An Elementary Treatise on Plane and Solid Geometry

Perpendicular to the diameter is a mean proportional between

its adjacent segments (185, 186),

Chords and secants of a circle drawn through a point are divid-
ed in reciprocal proportion (189, 190),

Tangent drawn to a circle through a point is a mean proportional
between the parts of the secant drawn through it (191),

To divide a line in extreme and mean ratio (192),

Case of similar polygons (193), .

To construct a polygon similar to a given one (194),

Case in which similar polygons are equal (195),

Ratio of the perimeters of similar polygons (197),

Ratio of the altitudes of similar triangles, parallelograms, and

trapezoids (198, 199); of their perimeters (200),

CHAPTER XII.

REGULAR POLYGONS,

Infinitely small arc same as its chord (203),

Circle a regular polygon of an infinite number of sides (204),

Infinitely small quantities which are neglected are infinitely

smaller than those which are retained (205), .

Regular polygons of the same number of sides are similar (206),

Ratio of the perimeters of regular polygons (207, 233),

Regular polygon may be inscribed (209),

To inscribe a regular polygon of double the number of sides of

a given inscribed regular polygon (210, 211), .

To inscribe a square (212); a polygon of 8, 16, 32, &c. sides

(213); a hexagon (214); a polygon of 12, 24, 48, &c. sides

(215); an equilateral triangle (216); a decagon (217); a poly.

gon of 20, 40, 80, &c. sides (218); a pentagon (219); a poly-

gon of 15 sides (220); of 30, 60, 120, &c. sides (221);

To circumscribe a circle about a regular polygon (222),

The centre of the regular polygon (223), .

To inscribe a regular polygon similar to a given one (227),

Sides of a regular polygon equally distant from its centre (228),

To inscribe a circle in a polygon (229),

To circumscribe about a circle a polygon similar to a given in.

scribed one (230, 231), .

Ratio of the homologous sides of similar regular polygons (232,
233),

Ratio of the circumferences of circles (234-238),

Ratio of circumference to diameter or л (237, 238),

CHAPTER XIII.

AREAS,

Equivalent figures, area (239); unit of surface (240),

Ratio of rectangles (241, 244),

Area of rectangle (242); of square (243),

Equivalent parallelograms (245, 246),

Area of parallelogram (247),

Ratio of parallelograms (248),

Ratio of parallelogram and triangle (249),

Equivalent triangles (250),

Area of triangle (251),

Ratio of triangles (252),

Area of trapezoid (253, 255),

Line joining middle points of sides of trapezoid (254),

Square upon the hypothen use of a right triangle equivalent to

the sum of squares upon its legs (256),

Square upon the diagonal of a square (258),

Ratios between the squares of hypothenuse and legs of right tri-

angle (259, 260),

To make a square equivalent to sum or difference of squares
(261-264),

To make a square in a given ratio to a given square (265),

Ratio of similar triangles (266, 267),

Ratio of circles (271),

To make a polygon or circle equivalent to the sum or to the dif-
ference of given similar polygons or circles (272-274),

To make a polygon or circle in a given ratio to a similar poly-

gon or circle (275, 276), .

Area of circumscribed polygon (277); of regular polygon (278), 60

Area of circle (279-281, 285),

Area of sector (283, 285); of segment (284),

Similar sectors and segments (286),

To convert a polygon into an equivalent triangle (289),

To find the quadrature of a parallelogram (290); of a triangle

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(291); of a circumscribed polygon (292); of a circle (293);
of any polygon (294),

To construct a figure similar to a given figure and equivalent to

a given figure (295, 296),

To construct a parallelogram, equivalent to a given square, and

having the sum (297), or the difference (298) of its base and

altitude equal to a given line,

To find the ratio π of circumference to diameter (301),

CHAPTER XIV.

ISOPERIMETRICAL FIGURES,

Isoperimetrical figures, maximum and minimum (302),

Maximum of isoperimetrical triangles of the same base (303),

Maximum of isoperimetrical polygons of the same number of

sides (304, 308),

Maximum of triangles formed with two given sides (305),

Maximum of polygons formed with sides all given but one (306), 92

Maximum of polygons formed with given sides (307),

Greatest of isoperimetrical regular polygons (309),

Maximum of isoperimetrical figures (310),

SOLID GEOMETRY.

CHAPTER XV.

PLANES AND SOLID ANGLES,

Three points not in the same straight line determine the posi-

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Cases in which a line is perpendicular to a plane (318, 324, 326,

Parallel lines perpendicular to plane (322); and converse (323), 100

Parallel lines in the same plane (327),

Line parallel to plane, parallel planes (328),

101

Cases in which a line is parallel to a plane (329, 335),

101

Cases in which planes are parallel (331, 332),

102

(338),

Lines between parallel planes (333, 336),

Intersection of two parallel planes by a third plane (334),

Solid angle (337).

Sum of two plane angles of a solid angle greater than the third

Sum of plane angles of a solid angle (339),

103

104

Solid angles formed by equal plane angles (340),

107

Equivalent solids (341); lamina (341),
Case in which solids are equivalent (342),
Polyedron, and its faces and sides (343); tetraedron, hexaedron,
octaedron. dodecaedron, icosaedron (344); prism, its base, its
convex surface, its altitude (345); right prism (346); triangu-
lar, quadrangular, pentagonal, hexagonal, &c. prism (347);
cylinder, its axis, right cylinder (318); generation of right
cylinder (349); parallelopiped, right parallelopiped (350);
cube, unit of solidity (351); volume, solidity (352),

Area of convex surface of right prism or cylinder (353),

Section of prism or cylinder by plane parallel to base (354),

Equivalent prisms and cylinders (355. 356, 367),

Ratio of parallelopipeds (357, 363, 364),