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ELEMENTS OF EUCLID

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INTRODUCTION.

Geometry is the Science of figured Space. Figured Space is of one, two, or three

dimensions, according as it consists of lines, surfaces, or solids. The boundaries

of solids are surfaces; of surfaces, lines; and of lines, points. Thus it is the

province of Geometry to investigate the properties of solids, of surfaces, and

of the figures described on surfaces. The simplest of all surfaces is the plane,

and that department of Geometry which is occupied with the lines and curves

drawn on a plane is called Plane Geometry; that which demonstrates the properties

of solids, of curved surfaces, and the figures described on curved surfaces,

is Geometry of Three Dimensions. The simplest lines that can be drawn on a

plane are the right line and circle, and the study of the properties of the point,

the right line, and the circle, is the introduction to Geometry, of which it forms

an extensive and important department. This is the part of Geometry on which

the oldest Mathematical Book in existence, namely, Euclid’s Elements, is written,

and is the subject of the present volume. The conic sections and other

curves that can be described on a plane form special branches, and complete

the divisions of this, the most comprehensive of all the Sciences. The student

will find in Chasles’ Aper¸cu Historique a valuable history of the origin and the

development of the methods of Geometry.

BOOK I.

THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND

PARALLELOGRAMS.

DEFINITIONS.

A point is that which has position but not dimensions.

A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness,

is a solid; that which has two dimensions, such as length and breadth, is a surface; and

that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor

a line; hence it has no dimensions—that is, it has neither length, breadth, nor thickness.

The Line.

A line is length without breadth.

A line is space of one dimension. If it had any breadth, no matter how small, it would

be space of two dimensions; and if in addition it had any thickness it would be space of three

dimensions; hence a line has neither breadth nor thickness.

iii. The intersections of lines and their extremities are points.

iv. A line which lies evenly between its extreme

points is called a straight or right line, such as AB.

If a point move without changing its direction it will describe a right line. The direction in

which a point moves in called its “sense.” If the moving point continually changes its direction

it will describe a curve; hence it follows that only one right line can be drawn between two

points. The following Illustration is due to Professor Henrici:—“If we suspend a weight by a

string, the string becomes stretched, and we say it is straight, by which we mean to express

that it has assumed a peculiar definite shape. If we mentally abstract from this string all

thickness, we obtain the notion of the simplest of all lines, which we call a straight line.”

Axioms.

“Elements of human reason,” according to Dugald Stewart, are

certain general propositions, the truths of which are self-evident, and which are

so fundamental, that they cannot be inferred from any propositions which are

more elementary; in other words, they are incapable of demonstration. “That

two sides of a triangle are greater than the third” is, perhaps, self-evident; but

it is not an axiom, inasmuch as it can be inferred by demonstration from other

propositions; but we can give no proof of the proposition that “things which are

equal to the same are equal to one another,” and, being self-evident, it is an

axiom.