186 Proceedings of the Royal Society of Edinburgh. [Sess. 
for the higher branches of astronomy and geodetics the usual seven-place 
tables are enough. But for the purpose of constructing new working 
tables it becomes necessary to carry the actual work further, both in the 
extent of the arguments and in the number of decimal places, and there- 
fore I determined on the formation of a table of logarithms to nine places 
for all numbers up to one million. But again, in order that such a table 
be true to the ninth place, the actual calculation must be carried still further 
— and to meet the cases in which the doubtful figures from say 4997 to 
5003 might occur in one million of cases, it became prudent to carry the 
accuracy even to the fifteenth place. And this limit of accuracy was 
further defined by the circumstance that there the differences of the third 
order just disappear. Even then it may happen that the doubt as to the 
figures which are to be rejected may not be cleared up, and it follows that 
a still more minute criterion should be at hand for use, and therefore the 
order of the work came to be as follows. 
“ In the first place, the computations of the logarithms of all numbers 
up to ten thousand, to twenty-eight (for twenty-five) places, was under- 
taken. At the outset, each logarithm of a prime number was computed 
twice, but as the work proceeded, it was judged advisable to have three 
distinct computations of each. The whole of this work is distinctly re- 
corded and indexed, so that every step in reference to any given number 
can at once be traced out. 
“ The idea was entertained of this work being ultimately extended to 
one hundred thousand, and the logarithms of the composite numbers from 
ten to twenty thousand were computed, spaces being left for those of 
intermediate prime numbers. 
“ By the addition of the logarithms thus obtained, those of the great 
majority of composite numbers from the limit one hundred thousand to 
one hundred and fifty thousand were computed, and the intervals were 
filled up by help of second differences. In this part of the work I was 
aided by my daughters. But, in all such separate additions, we are liable 
to sporadic errors, and in order to guard against these the whole of this 
work was redone by the use of the last two figures of the second differences; 
and thereafter the calculations were made by short interpolations of second 
differences all the way to three hundred and seventy thousand. Necessarily, 
on account of the occurrence of the minute final errors, the last, or fifteenth, 
figures cannot be trusted to within one or two units ; and after a very 
severe examination of the whole, it was found that in a very few instances 
this accumulation of last place inaccuracy extended even to five units ; and 
thus we are warranted in expecting that no last place error will be found 
