HAVING THEIR VALUES BETWEEN TWO PRESCRIBED LIMITS. 293 
here it may be permitted to me to make a few remarks on the usefulness of 
BURCKHARDT'S table. 
To the great majority of computers the knowledge of the divisors of large 
numbers is of no moment, and the author of a table of such divisors up to 
several millions may be regarded as a mere enthusiast. Not one among a 
hundred practised calculators may ever have occasion to consult such a work. 
Yet, for all that, there is no arithmetical table which, in proportion to the labour 
bestowed upon it, effects such a saving to those investigators who require its 
aid. For example, in the preceding search for a train of wheels to connect the 
indicators of solar and sidereal time, we have had to decompose many very large 
numbers. The examination of one of these, particularly when it turns out to 
be a prime, might have cost us a whole day. So irksome, too, is this operation, 
that had such an inquiry been imperative, we might even have preferred to 
prepare for it by first constructing the table of divisors. 
The method of differences, of so much use in the formation of many other 
tables, is here inapplicable; each number has to be examined by itself, the 
decomposition of one being no guide to the factors of the adjoining numbers. 
The user of the table has to rely implicitly on its accuracy. If the tabular 
statement be that a proposed number is divisible we can easily verify it by 
actual division ; were we to find it indivisible we should remain as ignorant of 
the factors of the proposed number as if we had had no table. And when the 
assertion is that the number is prime, we have no means of verification other 
than the tedious one of actual trial. 
Having had occasion to verify the divisibility of many hundred numbers, as 
shown in BuRcKHARDT’s table, I am desirous to bear testimony to its wondrous 
freedom from error. Only in one case have I found a mistake. In my copy 
of the work the number 854647 is marked as divisible by 7, which it is not, 
and the succeeding number 854651 is marked as prime, while it is divisible 
by 7. I have ascertained that 854647 is in reality prime. The work has been 
printed from movable, not from solid type, and, in all probability, the error 
has been caused by the withdrawal and misplacement of a type while at press. 
There is also another (unimportant) fault; in place of the title 6400, 6300 is 
printed. 
The processes above explained enable us to make a progressive list of all 
ratios expressible by numbers below a prescribed limit, and to select that por- 
tion of the progression which may be applicable to our purpose. By help of 
BurckHarpt’s table of divisors we are then able to decompose those numbers 
into their factors, and so to discover what of those ratios may be produced by 
the combination of wheels. 
Working from the exact ratio of solar to sidereal time, and extending the 
list on either side for numbers below 3 036 000, until we find a decomposable 
