PROPORTIONAL LOGARITHMS, also called logistic logarithms. [Tames.) Suppose it frequently required to calculate the fourth term of a proportion of which the first term is one given quantity, say A : that is, required a fourth proportional to A, p, and q. Common logarithmic calculation here requires three inspections of the table, one addition, and one eubtraction. But if A be always the same thing, a new table may be framed, which shall only require two inspections and one addition, as follows :—Opposite to p in the table, write log A —log p instead of log p, and call the former the proportional loga rithm ofip, which must be considered as the abbreviation of " logarithm of p proper to be used in proportions of which the first term is A." The rule then is,;—to find a fourth proportional to A, p, and q, add the proportional log of p to that of q, and the sum is the proportional logarithm of the answer. For log a. — log p, and log A — log q, added
together, give log A—log/Lg. • A ' which is, by definition, the proportional logarithm of p q ÷ A, the answer required.
In tables made to be used with the old Nautical Almanac, in which the moon'e motion was given for every three hours, A was made = =10800'; and p and q were given iu the table, not in seconds, but reduced to hours, minutes, and seconds. Thus the question 3° : 1° 23. 18' : : 14. 13. : x, could be answered, and x found, by two inspections and an addition. But the convenience of this table lay much more in the arrangement into hours, minutes, and secouds, than in the nature of the substitute for the logarithm : and since a similar arrangement is now made to accompany common tables of logarithms, it may be doubted whether the day of logistic logarithms be not past.