and in the solution of problems, unaided except by definitions

and principles.

The Practical Applications, or Exercises in Mensuration, given at the close of the volume, can be used at the discretion of the teacher, after the completion of Book III., in connection with the text, to which they refer, as the pupil progresses.

The course can be limited wholly to Plane Geometry by omitting all after Book IV.

In general, Geometry follows Algebra in a full Mathematical Course; but in a limited course, the first four books of this Manual may often be read with advantage by the Student, even before entering upon Higher Arithmetic, or Elementary Algebra.

The larger work, known as Greenleaf's Elements of Geometry, especially adapted to the wants of Higher Seminaries, and now extensively used in them, will continue to be published.

It is proper to state that this volume has been prepared by H. B. MAGLATHLIN, whose valuable labors have been before acknowledged in several books of the series.

BOSTON, May, 1873.

A Key to New Elementary Geometry has been prepared, and

published for the convenience of teachers, which will be mailed
prepaid, on receipt of 50 cents by the Publishers.

ELEMENTARY GEOMETRY.

BOOK I.

GENERAL PRINCIPLES.

PRELIMINARY DEFINITIONS.

1. Geometry is the science of position and extension. The elements of position are direction and distance.

The dimensions of extension are length, breadth, and thick

ness.

2. A Point is that which has position, without magnitude. 3. A Line is that which has length, without either breadth or thickness.

4. A Straight Line is one which

has the same direction in its whole ex- A

tent; as the line A B.

-B

The word line is frequently used to designate a straight line. 5. A Curved Line is one which

continually changes its direction; as C

the line CD.

D

The word curve is frequently used to designate a curved line.

6. A Surface is that which has length and breadth, without hight or thickness.

7. A Plane Surface, or simply a PLANE, is one in which any two points being taken, the straight line that joins them will lie wholly in the surface.

8. A Curved Surface is one that is not a plane surface, nor made up of plane surfaces.

9. A Solid, or Volume, is that which has length, breadth, and thickness.

ANGLES AND LINES.

10. An Angle is the difference in

the direction of two lines, which meet

at a point; as the angle A.

The point of meeting, A, is the ver

A

B

tex of the angle, and the lines A B, A C are the sides of the

angle.

An angle may be designated, not only by the letter at its vertex, as C, but when two or more angles have the same vertex, the letter at the vertex always occupies the middle place; as the angle A CD or D C B.

11. Two straight lines are said to be Perpendicular to each other, when their meeting forms equal adjacent angles; thus the lines A B and CD are perpendicular to each other.

A

C

Two adjacent angles, as CA B and BAD, have a common

vertex, as A; and a common side, as A B.

12. A Right Angle is one which is formed by a straight line and a perpendicular to it; as the angle CAB.

13. An Acute Angle is one which is less than a right angle; as the angle DEF.