Other famous geometers could also be named, but such men as Euclid, Archimedes, and Apollonius are enough to show that geometry was cultivated to a great extent by the philosophers of antiquity.  It progressively advanced, like philosophy itself, from the time of Thales until it had reached the perfection of which it was capable, when it became merged into astronomical science.  It was cultivated more particularly by the disciples of Plato, who placed over his school this inscription:  “Let no one ignorant of geometry enter here.”  He believed that the laws by which the universe is governed are in accordance with the doctrines of mathematics.  The same opinion was shared by Pythagoras, the great founder of the science, whose main formula was that number is the essence or first principle of all things.  No thinkers ever surpassed the Greeks in originality and profundity; and mathematics, being highly prized by them, were carried to the greatest perfection their method would allow.  They did not understand algebra, by the application of which to geometry modern mathematicians have climbed to greater heights than the ancients; but then it is all the more remarkable that without the aid of algebraic analysis they were able to solve such difficult problems as occupied the minds of Archimedes and Apollonius.  No positive science can boast of such rapid development as geometry for two or three hundred years before Christ, and never was the intellect of man more severely tasked than by the ancient mathematicians.

No empirical science can be carried to perfection by any one nation or in any particular epoch; it can only expand with the progressive developments of the human race itself.  Nevertheless, in that science which for three thousand years has been held in the greatest honor, and which is one of the three great liberal professions of our modern times, the ancients, especially the Greeks, made considerable advance.  The science of medicine, having in view the amelioration of human misery and the prolongation of life itself, was very early cultivated.  It was, indeed, in old times another word for physics,—­the science of Nature,—­and the physician was the observer and expounder of physics.  The physician was supposed to be acquainted with the secrets of Nature,—­that is, the knowledge of drugs, of poisons, of antidotes to them, and the way to administer them.  He was also supposed to know the process of preserving the body after death.  Thus Joseph, seventeen hundred years before the birth of Christ, commanded his physician to embalm the body of his father; and the process of embalming was probably known to the Egyptians before the period when history begins.  Helen, of Trojan fame, put into wine a drug that “frees man from grief and anger, and causes oblivion of all ills.”  Solomon was a great botanist,—­a realm with which the science of medicine is indissolubly connected.  The origin of Hindu medicine is lost in remote antiquity.  The Ayur Veda, written nine hundred years before Hippocrates was born, sums up the knowledge of previous periods relating to obstetric surgery, to general pathology, to the treatment of insanity, to infantile diseases, to toxicology, to personal hygiene, and to diseases of the generative functions.