Page:A History of Mathematics (1893).djvu/179

The invention of decimals is frequently attributed to Regiomontanus, on the ground that instead of placing the sinus totus, in trigonometry, equal to a multiple of 60, like the Greeks, he put it $\scriptstyle {=100,000}$. But here the trigonometrical lines were expressed in integers, and not in fractions. Though he adopted a decimal division of the radius, he and his successors did not apply the idea outside of trigonometry and, indeed, had no notion whatever of decimal fractions. To Simon Stevin of Bruges in Belgium (1548–1620), a man who did a great deal of work in most diverse fields of science, we owe the first systematic treatment of decimal fractions. In his La Disme (1585) he describes in very express terms the advantages, not only of decimal fractions, but also of the decimal division in systems of weights and measures. Stevin applied the new fractions "to all the operations of ordinary arithmetic."^[25] What he lacked was a suitable notation. In place of our decimal point, he used a cipher; to each place in the fraction was attached the corresponding index. Thus, in his notation, the number 5.912 would be $\scriptstyle {{\overset {\mathsf {0}}{5}}{\overset {\mathsf {1}}{9}}{\overset {\mathsf {2}}{1}}{\overset {\mathsf {3}}{2}}}$ or $\scriptstyle {5\bigcirc \!\!\!\!{\mathsf {0}}\,9\bigcirc \!\!\!\!{\mathsf {1}}\,1\bigcirc \!\!\!\!{\mathsf {2}}\,2\bigcirc \!\!\!\!{\mathsf {3}}\,}$. These indices, though cumbrous in practice, are of interest, because they are the germ of an important innovation. To Stevin belongs the honour of inventing our present mode of designating powers and also of introducing fractional exponents into algebra. Strictly speaking, this had been done much earlier by Oresme, but it remained wholly unnoticed. Not even Stevin's innovations were immediately appreciated or at once accepted, but, unlike Oresme's, they remained a secure possession. No improvement was made in the notation of decimals till the beginning of the seventeenth century. After Stevin, decimals were used by Joost Bürgi, a Swiss by birth, who prepared a manuscript on arithmetic soon after 1592, and by Johann Hartmann Beyer, who assumes the invention as his own. In 1603, he published at Frankfurt on the Main a Logistica