There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, are still obscured by the superstitions of gamblers. According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin "probabilis") meant "approvable", and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances." However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence. The earliest known forms of probability and statistics were developed by Middle Eastern mathematicians studying cryptography between the 8th and 13th centuries. Al-Khalil (717–786) wrote the "Book of Cryptographic Messages" which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. Al-Kindi (801–873) made the earliest known use of statistical inference in his work on cryptanalysis and frequency analysis. An important contribution of Ibn Adlan (1187–1268) was on sample size for use of frequency analysis. The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes). Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's "Ars Conjectandi" (posthumous, 1713) and Abraham de Moivre's "Doctrine of Chances" (1718) treated the subject as a branch of mathematics. See Ian Hacking's "The Emergence of Probability" and James Franklin's "The Science of Conjecture" for histories of the early development of the very concept of mathematical probability. The theory of errors may be traced back to Roger Cotes's "Opera Miscellanea" (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors.