178 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1914. 
factor, the modulus (a multiplication itself effected by means of 
logarithms). 
But it is worthy of mention that Napier himself, after having pro- 
duced his initial system of logarithms which we call Naperian, noted 
the utility, from a practical point of view, of adopting the common 
logarithm with the base 10. Death prevented him, however, from 
realizing this reform, of which he had merely confided the plan to 
his son, and it is to his friend, Henri Briggs, professor at Gresham 
College in London, that should be given the credit for making known 
for the first time in 1624 in his Arithmetica Logarithmica, these new 
logarithms whose use has become universal. 
Except those of the exact powers of 10, as 100, 1,000, etc., the 
common logarithms of all whole numbers are incommensurable 
numbers, that is, numbers represented decimally by an infinite suc- 
cession of figures. In order by their means, to obtain results of cal- 
culations closer and closer alike, it is necessary to obtain the loga- 
rithms themselves with a greater and greater number of decimals. 
It was with 14 decimals that Briggs had the patience himself to 
calculate the logarithms of whole numbers from 1 to 20,000, and 
from 90,000 to 100,000. The lacuna of from 20,000 to 90,000 was 
filled in in 1628 by the Dutch mathematician Vlacq in the second 
edition of the Arithmetica Logarithmica, published in Gouda. It 
was in the same year and in the same city that were given by the 
Englishman Gellibrand, in his Trigonometria Britannica, the first 
tables of logarithms of trigonometrical functions. It was at this 
date that logarithms were introduced into the ordinary usage of 
calculators. 
These early tables have constituted a treasure house from which 
the majority of subsequent editors have borrowed, those who, accord- 
ing to the degree of approximation which they had in view, have 
merely had to extract from them selected tables more restricted in 
decimals (falling to 5 and even to 4), in order to introduce improve- 
ments in general arrangement, the classification of the numbers, 
the readability, and finally their accuracy, for it must indeed be 
admitted that some few errors had crept into the first tables of 
Vlacq, but these mistakes have had at least the great advantage of 
establishing the character of those mere copies which otherwise 
might have been taken for originals. Such was the case with certain 
tables recovered in China in the course of the nineteenth century, 
to which the Mandarins, no doubt in good faith, had attributed 
great antiquity, but the presence therein of mistakes characteristic 
of Vlacq removed any doubt as to their real origin, and thus prevented 
Napier from being despoiled of his fame to the profit of some legend- 
ary figure rising suddenly from the depths of the history of the Celes- 
tial Empire. 
