NAPIER COMMEMORATIVE LECTURE. 47 
with which every schoolboy is now familiar, had been part 
of the mathematical apparatus of these days, the discovery 
of logarithms would not have tarried so long. But it was 
not till 1637 that Descartes introduced the modern notation 
a’, a*, a’, etc., which took the place of aa, aaa, aaaa, etc., 
and other similar, but to us, clumsy devices. Indeed the 
logarithms of Burgi and those of Napier, at any rate in 
their earliest form, were quite independent of the idea of a 
base at all. It was not till about 1750 that a systematic 
exposition of logarithms as exponents found a place in the 
text-books of algebra. 
Birgi’s two series are as follows :— 
10 x 0, 10 x], 10 x 2,...10 x m, ...(A.P.) 
8 8 1 8 1 2 8 1 n 
(be? 1° +.) 10° (1+ )",..-(G.P.) 
The terms of the A.P. he calls the red numbers, and he 
prints them in red in his tables. The terms of the G.P. he 
calls the black numbers, and they are printed in the ordinary 
type. 
* An extract from his tables is now given :— 
| 28 000 | 28 500 | 29 000 | 29 500 | 30 000 up to 31 500 
0 | 1323 11129 | 1329 74308 | 1336 40811 | 1342 10655 | 1349 83856 
Me | cs... 24362 | ...:.. 87605 | ...... BAN | cee 24086 | ...... 97355 
aay Ele 37593 | 1330 00904 | ...... 67541 | ...... 37518 | 1350 10854 
heal oe 50826 | ...... 14204 | ...... 80907 | ...... 56952 | ...... 24355 
“os 64061 | ...... 27506 | ...... 94267 | ...... 64387 | ...... 37858 
If we wish, for example, to multiply any two of the black 
numbers, we have only to look along the table to see what 
the corresponding red numbers are; these have to be added 
and the black number which corresponds to their sum read 
off. The required product will be this black number multi- 
plied by 10°. The closer the black numbers of the table 
are to each other, the more accurate an instrument does 
