CONSTRUCTION OF THE CANON. 29
sine and the table sine (by 40), either both limits or one or other of them, since they are almost equal, as is evident from the above example. Now these, or either of them, being found, add to them the limits above noted down, or else subtract (by 8, 10, and 35), according as the given sine is less or greater than the table sine. The numbers thence produced will be near limits between which is included the logarithm of the given sine.
Example.
Let the given sine be 9999975.5000000, to which the nearest sine in the table is 9999975. 0000300, less than the given sine. By 33 the limits of the logarithm of the latter are 25,0000025 and 25.0000000. Again (by 40), the difference of the logarithms of the given sine and the table sine is .4999712. By 35, subtract this from the above limits, which are the limits of the less sine, and there will come out 24.5000313 and 24.5000288, the required limits of the logarithm of the given sine 9999975.5000000. Accordingly the actual logarithm of the sine may be placed without sensible error in either of the limits, or best of all (by 31) in 24.5000300.
Another Example.
