An Elementary Treatise on Plane and Solid Geometry

confidence upon the one than upon the other. If there were an error involved in the latter theory, it must not only be infinitely small, but must remain infinitely small after all the magnifying processes to which it can possibly be subjected. But there is no error; for, if we suppose that there be an error which we may represent by A, since the aggregate of all the quantities neglected in arriving at the result is infinitely small, that is, as small as we choose, we may choose it to be smaller than A; and, therefore, the error A is greater than the greatest possible error which could be obtained, a manifest absurdity, but one which cannot be avoided as long as A is anything."

The term direction is introduced into this treatise without being defined; but it is regarded as a simple idea, and to be as incapable of definition as length, breadth, and thickness; and this innovation will probably be pardoned, when it is seen how much it contributes to the brevity and simplicity of demonstration, which I have every where studied.

BENJAMIN PEIRCE.

CONTENTS.

[The figures in parentheses refer to the articles.]

EXPLANATION OF SIGNS AND OF SOME USEFUL

TIONS IN THE DOCTRINE OF PROPORTIONS,

Introduction,

Axiom, Theorem, Problem, Lemma, Corollary, Scholium,

Hypothesis.

PLANE GEOMETRY.

CHAPTER I.

General Remarks and Definitions,

Geometry (1), point (2), line (4), surface (5), solid (6),

CHAPTER II.

The Point,

The position of a point (8), its direction (9), its distance (10),

CHAPTER III.

3

The Straight Line,

The direction of a line (11),

Straight line (12), broken line (13), curved line (13),

Plane (14),

5

Direction of a point of a straight line from preceding point (15), 5

Position of a straight line (16),

Shortest way from one point to another (18),

6

Definition of parallel lines (27),

Parallel lines cannot meet (28),

Angles which have their sides parallel (29),

External-internal, alternate-internal angles (30),

Interior angles on the same side (32),

Cases of parallel lines (31, 33, 35, 36),

Parallel lines perpendicular to a third line (34),

Lines parallel to a third line (36),

CHAPTER VI.

Perpendicular and Oblique Lines,

Only one perpendicular from a point to a line (37),

Oblique lines drawn from a point to a line (38, 39, 41),

Shortest distance from a point to a line (39),

Sum of two lines included by two other lines (40),

Distances of a point from the extremities of a line (42),

CHAPTER VII.

Sides and Angles of Polygons,

.

Plane figure, polygon, and its perimeter (43); triangle,

quadrilateral, pentagon, hexagon (44); equilateral, isosce-

les, and scalene triangle (45); right triangle and its hypoth-

enuse (46); square, rectangle, parallelogram, rhombus or

lozenge, trapezoid (47); diagonal (48),

14

Equilateral and equiangular polygon (49); polygons equi-
lateral and equiangular with respect to each other, and
their homologous sides and angles (50),
Three cases of equal triangles (51, 53, 61),

Equal sides and angles of the isosceles triangle (55, 58),
Equilateral and equiangular triangle (56, 59),

Line drawn from the vertex to the middle of the base of an
isosceles triangle (57),
Greater side of a triangle opposite the greater angle (62),
Case of equal right triangles (63),