61 
NAPIER COMMEMORATIVE LECTURE. 
which the corresponding differences of the logarithms for 
these multipliers are given. 
In the second the formula 
sin 26 = 2sin Osin (5 — 4) 
is used, but it must be noticed that this formula, in his 
notation, where the sines are lines in a figure of which the 
radius is r, must be written as 
Tr 
2 
It follows that, with Napier’s logarithms, 
sin 20 = sin @ sin G — 0), 
log sin 20 + log z= log sin 8 + log sin Coe 0). 
Thus the table of logarithms of sines can be extended to 
15°. Then applying the same formula, it can be continued 
to 7°30’, and so on indefinitely. 
Further from the logarithms of the sines of angles not 
less than 45° those of angles from 22°30’ to 45° can be 
obtained. From these again the interval 11°15’ to 22°30’ 
can be filled up, andsoon. Andin this way an independent 
investigation of the logarithms of the sines from 30° to 45° 
was available. 
Part of the first page of this table, as given in the 
Descriptio, is appended. 
Gr. 0 fo fe 
min Sinus. | Logarithmi| Differentiz| Logarithmi Sinus 
0 0 | Infinitum | Infinitum 0 | 10000000 | 60 
al 2909 | 81425681 | 81425680 1 | 10000000 | 59 
2 | 5818 | 74494213 | 74494211 2 | 9999998 | 58 
3 8727 | 70439560 | 70439560 4 | 9999996] 57 
4 11636 | 67562746 | 67562739 T | 9999993 | 56 
4) 14544 | 65331315 | 65331304 LIS | 9999989) oo 
27 | 78539 | 48467431 | 48467122 309 | 9999692) 33 
28 81448 | 48103763 | 48103431 332 | 9999668} 32 
29 | 84357 | 47752859 | 47752503 356 | 9999644) 31 
30 | 87265 | 47413852 | 47413471 381 |. 9999619 | 30° 
| min 
