66 | H. S. CARSLAW. 
he contrasts these artificial numbers with the natural 
numbers or sines. In the margin, presumably by Robert 
Napier, the word logarithm is inserted in the corresponding 
place. 
To this work was also added an Appendix ‘‘ On the 
Construction of another and better kind of Logarithms, — 
namely one in which the logarithm of unity is 0.’’ This is 
the system of logarithms referred to above in the quotation 
from the preface to the Rabdologia, and it is by the hand 
of Napier himself. It contains, moreover, ‘ propositions 
for the solution of spherical triangles by an easier method; 
with notes on them and on the above-mentioned Appendix 
by the learned Henry Briggs.”’ 
Napier’s Appendix begins with the words :— 
‘‘ Among the various improvements of Logarithms, the more 
important is that which adopts a cypher as the logarithm of 
unity, and 10,000,000,000 as the logarithm of either one tenth of 
unity or ten times unity. Then, these being once fixed, the log- 
arithms of all other numbers necessarily follow.” 
He gives three methods of finding these logarithms. 
According to the first, the procedure is as follows :— 
Divide the given logarithm of one-tenth, or of ten, namely 
10,000,000,000, by 5, ten times successively, and thereby 
obtain 2,000,000,000, 400,000,000, etc., down to 1024. Also 
divide the last by 2, ten times successively, and there will 
be produced 512, 256, etc., 4, 2, 1. 
The numbers which correspond to these logarithms are 
obtained by extractions of fifth roots and square roots. 
And, when these have been obtained, the same procedure 
can be employed with regard to them, and the table 
extended indefinitely. 
It appears from this account, that Napier possessed a 
method of extracting fifth roots. Briggs, to whom the 
