builders. A class of workers existed among them called "rope-stretchers," whose business was the marking out of the foundations of buildings. These men knew how to bisect an angle and also to construct a right angle. The latter was probably done by a method essentially the same as forming a right triangle whose sides are three, four and five units of length. Ahmes, in his treatise, has various constructions of the isosceles trapezoid from different data.

Thales (600 B. C.) enunciated the following theorems: If two straight lines intersect, the opposite or vertical angles are equal;

The angles at the base of an isosceles triangle are equal; Two triangles are equal if two sides and the included angle of one are equal to two sides and the included angle of the other;

The sum of the angles of a triangle equals two right angles;

Two mutually equiangular triangles are similar.

Thales used the last of these theorems to measure the height of the great pyramid by measuring the length of the shadow cast by the pyramid and also measuring the length of the shadow of a post of known height at the same time and making a proportion between these quantities.

Pythagoras (525 B. C.) and his followers discovered correct formulas for the areas of the principal rectilinear figures, and also discovered the theorems that the areas of similar polygons are as the squares of their homologous sides, and that the square on the hypotenuse of a right triangle equals the sum of the squares on the other two sides. The latter is called the Pythagorean theorem. They also discovered how to construct a square equivalent to a given parallelogram, and to divide a given line in mean and extreme ratio.

To Eudoxus (380 B. C.) we owe the general theory of proportion in geometry, and the treatment of incommensurable quantities by the method of Exhaustions. By the use of these he obtained such theorems as that the areas of two circles are to each other as the squares of their radii, or of their diameters.

In the writings of Hero (Alexandria, 125 B. C.) we first find the formula for the area of a triangle in terms of its sides, K-Vs(s—a) (s—b) (s--c). Hero also was the first to place land-surveying on a scientific basis.

=

It is a curious fact that Hero at the same time gives an incorrect formula for the area of a triangle, viz., K=1a(b+c), this formula being apparently derived from Egyptian sources.

Xenodorus (150 B. C.) investigated isoperemetrical figures.

The Romans, though they excelled in engineering, apparently did not appreciate the value of the Greek geometry. Even after they became acquainted with it, they continued to use antiquated and inaccurate formulas for areas, some of them of obscure origin. Thus, they used the Egyptian formula for the area of a quadrilateral, a+bc+d

K atbed. They determined the area of an equilat

= 2

2

eral triangle whose side is a, by different formulas, all

13a2
30

incorrect, as K= K=(a2+a), and Ka2.

853. The Circle. Thales enunciated the theorem that every diameter bisects a circle, and proved the theorem that an angle inscribed in a semicircle is a right angle.

To Hippocrates (420 B. C.) is due the discovery of nearly all the other principal properties of the circle given in this book.

The Egyptians regarded the area of the circle as equivalent to of the diameter squared, which would make π= 3.1604.

The Jews and Babylonians treated as equal to 3.

Archimedes, by the use of inscribed and circumscribed regular polygons, showed that the true value of lies between 3 and 31; that is, between 3.14285 and 3.1408.

The Hindoo writers assign various values to л, as 3, 3, √10, and Aryabhatta (530 A. D.) gives the correct approximation, 3.1416. The Hindoos used the formula √2—√4—AB2 (See Art. 468) in computing the numerical value of л.

Within recent times, the value of 7 has been computed to 707 decimal places.

The use of the symbol for the ratio of the circumference of a circle to the diameter was established in mathematics by Euler (Germany, 1750).

HISTORY OF GEOMETRIC TRUTHS. SOLID GEOMETRY

854. Polyhedrons. The Egyptians computed the volumes of solid figures from the linear dimensions of such figures. Thus, Ahmes computes the contents of an Egyptian barn by methods which are equivalent to the use

3c

of the formula V=axb. As the shape of these barns is not known, it is not possible to say whether this formula is correct or not.

Pythagoras discovered, or knew, all the regular polyhedrons except the dodecahedron. These polyhedrons were supposed to have various magical or mystical properties. Hence the study of them was made very prominent.

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Hippasus (470 B. C.) discovered the dodecahedron, but he was drowned by the other Pythagoreans for boasting of the discovery.

Eudoxus (380 B. C.) showed that the volume of a pyramid is equivalent to one-third the product of its base by its altitude.

E. F. August (Germany, 1849) introduced the prismatoid formula into geometry and showed its importance.

855. The Three Round Bodies. Eudoxus showed that the volume of a cone is equivalent to one-third the area of its base by its altitude.

Archimedes discovered the formulas for the surface and volume of the sphere.

Menelaus (100 A. D.) treated of the properties of spherical triangles.

Gerard (Holland, 1620) invented polar triangles and found the formulas for the area of a spherical triangle and of a spherical polygon.

856. Non-Euclidean Geometry. The idea that a space might exist having different properties from those which we regard as belonging to the space in which we live, has occurred to different thinkers at different times, but Lobatchewsky (Russia, 1793-1856) was the first to make systematic use of this principle. He found that if, instead of taking Geom. Ax. 2 as true, we suppose that through a given point in a plane several straight lines may be drawn parallel to a given line, the result is not a series of absurdities or a general reductio ad absurdum; but, on the contrary, a consistent series of theorems is obtained giving the properties of a space.

III. REVIEW EXERCISES

EXERCISES. CROUP 86

REVIEW EXERCISES IN PLANE GEOMETRY

Ex. 1. If the bisectors of two adjacent angles are perpendicular to each other, the angles are supplementary.

Ex. 2. If a diagonal of a quadrilateral bisects two of its angles, the diagonal bisects the quadrilateral.

Ex. 3. Through a given point draw a secant at a given distance from the center of a given circle.

Ex. 4. The bisector of one angle of a triangle and of an exterior angle at another vertex form an angle which is equal to one-half the third angle of the triangle.

Ex. 5. The side of a square is 18 in. Find the circumference of the inscribed and circumscribed circles.

Ex. 6. The quadrilateral ADBC is inscribed in a circle. The diagonals AB and DC intersect in the point F. Arc AD = 112°, arc AC 108°, LAFC=74°. Find all the other angles of the figure.

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Ex. 7. Find the locus of the center of a circle which touches two given equal circles.

Ex. 8. Find the area of a triangle whose sides are 1 m., 17 dm., 210 cm.

Ex. 9. The line joining the midpoints of two radii is perpendicular to the line bisecting their angle.

Ex. 10. If a quadrilateral be inscribed in a circle and its diagonals drawn, how many pairs of similar triangles are formed?

Ex. 11. Prove that the sum of the exterior angles of a polygon (Art. 172) equals four right angles, by the use of a figure formed by drawing lines from a point within a polygon to the vertices of the polygon.