2l8 
Mr. Knight on the construction 
and L( 1 +#, ='A j -A -- A- A- — +| 
As for 'A it may be evidently taken at pleasure ; and innu- 
merable systems of logarithms may be formed by assigning 
different values to it, for 
'A L(i -f- x) -j- 'AL( i -j-jy) — ? AL[ (i 
which expresses that, if every logarithm in a system be mul- 
tiplied by the same constant quantity, the products will still 
form a system of logarithms to the same numbers. 
Cor. By an easy transformation of L(i -f- x), we get for 
Brigg's logarithms, M being the modulus. 
ever the logarithm of a fraction is spoken of in the following 
proposition, it is supposed to be found by this series. 
2. How are we to begin, in forming a table of logarithms? 
Delambre (Preface to Borda, p. 75) says, that we should 
begin at 10,000 ; and the same writer (Memoires de l’Institut, 
Tome cinq. p. 65), speaking of the great French Tables, says 
that the logarithms of primes under 10,000 were calculated 
directly by series, and those of numbers above 10,000 by 
six orders of differences. 
Now it is not easy to see, why any of the logarithms in the 
lower half of the Table, except those of the numbers 2 and 3, 
should be computed directl : A v they may be got, each 
by a single subtraction, from those in the upper half. Sup- 
pose, for instance, there had been found directly the loga- 
rithms oi numbers from 100,000 to 200,000; those of numbers 
down to 30,000 are found by merely subtracting the loga- 
rithm of 2, successively, from those of all the even numbers ; 
