198 On the Stanhope Tewperament 



When I have .spoken above, of adding tuneable intervals to 

 each other, of subtracting them from others and dividing 

 them into aliquot parts, I did not, as in the case of strings, 

 consider them as lineally or numerically added, &c., but 

 that the ratios, of which mathematicians have demonstrated 

 musical intervals to consist, are to be added, subtracted, or 

 divided. In managing numerical ratios, multiplication of 

 the terms answers to addition, division to subtraction, and 

 extracting of roots, or raising of powers of the terms, to divi- 

 sion or multiplication of the ratios. Thus in the 6th column 

 of Table I, line 1, when a V, whose ratio is f, is to be added 

 to a 4th or », in order to obtain an VIII or -i^ : if the two 

 fractions were to be reduced to a commoH denominator and 

 added, we should have -^ + -^-^f or -}-|- instead of-j-V; but 

 by considering these fractions as measures of ratios, and of 

 course multiplying them together, we have -*- x \ equal -^ 

 = i, or the true octave. Again, if the V or ^ were required 

 to be deducted from the VIII or i (line 8), we should, pro- 

 ceeding arithmetically, have f — f , equal to — ■^, instead 

 of the geometrical subtraction of ratios, which gives -^ X ^ 

 (for division of fractions is performed by reversing the divisor 

 and then multiplying), equal to I, which is the true 4th. 



Logarithms have the peculiar anol inestimable property of 

 numerically ox lineally measuring or representing ratios; and 

 when we read in a table of logarithms that 0-3010300 is the 

 logarithm of 2, we are to understand that 0*3010300 is the 

 numerieal measure in this table of the ratio \, 0*477 1213 the 

 measure of ^, he. And \, divided by -f , or f x \, being equal 

 . j-J.the ratio answering to aV will be, 0-301 0300 — 0-47 7 121 3 

 = 9-8239087 ; observing, that 10 may generally be borrowed 

 or cast away in the index, or whole number of a logarithm ; 

 and even the index sometimes omitted altogether, as is done in 

 these tables. In this njanner the numbers called logarithms, 

 in columns 7 and 5, in the Tables I and II respectively, have 

 been deduced from, or measure the ratios (to -|- or C) ex- 

 pressed by the numerical fractions in the preceding columns j 

 and are also the measures of the decimal ratios in their suc- 

 ceeding columns to the common denominator 1, as before 

 ob?er\ cd ; the ratios in the fi and 8, and in the 4 and 6 co- 

 lumns, 



