CONSTRUCTION OF
TRACT 21.
364
the like process is repeated for the proportion between thenumbers lOOOand 500, from which arises 69314.7193 for thelogarithm of 500 ; which he also calls the logarithm of dupli-cation, being the measure of the proportion of 2 to 1.
Then from the foregoing he derives all the other logarithmsin the chiliad, beginning with those of the prime numbers 1,2, 3, 5, 7, &c, in the first 100. And first, since 1024, 512,256, 128, 64, 32, 16, 8, 4, 2, 1, are all in the continued pro-portion of 1000 to 500, therefore the proportion of 1024 to1 is decuple of the proportion of 1000 to 500, and conse-quently the logarithm of 1 would be decuple of the logarithmof 500, if 0 were taken as the logarithm of 1024; but since thelogarithm of 1024 is applied negatively, thq logarithm of 1must he diminished by as much: diminishing therefore 10times the log. of 500, which is 693147.1928, by 2371.6526,the remainder 690775.5422 is the logarithm of 1 , or of 100 . 00 ,
Nos.
100
230258.5141
10
460517.0282
1
690775.5422
.1
921034.0563
.01
1151292.5703
.001
1381551.0844
0001
1611809.5985
which is set down in the table.
And because 1, 10, 100, 1000, arecontinued proportionals, thereforethe proportion of 1000 to 1 is tripleof the proportion of 1000 to 100,andconsequently $ of the logarithm of 1is to be set for the logarithm of 100,viz. 230258.5141, and this is also thelogarithm of decuplication, or of theproportion of 10 to 1. And hence,
multiplying this logarithm of 100 successively by 2, 3, 4,5, 6,and 7, there arise the logarithms to the numbers in the de-cuple proportion, as in the margin.
Also if the logarithm of dupli-cation, or of the proportion of 2to 1, lie taken from the logarithmof 1, there will remain the loga-rithm of 2; and from the logarithmof 2 taking the logarithm of 10,there remains the logarithm ofthe proportion of 5 to 1; which
Log. of 1
690775.5422
of 2 to 1
69314.7193
log. of 2
621460.8229
log. of 10
460517.0281
of 5 to 1
160943.7948
log. c.f 5
,529831.7474