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has all its angles right angles, but not all its sides, only the opposite ones, equal to each other, as fig. 2.

(3) A rhombus, or lozenge, is a parallelogram, which has all its sides equal, but none of its angles is a right angle, as fig. 3.

(4) A rhomboid is a parallelogram, which has its opposite sides, and not all its sides, equal; and none of its angles is a right angle, as fig. 4.

A parallelogram is generally denoted or expressed by by giving the four letters in order which are placed at the four angular points. Thus the parallelogram here traced would be called the parallelogram ABCD, or ADCB, or B BADC, &c. whichever we please.

14. A diagonal, or diameter, of a parallelogram is the straight line joining two of its opposite angular points. Thus AC, and BD are the diagonals, or diameters, of the parallelogram ABCD, in the preceding fig.

Also the side BC, upon which the parallelogram may be supposed to stand, is sometimes called its base.

15. A plane surface bounded by four straight lines of which two only are parallel, is called a trapezium, as ABCD, where AD is parallel to BC, but AB is not parallel to CD.

B

D

16. CIRCLES. A circle is a plane surface bounded by a curved line, such that every point in this line is equally distant from a certain point within the figure called the centre of the circle.

The curved boundary is called the circumference of the circle; and the straight line which measures the distance from the centre to the circumference is called the radius of the circle. Any straight line drawn through the centre and terminated both ways by the circumference is called a diameter of the circle.

Thus, in the fig. here traced, the area or surface in the

plane of the paper bounded by the curved line ABCDA is a circle, when from the centre O all straight lines to the circumference, as OA, OB, OC, OD, are equal to each other.

Any one of the lines OA, OB, OC, OD is the radius, and any radius, as AO, extended in the same straight line to meet the circumference in D, that is AD, is a diameter, of the circle.

D

17. Hence it is plain, that a circle may be traced by means of a string, one end of which is kept fixed in a certain point as the centre, while the other is made to revolve and trace out the circumference, the string being kept perfectly tight. The same thing is also done by the ordinary compasses.

18. A semi-circle is the half of a circle, bounded by the half of the circumference and the diameter joining its extremities.

A quadrant is the quarter of a circle, or the half of a semi-circle, bounded by the fourth-part of the circumference and two radii joining its extremities with the

centre.

Thus fig. 1 is a semicircle, and fig. 2 is a quadrant,

c

C

2

A

B

where O is the centre of the circle in each case; and whilst ACB is half of the whole circumference in the former, it is a quarter of it in the latter.

An arc of a circle is a portion of the circumference.

It may be observed here, that although two letters are sufficient to express a straight line, three or more are generally required for a curved line; and for an obvious reason, because between any two points there is only one straight line, but an infinite number of crooked lines, so that the extreme points entirely determine the former but not the latter.

19. It will be found, hereafter, that we often, for shortness, call the circumference of a circle the circle, which, though convenient, is not a correct way of speaking. In the same manner it is not unusual to hear persons speak of a triangle, square, or other plane surface, when, in fact, they mean no more than the boundary of the figure in each case.

Let it, then, be borne in mind, that in strictness a circle does not consist of one curved line merely, called the circumference, but that it is the whole inner area bounded by that line.

So, again, a triangle does not consist of three straight lines called sides, but is the whole inner area bounded by those sides. And similarly with respect to other plane surfaces.

20. EUCLID's 'Postulates' must now be admitted as truths to be granted without proof, viz.

I. A straight line may be drawn on a given plane surface from any one point to any other point.

II. A terminated straight line may be 'produced', that is, extended, to any length in a straight line.

III. A circle may be described' with any centre, and any given length, or line, for its radius.

Granted that we can do these three things, and we will assume nothing further in the construction and treatment of Geometrical figures.

In 'describing' a circle, by the third Postulate, we trace out the circumference which is the boundary of the circle. Of course we can trace a part, as well as the whole, that is, any arc of the circle.

21. EQUALITY of LINES, AREAS, and ANGLES.

It is evident that magnitudes which coincide in every part are equal to one another. This is a received axiom which admits of no dispute. It is the simplest notion we have of equality.

Hence the two straight lines AB, and CD, are equal to one another, if, when CD is placed upon AB, so that the point C is

upon

A and CD upon AB, the point Dis found to coincide with the point B.

A

C

B

D

In like manner two areas are equal to one another, when they can be made to coincide in every part, that is, when one can be made exactly to cover the other, and no more. For example, all the pages of this book are exactly equal to one another. But areas may be equal also, when they are not exactly alike, (as the pages of the same book are,) but can be made so by a different arrangement of the parts of one or both. For it is evident, that this page might be cut up into many parts, without at all altering the total area; and those parts might be arranged so as to form a great variety of plane figures having precisely the same area, but with a different boundary. Thus, if we have a square and a triangle, and we can cut up the square, so as with the parts exactly to cover the triangle, the area of the square is equal to that of the triangle. Or, again, two triangles, which to the eye appear unequal, may yet be equal, and shall be so, if by a different arrangement of parts they can be made to coincide.

The Equality of Angles has been already defined in (8).

22. ADDITION, SUBTRACTION, &c. of LINES, AREAS, and ANGLES.

A

It follows from (21) that lines, areas, and angles, may be added together, subtracted from each other, multiplied, or divided, like other magnitudes. Thus c

D

B

E

P

if AB, CD be two straight lines equal to one another, 'produce' CD indefinitely towards D, then by applying AB to it so that A is upon D, and AB upon DE, we find DE equal to AB, and .. CE is plainly twice AB. Again, if ÊF= AB, then CF = three times AB, and so on. And thus we multiply the line AB. Obviously also CD is one-third of the line CF; that is, a line may be divided. Again, that lines may be added or subtracted is plain enough, for AB + CD=CD+DE = CE; and AB taken from CE = CD.

The same principle, viz. that 'magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another', leads to the conclusion that, in like manner areas and angles may be added, subtracted, multiplied, or divided.

Thus, for instance, if AB, BD be in the same straight line, so that ABC, BCD, ACD are three distinct triangles, it is plain that the two areas ABC, BCD, exactly cover the same space as the area ACD, and . the two areas ABC, BCD may be added together, and their sum will be the area, or triangle, ACD.

Similarly, if the area BCD be subtracted from the area ACD, the difference is the area ABC.

B

Again, if area ABC= area BCD, then area ACD is double of the area ABC; and area ABC-half of area ACD. Angles likewise are magnitudes which may be added, subtracted, &c. Thus, ACB+ BCD 2 ACD. And 4 BCD taken from ACD leaves ACB. Also if < ACB = 4 BCD, then ▲ ACD is double of 4 ACB.

L

L

QUESTIONS ON THE PRECEDING DEFINITIONS, &c.

(1) What does Geometry treat of? To what properties of bodies is it restricted?

(2) Define a 'line'; can it be exhibited in practice? If not, why not?

(3) How many different kinds of lines are there? Give an example of each.

(4) Define a point'. Can it be exhibited to the eye? If not, why not? Give an example of a 'point'.

(5) Define 'superficies', 'surface', or 'area'. Give an example.

(6) How many kinds of 'surfaces' are there? Give an example of each. By what general rule are they distinguished from each other?

(7) What is meant by the line AB'? Is it the same as the line BA?

(8) What is an 'Angle'? Is it a magnitude admitting of increase or decrease? Exhibit two angles; and say which is the greater, and why.

(9) What is meant by the angle ABC'? Is it the same as 'the angle CBA'? Is it the same as 'the angle ACB'?

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