TRACT 21.

LOGARITHMS.

345

more than easy subtractions; which proportionals nearly coin-cide with all the natural sines from 90* down to 30°.

To obtain the logarithms of all those proportionals, he de-monstrates several properties and relations of the numbersand logarithms, and illustrates the manner of applying them.The principal of these properties areas follow : 1st, that thelogarithm of any sine is greater than the difference betweenthat sine and the radius, but less than the said difference whenincreased in the proportion of the sine to radius* ; and 2dly,that the difference between the logarithms of two sines, is lessthan the difference of the sines increased in the proportion ofthe less sine to radius, but greater than the said difference ofthe sines increased in the proportion of the greater sine toradius f.

Hence, by the 1st theorem, the logarithm of 10000000, theradius or first term in the first table, being 0, the logarithmof 9999999, the 2d term, will be between 1 and T0000001,and will therefore be equal to T00000005 very nearly?; andthis will be also the common difference of all the terms orproportionals in the first table; therefore, by the continualaddition of this logarithm, there will be obtained the loga-rithms of all these 100 proportionals; consequently? 100 timesthe said first logarithm, or the last of the above sums, will

* By this first theorem, t being radius, the logarithm of the sine s is between

r—s and -—-r; and therefore, when s differs but little from r, the logarithm of ts

will be nearly equal to

the arithmetical mean between tire limits

r—s and -—-r; but still nearer to (r—s)*/— or- A /">thegeometricalmean

S S J

between the said limits.

By this second theorem, the difference between the logarithms of the two

S— s S —s

sines S and j, lying between the limits - r and —— r, will, when those sines

s S

differ but little, be nearly equal to or — * ■ ' V* ^ —- r, their arithmetic

,1L

2Sr

cal mean ; or nearly -~r, the geometrical mean ; or nearly = -— -2 r> by sub-V bj S + s

stituting in the last denominator, \ (S + i) for ^Sj, to which it is nearly equal.