limit, and 100.0004950 for the less limit, between which the required logarithm of the given sine is included.
42.Hence it follows that the logarithms of all the proportionals in the Second table may be found with sufficient exactness, or may be included between known limits differing by an insensible fraction.
Thus since the logarithm of the sine 9999900, the first proportional of the Second table, was shown in the precedes, example to lie between the limits 100.0005050 and 100.0004950; necessarily (by 32) the logarithm of the second proportional will lie between the limits 200.0010100 and 200,0009900; and the logarithm of the third proportional between the limits 300.0015150 and 300.0014850, &c. And finally, the logarithm of the last sine of the Second table, namely 9995001.222927, is included between the limits 5000.0252500 and 5000.0247500. Now, having all these limits, you will be able (by 31) to find the actual logarithms.
43.To find the logarithms of sines or natural numbers not proportionals in the Second table, but near or between them; or to include them between known limits differing by an insensible fraction.
Write down the sine in the Second table nearest the given sine, whether greater or less. By 42 find the limits of the logarithm of the table sine. Then by the rule of proportion seek for a fourth proportional, which shall be to radius as the less of the given and table sines is to the greater. This may be done in one way by multiplying the less sine into radius and dividing the product by the greater. Or, in an easier way, by
