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29. The word, therefore, or hence, frequently occurs. these words, the sign.. is generally used.

To express either of

30. If the quotients of two pairs of numbers, or quantities, are equal, the A с

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Ꭰ ; then, A is to B as C to D. And the abbreviations of the proportion is, A : B :: C: D; it is sometimes written A: B=C: D.

quantities are said to be proportional: thus, if

B

DEFINITIONS.

1. "A POINT is that which has position, but not magnitude." (See Notes.)

2. A line is length without breadth.

"COROLLARY. The extremities of a line are points; and the intersections "of one line with another are also points."

3. "If two lines are such that they cannot coincide in any two points, with"out coinciding altogether, each of them is called a straight line." "COR. Hence two straight lines cannot inclose a space. Neither can two "straight lines have a common segment; that is, they cannot coincide "in part, without coinciding altogether."

4. A superficies is that which has only length and breadth.

'COR. The extremities of a superficies are lines; and the intersections of

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one superficies with another are also lines."

5. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

6. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

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N. B. When several angles are at one point B, any one of them is ex'pressed by three letters, of which the letter that is at the vertex of the an'gle, that is, at the point in which the straight lines that contain the angle 'meet one another, is put between the other two letters, and one of these 'two is somewhere upon one of those straight lines, and the other upon the 'other line: Thus the angle which is contained by the straight lines, AB, 'CB, is named the angle ABC, or CBA; that which is contained by AB,

*The definitions marked with inverted commas are different from those of Euclid.

'BD, is named the angle ABD, or DBA; and that which is contained by 'BD, CB, is called the angle DBC, or CBD; but, if there be only one an'gle at a point, it may be expressed by a letter placed at that point; as the angle at E.'

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7. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other, is called a perpendicular to it.

8. An obtuse angle is that which is greater than a right angle.

9. An acute angle is that which is less than a right angle.

10. A figure is that which is enclosed by one or more boundaries.-The word area denotes the quantity of space contained in a figure, without any reference to the nature of the line or lines which bound it.

11. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

12. And this point is called the centre of the circle.

13. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

14. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

15. Rectilineal figures are those which are contained by straight lines.

16. Trilateral figures, or triangles, by three straight lines.

17. Quadrilateral, by four straight lines.

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

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

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

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21. A scalene triangle is that which has three unequal sides.
22. A right angled triangle is that which has a right angle.

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

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

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

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

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

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

29. All other four sided figures besides these, are called trapeziums. 30. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.

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

AXIOMS.

1. THINGS which are equal to the same thing 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.

S. 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.

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.

10. All right angles are equal to one another.

11. "Two straight lines which intersect one another, cannot be both pa"rallel to the same straight line."

PROPOSITION I. PROBLEM.

To describe an equilateral triangle upon a given finite straight line.

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

From the centre A, at the distance AB, describe (3. Postulate) the circle BCD, and from the centre B, at the distance BA, describe the circle ACE; and from the point C, in which the circles cut one another, draw the straight lines (1. Post.) CA, CB to the points A, B; ABC is an equilateral triangle.

Because the point A is the centre of the circle BCD, AC is equal

(11. Definition) to AB; and because the point B is the centre of the circle ACE, BC is equal to AB: But it has been proved that CA is equal to AB; therefore CA, CB are each of them equal to AB; now things which are equal to the same are equal to one another, (1. Axiom); therefore CA is equal to CB; wherefore CA, AB, CB are equal to one another; and the triangle ABC is therefore equilateral, and it is described upon the given straight line AB.

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B

PROP. II. PROB.

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 the point A, a straight line equal to BC.

From the point A to B draw (1. Post.) the straight line AB; and upon it describe (1. 1.) the equilateral triangle DAB, and produce (2. Post.) the straight lines DA, BD, to E and F; from the centre B, at the distance BC, describe (3. Post.) the circle CGH, and from the centre D, at the distance DG, describe the circle GKL, AL is equal to BC.

K

H

D

B

E

G

Because the point B is the centre of the circle CGH, BC is equal (11. Def.) to BG; and because D is the centre of the circle GKL, DL is equal to DG, and DA, DB, parts of them, are equal; therefore the remainder AL is equal to the remainder (3. Ax.) BG: But it has been shewn that BC is equal to BG; wherefore AL and BC are each of them equal to BG; and things that are equal

F

E

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