the Temperament of Mufical lrflruments. 249 
£or the lafc term of the feries ; therefore the problem is reduced 
to the finding out of eleven mean proportionals between the 
two numbers 13286025 and 103797,0703 1 2 
It is demonftrated in almoft every treatife on algebra and 
arithmetic, that in a geometrical progreflion, as the above- 
mentioned one, the firft or fmalleft extreme is to the laft or 
greateft extreme as unity is to a power of the ratio, the index 
of which is equal to the number of terms lefs one. Hence, 
in our cafe, in which the number of terms, including the 
two extremes, is 13, we fhall have 1 0379750703 1 25 : 
13286025 :: 1 : R‘% from which the ratio is found by dividing 
the fecond number by the firft, and extracting the twelfth 
root from the quotient, viz. — — — 12 8 ; and 
1 I0 3797>°7°3 12 5 
1 281- = 1,4983069, which is the ratio fought. 
The ratio having been afcertained, the fucceftion of tem- 
pered fifths is thus eafily determined ; viz. divide the length 
of the whole firing by this ratio, and the quotient gives the 
firft tempered fifth ; divide this fifth by the fame ratio, and 
the quotient gives the fecond tempered fifth ; divide this fe- 
cond fifth by the fame ratio, and fo on till the laft fifth, which 
comes out equal to 103797, 21735 (fee fig. 3.) which is fo 
nearly equal to the length of the feventh oCtave, that the dif- 
ference is truly infignificant ; but, if greater accuracy were 
required, we need only extract the proper root of 128 to a 
greater number of decimals. 
Fig. 3. fhews the divifions of the firing XZ tempered in 
the above-mentioned manner; viz. the fucceffive fifths have 
been afcertained firft, and then, by taking their oCtaves, the 
whole fet of divifions has been completed. By comparing 
this figure with fig. 2. one may ealily perceive, how fmali is 
; the 
