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PART FIRST.

SECTION I.-The properties of straight lines and circular lines.

6. Between two assumed points (as A and B in

Fig. 1. Jog. 1), several lines inay be drawn. Among these there

Fig. 2.

may be one which tends the same way in all its parts;
this is the straight line. We therefore define a straight
line to be, one which has the same direction throughout its
whole extent. It is evident, that the straight line is the
shortest which can be drawn between two points. It is
also evident, that between two points only one straight line
can be drawn.
7. A line which continually changes its direction is
called a curve line.
Curve lines admit of great variety in the character of
their curvature, whereas among straight lines, the only
variety is that of length.
Of the various curves, the only one whose properties
are made the subject of elementary geometry, is the cir-
cular curve. This is the most simple of all curves, its
curvature being the same in every part. It is described
by one extremity of a straight line, revolving about the
other extremity which remains fixed. In fig. 2, if the
straight line AC be made to revolve about the point C
which does not change its position, the point A will de-
scribe the circular curve ABDA. This line, in refer-
ence to the circle ABDA, is called its circumference ; and
it is manifest that every point in this curve is equally dis-
tant from the point C, called the centre of the circle. The
lines AC, BC, DC, &c, which measure this distance, are
called radii of the circle. A straight line passing
through the centre and terminating in the circumfer-
ence, is called a diameter. Any portion of this curve,
as BD, is called an arc of the circle.
In the same circle are the radii equal 7 and why
How does the diameter compare with the radius'
8. A surface to which a straight line can be applied
in every direction, so as to touch the surface through the
whole extent of the line, is called a plane surface, or simply
a plane. A surface which is neither a plane surface nor
composed of planes, is called a curve surface. Such is
the surface of a ball, a roll of paper, &c. -

9. Tico straight lines which have the same direction in space, are called parallel lines (fig. 3). As they have Fig. * the same direction, they can neither approach towards, nor diverge from each other ; parallel lines, therefore, must be throughout at the same distance from each other ; and, however far produced, can never meet. 10. Where two straight lines taken in the same plane are not parallel, they must incline the one to the other, and are said to make an angle with each other (fig. 4). Fig. 4. An angle, then, is the inclination of one line to another. The magnitude of the angle depends upon the degree of inclination, and not at all upon the distance or length of the lines. * 11. The nearer two lines are to being parallel, the smaller the angle, which becomes nothing when the lines become parallel. It increases as the lines diverge ; and when this inclination is the same on both sides (as AB and CD, fig. 5), the angles are called right angles. Fig. 5. In this case, the line AB is said to be perpendicular to the line CD. CD is also perpendicular to A.B. When two lines are neither parallel nor perpendicular to each other, they are said to be oblique to each other. Remark. We have, then, in geometry, four kinds of magnitudes. 1. That of a line. 2. That of a surface. 3. That of a body. 4. That of an (inclination or) angle. Angles may be added, subtracted, multiplied, and divided, like other magnitudes or quantities. 12. Two lines which form an angle in a plane, either meet, or being produced, will meet each other in some point, the point E, for instance, as in fig. 6. This point Fig. 6. of meeting is called the vertex of the angle. We express the angle, usually, by three letters, thus, the angle AEC. When there is only one angle at the same vertex, we express it by a single letter ; as the angle E (fig. 7), that Fig. 7. is, the angle whose vertex as at E. 13. Suppose the line AB perpendicular to CD (fig. Fig. S. 8) to move about the point B, into the position A/B, the angle A'BC will be greater than a right angle, and the angle A'BD, will be less than a right angle; but the amount of angular space is not changed by this movement; the angle ABA' is taken from one of the angles and added to the other ; the sum of the two angles is the same as before. Hence we say, that, when one straight line meets another straight line, the sum of the two angles is equal to two right angles. -

Fig. 8.

Fig. 10.

The sum of the angles being equal to two right angles, the exterior lines CB, BD, must be in the same straight line; otherwise something must be added to or subtracted from the angular space to make CBD a straight line ; which would make the sum of the two angles greater or less than two right angles. . . Remark. An angle which is less than a right angle is called an acute angle ; as A/BD. An angle greater than a right angle is called an obtuse angle, as the angle A/BC. Both obtuse and acute angles are called oblique angles. 14. If other lines were drawn to the same point B, on the same side of CD, as in figure 9; the whole amount of angular space, would evidently be neither increased nor diminished by it ; but would still be equal to two right angles. And we hence derive this general truth, that, I When a straight line is met by several other straight lines, in the same point, the sum of all the angles on the same side of it, is equal to two right angles. i 15. If all these lines be produced through B, it is manifest that the angular space below the line CD, will be cqual to the angular space above ; therefore, The sum of all the angles made by several straight lines diverging from the same point, is equal to four right angles. 16. As a straight line has the same direction in all its parts (6), two straight lines must have the same inclination in every part (fig. 10); therefore if they intersect each other, the angles which are opposite, at the vertex, (called vertical angles,) are equal. Each of the two angles AEC and DEB, expresses the inclination of the two straight lines AB and CD, to each other ; they are therefore equal angles. If the two lines are perpendicular to each other, the two angles on the same side of either of them, are equal; but these are respectively equal to their vertical angles (16); it is therefore evident that all right angles

are equal.

Of Parallel Lines.

17. The properties of parallel lines are deduced from this fundamental proposition. Two straight lines which

Jave the same direction in space, must be equally inclined to the same straight line which meets them. This proposition results directly from the definition of a straight line. If two straight lines have different inclinations to the same straight line, that is, to the same direction, they must be inclined to each other, and therefore cannot be parallel. 18. Suppose the straight line AB (fig. 11) to intersect Fig. 11. the two parallel straight lines CD and FG ; it must have the same inclination to both. This inclination is expressed by either of the angles AEC, AHF; they are therefore equal. As these angles have correspondent situations in relation to these two intersections, they are called corresponding angles.* And as nothing in this analysis depends upon any particular inclination of AB to the parallels, we derive this general truth : When a straight line intersects parallel straight lines, the corresponding angles are equal. 19. The sum of the two angles AEC and CEH, is equal to two right angles (13); if instead of the angle AEC, we substitute its equal EHF (18), we shall have the sum of the angles CEH and EHF, equal to two right angles. These angles are called interior on the same side of the cutting line ; and we say that—When a straight line cuts two parallels, the sum of the two interior angles, on the same side, is equal to two rightangles. And as the sum of the four angles on the same side of the cutting line, or secant, must be equal to four right angles. The sum of the exterior angles on the same side, is equal to two right angles. 20. The angle EHF is equal to its corresponding angle AEC (18), and the angle DEH is equal to the angle AEC, being vertical with it (16). Therefore the two angles, DEH and EHF, (each equal to AEC,) are equal to one another. These angles are internal with respect to the parallels, but on alternate sides of the cutting line. Whence we derive the general truth,-A straight line o parallels, makes the alternate-internal angles equal. 21. The angle GHB is equal to the angle DEH (18), and therefore equal to AEC, vertical with DEH. These

* They are sometimes called internal-external angles.

Fig. 11.

Fig. 12.

two angles AEC and GHB, are exterior with respect to
the parallels, and on alternate sides of the secant ; they
are therefore called alternate-eaternal; and we have
this general proposition,-- When a straight line intersects
two parallel straight lines, the alternate-eaternal angles
are equal.
22. These five propositions may be summarily enun-
ciated as follows: When a straight line meets two par-
allel straight lines,
(1.) The corresponding angles are equal.
(2.) The sum of the two interior angles upon the
same side, is equal to two right-angles.
(3.) The sum of the two exterior angles on the same
side is equal to two right-angles.
(4.) The alternate-internal angles are equal to each
other ; and
(5.) The alternate-external angles are equal to each
other.
23. If we incline one of the parallels in question, FG
for instance, the other two lines remaining the same, it
will change the magnitude of each of the angles at H ;
and neither of the five propositions will be true. We have
therefore the converse of these propositions; and we say
that, when a straight line meets two straight lines, and
the corresponding angles are equal ; or the sum of the
two interior angles on the same side, or the sum of the
two exterior, is equal to two right angles ; or the alternate-
internal angles equal to each other, or the alternate-eater-
nal angles equal to each other, the two lines must be par-
allel.
Cor. If the two interior angles are less than two right-
angles, the lines incline towards each other, and will
meet on that side, if produced sufficiently far. If the
sum of the interior angles on one side is greater than
two right angles, the sum of those on the other side, is
less than two right-angles (13), and therefore the lines
will meet on that side, if produced sufficiently far.
24. Let the two angles ABC, DEF, (fig. 12) have the
side AB, parallel to DE, and BC parallel to EF; pro-
duce BC to I, and also produce DE till"it meet BI in H.
Then on account of the parallels EF and BI, the cor-
responding angles DEF and DHI, are equal (22); and
with reference to the parallels DH and AB, the corre-
sponding angles DHI and ABC are equal ; that is, ABC

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