Of the points in which the author claims special originality, attention is called to Propositions XVIII. (including its Corollaries) and XX. of Book I.; the definition and consequent discussion of Similar Polygons (II. 52-58, 76-78); the use made of Proposition X., of Book III., in subsequent demonstrations; and the definition and consequent discussion of Similar Solids (VII. 78-82).

For the introduction of the terms "Normal to a Plane," and "Aspect of a Plane," the author is indebted to JAMES MILLS PEIRCE, Professor of Mathematics in Harvard University. By the use of these terms the author is enabled to extend to planes the same idea as is used in the definition and treatment of lines and of angles in Book I. For a discussion of the word "Aspect," as applied to planes, those interested are referred to several articles in the London journal, "Nature," for the years 1871-72, and specially to an article, by Professor J. M. PEIRCE, on p. 102, Vol. V., of the same journal.

W. F. B.

CAMBRIDGE, MASS., April, 1877.

ELEMENTARY GEOMETRY.

INTRODUCTORY DEFINITIONS.

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.

A

B

A straight line has two directions exactly opposite, of which

either may be assumed as its direction.

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

1

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.

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

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

23. If equals are multiplied by equals, the products are equal. 24. If equals are divided by equals, the quotients are equal. 25. Like powers and like roots of equals are equal.

26. The whole of a magnitude is greater than any of its parts. 27. The whole of a magnitude is equal to the sum of all its parts.

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

29. A straight line is the shortest distance between two points.