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AG H. S. CARSLAW. 
progress in his tables many years before the publication of 
Napier’s canon; but he did not publish them till 1620, and 
then only in part. By that time the value of Napier’s 
discovery was already recognised, and the labours of the 
astronomer were shortened so much that, as has been said, 
the length of his life was many times multiplied. 
Biirgi’s work was entitled Arithmetische und Geo- 
metrische Progresstabulen, sambt grundlichen unterricht, 
wie solche nutzlich in allerley Rechnungen zu gebrauchen 
und verstanden werden sol.’ But the explanation promised 
in the title was not included in the book. About 1850 a 
copy of his MSS. was found in Dantzig, containing the 
missing description of the tables. From this MSS. it is now 
clear that he was very near indeed to anticipating Napier; 
for had he had the pen of a ready writer, or had he been a 
man of a different temperament, there can be little doubt 
that his tables would have seen the light earlier. 
I have already remarked that the idea from which both 
Biirgi and Napier started was the same. We see it in the 
title of Burgi’s book, Arithmetische und Geometrische 
Progresstabulen. Two progressions, an arithmetical and 
a geometrical, are calculated. Multiplication and division 
in the one correspond to addition and subtraction in the 
other. For example, consider the series 
1, 2,85 c46 jG, Tepes LD: seems (A.P.) 
2, 4, 8, 16, 32, 64, 128, 256, ...... 32768...(G.P.). 
The product of 128 and 256 in the G.P. is 32768, as this is 
the number corresponding to the sum of 7 and 8 inthe A.P. 
This idea was not new. Aristotle had used it long ago, 
and, since his time, several mathematicians had returned 
to it. But none of them had fully realised its possibilities, 
nor did any of them conceive or execute the plan of com- 
puting a pair of corresponding series, sufficiently dense, to 
be of practical use in calculation. If the index notation, 
