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24. Trilateral figures, or triangles, are contained by three straight lines.

25. Quadrilateral figures by four straight lines.

26. Multilateral figures, or polygons, by more than four straight lines.

27. Of three-sided figures, an equilateral triangle is that which has three equal sides.

28. An isosceles triangle is that which has two sides equal.

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29. A scalene triangle is that which has three unequal sides. 30. A right angled triangle is that which has a right angle. 31. An obtuse angled triangle is that which has an obtuse angle.

32. An acute angled triangle is that which has three acute angles,

Note. A triangle which has no right angle is often called an oblique angled triangle.

ED.

33. Of quadrilateral figures, a square is that which has one right angle, and all its sides equal.

34. A rectangle is that which has all its angles right angles, but has not all its sides equal.

35. A parallelogram is that which has its two opposite sides parallel.

36. A trapezium is a four-sided figure, of which the opposite sides are not parallel; and the diagonal is the straight line joining two of its opposite angles.

ED.

37. A straight line joining two opposite angles of any quadrilateral figure, or two opposite angles of any polygon, is called a diagonal.

ED.

38. In a right angled triangle the side opposite to the right angle is called the hypothenuse; and of the other two sides one is called the base and the other the perpendicular, according to the position in which the triangle is described. Also, the two sides are sometimes called the legs.

ED.

39. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.

40. That part of Geometry which treats of the measurement and position of plane figures, of straight lines, and of rectilineal angles, is called Plane Geometry.

POSTULATES.

1. LET it be granted that a straight line may be drawn from any one point to any other point.

2. That a terminated straight line may be produced to any length in a straight line.

3. That a circle may be described from any centre, at any distance from that centre.

4. That a straight line which meets one of two parallel straight lines may be produced till it meet the other.

ED.

5. If there be two equal straight lines, and if any figure whatever be constituted on one of them, a figure exactly similar to it may be constituted on the other.

AXIOMS.

1. THINGS which are equal to the same thing are equal to one another. Also things which are equal to equal things 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. Things which are doubles of the same thing are equal to one another.

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

S. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. 9. The whole is greater than its part.

A. The whole is equal to all its parts taken together. ED. B. It is impossible for the same thing to have, at the same time, two qualities which are inconsistent with each other. ED.

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PROPOSITION I. PROBLEM.

To describe an equilateral triangle on a given finite straight line. See Note.

Let AB be the given straight line; it is required to describe an equilateral triangle on it.

From the centre A, at the distance AB, describe the circumference of the circle BCD (3 Postulate); and from the centre B, at the distance BA, describe the circumference of the circle ACE. Then, because the circumference of each circle passes through the cen-. tre of the other, the two cir

C

D A

E

cles will cut each other in some point C. From the point C draw the straight lines CA, CB, to the points A, B (1 Post.); ABC will be an equilateral triangle.

Because the point A is the centre of the circle BCD, AC is equal to AB (20 Definition); and because the point B is the centre of the circle ACE, BC is equal to BA: therefore CA and CB are each of them equal to AB. But things which are equal to the same thing are equal to one another (1 Axiom); therefore CA is equal to CB; wherefore CA, AB, BC are equal to one another; and they form a triangle ABC; therefore the triangle ABC is equilateral; and it is described on the given straight line AB. Which was required to be done.*

PROPOSITION II. PROBLEM.

From a given point to draw a straight line equal to a given straight line.

Let A be the given point, and BC the given straight line; it is required to draw from A a straight line equal to BC.

*This demonstration is full and minute.

Join the points A, B, and on ARB describe the equilateral triangle DAB (Prop. 1). From the centre B, at the distance BC, describe the circle CGH (3 Post.), and produce DB to G (2 Post.). From the centre D, at the distance DG, describe the circle GKL, and produce DA to L. AL will be equal to BC.

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G

F

Because the point B is the centre of the circle CGH, BC is equal to BG (20 Def.); and because D is the centre of the circle GKL, DL is equal to DG. But DA, BD, parts of them, are equal (27 Def.); therefore the remainder AL is equal to the remainder BG (3 Ax.). But BC is equal to BG; wherefore AL and BC are each of them equal to BG; therefore the line AL is equal to BC (1 Ax.); and AL is drawn from the given point A, as was to be done.

PROPOSITION III. PROBLEM.

From the greater of two given straight lines to cut off a part equal to the less.

Let AB and C be two given straight lines, whereof AB is the greater. It is required to cut off from AB a part equal to C.

From the point A draw the straight line AD equal to C (Prop. 2.); and from the centre A, at the distance AD, describe the circle DEF, cutting AB in E (3 Post.). Then AE is equal to C.

D

A

E B

F

Because A is the centre of the circle DEF, AE is equal to AD (20 Def.). But the straight line C is equal to AD; whence AE and C are each of them equal to AD; wherefore AE is equal to C (1 Ax.). Therefore, from AB, the greater of two straight lines, a part AE has been cut off equal to C, the less. Which was to be done.

PROPOSITION IV. THEOREM.

IF two triangles have two sides of one triangle equal to two sides of the other, each to each; and have also the angles contained by those sides equal to each

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