LOGARITHMS—D OCAGNE, 177 
which in themselves are certainly of great worth though without 
true practical usefulness, it would not be better to initiate them from 
the start in the handling of logarithms, reserving the theory for later 
explanation; this question, unless it be decided a priori, should at 
least be seriously examined. 
The great usefulness of logarithms was everywhere manifest from 
the start, particularly in trigonometric calculations involving sines, 
cosines, tangents, cotangents, etc., as in astronomy, especially navi- 
gation, so that to the tables of logarithms of numbers there were 
promptly added those of trigonometric functions. 
But it would still be underestimating the exceptional importance 
of logarithms to consider them only from this utilitarian point of 
view, however important it may be. In reality, the contribution 
of this new invention has been found to constitute, not only in the 
domain of simple calculation, but also in that of pure mathematics 
considered under the form of algebra, an acquisition of the very 
first order, the initiator of great progress, both in itself, and because 
of the unexpected generalities of which it has been the source. There 
is no need to dwell here on this side of the subject, which to mathe- 
maticians is the most captivating, but it is not fitting in this rapid 
explanation of the whole subject to leave it entirely in the shade. 
When a system of logarithms is once formulated there can evi- 
dently be deduced from it an infinite number of others by multiply- 
ing all the logarithms of the first system by the same factor, what- 
ever it may be. What characterizes each of these systems is the 
number which takes for a logarithm, unity, a number which is called 
the base of the system. The simplest system employed for all ordi- 
nary uses, and which for this reason is given the name ‘‘common,”’ is 
that of which the base is 10. But strange to say, this is not the 
system of which Napier at first dreamed; that one had for a base a 
certain incommensurable number which, like the well-known number 
x (ratio of the circumference to the diameter), belongs to the class 
of numbers which mathematicians call transcendant,! a number 
equally designed, however, for universal use by a special notation, 
and designated by the letter e. Now, and this is somewhat remarka- 
ble, it is precisely this initial system of Napier which in the domain 
of analysis plays a réle indeed fundamental; it is these Naperian 
logarithms which enter directly into the speculations of the mathe- 
-matician, giving them the name of natural logarithms, as it is the 
common logarithms which constitute the daily implement used by 
the calculator; the passage from one to the other, moreover, is 
made easier since it is again a matter of multiplication by a constant 
1 These numbers are those which no algebraical equation with integral coefficients can take for a root. 
Numbers such as -/2 are indeed incommensurable (as a result expressible only with an infinite number of 
figures), but they are not transcendant, -/2 in particular being a root of the equation 2—2=0, 
73176°—sM 1914 12 
