524 . THE POPULAR SCIENCE MONTHLY. 



sions, wliether it is employed in representing in figures the ar- 

 rangement of the leaves of a plant, or in formulating the law of 

 gravitation, or the law of the enfeeblement of sound and light in 

 the ratio of the square of the distance. It is mathematics that 

 points out perturbations, and imposes limits upon physical laws. 

 It is mathematics that serves to direct the intelligence, whether 

 by showing it how to correct an error of observation, such as 

 Newton had to contend against when he was inquiring if the law 

 of gravitation was applicable to the motions of the moon, or by 

 helping it to ascertain that not all the causes on which a phe- 

 nomenon depends have been included in a formula to which the 

 facts refuse to adapt themselves. In short, mathematics is a bea- 

 con-light and a means of verification. It inspires all the more 

 confidence because it is the only science that has never had to 

 change its direction, from Euclid to Galileo, from Newton and 

 Huygens to Laplace and Lagrange. 



It gives form as well as foundation to knowledge. No draughts- 

 man can hold a contest with graphic geometry in any question of 

 figuring the relations of different phenomena that are functions 

 of one another. 



Mathematics is the draughtsman of thought. The beauty of the 

 formulas by which it has been the prophetic guide for the other 

 sciences can not be forgotten ; as in the discovery, for example, of 

 the series of homologous compounds by which chemistry has been 

 so greatly enriched. This science, which has produced so quick a 

 revolution in the conscience, force, and art of life, owes to mathe- 

 matics the concept of valences, and consequently the knowledge 

 of the mechanism of substitutions, the variations of which are so 

 infinite as to belie the maxim, " There is nothing new under the 

 sun." 



It is mathematics that renders the honors of positive knowledge 

 to ancient times, in that j)eriod which, besides having become 

 classic for its art, also laid the foundations of science, and made 

 the Greeks masters of the true as well as of the beautiful ; whether 

 with Pythagoras, Euclid, and Archimedes it established the bases 

 of geometry and mechanics, with Aristotle founded natural his- 

 tory, with Hippocrates introduced the art of observing and ques- 

 tioning, or with Plato made the method of discussion an art of 

 reflecting ideas in the mirror of facts. 



Hence, the person who sees only labor lost in philosophical 

 researches, is in error. Philosophy (I am speaking of speculative 

 philosophy) has not gone on a journey from which it has not re- 

 turned, but, having traversed heaven and earth with vigorous 

 will, has come back to tell that it has not succeeded with its 

 a 'priori theories in solving the problems which impose themselves 

 uj^on all thinking men. That is the confession which Dr. Faust 



