CIVILIZATION AND SCIENCE. 533 



Now, this I hold to be a serious error. The influence of mathe- 

 matics as an educating force is not fully exerted till the student passes 

 from these elementary studies to analytical geometry. No doubt, even 

 simple geometry and algebra accustom the mind to strict quantitative 

 reasoning, and to assuming as true nothing but axioms or demonstrated 

 propositions. But the representation of functions by curves or sur- 

 faces opens a new world of ideas, and teaches us the use of one of the 

 most fruitful methods whereby the human mind has increased its own 

 powers. What the invention of this method by Viete and Descartes 

 was to mankind, that will initiation into it still be to every mind that 

 has any turn for such studies namely, an illumination marking an 

 epoch in life. This method has its roots in the profoundest depths of 

 the human intellect, and hence is of far higher importance than the 

 most ingenious analytical processes which are applicable only to a par- 

 ticular case. True, trigonometry is analytical geometry ; as taught in 

 the gymnasia, trigonometry, like stereometry, as both these terms indi- 

 cate, has to do rather with mensuration, and its use is restricted to a 

 certain class of problems. On the other hand, between any two quan- 

 tities whatsoever, of which the one can be regarded as dependent on 

 the other, there never exists a relation so complicated but that it may 

 be represented by a curve ; of this fact Quetelet has furnished an in- 

 structive demonstration as, for example, where he represents by curves 

 criminal tendency, literary talent, etc. This mode of representing the 

 mutual dependence of things is of as much advantage to the govern- 

 ment functionary and the political economist as to the physicist and 

 the meteorologist. 



But in medicine it is indispensable. In the preface to my " Unter- 

 suchungen uber thierische Elektricitiit," which bears the date of March, 

 1848, I spoke in commendation of it as a means of bringing mathe- 

 matics to bear on physiology, even in cases where the complexity is so 

 great as to preclude the possibility of measuring, of weighing, or of 

 calculating time. I then first laid an absciss-axis in a nerve, while 

 Ludwig made the blood-circulation itself trace in curves its variations 

 of pressure, and Helmholtz made the muscle in like manner trace its 

 own contractions. Nowadays, thanks mainly to the labors of Marey, 

 there is scarcely any department of experimental physiology or pathol- 

 ogy that does not yield, through the graphical method, results of high 

 importance. But, as our students of medicine may have quit the gym- 

 nasium without ever having so much as heard of a system of coordi- 

 nates, I am compelled, at the opening of my lectures on physiologj', 

 to make my hearers acquainted with the elements of analytical 

 geometry. 



From the reasons assigned for the above-quoted decision of the min- 

 istry, whereby conic sections are excluded from the gymnasium course 

 of study, it is plain that its author was unacquainted with the general 

 scope of the branch of science he put under ban, and that he considered 



