March i6, 1922J 



NATURE 



331 



whole ; but the essential difficulty remains of piecing 

 together the broken stones. 



In our first glimpse of ancient mathematics two 

 great figures, Pythagoras and Plato, stand out above 

 all the rest : not as the greatest mathematicians, but 

 as the greatest landmarks in history and tradition, 

 much like Descartes and Leibniz in another age. 

 From Plato on, the story runs fairly smooth ; we 

 leave tradition behind and enter upon history ; mathe- 

 matics becomes a science of its own arid slowly dis- 

 entangles itself from philosophy ; the " polymath " 

 is giving place to ^he specialist. With Euclid we 

 enter on the " Golden Age " of geometry cradled in 

 Plato's Academy, and though some of the minor men 

 (like Nicomedes and Diodes) remain shadowy figures, 

 we have a solid inheritance in Euclid and Apollonius 

 and in Archimedes — " ordine quidem tertius " (as 

 Torelli said), " dignitate facile princeps." On the 

 physical side the astronomy of Eudoxus, the Optics 

 of Euclid, the astronomy again of Aristarchus — the 

 " Copernicus of antiquity " — may be taken as " nails 

 in a sure place " ; and when we come to the " Silver 

 Age " of Diophantus, Pappus, Ptolemy and Theon, we 

 have a wealth of historical material. 



But up to Plato's time it is a very different story, 

 for the simple reason that very few of the earlier 

 philosophers and mathematicians (say down to 

 Archytas) ever wrote at all. Their teaching was 

 private or even secret, and only oral tradition carried 

 it on. Two things men will continue to discuss but 

 never know for sure — the debt the first Greek mathe- 

 maticians owed to Egypt and the East, and the real, 

 actual attainments in mathematics of men like Thales, 

 Pythagoras, Archytas, Democritus and Plato. Pytha- 

 goras is the strangest case of all, for no wise man's 

 name since Solomon's has so fascinated the world. 

 Tradition, and tradition alone, tells us that he invented 

 the forty-seventh proposition, and tells it no more confi- 

 dently than the Dancing Dervishes of Constantinople tell 

 you they are the descendants of his holy Brotherhood ! 

 Where there is nothing but tradition to go upon, 

 the historian is lost. The imaginative man begins 

 putting two and two together, arguing what must 

 have been known in order to know this or that, and 

 what must have followed as soon as this or that was 

 understood ; and so the story grows. The construc- 

 tion of the Pentagon " must " have implied a know- 

 ledge of the Golden Section, and this and the 

 " Theorem " itself must have involved the concept 

 of the Irrational ; .the triangle of the Pentalpha and 

 its gnomons must have led on to the logarithmic 

 spiral — even to the Lima9on of Pascal ! In some 

 such way Naber, for instance, discourses on what 

 Pythagoras " must " have known and what he may 

 NO. 2733, VOL. 109] 



have known ; it is pleasant, even suggestive, reading, 

 but it is a long, long way from history. At the other 

 end stand the sceptical critics, to be taken more 

 seriously— like Eva Sachs, whose book, by the way, 

 on " Die fiinf platonischen Korper," Heath does not 

 seem to quote. She, for example, holds that up to 

 Philolaus we know nothing at all ; that he and later 

 Pythagoreans were chiefly bent on ascribing each new 

 thing to the old master of their school ; that there is 

 no proof that Pythagoras knew anything of irrationals, 

 little that he was acquainted with the regular solids, 

 and none at all that he associated them, more Platonico, 

 with a theory of the Elements. Between such opinions 

 Sir Thomas Heath steers a careful middle course, and 

 what he has to tell he attributes to " the Pythagoreans " 

 rather than to Pythagoras. 



Again, as to alien sources behind early Greek mathe- 

 matics, one school will tell you that Greece had nothing 

 or next to nothing to gain from the land-surveyors or 

 " rope-stretchers " of Eg}^pt, nor from Mesopotamian 

 astronomers and calendar-makers. Others eagerly 

 pick up little stray hints of a community or descent of 

 ancient learning. They remind us 'that it was in Ionia 

 that Greek philosophy arose and the special sciences, 

 even medicine, began — in a " melange de races 

 d'emigres, d'origine diverse," as Heiberg called the 

 lonians the other day ; that Ionia was in close and 

 constant touch with Lydia ; and that Ionian science 

 appeared after a clash of empires and fall of cities, as 

 a later Renaissance followed the fall of Constantinople. 

 Or they catch hold of straws which point, or seem to 

 point, to Far Eastern intercourse, such (for instance) 

 as that mode of reckoning by myriads and myriads of 

 myriads which the Japanese are said to have used 

 about the time (say) of Thales, which exists in China 

 to this day, and seems identical with Archimedes' 

 famous numeration of the Arenarius — where he began 

 by supposing a myriad grains of sand in the space of 

 a poppy-seed (or rather surely a poppy-^eai), and 

 went on to myriad-myriads of units, and of orders, 

 and of periods. Heath discusses the broad question 

 briefly and fairly, and is content in the end to agree 

 (as we all must) with Plato, that whatsoever the Greeks 

 had borrowed, were it much or little, they it was who 

 improved on it and carried it towards perfection. The 

 Greeks at least knew well what not to borrow, and 

 striking above all else is their choice of themes. It is 

 they who best exemplify what Sir John Herschel laid 

 down (perhaps rightly) as a general proposition-— that 

 men delight to escape from the trammels of earth : 

 that not practical problems but " the abstractions of 

 geometry, the properties of numbers, the movements 

 of the celestial spheres, whatsoever is abstruse, remote 

 and extramundane, become the first objects of infant 



