13. A term is the extremity of any thing.

14. A figure is that which is contained under one or

more terms.

15. A circle is a plane figure contained by one line, which is called the circumference, to which all right lines drawn from one point within the figure are equal to one another.

16. And this point is called the centre of the circle. 17. A diameter of a circle is a certain right line drawn through the centre, and terminated both ways by the circumference of the circle, and divides the circle into two equal parts.

18. A semicircle is the figure contained by the diameter, and the part of the circumference cut off by the diameter.*

19. Rectilineal figures are those which are contained by right lines.

20. Triangles are such as are contained by three right lines.

21. Quadrilateral, by four right lines.

22. Multilateral figures, or polygons, by more than four right lines.

23. Of trilateral figures, an equilateral triangle is that which has three equal sides.

24. An isosceles triangle is that which has only two equal sides.

25. A scalene triangle is that which has three unequal sides.

26. Of three sided figures, a right angled triangle is that which has a right angle.

* The segment of a circle which is defined in this place, I have purposely omitted, as being of no use, until the third book, where the definition is repeated; instead of this Proclus has given in his Commentaries the following. The centre of the semicircle is the same with that of the circle; but as this is never used in the Elements, I have thought proper to reject it likewise.

27. An obtuse angled triangle is that which has an obtuse angle.

28. An acute angled triangle is that which has three acute angles.

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

30. An oblong is that which has its angles right angles, but all its sides not equal.

31. A rhombus has its sides equal, but its angles not right angles.

32. A rhomboid has its opposite sides and
angles equal to one another, but all

its sides are not equal, nor its angles
right angles.

ᄆ

33. All other four sided figures besides these are called trapeziums.

34. Parallel right lines are those which are in the same plane, and being infinitely produced either way, do not meet one another.*

POSTULATES.

1. Grant, that a right line may be drawn from any one point to any other point.

2. That a finite right line may be produced directly forwards.

3. That a circle may be described with any distance and from any centre.

4. That all right angles are equal to one another.† 5. That if a right line falling on two right lines make

the interior angles at the same parts less than two right angles; these right lines being continually produced shall meet on that side where the angles are less than two right angles.

6. That two right lines cannot inclose a space.

*Newton in lemma 22, book 1, of his Principia, says, that parallels are such lines as tend to a point infinitely distant.

+ For a demonstration of this, see Legendre's Geometry, proposition 1, book 1.

AXIOMS.

1. Things which are equal to the same 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 double of the same, are equal to one another.

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

8. Things which mutually agree with one another, are equal to one another.

9. The whole is greater than its part.

PROPOSITION I.

PROBLEM.

Upon a given finite right line to describe an equilateral triangle

Let AB be the given finite right line; it is required upon AB to describe an equilateral triangle. From the centre A with the distance AB de

scribe the circle BCD:a and again from the centre B, with the dis

tance BA, describe the circle ACE, D and from the point c in which the circles cut one another, draw the right lines CA, CR, to the points A, B.

b

a Post. 3.

b Post. 1.

Therefore because A is the centre of the circle DBC, AC will be equal to AB. Again, because в is the centre Def. 15. of the circle CAE, BC will be equal to BA: but it has been shown that CA is equal to AB: therefore CA, CB, are each of them equal to AB. And things which are equal to the same are equal to one another. Whence ca is equal to CB; wherefore the three, CA, AB, BC, are equal to one another; and, consequently, the triangle ABC is equilateral, and it is described upon the given finite right line AB. Q. E. F.

PROPOSITION II.

PROBLEM.

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

Let A be the given point, and BC the given right line : it is required to draw from the point a a right line equal to the given right line BC. Draw the right line ac from the point A to c, and upon it describe the equilateral triangle DAC, and produce the right lines DA, DC, to E and F, and with centre

C,

a

and distance BC, describe the

K B

D

a 1. 1.

G

Eb Post. 2.

circle BGH.C Again, with centre D, and distance DG, Post. 3. describe the circle GKL: therefore because the point c

is the centre of the circle BGH, BC will be equal to Def. 15. CG. Again, because D is the centre of the circles GKL,

• Ax. 3.

a 2.1.

DL will be equal to DG and DA DC parts of them are equal: therefore the remainder AL is equal to the remainder CG. But it has been shown that BC is equal to CG. Wherefore each of them, AL, BC, is equal to CG. And things which are equal to the same thing are equal to one another. Whence AL is equal to BC. Therefore from a given point, AL has been drawn, &c. g. E. F.*

PROPOSITION III.

PROBLEM.

Two unequal right lines being given, to cut off from the greater a part equal to the less.

Let AB and c be two unequal given right lines of which AB is the greater: it is required to cut off from the greater, AB, a right line equal to

a

F

D

A

E B

c, the less. Draw from the point ▲ a right line, AD, equal to c; and from the centre, A, with the distance AD, b Post. 3. describe the circle DEF. And because A is the centre of the circle DEF, AD will be equal to AE. But AD is also equal to c. Therefore each of them, AE, C, will be equal to AD. Wherefore AE is also equal to c. Therefore two unequal right lines being given, &c.† Q. E. F.

Ax. 1.

PROPOSITION IV.
THEOREM.

If two triangles have two sides equal to two sides, each to each; and have also one angle equal to one angle, viz. that which is contained by the equal right lines: then shall the base of the one be equal to the base of the other; and

A,

* This proposition may be divided into a variety of cases according to the different positions of the point a, although the construction and demonstration will, in every respect, be the same. Proclus remarks that some performed it by taking the line AL with a pair of compasses; but he by no means approved of the method, as those who thus reason, he says, beg in the very beginning.

+ Some persons perform this proposition by taking the less line in the compasses, and with one leg in either extremity of the greater, cutting off with the other leg the part required: this, though correct in its operation, is certainly not geometrical, and would come rather under the class of postulates, than a demonstrable proposition.