232 A SHORT HISTORY OF SCIENCE 



fundamental concepts : function, continuity, limit, derivative, in- 

 finitesimal, on which our modern mathematics has been built up. 

 Descartes, Newton and Leibnitz are soon to make their revolu- 

 tionary discoveries in analytic geometry and the calculus. 



We have seen that up to about 1500 the chief stages in the de- 

 velopment of mathematics have been the introduction and im- 

 provement of Arabic arithmetic for commercial purposes (though 

 accounts were kept in Roman numerals until 1550 to 1650), the 

 rediscovery of Greek geometry, and the improvement of trigo- 

 nometry in connection with its increasing use in astronomy, navi- 

 gation and military engineering. The development of science has 

 been powerfully promoted by the general intellectual emancipation 

 of the Renaissance, while mathematical progress, beginning earlier, 

 has been both a cause and a consequence of the general advance. 

 The diffusion and the preservation of scientific knowledge have 

 derived immense advantage from the new art of printing and from 

 expanding commercial intercourse. Algebra, almost helpless in 

 Greek times because, for lack of proper symbolism, expressed only 

 in geometrical or rhetorical form, has been converted by a 

 process of abbreviation, at first into a syncopated form, inter- 

 mediate between the rhetorical and our modern purely symbolic 

 notation. 



PACIOLI. The earliest printed book on arithmetic and algebra 

 was published at Venice in 1494 by Lucas Pacioli, a Franciscan 

 monk born in Tuscany about 1450. Rules are here given for the 

 fundamental operations of arithmetic, and for extracting square 

 roots. Commercial arithmetic is treated at considerable length 

 by the newer algoristic or Arabic methods. The method of arbi- 

 trary assumption corrected by proportion is used effectively, for 

 example : 



To find the original capital of a merchant who spent a quarter 

 of it in Pisa and a fifth of it in Venice, who received on these transac- 

 tions 180 ducats, and who has in hand 224 ducats. 



Assume that his original capital was 100 ducats ; then the surplus 

 would be 100 25 20 = 55, but this is ^ of his actual surplus 

 224 180, therefore his original capital was f of 100 = 80 ducats. 



