373 
of Edinburgh, Session 1873 - 74 . 
of the great “ Table des Diviseurs,” by Burckhardt. The facility 
afforded by this admirable work for finding convenient formulas of 
approximation, determined me to persevere in the construction of 
the large table ; and, putting aside all my previous calculations, I 
arranged a comprehensive scheme for recording each step of the 
process, so that it might serve as occasion might arise to facilitate 
subsequent steps, and so that any suspected error might be traced 
to its source. By this means the progress of the work was effec- 
tually secured, because each little addition took its proper place, at 
however long an interval of time it might happen to be made. 
Without going into the details of the procedure, it is enough to 
mention here that the logarithms of prime numbers up to 3600, 
and of many others occurring incidentally, have been computed to 
twenty-eight places with the view of being exact to twenty-five, 
and that the logarithms of all their products under 10,000 have 
been tabulated ; and, by help of these, tables have been made to 
fifteen places of the logarithms of all numbers from 300,000 to 
320,000, with their first and second differences. These, filling in 
all twenty-four quarto volumes, are laid on the Society’s table. 
Henry Briggs computed to fourteen places the logarithms of all 
numbers up to 20,000, and of numbers from 90,000 to 100,000 ; so 
that Ylacq, in shortening them to ten places, was safe from error 
excepting in one or two rare cases. But when Ylacq set himself 
to fill in the intermediate 70,000, he sought to lessen the labour 
by using only twelve places, thus making his last figure insecure 
in many more cases ; and, moreover, the process followed by him 
wanted the quality of self-verification. On these accounts I sus- 
pected the occurrence of last-place errors in Ylacq’s part of the 
table. Seeing that each tenth logarithm of my own computation 
from 200,000 to 300,000, should agree with Vlacq’s from 20,000 
to 30,000, the comparison was made, and the result was the dis- 
covery of forty-two errors in this single myriad — an exceedingly 
small number when the nature of the process is considered, but a 
very large number to have escaped detection for two centuries and 
a half. At the same rate for each of the remaining six myriads, 
we may expect a total of nearly three hundred errors. 
In 1658, that is thirty years after Ylacq, John Newton published 
a translation of Gellibrand’s “ Trigonometria Britannica,” in which 
