Page:The Construction of the Wonderful Canon of Logarithms.djvu/56

32 CONSTRUCTION OF THE CANON.  in the first column of the Third table may be found with sufficient exactness, or may be included between known limits differing by an insensible fraction.

For, since (by 43) the logarithm of 9995000, the first proportional after radius in the first column of the Third table, is 5001.2485387 with no sensible error; therefore (by 32) the logarithm of the second proportional, namely 9990002.5000, will be 10002.4970774; and so of the others, proceeding up to the last in the column, namely 9900473.57808 the logarithm of which, for a like reason, will be 100024.9707740, and its limits will be 100024.9657720 and 100024.9757760.

Write down the sine in the first column of the Third table nearest the given sine, whether greater or less. By 44 seek for the limits of the logarithm of the table sine. Then, by one of the methods described in 43, seek for a fourth proportional, which shall be to radius as the less of the given and table sines is to the greater. Having found the fourth proportional, seek (by 43) for the limits of its logarithm from the Second table. When these are found, add them to the limits of the logarithm of the table sine found above, or else subtract them (by 8, 10, and 35), and the limits of the logarithm of the given sine will be brought out.

HUS, let the given sine be 9900000. The proportional sine nearest it in the first column