11. AN ACUTE ANGLE is an angle which is less than

a right angle.

12. A PLANE FIGURE is a plane surface which is bounded on all sides by one or more lines.

SCHOLIUM. The bounding line of a plane figure is termed its perimeter, and the space which is contained within the same is termed the area of the figure.

13. A CIRCLE is a plane figure bounded by one curved line, which is such that all straight lines drawn from it to a certain point within the figure are equal.

14. The CIRCUMFERENCE of a circle is the curved line by which it is bounded.

15. The CENTER of a circle is a point within the figure equally distant from its circumference.

16. A RADIUS of a circle is a straight line drawn from the center to the circumference.

17. A DIAMETER of a circle is a straight line drawn through the center and terminated both ways by the circumference.

SCHOLIUM. Thus the curved line ABCDF is the circumference of a circle, of which E is the center, BD a diameter, and AE a radius. Any portion of the circumference, as AFD, is termed an arc of the circle; the straight line AD joining its extremities is termed its chord; and the figure ADF contained by the arc and its chord is termed a segment. The space contained by two arcs of circles of different radii is termed a lune, as GHI.

18. A RECTILINEAL FIGURE is a plane surface, bounded on all sides by straight lines.

SCHOLIUM. The straight lines by which a rectilineal figure is bounded are termed its sides, which are said to contain the figure.

19. A TRIANGLE is a rectilineal figure which is bounded by three straight lines.

SCHOLIUM. For convenience one of the lines by which a triangle is contained is termed the base of the triangle, the other two lines being termed its sides, and the point in which the two sides meet is termed the

vertex.

20. An EQUILATERAL TRIANGLE is a triangle which has three sides equal.

21. An ISOSCELES TRIANGLE is a triangle which has only two sides equal.

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A

SCHOLIUM. When all the three sides of a triangle are unequal, it is sometimes termed a scalene triangle.

22. A RIGHT-ANGLED TRIANGLE is a triangle two sides of which form a right angle.

SCHOLIUM. In a right-angled triangle the third side opposite to the right angle is termed the hypotenuse; and in any triangle any side is said to subtend the angle opposite to it; thus the hypotenuse subtends the right angle.

23. An OBTUSE-ANGLED TRIANGLE is a triangle two sides of which form an obtuse angle.

24. An ACUTE-ANGLED TRIANGLE is a triangle the sides of which form three acute angles.

25. A QUADRILATERAL FIGURE is a rectilineal figure which is bounded by four straight lines.

SCHOLIUM. A straight line drawn from any two opposite angles of a quadrilateral figure is termed a diagonal.

26. A PARALLELOGRAM is a quadrilateral figure whose opposite sides are parallel.

SCHOLIUM. If a diagonal AC be drawn to any parallelogram ABCD, and lines GH and EF be drawn respectively parallel to two contiguous sides of the same, so as to intersect in some point K of the diagonal, the parallelogram will be divided into four parallelograms, two of which, AEKH and KGCF, are said to be about the diagonal, and the other two of which, EBGK and HKFD, are termed the complements of the former.

B

H

For brevity parallelograms are frequently designated by only two letters placed at the opposite corners, as the parallelogram EG, instead of EBGK. 27. A RECTANGLE is a parallelogram two of whose sides form a right angle.

SCHOLIUM. As a rectangle is contained under four lines, two of which, AB and BC, are equal to the other two, CD and AD, it is designated as the rectangle under those two lines; thus the rectangle ABCD would be termed the rectangle under AB and BC.

28. A SQUARE is a quadrilateral figure which has all its sides equal, and two sides of which form a right angle.

SCHOLIUM. As a square is contained by four equal lines, upon either of which it may be conceived as constructed, it is designated as the square on one of those lines, as the square on the line AB.

29. A POLYGON is a rectilineal figure which is bounded by more than four sides.

SCHOLIUM. In any rectilineal figure, ABCDEF, the angles formed by its several sides on the inner side and distinguished by being shaded, are termed the internal or interior angles; when the internal angle is less than two right angles, it is termed a salient angle, as the angle E; but when greater, it is termed a reëntrant angle, as the internal angle at F. If any side is produced, the angle which its production makes with the contiguous side is called the exterior or external angle. Thus, if the side AB is produced to G, the angle CBG is termed the external angle at B.

E

When two straight lines intersect, as AB and CD, the two opposite equal angles, as AEC and DEB, are termed vertical angles; while those contiguous, as AEC and CEB, are termed adjacent angles.

When a straight line, as AB, intersects two other straight lines, as CD and EF, the angles CGH and GHF are said to be alternate angles, as are also DGH and GHE.

B

Let it be granted:

POSTULATES.

1. That a straight line may be drawn from any point to any other point.

2. That any finite straight line may be extended, or produced, to any length.

3. That a circle may be described from any center with any radius.

SCHOLIUM. A Postulate is a problem the solution of which is self-evident, and therefore requiring no demonstration; it will be observed that they are only subsequently employed in the construction of theorems or the solution of problems, but never in the demonstration.

The third postulate points out the restricted use allowed to the compasses, namely, only to describe circles, but never to measure or carry distances. The compasses must be conceived as closing whenever removed from actual contact with the surface of the paper.

AXIOMS.

1. Magnitudes which are equal to the same are equal to one another.

SCHOLIUM. This axiom is frequently employed in the Elements under the form, "Magnitudes which are equal to equals are equal to one another." 2. If equals be added to equals, the wholes are equal.

3. If equals be taken from equals, the remainders are equal.

4. If equals be added to unequals, the wholes are unequal.

5. If equals be taken from unequals, the remainders are unequal.

6. Magnitudes which are double of the same are equal to one another.

7. Magnitudes which are halves of the same are equal to one another.

SCHOLIUM. A similar extension may be given to the 6th and 7th axioms as that given above to the 1st by substituting "equals" for "the same."

8. Magnitudes which coincide with one another are equal to one another.

SCHOLIUM. The converse of this axiom is sometimes made use of in the elements, namely, "Magnitudes which are equal coincide with one another when similarly placed."

9. The whole is greater than its part.

10. Two straight lines cannot enclose a space.

SCHOLIUM. This axiom may be otherwise expressed, namely, "If two straight lines coincide in two points, they coincide when produced."

11. All right angles are equal to one another.

SCHOLIUM. Angles being a species of magnitude, this axiom is a particular case of the 8th.

12. Through the same point two straight lines cannot be drawn parallel to the same straight line.

SCHOLIUM. This axiom is substituted for that of Euclid, not only as being more self-evident, but because the latter being the converse of the 17th proposition, it was considered that it ought to be demonstrated, which has been done after the 29th proposition.

The axioms are self-evident theorems, that is, theorems which do not admit of being demonstrated, but the truth of which is nevertheless so apparent as to be instantly admitted. No theorem should be considered as an axiom simply because it is self-evident, but only when it will not admit of being demonstrated by means of arguments founded on still simpler theorems; for it is desirable that the number of axioms should be reduced as much as possible, for which reason the 20th proposition, and some others, although as self-evident as any of the axioms, are demonstrated at length.

EXPLANATORY REMARKS.

A proposition in geometry is something either proposed to be solved or proved; and is consequently divided into two kinds, a problem or a theorem.

A problem proposes something to be done; while a theorem makes an assertion, of which it proposes to demonstrate the truth.

The statement of the thing to be done, or of the assertion to be proved, is termed the enunciation of the proposition. In the following work the enunciation is distinguished by a bolder type.

The enunciation of a problem may be divided into two parts, the data or things given, and the quæsita or things sought to be done. The former is distinguished by being printed in italics.

In like manner the enunciation of a theorem may be divided into two parts, the hypothesis, and the consequence, which it is to be proved results from that hypothesis. The former is distinguished from the consequence by being in italics.

The solution of a problem is the mode in which the quæsita are found, or the thing proposed to be done is accomplished, and is always performed by means of some other problems the truth of which has been either admitted or proved previously.

The construction of a theorem is certain things which may be required to be done by means of problems, in order to admit of the truth of the theorem being demonstrated.

The demonstration either of a problem or theorem is a succession of arguments logically deduced from theorems already admitted or proved, by means of which the truth of the solution of a problem or the assertion of a theorem are undeniably established.

A lemma is a proposition of no importance in itself, merely introduced for the purpose of demonstrating some other proposition.

A corollary is a proposition the truth of which immediately follows from that to which it is affixed.

A scholium is a note or remark appended to any proposition by way of explanation or elucidation.

In the marginal references the following abbreviations are employed:Ax. 1, signifies the first axiom.

Post. 1, signifies the first postulate.

Def. 15, signifies the 15th definition.

I. 31, signifies the 31st proposition of the first book.