March i6, 1922] 



NA TURE 



ZZTi 



The Archimedean bodies, by the way, Heath deals 

 with pretty fully, but he might perhaps have told us 

 that one of these (Kepler's " truncated octahedron "), 



d to have been known to Plato, is no other than 

 t tetrakaidekahedron which Lord Kelvin showed 

 be (with a slight modification) the typical " cell " 

 a homogeneous froth. He might even perha4)s 



,ve told us a little of how Kepler (true disciple of 

 agoras and of Plato) used both Platonic and 

 Vrchimedean bodies in that treasure-house of elegant 

 geometry, the " Harmonice Mundi " ; and how he 

 showed that not only the five Platonic bodies (as Euclid 

 knew) but also Archimedes' thirteen were all there are, 

 the complete series of their respective families. 



There is a vast deal of information in Heath's 

 book, clearly set forth and orderly arranged ; we 

 have nothing to compare with it in English, and Gino 

 Loria's " Scienze esatte " is its only serious rival 

 abroad. I am inclined to think that Loria paints 

 history with a broader brush, while Heath excels 

 in "his account of individual mathematicians ; but 

 I cannot help thinking, that Heath, who has attained 

 such complete and acknowledged success in his editions 

 of Euclid and the rest, must have found that in this 

 history he had struck a harder task than any he had 

 tried before. We may know more of the history of 

 mathematics than of any other science, but the lacunae 

 are immense, and tradition is poor material for the 

 historian. Moreover the historical aspect is somewhat 

 uncongenial to the mathematician, if only because (as 

 Eva Sachs says) history deals with das Werden, and 

 mathematics with das Sein ! 



When Sir Thomas Heath deals with Euclid, Apol- 

 lonius, Archimedes, Diophantus, Hero or Pappus, 

 he gives us in a few pages all we could expect by 

 way of epitome of the trend, the method and the 

 results of their labours. But his book pursues its 

 steady, instructive course with little digression, 

 allusion or anecdote, and with curiously little biblio- 

 graphical information such as he puts abundantly 

 into his other books. Surely one of the objects of a 

 text-book is to guide the student to what it does not 

 and cannot contain ! Some of us, I think, would have 

 liked a little more digression or even gossip. When 

 Sir Thomas has told us that the Pentalpha was the 

 Pythagorean symbol of Health he is well-nigh done; 

 but Chasles gives us a dozen pages of learned gossip 

 upon it, traces it through Boetius and Thomas Brad- 

 wardine and the Margarita Philosophica and Father 

 Kircher to Kepler himself, and ends with Poinssot's 

 '' Memoire sur les Polygones " ! The Shoemaker's 

 vnife is a beautiful and simple construction in easy 

 t,eometry, of great antiquity — an ancient proposition, 

 Pappus calls it — ^and Heath tells us doubtless all that 



NO. 2733, VOL. 109] 



is essential for us to know ; but a short footnote might 

 have told us how Jacob Steiner investigated and 

 elaborated it, or how J. S. Mackay epitomised its many 

 properties, or how Sir Thomas Muir added a pretty 

 corollary. 



Again (as a random instance) Heath discusses at 

 length the simple but important rule of Thymaridas 

 (simplicity itself in our notation) for solving 

 certain simultaneous equations, where the sum of 

 Xj + Xg-h . . . x;„ is known, and also the successive 

 sums of x^ + x^, x-^ + x^, etc.; but of Thymaridas 

 he only tells us that he was " an ancient Pythagorean, 

 probably not later than Plato's time." If we be 

 limited to a phrase I do not know that we could say 

 a safer thing ; but why should we not have some 

 little sign-post, even a footnote, to Tannery's dis- 

 cussion (in " L'Arithmetique pythagorienne ") on who 

 Thymaridas was and when he lived, or to the many 

 discussions by Cantor, Martin, Nesselmann, and even 

 Fabricius ; for " il y a un assez grand interet historique 

 a determiner I'age ou vivait Thymaridas." Heath 

 tells us that this rule of his was called by the special 

 name of €7rav^7///a, and he translates it " the ' flower ' 

 or ' bloom ' of Thymaridas." He qualifies this by a 

 parenthetic remark that the name was not, after all, 

 confined "to this particular proposition, but what it 

 really means he does not explain ; Tannery, I think, 

 has shown fairly clearly that it was a name (" pour 

 ainsi dire ") " pour les mati^res non exigees du pro- 

 gramme de I'arithmetique pour les etudiants en 

 philosophic." 



It was again Thymaridas who defined (as Heath 

 tells us) " a unit as ' limiting quantity,' " Trepait'ova-a 

 TTOfroTTys. It was a very important definition, 

 but was it not a definition of " unity " rather 

 than of " a unit," and is a limiting quantity a fair 

 and full translation of Troo-orr/s' ? Turn towards the 

 other end of the volume, to ground that is peculiarly 

 Heath's own, and see Euclid's famous definition (V. 3) 

 of ratio, which Heath renders " a sort of relation in 

 respect of size (Tri/AtKorr/s) between two magnitudes 

 of the same kind." I cannot help thinking that, 

 between the two, we lose the fine and even crucial 

 distinction between ttoo-ot^/'j and mjXiKoTij^. The one 

 mathematician was talking of a relation between 

 numbers, the other of a ratio between any two magni- 

 tudes ; I think they both picked their words accord- 

 ingly, and I should like at least to give them the benefit 

 of the doubt. Of Euclid's definition Heath tells us 

 that " it was probably inserted for completeness' 

 sake, and in order merely to aid the conception of a 

 ratio." All the same, I should rather like to hear 

 what Barrow had to say of its metaphysical character ; 

 or what an older school meant when they translated 



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