OP ARTS AND SCIENCES. 167 



as discoursed in Timaeus," is of special value as a statement — in great 

 part intelligible, in spite of the corruption of the text — of the ancient 

 conceptions of the nature of the Pythagorean scale. He points out 

 that this Timaean " quaternary," * as he terms it, contains two series, each 

 commencing with the unit as their common original ; next after the 

 unit come, respectively, two and three, the first plane t numbers ; 

 then four and nine, the first squares ; and, lastly, eight and twenty- 

 seven, the first cubes. The relations of these numbers are exhibited 

 in the following scheme : — 



In this manner similar numbers, says Plutarch, are joined together, 

 and they " will produce other remarkable numbers, as well by addition 

 as multiplication. By addition thus : two and three make five, four 



bringt er hiervon zunachst die Zahlen mit einander in Zusammenhang, welche 

 5t7rA.o<r4a Smimij/uoTo (Octaven) und TpnrKdaia Siao-T^j/xoTa (Duodecimen) bilden. 



SiTrXacria Sjost. 1 2 4 8 



rpiirKdaia ^laffT. 1 3 9 27 



* This was the second numerical Tetractys of the Pythagoreans. The 

 Tetractys was the root or source of all things, irayav aevdov <pv<Tecos (Carm. 

 Aur. 48.) The first was called the Tetractys of the Ten, and was composed of 

 1, 2, 3, and 4, the sum of which is the perfect number 10. The second was 

 double, 1, 2, 4, 8 and 1, 3, 9, 27. Each first number represents the point ; the 

 second, the line ; the third, the plane ; the fourth, the solid. The whole Tetractys 

 is 1, 2, 3, 4, 8, 9, 27 ; the sum of the first six numbers being equal to the seventh. 

 The sacred Seven includes the whole series. See Boeckh, Op. cit. p. 142. 



Hierocles, in his comment on the Golden Verses says, koI ovk ecrriv uveTv 6 /xi) 

 rrji TerpaKTvos, i>s pi(r]s Kol apxvs, ^pTrfrai. p. 230; ed. 1654. 



t The ancient arithmeticians used the term plane numbers for those which 

 were the products of two prime numbers. 



