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 of years upon numeral notations before they happened to strike upon the so-called "Arabic notation." In the simple expedient of the cipher, which was introduced by the Hindoos about the fifth or sixth century after Christ, mathematics received one of the most powerful impulses. It would seem that after the "Arabic notation" was once thoroughly understood, decimal fractions would occur at once as an obvious extension of it. But "it is curious to think how much science had attempted in physical research and how deeply numbers had been pondered, before it was perceived that the all-powerful simplicity of the 'Arabic notation' was as valuable and as manageable in an infinitely descending as in an infinitely ascending progression."[28] Simple as decimal fractions appear to us, the invention of them is not the result of one mind or even of one age. They came into use by almost imperceptible degrees. The first mathematicians identified with their history did not perceive their true nature and importance, and failed to invent a suitable notation. The idea of decimal fractions makes its first appearance in methods for approximating to the square roots of numbers. Thus John of Seville, presumably in imitation of Hindoo rules, adds $$\scriptstyle{2n}$$ ciphers to the number, then finds the square root, and takes this as the numerator of a fraction whose denominator is 1 followed by $$\scriptstyle{n}$$ ciphers. The same method was followed by Cardan, but it failed to be generally adopted even by his Italian contemporaries; for otherwise it would certainly have been at least mentioned by Cataldi (died 1626) in a work devoted exclusively to the extraction of roots. Cataldi finds the square root by means of continued fractions—a method ingenious and novel, but for practical purposes inferior to Cardan's. Orontius Finaeus (died 1555) in France, and William Buckley (died about 1550) in England extracted the square root in the same way as Cardan and John of Seville.