the death of Archimedes, and had given orders to spare his

house and person.

The Romans built a splendid monument over his grave, upon which, according to his wish, was engraved a sphere inscribed in a cylinder in memory of his discovery that the inscribed sphere is two thirds of the cylinder, and that the area of the surface of the sphere equals the area of four great circles. [§§ 123, 701.]

Descartes (1596–1650) was a French philosopher as well as a great mathematician. Descartes was among the first to apply algebra to geometry. His great work was founding the science of Analytic Geometry.

Euclid, c 300 B. C. Euclid's greatest activity was in the reign of the first Ptolemy, 306–283 B. C. Euclid was a student of the Platonian philosophy, and an eminent writer on geometry. He gave to the world the first scientific text-book, "Elements," in thirteen books, which is still a standard text-book in many English schools. Ptolemy once asked Euclid if geometry could not be mastered by an easier process than by studying his "Elements." The answer was, "There is no royal road to geometry." § 225 is the 47th Prop. in Euclid's "Elements," and is often referred to as the 47th of Euclid.

Euler, 61707, d 1783, one of the greatest of modern mathematicians. He was a Swiss. He solved in three days mathematical problems which eminent mathematicians had said would require months. His intense study caused him to go blind and he dictated his "Elements of Algebra" to his servant, who was quite ignorant of mathematics. This work is still considered one of the best of its class. Besides making many discoveries in geometry, he did much work in higher mathematics and astronomy. [SS 277, 540.]

Hero, c 190 B. C. He was a practical surveyor of Alexandria and made many mechanical inventions, among them being an instrument resembling a modern theodolite. This mathematician wrote a commentary on Euclid's "Elements."

Among other formulas, he developed the one which expresses the area of a triangle in terms of its sides.

Johannes Kepler, b 1571, d 1630, was a great student of science and many publications are from his pen. He has enriched pure mathematics as well as astronomy.

He was one of the first great mathematicians to make extensive use of logarithms. He conceived the circle to be composed of an infinite number of triangles with their com mon vertices at the center and their bases in the circumference, and the sphere to consist of an infinite number of pyramids. He made a study of the ellipse, parabola, and hyperbola; he gave to the world his laws concerning the movements of heavenly bodies.

Legendre (le zhondr), b 1752, d 1833, one of the most eminent of modern mathematicians. Besides many valuable additions to higher mathematics, he gave to the world one of the most celebrated works on geometry. It is called the "Eléments de Géometrie." [$.05.]

Sir Isaac Newton (1642--1727) was a great English philosopher and mathematician. Newton was a student of geometry, but his works are on higher mathematics, and applications of algebra and geometry to the solution of astronomical problems. He discovered the Binomial Theorem.

Blaise Pascal, b 1623, d 1662, lived in Paris, where his father taught him. The father would not trust his son's education to others. Blaise Pascal's genius in geometry showed itself when he was only twelve years old. The father tried to keep mathematical work from his son till he had learned Latin and Greek, but with charcoal and paving tiles the boy studied the methods of drawing the circle and the equilateral triangle.

He discovered for himself the sum of the three angles of a triangle. Pascal's genius was so great for geometry that at sixteen he wrote a treatise on conics which had not been

equaled since the time of Archimedes. Pascal was a great student in other subjects.

Plato, who lived about 400 B..C., was the founder of a school of philosophy. Plato studied mathematics and gave the analytic method of attacking a proposition in geometry. The Platonic Bodies are so named because of the study given them in Plato's school. [§ 456]

Pythagoras, c 580-500 B. C. To Pythagoras is attributed the important theorem that the square on the hypotenuse of a right-angled triangle equals the sum of the squares on the other two sides. He probably learned the special truth when the sides are 3, 4, 5, respectively, from the Egyptians, and then developed the general truth. The Pythagorean school of mathematics taught that the plane about a point is completely filled by six equilateral triangles, four squares, or three regular hexagons. Pythagoras called the circle the most beautiful of all plane figures and the sphere the most beautiful of all solid figures. The star-shaped pentagram was used as a sign of recognition by the Pythagoreans, and was called by them Health.