propositions, which are of no use, unless for demonstrating others; and so arrange the propositions, that all the affections of the same kind can be demonstrated together.

I conclude with expressing my earnest wish and hope, that this work may be highly instrumental, in promoting and diffusing useful and interesting knowledge.

University of Maryland, Baltimore, May 1st, 1822.

EUCLID'S ELEMENTS OF GEOMETRY.

GEOMETRY, is that science, whereby we compare together such quantities, as have extension.

OBSERVATIONS.

A Proposition, proposes something to be done or demonstrated. A Problem, proposes something to be done, which, when taken for granted as obvious or self-evident, is called a postulate.

A Theorem, proposes something to be demonstrated, which, when taken for granted as obvious or self-evident, is called an axiom.

A Lemma, is something demonstrated, in order to prove something which follows.

A Corollary, is something drawn, as an inference, from a preceding proposition.

A Scholium, is a remark or remarks on something which preceded.

The rules of mathematical reasoning, whereby we should be directed in the prosecution of this science, and which have been formed after mature consideration, are few and obvious; and are as follow:

1. The principles assumed, whether practical or theoretical, under the appellation of postulates and axioms, ought to be as few and as simple as possible.

2. Nothing ought to be assumed, in any construction or demonstration, but these principles.

Note. When in the quotations you meet two numbers, the first shews the proposition, and the second the book.

Also

Post. denotes Postulate; Ax. Axiom; Def. Definition; B. Book; Constr. Construction; Hyp. Hypothesis, or supposition; contra hyp. contra hypothesin, or contrary to the supposition.

BOOK I.

DEFINITIONS.

1. A point, is that which has no part. 2. A line, is length without breadth. 3. The extremes of a line, are points.

4. A right line, is that which lieth equally between its points: 5. A superficies, is that which hath length and breadth only. 6. The extremes of a superficies, are lines.

7. A plain superficies, is that which lieth equally between its lines.

8. The distance of two points from each other, is a right line drawn from one of them to the other.

9. A figure, is that which is inclosed by one or more boundaries.

10. A circle, is a plain figure, bounded by one line (ADBA), every where equally distant from a point (C) within it.

11. That point (C), is called the centre of the circle. 12. The bounding line (ADBA), is called its circumference or periphery.

13. A diameter of a circle, is a right line, passing through the centre, and terminated both ways by the circumference, (as AB).

14. A radius of a circle is a right line, drawn from the centre to the circumference, (as CD).

15. A semicircle, is a figure, contained by a diameter of a circle, and the part of the circumference which is cut off thereby, (as ADB).

16. A plain angle, is the inclination of two lines to each other in a plain, which meet together, but are not in the same direction.

17. A rectilineal angle, is the inclination of two right lines, meeting each other, and not being in the same right line.

18. The legs of an angle, are the lines, which form the angle. 19. The vertex of an angle, is the point, in which the legs 'meet each other.

An angle is designated either by one letter placed at its vertex, as E; or by three letters, of which the middle one is at the vertex, the other two somewhere in the legs; thus, the angle formed by the lines DB, BC, meeting in B, is called the angle DBC or CBD.

20. When one right line (AB), standing on another (CD), makes the angles (ABC, ABD) on each side of the insisting line (AB) equal, each of these equal angles, are called right angles, and the insisting line, is said to be perpendicular to the other.

21. The distance of a point from a right line, is a perpendicular drawn from the point to the right line.

22. An angle (EBC), which is greater than a right angle, is called obtuse.

23. An angle (EBD), which is less than a right angle, is called acute.*

* Mathematicians have supposed the whole circumference of a circle to be divided in 360 equal parts, called degrees, each degree into 60 equal parts, called minutes. and each minute into 60 equal parts, called seconds, &c. and a circle being described from the vertex of an angle, as a centre, an angle is said to be of as many degrees, minutes, seconds, &c. as are contained in the arch intercepted between the legs of the angle. Thus, see fig. to Def. 10, above, the angle DCB, is said to be of as many degrees, minutes, seconds, &c. as are,

24. A rectilineal figure, is a plain one, bounded by right lines. 25. A triangle, is a plain figure, bounded by three right

lines.

26. A quadrilateral figure, or quadrangle, is one, bounded by four right lines.

27. Plain figures, bounded by more than four right lines, are called polygons.

28. Of triangles, those, whose three sides are equal,are called equilateral.

AAA

29. An isosceles triangle, is one, which has only two equal sides.

30. A scalene triangle, one, which has three unequal sides 31. A right-angled triangle, is one, which has one right angle,

440

32. An obtuse angled triangle, one, which has one obtuse angle. 33. An acute-angled triangle, one, which has three acute angles.

34. Parallel right lines, are such as, being in the same plain, would never meet, though ever

so much produced both ways.

35. Of quadrilateral figures or quadrangles, a parallelogium, is one, whose opposite sides are parallel.

36. A square, is one, which has all its sides equal, and all its angles right.

contained in the arch BD, as will be more fully explained in the tract on plain trigonometry in this work.

This note is inserted, to give beginners a more correct idea of the magni tude of angles, about which they are apt at first to be puzzled. And it chiefly for the sake of setting them right in this particular, that it was thought expedient, to place the definition of a circle before that of an angle, contrary, to the usual practice.