PROGRESS OF MATHEMATICS AND MECHANICS 231 



beginning, Arabic numerals were known, but the mathematics 

 even of the universities hardly extended beyond the early books 

 of Euclid and the solution of simple cases of quadratic equations 

 in rhetorical form. At the end of the period the foundations 

 of modern mathematics and mechanics were securely laid. 



AIMS AND TENDENCIES OF MATHEMATICAL PROGRESS. - - In 

 the centuries just preceding, the chief applications of mathematics 

 had connected themselves with the relatively simple needs of trade, 

 accounts and the calendar, with the graphical constructions of the 

 architect and the military engineer, and with the sines and tangents 

 of the astronomer and the navigator. During the period in ques- 

 tion some of these applications became increasingly important, 

 and at the same time mathematics was more and more cultivated 

 for its own sake. Mathematicians became gradually a more and 

 more distinctly differentiated class of scholars ; mathematical text- 

 books took shape. The beginnings of this evolution have been 

 dealt with already ; its further progress is now to be traced. 



The larger achievements and tendencies of the period in mathe- 

 matical science were the following : 



In Arithmetic, decimal fractions and logarithms were introduced, 

 regulating and immensely simplifying computation; a general 

 theory of numbers was developed ; in Algebra, a compact and ade- 

 quate symbolism was worked out, including the use of the signs 

 -f, T-, X, , =, (), V , and of exponents; equations of the 

 third and fourth degree were solved, negative and imaginary roots 

 accepted, and many theorems of our modern theory of equations 

 discovered. 



In Geometry, the computation of TT was carried to many deci- 

 mals, the beginnings of projective geometry were made, and a 

 so-called method of indivisibles developed, foreshadowing the 

 integral calculus; in plane and spherical Trigonometry, the 

 theorems and processes now in use were worked out, and extensive 

 tables computed. 



In Mechanics, ideas about force and motion, equilibrium and 

 centre of gravity, were gradually clarified. 



Underlying some of these new developments are the dawning 



