became conspicuous for numerical skill. The first important arithmetical work of English authorship was published in Latin in 1522 by Cuthbert Tonstall (1474–1559). He had
studied at Oxford, Cambridge, and Padua, and drew freely from the works of Pacioli and Regiomontanus. Reprints of
his arithmetic appeared in England and France. After
Recorde the higher branches of mathematics began to be studied. Later, Scotland brought forth Napier, the inventor of logarithms. The instantaneous appreciation of their value
is doubtless the result of superiority in calculation. In Italy, and especially in France, geometry, which for a long time had
been an almost stationary science, began to be studied with success. Galileo, Torricelli, Roberval, Fermat, Desargues, Pascal, Descartes, and the English Wallis are the great revolutioners of this science. Theoretical mechanics began to be
studied. The foundations were laid by Fermat and Pascal for the theory of numbers and the theory of probability.
We shall first consider the improvements made in the art of calculating. The nations of antiquity experimented thousands
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 ciphers to the number, then finds the square root, and takes this
as the numerator of a fraction whose denominator is 1 followed by 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