Page:The Construction of the Wonderful Canon of Logarithms.djvu/55

CONSTRUCTION OF THE CANON. 31 multiplying the difference of the sines into radius, dividing this product by the greater sine, and sub- tracting the quotient from radius.

Now since (by 36) the logarithm of the fourth proportional differs from the logarithm of radius by as much as the logarithms of the given and table sines differ from each other; also, since (by 34) the former difference is the same as the loga- rithm of the fourth proportional itself; therefore (by 41) seek for the limits of the logarithm of the fourth proportional by aid of the First table; when found add them to the limits of the logarithm of the table sine, or else subtract them (by 8, 10, and 35), according as the table sine is greater or less than the given sine; and there will be brought out the limits of the logarithm of the given sine.

HUS, let the given sine be 9995000.000000, To this the nearest sine in the Second table is 9995001.222927, and (by 42) the limits of its logarithm are 5000.0252500 and 5000.0247500. Now seek for the fourth proportional by either of the methods above described; it will be 9999998. 7764614, and the limits of its logarithm found (by 41) from the First table will be 1.2235387 and 1.2235386. Add these limits to the former (by 8 and 35), and there will come out 5001.2487888 and 5001.2482886 as the limits of the logarithm of the given sine. Whence the number 5001.2485387, midway between them, is (by 31) taken most suitably, and with no sensible error, for the actual logarithm of the given sine 9995000.

44.Hence it follows that the logarithms of all the tionals