_Logistic or Proportional Logarithms._--The old name for what are now called ratios or fractions are _logistic numbers_, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) = log a - log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign. The first table of this kind appeared in Kepler's work of 1624 which has been already referred to. The object of a table of log (a/x) is to facilitate the working out of proportions in which the third term is a constant quantity a. In most collections of tables of logarithms, and especially those intended for use in connexion with navigation, there occurs a small table of logistic logarithms in which a = 3600´´ (= 1° or 1^h), the table giving log 3600 - log x, and x being expressed in minutes and seconds. It is also common to find tables in which a = 10800´´ (= 3° or 3^h), and x is expressed in degrees (or hours), minutes and seconds. Such tables are generally given to 4 or 5 places. The usual practice in books seems to be to call logarithms logistic when a is 3600´´, and proportional when a has any other value. _Addition and Subtraction, or Gaussian Logarithms._--_Gaussian logarithms_ are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a ± b) by only one entry when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli in a book entitled _Supplément logarithmique_, printed at Bordeaux in the year XI. (1802/3); he calculated a table to 14 places, but only a specimen of it which appeared in the _Supplément_ was printed. The first table that was actually published is due to Gauss, and was printed in Zach's _Monatliche Correspondenz_, xxvi. 498 (1812). Corresponding to the argument log x it gives the values of log (1 + x^-1) and log (1 + x). _Dual Logarithms._--This term was used by Oliver Byrne in a series of works published between 1860 and 1870. Dual numbers and logarithms depend upon the expression of a number as a product of 1.1, 1.01, 1.001 ... or of .9, .99, .999.... In the preceding _résumé_ only those publications have been mentioned which are of historic importance or interest.[14] For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the article TABLES, MATHEMATICAL.