Page:Works of Plato his first fifty-five dialogues (Taylor 1804) (Vol 2 of 5) (IA Vol2worksofplato00plat).pdf/450

 440

INTRODUCTION TO THE TIM^US.

The firſt numbers of the foul are thefe: i, 2, 3, 4, 9, 8, 27 ; but the other numbers are,

6 8 9 12

9 12 18

16

27

18

36

24

54 81

32 36 48

108 162

But in order to underftand thefe numbers mathematically, it is neceffary to know, in the firft place, what is meant by arithmetical, geometrical, and harmonic proportion.

Arithmetical proportion, then, is when an equal ex-

cefs is preferved in three or more given numbers ; geometrical, when num¬ bers preferve the fame ratio; and harmonic, when the middle term is exceeded by the greater, by the fame part of the greater as the excefs of the middle term above the leffer exceeds the leffer. Hence, the numbers 1, 2, 3, are in arithmetical proportion ; 2, 4, 8, in geometrical, fince as 2 is to 4, fo is 4 to 8 ; and 6, 4, 3, are in harmonic proportion, for 4 is exceeded by 6 by 2, which is a third part of 6, and 4 exceeds 3 by 1, which is the third part of 3.

Again, fefquialter proportion is when one number contains

another and the half of it befides, fuch as the proportion of three to 2 ; but fefquitertian proportion takes place when a greater number contains a leffer, and befides this, a third part of the leffer, as 4 to 3 ; and a fefquioclave ratio is when a greater number contains a leffer one, and an eighth part of it befides, as 9 to 8 ; and this proportion produces in mufic an entire tone, which is the principle of all fymphony.

But a tone contains five fympho-

nies, viz. the diateffaron, or fefquitertian proportion, which is compofed from two tones, and a femitone, which is a found lefs than a tone ; the diapente, or fefquialter proportion, which is compofed from three tones and a femitone ; the diapafon, or duple proportion, i. e. four to two, which is compofed from fix tones ; the diapafon diapente, which confiffs of nine tones and