The theory of probability, though undoubtedly true, requires very careful application. Not only is it a branch of mathematics in which oversights are frequently committed, but it is a matter of great difficulty in many cases, to be sure that the formula correctly represents the data of the problem. These difficulties often arise from the logical complexity of the conditions, which might be, perhaps, to some extent cleared up by constantly bearing in mind the system of combinations as developed in the Indirect Logical Method. In the study of probabilities, mathematicians had unconsciously employed logical processes far in advance of those in possession of logicians, and the Indirect Method is but the full statement of these processes. It is very curious how often the most acute and powerful intellects have gone astray in the calculation of probabilities. Seldom was Pascal mistaken, yet he inaugurated the science with a mistaken solution.[121] Leibnitz fell into the extraordinary blunder of thinking that the number twelve was as probable a result in the throwing of two dice as the number eleven.[122] In not a few cases the false solution first obtained seems more plausible to the present day than the correct one since demonstrated. James Bernoulli candidly records two false solutions of a problem which he at first thought self-evident; and he adds a warning against the risk of error, especially when we attempt to reason on this subject without a rigid adherence to methodical rules and symbols. Montmort was not free from similar mistakes. D’Alembert constantly fell into blunders, and could not perceive, for instance, that the probabilities would be the same when coins are thrown successively as when thrown simultaneously. Some men of great reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste Comte,[123] Poinsot, and J. S. Mill,[124] have so far misapprehended the theory, as to question its value or even to dispute its validity. The erroneous statements about the theory given in the earlier editions of Mill’s *System of Logic* were partially withdrawn in the later editions. [121] Montucla, *Histoire des Mathématiques*, vol. iii. p. 386. [122] Leibnitz *Opera*, Dutens’ Edition, vol. vi. part i. p. 217. Todhunter’s *History of the Theory of Probability*, p. 48. To the latter work I am indebted for many of the statements in the text. [123] *Positive Philosophy*, translated by Martineau, vol. ii. p. 120. [124] *System of Logic*, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61.