23. Trilateral figures, or triangles, are contained by

three straight lines.

24. Quadrilateral figures are contained by four straight lines.

25. Multilateral figures, or polygons, are contained by more than four straight lines.

26. An equilateral triangle has all its sides equal.

27. An isosceles triangle has two equal sides.

28. A scalene triangle has three unequal sides.

29. A right angled triangle is that which has one right angle.

30. An obtuse angled triangle is that which has one obtuse angle.

31. An acute angled triangle has all its angles acute.

Δ.

32. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles.

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

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

35. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor are its angles right angles.

36. All other four-sided figures besides these are called trapeziums.

37. Parallel straight lines are such as, being in the same plane, and being produced ever so far both ways, do

not meet.

38. A parallelogram is a four-sided figure whose opposite sides are parallel.

39. A postulate requires us to admit the possibility of doing something, without being shown how to do it.

40. A proposition is a distinct portion of science, and is either a problem or a theorem.

41. A problem is an operation proposed to be performed. 42. A theorem is a truth which it is proposed to prove. 43. A lemma is a preparatory proposition to render what follows more easy.

44. A corollary is an obvious consequence resulting from a preceding proposition.

45. A scholium is an observation, or remark upon something preceding it.

46. An axiom is a self-evident truth.

47. The side opposite to the right angle of a right-angled triangle is called the hypotenuse; one of the sides about the right angle is called the base; and the remaining side is called the perpendicular.

48. In a triangle which is not right-angled, any side may be called the base; the intersection of the other two sides is called the vertex; and the angle at that point the vertical angle.

49. The space contained within a figure is called its surface; and in reference to that of another figure with which it is compared, is called its area.

50. A polygon is a figure contained by more than four straight lines; when its sides are all equal, and also its angles, it is called a regular polygon.

51. A polygon of five sides is called a pentagon; that of six sides, a hexagon; that of seven sides, a heptagon; that of eight sides, an octagon; that of nine sides, a nonagon; that of ten sides, a decagon; that of eleven sides, an undecagon; that of twelve sides, a dodecagon; that of fifteen sides, a quindecagon.

POSTULATES.

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

2. Let it be granted that a terminated straight line may be produced to any length in a straight line.

3. Let it be granted that a circle may be described from any centre, and with any radius.

AXIOMS.

1. Things that are equal to the same thing are equal to each other.

2. If equals be added to equals, the sums are equals. 3. If equals be taken from equals, the remainders are equals.

4. If equals be added to unequals, the sums are unequals.

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

6 Things which are double of the same, or equal things, are equal to one another.

7. Things which are halves of the same, or equal things, are equal to one another.

8. 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, and equal to all its parts taken together.

10. Two straight lines cannot inclose a space.

11. All right angles are equal to one another.

12. If two magnitudes be equal, and one of them be greater than a third, the other is also greater than the third. 13. If two quantities be equal, and one of them be less than a third, the other is also less than the third.

14. If there are three magnitudes, such that the first is greater than the second, and the second greater than the third, much more is the first greater than the third.

15. If there are three magnitudes, such that the first is less than the second, and the second less than the third, much more is the first less than the third.

16. Through the same point there cannot be drawn two straight lines parallel to the same straight line without coinciding.

EXPLANATION OF SYMBOLS.

PROPOSITION I.-THEOREM.

The angles ACD and DCB, which one straight line, DC, makes with another, AB, on one side of it, are either two right angles, or are together equal to two right angles. A If the LS ACD and DCB be equal,

each of them is a L, (Def. 9.); but if they are not equal, conceive CE to be drawn to AB, then the Ls ACE and but the three Ls ACE, ECD, and the two Ls ACE and ECB, and also and DCB; .. the two Ls ACD and the two Ls ACE and ECB, but the two Ls ACE and ECB are two rLs; .. the two Ls ACD and DCB are together equal to two 'Ls.

ECB are two Ls;
DCB, are together:
to the two Ls ACD
DCB are together

Q. E. D. Cor. 1. All the angles that can be formed at the point C, in the straight line AB, on one side of it, are together equal to two right angles.

For the LACD is the two Ls ACE and ECD, so that the sum of the Ls is not increased by drawing the | EC, and in the same manner it may be shown the sum of the Ls would not be increased by drawing any number of Is through the point C.

Cor. 2. All the angles formed at the point C, on the other side of the line, by any number of lines meeting in C, will also be equal to two right angles.

Cor. 3. Hence all the angles formed round a point, by any number of lines meeting in it, are together equal to two right angles.

SCHOLIUM. For the purposes of calculation, the circumference of every circle is supposed to be divided into 360 equal parts, called degrees, and each degree is supposed to be divided into 60 equal parts, called minutes, and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are distinguished by the following marks:-7° 3′ 24′′, which is read 7 degrees, 3 minutes, and 24 seconds.

B

In the same manner all the angles round about a point are divided into the same number of degrees, minutes, and seconds. Since then the circle entirely surrounds its centre, and is similarly situated to it in every direction, the portion of the circumference intercepted between two lines drawn from the centre to the circumference, is the measure of the angle

A

at the centre; thus the angle AOB is measured by the intercepted arc AB, and the angle COB is measured by the arc CB.

Since all the angles round a point are (Cor. 3.) equal together to four right angles, and also to 360°, the numerical measure of a right angle is 90°.

PROPOSITION II.-THEOREM.

If, at a point B, in a straight line AB, two other straight lines, CB and BD, on opposite sides of AB, make the adjacent angles ABC and ABD together equal to two right angles, these two

B

A

straight lines are in one and the same straight line.

D

= two

=

For if BD be not in the same | with CB, let BE be in the same with it; then since CBE is a |, and AB makes Ls with it, the two Ls ABC and ABE are together Ls; but the two Ls ABC and ABD are also together two Ls by supposition; .. the two Ls ABC and ABE are the two Ls CBA and ABD; take away the common angle ABC, and there remains the LABE the LABD, the less the greater, which is impossible; .. BE is not in the same with CB, and in the same manner it can be shown that no can be in the same | with CB, except BD, which therefore is in the same with it.

Q. E. D.

E

For the two Ls AEC and AED, which the AE makes with the | CD, are together equal to two Ls; and the two Ls AED and DEB, which the | DE makes with the | AB, are also together equal to two Ls; the two Ls AEC and AED, are together equal to the two Ls AED and DEB; take from each the common LAED, and there remains the LAEC DEB; in the same manner it may be demonstrated that the two Ls AED and CEB are equal. Q. E. D.