Now, this I hold to be a serious error. The influence of mathematics 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 surfaces 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 Viète 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 particular case. True, trigonometry is analytical geometry; as taught in the gymnasia, trigonometry, like stereometry, as both these terms indicate, has to do rather with mensuration, and its use is restricted to a certain class of problems. On the other hand, between any two quantities 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 instructive 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 government functionary and the political economist as to the physicist and the meteorologist.
But in medicine it is indispensable. In the preface to my "Untersuchungen über thierische Elektricität," which bears the date of March, 1848, I spoke in commendation of it as a means of bringing mathematics 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 pathology that does not yield, through the graphical method, results of high importance. But, as our students of medicine may have quit the gymnasium without ever having so much as heard of a system of coördinates, I am compelled, at the opening of my lectures on physiology, to make my hearers acquainted with the elements of analytical geometry.
From the reasons assigned for the above-quoted decision of the ministry, 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