4. MATHEMATICS. 
circumference, as ab (pl. 1, fig.40). Every chord cuts from a circle a segment, 
as bac (fig. 41). The part of acircle included between two radii and an 
are is called a sector (fig. 42). The angle formed by two radii is called a 
centre angle, as bac (fig. 42). An angle formed by two chords meeting 
in the line of the cireumference, is called an inscribed angle (ig. 44). 
When a straight line, produced at pleasure, touches a circumference onlv 
at one point, it is called a tangent, e. g. be ( fig. 46); any such straight 
line, however, which either immediately or when produced cuts the 
circumference in two points, is caued a secant, ab (fig. 46). A 
rectilineal figure is said to be inscribed in a circle when all its sides are 
chords (fig. 45); it is said to be circumscribed about a circle when all its 
sides are tangents (fig. 54). Two or more circles are said to be concentric 
when they have a common centre (ig. 47); circles of different centres are 
excentric. Two excentric circles touch one another when their circum- 
ferences have only one point in common: this point may be either on the ex- 
terior ( fig. 53) or on the interior of the circumference. In the first case the 
sum, in the second the difference of their radii, will be the distance between 
their centres; in both cases the centres and the point of tangency will be in 
the same straight line. Two excentric circles cut each other (fig. 48) 
when their circumferences have two points in common. LEach point of 
intersection forms a triangle with the two centres. 
2. OF THE POSITION OF STRAIGHT LINES IN THE SAME PLANE. 
Only one straight line can be drawn between two given points, so that 
the position and direction of a line is completely determined by these points. 
On the other hand, innumerable curved lines are possible between two 
points. A straight line is the shortest distance between two given points. 
Hence it follows that in a triangle each of the sides is less than the 
sum, but greater than the difference of the two others. If two triangles 
have the same base, so that the one lies entirely within the other, the outer 
has a greater perimeter than the inner (pl. 3, fig. 82). Two straight 
lines on one plane may intersect each other, either directly or when 
produced ; they can, however, have but one point in common, or they may 
never meet even when produced. In the first case, they converge and form 
an angle; in the second, they are parallel. If two lines, whether parallel or 
not, as kl, mn, or op, qr (fig. 1), are intersected by a third straight line, 
st, then there will be eight angles formed,—four internal and four external : 
of these, the two internal or two external angles which lie on opposite 
sides of the secant line, and are not ailjacent to each other, are called alternate 
angles, as a@andh, cand f, cand u, mand n. ‘Then again, two angles, one inter- 
nal and the other external, lying on the same side of the secant without being 
adjoining, are called opposite angles; e.g.a andi, c and g,k and 0, m and n. 
When parallel lines are intersected by a third straight line, each two of the 
alternate angles, as well as of the opposite, are equal to each other. In every 
triangle, the sum of all the angles is equal to two right angles. The angles 
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