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4.1 POSITIONAL NUMBER SYSTEMS 185 binary system of arithmetic and metrology. I know I have nature on my side; if I do not succeed to impress upon you its utility and great importance to mankind, it will reflect that much less credit upon our generation, upon our scientific men and philosophers. Nystrom devised special means for pronouncing hexadecimal numbers; e.g., (B0160)1e, was to be read ?ybong, bysanton. His entire system was called the Tonal System, and it is described in J. Franklin Inst. 46 (1863), 263-275, 337-348, 402-407. A similar system, but using radix 8, was worked out by Alfred B. Taylor [Proc. Amer. Pharmaceutical Assoc. 8 (1859), 115-216; Proc. Amer. Philosophical Sot. 24 (1887), 296-3661. Increased use of the French (metric) system of weights and measures prompted extensive debate about the merits of decimal arithmetic during that era; indeed, octal arithmetic was even being proposed in Prance [J. D. Colenne, Le systkme octaval (Paris: 1845); Aim6 Mariage, Numeration par huit (Paris: Le Nonnant, 1857)]. The binary system was well known as a curiosity ever since Leibniz s time, and about 20 early references to it have been compiled by R. C. Archibald [A&&f 25 (1918), 139-1421. It was applied chiefly to the calculation of powers, as explained in Section 4.6.3, and to the analysis of certain games and puzzles. Giuseppe Peano [Atti della R. Accademia delle Scienze di Torino 34 (1898), 47- 55] used binary notation as the basis of a logical character set of 256 symbols. Joseph Bowden [Special Topics in Theoretical Arithmetic (Garden City: 1936), 491 gave his own system of nomenclature for hexadecimal numbers. The book History of Binary and Other Nondecimal Numeration by Anton Glaser (privately printed, 1971) contains an informative and nearly complete discussion of the development of binary notation, including English translations of many of the works cited above. Much of the recent history of number systems is connected with the devel- opment of calculating machines. Charles Babbage s notebooks for 1838 show that he considered using nondecimal numbers in his Analytical Engine [cf. M. V. Wilkes, Historia Math. 4 (1977), 4211. Increased interest in mechanical devices for arithmetic, especially for multiplication, led several people in the 1930s to consider the binary system for this purpose. A particularly delightful account of such activity appears in the article Binary Calculation by E. William Phillips [Journal of the Institute of Actuaries 67 (1936), 187-2211 together with a record of the discussion that followed a lecture he gave on the subject. Phillips began by saying, The ultimate aim [of this paper] is to persuade the whole civilized world to abandon decimal numeration and to use octonal [i.e., radix 8] numeration in its place. Modern readers of Phillips s article will perhaps be surprised to discover that a radix-8 number system was properly referred to as octonary or octonal, according to all dictionaries of the English language at that time, just as the radix-10 number system is properly called either denary or decimal ; the word octal did not appear in English language dictionaries until 1961, and it apparently originated as a term for the base of a certain class of vacuum tubes. The word Lhexadecimal, which has crept into our language even more recently, is a mixture of Greek and Latin stems; more proper terms would be