1889 - 90 .] Lord McLaren on Equation of n th Degree. 
271 
completeness ; but in point of utility there is very little difference. 
Supposing that an equation of the 5th degree could be solved 
algebraically so that x is expressed as the 5th root of a function of 
known quantities, a , b , and c. If we write for a , 5, and c their 
values, and proceed to find the 5th root of the function, there is no 
known method of obtaining the root except by the use of logar- 
ithms. Now the extraction of the 5th root by the division of a 
logarithm is equivalent to the solution of the transcendental equa- 
lly 
tion a= 10 5, , which is only possible because tables of values of y to 
the argument x have been computed. 
The solution which I here offer is obtained by the use of 
differential logarithms, and is not different in character from that 
which I have considered. 
It is not an approximate solution, as I understand the expression; 
because the value found for x is obtained by one operation, and is 
incapable of being made more accurate by applying corrections to it. 
The value found for x is, however, only true on the assumption 
that the tabular values^ are true values. But this limitation of the 
accuracy of the solution is not a consequence of any imperfection 
of the method, but results solely from the fact that the tabular 
quantities which are assumed to be known are quantities which do 
not admit of exact numerical expression, or wdiich, in other words, 
cannot be completely expressed in a series of powers of the number 
ten. 
Method of Logarithmic Differences. — These functions, which are 
sometimes called Gaussian logarithms, were really invented, or first 
computed, by Lionelli (i Supplement Logarithmique, Bordeaux, 
1802-3 ; German translation, 1806). Gauss’s table was published 
in 1812, with the knowledge, on Gauss’s part, of what Lionelli had 
done. It may be convenient here to point out how these functions 
are constructed, and how the tables are used : — 
Let u = log (ci + b) ; then u - log a = Su. The quantity 8u is the 
logarithmic difference, or tabular quantity, which has to be added to 
log a, to make up u , or log (a + b). In this paper the logarithmic 
differences are denoted by the symbol, Aoy“; i.e., log 
the 
Greek character being used to distinguish these from common 
logarithms. 
