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strated, include those that are usually inserted, but whose demonstration in this work has been omitted. In some of these Exercises references are given to the necessary propositions; in some suggestions are made; and in a few cases the figure is constructed as the proof will require.

A sixth Book of Problems of Construction is added, which is followed by Problems for the pupil to solve. This Book, or any part of it, if thought best, can be taken immediately after completing Book III.

CAMBRIDGE, MASS., April, 1872.

W. F. B.

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PLANE GEOMETRY.

INTRODUCTORY DEFINITIONS.
UCTOR

1. Mathematics is the science of quantity.

2. Quantity is that which can be measured; as distance, time, weight.

3. Geometry is that branch of mathematics which treats of the properties of extension.

4. Extension has one or more of the three dimensions, length, breadth, or thickness.

5. A Point has position, but not magnitude.

6. A Line has length, without breadth or thickness.

7 A Straight Line is one whose direction

is the same throughout; as A B.

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A straight line has two directions exactly opposite, of which either may be assumed as its direction.

The word line, used alone in this book, means a straight line.

8. Corollary. Two points of a line determine its position.

9. A Curved Line is one whose direction

is constantly changing; as CD.

C

10. A Surface has length and breadth, but no thickness.

D

11. A Plane is such a surface that a straight line joining any two of its points is wholly in the surface.

12. A Solid has length, breadth, and thickness.

13. Scholium. The boundaries of solids are surfaces; of surfaces, lines; the ends of lines are points.

14. A Theorem is something to be proved.

15. A Problem is something to be done.

16. A Proposition is either a theorem or a problem.

17. A Corollary is an inference from a proposition or state

ment.

18. A Scholium is a remark appended to a proposition.

19. An Hypothesis is a supposition in the statement of a proposition, or in the course of a demonstration.

20. An Axiom is a self-evident truth.

AXIOMS.

1. If equals are added to equals, the sums are equal.

2. If equals are subtracted from equals, the remainders are equal.

3. If equals are multiplied by equals, the products are equal. 4. If equals are divided by equals, the quotients are equal. 5. Like powers and like roots of equals are equal.

6. The whole of a magnitude is greater than any of its parts. 7. The whole of a magnitude is equal to the sum of all its parts.

8. Magnitudes respectively equal to the same magnitude are equal to each other.

9. A straight line is the shortest distance between two points. sa san

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