Essentials of Solid Geometry

271. Geometry in the East. The East did little for geometry, although contributing considerably to algebra. The first great Hindu writer was Aryabhatta, who was born in 476 A.D. He gave the very close approximation for π which we express in modern notation as 3.1416. The Arabs, about the time of the Arabian Nights tales (800 A.D.), did much for mathematics by translating the Greek authors into their own language and by bringing learning from India. Indeed, it is to the Arab mathematicians of the ninth and tenth centuries that modern Europe owes its first knowledge of the Elements of Euclid. The Arabs, however, contributed nothing of importance to geometry.

272. Geometry in Europe. In the twelfth century Euclid was translated from the Arabic into Latin, since Greek manuscripts were not then at hand, or were neglected because of ignorance of the language. The leading translators were Adelard of Bath (1120), an English monk who had learned Arabic in Spain or in Egypt; Gherardo of Cremona, an Italian monk of the twelfth century; and Johannes Campanus (about 1250), chaplain to Pope Urban IV.

In the Middle Ages in Europe nothing worthy of note was added to the geometry of the Greeks. The first Latin edition of Euclid's Elements was printed in 1482, and the first English edition in 1570.

273. Important Propositions. A few facts concerning some of the important propositions will be found of interest.

The theorem which asserts that the base angles of an isosceles triangle are equal is said to have been first proved by Thales, about 575 B.C. This theorem represented the usual limit of instruction in geometry in the Middle Ages, and probably on this account was called the pons asinorum (the bridge of fools); that is, it formed a kind of bridge across which fools could not pass. Roger Bacon,

about 1250, called it the fuga miserorum (the flight of the miserable ones) because they fled at the sight of it.

The second of the congruence theorems is also attributed to Thales, who is said to have used it in measuring the distance from the shore to a ship.

The proposition which relates to the sum of the angles of a triangle is referred to by one of the later Greek writers in these words: "The ancients investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle." It is interesting to see that we do not have to take this long method of proving this simple proposition today. It is said that one of the earlier writers, Eudemus, who lived about 335 B.C., attributed the theorem to the Pythagoreans.

Perhaps the earliest records of the Pythagorean Theorem are found in Egyptian and Chinese works which are of uncertain dates, but were apparently written before 1000 B. C., or long before Pythagoras lived. In the Chinese work the statement reads: "Square the first side and the second side and add them together; then the square root is the hypotenuse." The theorem, however, was not proved in either of these works.

274. The Three Famous Problems. The Greeks very early found three problems which they could not solve. The first was that of trisecting any given angle, the trisection problem; the second was that of constructing a square equivalent to a given circle, -the quadrature problem; and the third was that of constructing a cube that should have a volume twice that of a given cube,— the duplication problem. All three are easily solved if we allow other instruments than the ruler and compasses, but they cannot be solved by the use of these two instruments alone.

VII. IMPORTANT FORMULAS

275. Notation. The following notation is used in the formulas of plane geometry: