CHAPTER XIII 

 BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 



.... All the sciences which have for their end investigations 

 concerning order and measure, are related to mathematics, it being 

 of small importance whether this measure be sought in numbers, forms, 

 stars, sounds, or any other object ; that, accordingly, there ought to 

 exist a general science which should explain all that can be known about 

 order and measure, considered independently of any application to a 

 particular subject, and that, indeed, this science has its own proper 

 name, consecrated by long usage, to wit, mathematics. And a proof 

 that it far surpasses in facility and importance the sciences which de- 

 pend upon it is that it embraces at once all the objects to which these 

 are devoted and a great many others besides. ... Descartes. 



As long as algebra and geometry proceeded along separate paths, 

 their advance was slow and their applications limited. But when 

 these sciences joined company, they drew from each other fresh vitality 

 and thenceforward marched on at a rapid pace toward perfection. 



Lagrange. 



The application of algebra has far more than any of his meta- 

 physical speculations, immortalized the name of Descartes, and con- 

 stitutes the greatest single step ever made in the progress of the 

 exact sciences. Mill. 



The idea of coordinates which forms the indispensable scheme for 

 making all processes visible, with its many-sided and stimulating 

 applications in all branches of daily life, whether medicine, physi- 

 cal geography, political economy, statistics, insurance, the technical 

 sciences the first beginnings of the calculus in their historical 

 evolution, the development of the ideas of function and limit in 

 connection with the elementary theory of curves, these are things 

 without which in the present day not the slightest comprehension of 

 the phenomena of nature can be attained, of which, however, the 

 knowledge enables us as by magic to gain an insight with which in 

 depth and range, but above all in certainty, scarcely any other can be 

 compared. Voss. 



How many celebrate the names of Newton and Leibnitz ! How few 

 have a real appreciation of that which these men have created of 

 permanent value ! Here lie the roots of our present-day knowledge, 

 here the true continuation of the strivings of antique wisdom. 



Lindemann. 

 T 273 



