The construction of the wonderful canon of logarithms
The construction of the wonderful canon of logarithms
The construction of the wonderful canon of logarithms
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1<br />
CONSTRIJCTION OF THE CaNON, 3<br />
multiplying <strong>the</strong> difference <strong>of</strong> <strong>the</strong> sines into radius,<br />
dividing this product by <strong>the</strong> greater sine, and subtracting<br />
<strong>the</strong> quotient from radius.<br />
Now since (by 36) <strong>the</strong> logarithm <strong>of</strong> <strong>the</strong> fourth<br />
proportional differs from <strong>the</strong> logarithm <strong>of</strong> radius<br />
by as much as <strong>the</strong> <strong>logarithms</strong> <strong>of</strong> <strong>the</strong> given and<br />
table sines differ from each o<strong>the</strong>r ; also, since (by<br />
34) <strong>the</strong> former difference is <strong>the</strong> same as <strong>the</strong> logarithm<br />
<strong>of</strong> <strong>the</strong> fourth proportional itself; <strong>the</strong>refore<br />
(by 41) seek for <strong>the</strong> limits <strong>of</strong> <strong>the</strong> logarithm <strong>of</strong> <strong>the</strong><br />
fourth proportional by aid <strong>of</strong> <strong>the</strong> First table ; when<br />
found add <strong>the</strong>m to <strong>the</strong> limits <strong>of</strong> <strong>the</strong> logarithm <strong>of</strong><br />
<strong>the</strong> table sine, or else subtract <strong>the</strong>m (by 8, 10, and<br />
35), according as <strong>the</strong> table sine is greater or less<br />
than <strong>the</strong> given sine ; and <strong>the</strong>re will be brought<br />
out <strong>the</strong> limits <strong>of</strong> <strong>the</strong> logarithm <strong>of</strong> <strong>the</strong> given sine.<br />
THUS,<br />
Example.<br />
let <strong>the</strong> given sine be 9995000.000000.<br />
To this <strong>the</strong> nearest sine in <strong>the</strong> Second table<br />
is 9995001.222927, and (by 42) <strong>the</strong> limits <strong>of</strong> its<br />
logarithm are 5000.0252500 and 5000.0247500.<br />
Now seek for <strong>the</strong> fourth proportional by ei<strong>the</strong>r <strong>of</strong><br />
<strong>the</strong> methods above described ; it will be 9999998.<br />
7764614, and <strong>the</strong> limits <strong>of</strong> its logarithm found (by<br />
41) from <strong>the</strong> First table will be 1.2235387 and<br />
1.2235386. Add <strong>the</strong>se limits to <strong>the</strong> former (by 8<br />
and 35), and <strong>the</strong>re will come out 5001.2487888 and<br />
5001.2482886 as <strong>the</strong> limits <strong>of</strong> <strong>the</strong> logarithm <strong>of</strong> <strong>the</strong><br />
given sine. Whence <strong>the</strong> number 5001.2485387,<br />
midway between <strong>the</strong>m, is (by 31) taken most<br />
suitably, and with no sensible error, for <strong>the</strong> actual<br />
logarithm <strong>of</strong> <strong>the</strong> given sine 9995000.<br />
44, Hence itfollows that <strong>the</strong> <strong>logarithms</strong> <strong>of</strong> all <strong>the</strong> propor-<br />
D 4 tionals