philobiblon, c. xiii, where it was somewhat oddly translated by J. B. Inglis in 1832 “How many scholars has the Helleflight of Euclid repelled!”
I. 47.¶
The Pythagorean proposition about the square on the hypotenuse has taken even a deeper hold of the minds of men, and has been known by a number of names.
The Theorem of the Bride (θεώρημα τῆς νύμφης).
This name is found in a MS. of Georgius Pachymeres (1242–1310) in the Bibliothèque Nationale at Paris; there is a note to this effect by Tannery (La Géométrie grecque, p. 105), but, as he says nothing more, it is probable that the passage gives the mere name without any explanation of it. We have, however, much earlier evidence of the supposed connexion of the proposition with marriage. Plutarch (born about 46 A.D.) says (De Iside et Osiride 56, p. 373 F) “We may imagine the Egyptians (thinking of) the most beautiful of triangles (and) likening the nature of the All to this triangle most particularly, for it is this same triangle which Plato is thought to have employed in the Republic, when he put together the Nuptial Figure (γαμήλιον διάγραμμα)”—διάγραμμα, though literally meaning “diagram” or “figure,” was commonly used in the sense of “proposition”—“and in that triangle the perpendicular side is 3, the base 4, and the hypotenuse, the square on which is equal to the sum of the squares on the sides containing (the right angle), 5. We must, then, liken the perpendicular to the male, the base to the female and the hypotenuse to the offspring of both….For 3 is the first odd number and is perfect, 4 is the square on an even side, 2, while the 5 partly resembles the father and partly the mother, being the sum of 3 and 2.”
Plato used the three numbers 3, 4, 5 of the Pythagorean triangle in the formation of his famous Geometrical Number; but Plato himself does not call the triangle the Nuptial Triangle nor the number the Nuptial Number. It is later writers, Plutarch, Nicomachus and Iamblichus, who connect the passage about the Geometrical Number with marriage; Nicomachus (Introd. Ar., II, 24, 11) merely alludes to “the passage in the Republic connected with the so-called Marriage,” while Iamblichus (In Nicom., p. 82, 20 Pistelli) only speaks of “the Nuptial Number in the Republic.”
It would appear, then, that the name “Nuptial Figure” or “Theorem of the Bride” was originally used of one particular right-angled triangle, namely (3, 4, 5). A late Arabian writer Behā-ad-dīn (1547–1622) seems to have applied the term “Figure of the Bride” to the same triangle; the Arabs therefore seemingly followed the Greeks. The idea underlying the use of the term, first for the triangle (3, 4, 5), and then for the general theorem of I. 47, seems to be roughly that of the two parties to a marriage becoming one, just as the two squares on the sides containing the right angle become the one square on the hypotenuse in the said theorem.
The “Bride’s Chair.”
The origin of this name is more obscure. It must presumably have been suggested by a supposed resemblance between the figure of the proposition and such a chair. D. E. Smith (History of Mathematics, II, pp. 289–90) remarks that the “Bride’s Chair” may be so-called “because the Euclid figure is not unlike the chair which a slave carries on his back and in which the Eastern bride is sometimes transported to the ceremony,” and he cites a note from Edouard Lucas’ Récréations Mathématiques, II, p. 130: “La démonstration que nous venons de donner du théorème de Pythagore sur le carré de l’hypoténuse ne diffère pas essentialement de la démonstration hindoue, connue sous le nom de la Chaise de la petite mariée, que l’on rencontre dans l’ouvrage de Bhascara (Bija-Ganita, § 146).” The figure of Bhāskara is not that of Euclid but that shown at the top of p. 355 above; I have however not been able to find the name “Bride’s Chair” in Colebrooke’s translation of the work of Bhāskara.
Notwithstanding the apparent frivolity of the setting, I venture to suggest that light may be thrown on the question by a very modern version of the “Bride’s Chair” which appeared during or since the War in La Vie Parisienne. The illustration represents Euclid’s figure for I. 47 and, drawn over it, as on a frame, a poilu in full fighting kit carrying on his back his bride and his household belongings. Roughly speaking, the soldier is standing (or rather walking) in the middle of the large square, his head and shoulders are bending to the right within the contour of one of the small squares, while the lady, with mirror and powder-puff in action, is sitting with her back to him in the right angle between the two smaller squares ($HAG$ in the figure on p. 349 above)[1]. I am informed by Sir George Greenhill that there was also an earlier version “showing the chair as it is in use to-day in Cairo and Egypt, the earliest version of a taxi-chair, a pattern as early as Euclid and suggesting the nickname of the proposition.” This recalls to my mind the remark of a friend to whom I mentioned the subject and showed the figure of the proposition; he observed at once on seeing it “But I should have said it was more like a sedan chair,” the large square suggesting to him the actual chair and the two smaller squares the two bearers.
Dulcarnon.
This name for I. 47 appears, as above mentioned, in Chaucer’s Troilus and Criseyde, III, ll. 930–3, where Criseyde says:
Billingsley, too, in his edition of Euclid (1570) observes of I. 47 that “it hath bene commonly called of barbarous writers of the latter time Dulcarnon.”
Dulcarnon (see Skeat’s note ad loc.) seems to represent the Persian and Arabic du ‘lkarnayn, lit. two-horned, from Pers. du, two, and karn, horn. The name was applied to I. 47 because the two smaller squares stick up like two horns and, as the proposition is difficult, the word here takes the sense of “puzzle”; hence Criseyde was “at dulcarnon” because she was perplexed and at her wit’s end.
Francisci tunica = “Franciskaner Kutte,” “Franciscan’s cowl.”
This name is quoted by Weissenborn (Die Uebersetzungen des Euklid durch Campano und Zamberti, p. 42) as given in a Geometrie by one Kunze. The name is quite appropriate, one of the squares representing the hood thrown back.
III. 7, 8.¶
I have already mentioned the names “Goose’s Foot” (Pes anseris) and “Peacock’s Tail” (Cauda pavonis) applied, suitably enough, to these propositions respectively. They come from Luca Paciuolo’s edition of Euclid published in 1509 (vide Weissenborn, ibid.).