of Logarithmic Tables. 233 
terms of L . «, with the sign changed. L . $"••*» is expanded, 
by means of the logarithmic series, in a similar manner. 
Prop. VIL 
13. To calculate a Table of Logarithmic shies, or cosines. 
It is quite evident, that, if, in the fractional products, in 
Propositions II, and III, instead of a?, -l, x— -s, &c. we had 
used successively cos. x, cos. ( x — w), cos. (a?— 2 u), & c. or 
sin. x, sin. {x— u sin. (x — qu), See., the reasonings made use 
of would have been equally applicable; and that the whole 
methods given in Proposition III, IV, V, including the gene- 
ral expressions for L . «"•••«, L $"••••*, (but not the expansions 
of the said expressions) will hold good here, after we have 
made the above mentioned substitutions. Thus if it is L. cos. 
x which we are calculating, we shall have 
T / cos. X \ . T , T I cos. (x — u) cos. + t /» 
^H cos. (*-* - ) ] ; L • « =L . co ;, a - — -J ; L.*" = 
y / COS, (x + zu) COS. 3 X \ 
(cos. (x—u.) cos. 3 (.r + «)j 
These are the logarithms of numbers converging conti- 
nually towards unity, and must be found by the form for 
