Page:Popular Science Monthly Volume 33.djvu/540

524, whether it is employed in representing in figures the arrangement 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 phenomenon depends have been included in a formula to which the facts refuse to adapt themselves. In short, mathematics is a beacon-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 draughtsman 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 mathematics 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 period 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 history, with Hippocrates introduced the art of observing and questioning, 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 returned, 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 upon all thinking men. That is the confession which Dr. Faust