elegance, and distinctness of form. The advantages, for instance, of a full, clear, free typography, are not liable to be overestimated. The force otherwise expended in disengaging the idea from its dress, and one idea from another, is thus at once concentrated upon the thing and its relations. It is not an accident, but a law, that in proportion as there is attention to the sign there is inattention to the thing signified.

From first to last I have kept in view the student of average ability, and have written in the continual presence of those difficulties which I have had to meet as an instructor. As nearly as possible I have written as I have found myself obliged to teach, and thus hope to have furnished a thoroughly intelligible companion text. Tastes and habits are determined in the privacy of preparation. If perplexities are experienced here which must await the explanations of the class-room, the effect cannot be other than harmful.

So many writers have been consulted that an acknowledgment of indebtedness would be little else than the catalogue of a library. It will be sufficient to observe that the volume would have been different, as respects its matter, its spirit, and its form, had not the able treatises of Todhunter, Chauvenet, Olney and Schuyler appeared before it, as well as the several publications of the English Association for the Improvement of Geometrical Teaching; but that its differential features from these and others now current abundantly justify its existence, is confidently believed. Every man builds upon his predecessors, and all systems are but approximations to the ideal which no one has ever actually reached.

It remains only to express my sense of obligation to my able successor in the chair of mathematics, Dr. E. Fraunfelter, who has kindly read the proof-sheets and furnished me with many valuable suggestions.

Columbus, Ohio, June 15, 1883.

THE AUTHOR.

PART I.

PLANE GEOMETRY.

CHAPTER I.

THE SUBJECT-MATTER.

EVERY object that we can touch or see, occupies a portion of space, is extended, and has some outline.

For example, conceive this book to be suspended in mid-air: it has a certain length, width, and thickness, and a particular shape, and is distinguished by a particular form.

If, now, the book be removed, the space which it filled will still remain; and we can imagine it to have, as it does, the same form as the book had. It is not a material thing, like the book, but immaterial, yet having the same shape and the same extent.

This portion of space is called a Geometric solid, in distinction from the book itself, which is a Physical solid. A Geometric solid, then, is the space occupied by a physical solid, and has: first, three dimensions,- length, breadth, and thickness; second, a definite form, its form being the same as that of the physical solid.

Now, let the annexed diagram represent a geometric solid, or portion of space. The boundaries which separate this from all surrounding space are called Surfaces. As the boundary is not a part of the things separated, which are nevertheless in immediate contact, it is plain that a surface has no thickness, only length and breadth.

It is clear, also, that the bounding surfaces are themselves limited, or bounded. The limits or edges of surfaces are called Lines, which being no part of the surface, but merely its limits, have length only.

The limits, or extremities, of lines are called Points. As they mark merely the limits or ends of lines, and thus are no part of the lines themselves, it is evident that they have no length,they are not long nor wide nor thick.

Points may be considered without reference to lines, merely as positions in space. Lines may be considered without reference to surfaces, merely as extension in one dimension - length.

Surfaces may be considered without reference to geometric solids, merely as extension in two dimensions-length and breadth.

In reference to their extent, lines, surfaces, and solids are called magnitudes; in reference to their shape, figures. Figures formed by straight lines are called rectilinear.

It is thus seen that Geometry deals, not with material, but with immaterial, things. It treats of physical bodies by considering the space which they fill; and the student should distinctly realize and remember that, while points, lines, surfaces and geometric solids can be indicated and represented only in a material way, yet the points, lines, surfaces and solids of Geometry are purely ideal, though we know them to exist, and endeavor to make them apparent to the eye. For instance, while any line he may draw will have some breadth, and even thickness, or be otherwise imperfect, and, so far, fail to represent the true line, yet his reasoning should proceed on the supposition that the drawing is perfect that the line drawn has no breadth.

DEFINITIONS.

These considerations make clear the following definitions: 1. A Point is that which has position, but no extension.

2. A Line is that which has extension in one dimension

length.

3. A Surface is that which has extension in two dimensions

-length and breadth.

4. A Geometric Solid, or Volume, is that which has extension in three dimensions — length, breadth, and thickness.

5. Geometry, which treats of the properties and relations of Points, Lines, Surfaces, and Volumes, may be defined as The Science of Position, Extension, and Form.

LINES.

A Straight Line, or a Line simply, is one whose direction does not change.

It is designated by letters placed at two of its points, or by a small italic letter placed near its

middle. Thus, the line A B or the A

a

B

line a. We may assign to it either direction—from B to A, or from A to B.

ANGLES.

When the legs of a pair of compasses are opened, they point in different directions, and are said to make an angle with each other. If, now, we imagine the legs to be replaced by geometrical lines, we shall have an angle as understood in Geometry.

1. An Angle, therefore, is the differ

ence in direction of two lines which actually do meet, or would if sufficiently prolonged

Thus the opening between the lines A B and A C, at the point A, is an angle.

A

B

C

2. The lines which form the angle are called Sides; the point at which the sides meet is called the Vertex. Thus, A B and A C are the sides of the angle, and A the vertex.

3. The degree of opening evidently remains the same, what

ever be the length of the sides; that is, the magnitude, or size, of an angle depends upon the position of the sides, not their length.

4. If there is but one angle at a point, it may be read by one letter or by three, special care being taken to read the letter at the vertex between the other two. Thus, the above angle may be read, angle B A C, or C A B, or simply angle A.

read by three letters; as, the angle B CD, or the angle G C F.

a

bc

d

6. It is sometimes convenient to denote angles by small italics placed between the sides and near the vertex, in which case the single letter sufficiently indicates what angle is meant; as, the angles a, b, c, and d.

7. Adjacent Angles are those which have a common vertex and a common side between them. Thus, in the last figure, the angles a and b are adjacent; and, in the previous figure, the angles ECF and FCG, having the common vertex C and the common side

A

B

D

CF, are adjacent; also, in the annexed

figure, the angles A CB and B CD.

8. When one straight line meets another straight line so as to make the adjacent angles equal, each of these angles is a Right Angle, and the first line is said to be perpendicular to the

A

D

second. Thus, if the two adjacent angles A CD and BCD, formed by the line C D meeting A B, are equal, each is a right angle, and D C is perpendicular to A B.