332 



NA TURE 



[March i6, 1922 



science." The useful applications, mechanical inven- 

 tionSj follow later on ; and David Hume put it neatly 

 when he remarked that " we cannot reasonably expect 

 a piece of woollen cloth to be brought to perfection in 

 a nation that is ignorant of astronomy." 



Two things attract the general student, I think, 

 more than any others connected with Greek mathe- 

 matics — the Pythagorean arithmetic and the matter 

 of the Platonic bodies. The former begins with the 

 notion, characteristically Greek rather than peculiarly 

 Pythagorean or Platonic, of arithmetic as something 

 apart from mere calculation or the doing of sums 

 — as, in short, a " theory of numbers " ; and not the 

 least curious thing about it is that that arithmetic 

 was studied (or so it is said) for generations before the 

 Greeks had signs, even alphabetic ones, for the numerals. 

 We may go to Sir Thomas Heath for a clear and full 

 account of all the curious sorts of numbers, figurate 

 and other, odd and even, square, triangular, and 

 pyramidal, friendly numbers, perfect numbers, and 

 so forth, which fertile imagination along with true 

 mathematical insight was able to discover. It is not 

 without a deep meaning, I believe, that we find on the 

 very threshold of Mathematics this instinct for the 

 symmetry of numbers, this sense of the intrinsic beauty, 

 the comparative perfection, of one number or another. 

 It is the way the calculating boy begins ; we had it 

 exemplified in the highest degree, only the other day, 

 in Ramanujan's extraordinary but too short-lived 

 talent. The theme, in Greek hands, leads on and on 

 by many roads. By way of the " means," it is at 

 the root of the theory of music itself, of the " acous- 

 matic " side of the Pythagorean philosophy ; by the 

 theory of " gnomons " it is close-linked Avith the 

 " theorem " of Pythagoras ; it carries us, though 

 more in contrast than identity, to the Euclidean 

 treatment of arithmetic ; and at last it brings us 

 straight to the Neo-pythagoreans, to the " Theo- 

 logumena," and to later writers down even to Kircher, 

 who dealt more and more extravagantly (much after 

 the fashion of the Cabbala) " de abditis numerorum 

 mysteriis " — with the physical, the " ethical " and the 

 " theological " properties of numbers. 



The other matter, that of the Platonic bodies, is 

 a long story. We know that Plato did not discover 

 them but we may still be curious to know whether 

 Pythagoras did ; and here we must distinguish the 

 mathematical side of the question from the physico- 

 philosophical one — from the deeper meaning which 

 Plato and others found in these five symmetries. It 

 has been amply shown, I think, that their association 

 with the elements was not due to Pythagoras, and it is 

 not likely that Pythagoras knew very much even about 

 their construction and properties. To suppose him 

 NO. 2733, VOL. 109] 



to have understood these is to credit him with too 

 much ; the main teaching of Euclid would have been 

 already his : " Le cadre etait deja celui," as Tannery 

 says, " que remplissait les Elements [d'Euclide]." 

 But who were the great mathematicians who investi- 

 gated them } Theaetetus was probably the outstanding 

 man — he who '' described ^^ that is constructed, the five 

 solids, according to Suidas — though Heath hesitates 

 (curiously) over the meaning of eypaij/e. Heath quotes 

 also the EuClid-scholion that Theaetetus added the 

 octahedron and icosahedron to the other three, but I 

 do not think he mentions that this has (or so it might 

 seem) a textual flaw ; for surely the octahedron was as 

 old as the Pyramids, while the dodecahedron would be 

 one of the last, probably the last of all, to be con- 

 structed and explained. Plato, then, may have 

 taken his mathematics in this matter from Theaetetus, 

 partly (some would say) from Democritus, and some- 

 thing more straight from Leucippus. So with their 

 help was built up Plato's theory, fanciful no doubt 

 but very beautiful, of these five figures, all inscribable 

 in spheres, not really solids but hollow shells with 

 filmy surfaces, made out of tiny triangles — as the gold- 

 beater begins with little three-sided patins of gold — 

 the figures being, as it were, molecules with the facets 

 for atoms, and the whole forming a sort of foamy, 

 cellular structure, like a froth of soap-bubbles, out of 

 which to build the material of an harmonious world. 

 Indeed, one wonders whether Plato had not in his mind's 

 eye the homely but exquisite configuration of a froth 

 of soap-suds ! 



The theme is kindred to our last. For just as an 

 arithmetic grew up regardless of practical reckoning 

 and dealing only with the symmetrical properties of 

 numbers, so did a geometry arise which thought nothing 

 of practical mensuration, only of the abstract properties, 

 the essential symmetries, of planes and solids. This 

 geometry, which studied the triangle, the square, the 

 pentagon,etc., then the "Platonic" and "Archimedean" 

 bodies, the regular and semi-regular solids, the perfect, 

 the less perfect and the imperfect geometrical forms, 

 was own sister to that arithmetic which investigated 

 the triangles, squares, polygons, pyramids and cubes, 

 the " perfections and imperfections," which lie hidden 

 among the mysterious properties of numbers. And 

 all the while these theoretical studies of configuration 

 were being applied along somewhat narrow but very 

 important lines to music, to optics and to astronomy, 

 as we should say to problems of sound, light and 

 periodic motion — in short, to the three great recognised 

 groups of harmonious natural phenomena. This, then, 

 in a word, was the concept of Greek mathematics as it 

 occupied the wit of man, the intellect of philosophers, 

 for just a thousand years. 



