10 MATHEMATICS. 
shape, e. g. the triangle abe (fig. 61) into a quadrilateral similar to the 
given quadrilateral defg. 7. To divide a triangle into a certain number 
of equal parts by lines proceeding from one angle. 8. To cut off from a 
triangle a certain portion, as for instance a third, by means of a line which 
is to proceed from a given point, d, in one of its sides (fig. 62).. 9. Toseut 
off from a triangle, abc, a certain part, as a third, by lines proceeding from 
a given point, d, within the triangle (fig. 63). 10. From a triangle, abe, 
to cut off a certain portion, by a line parallel to one of the sides (fig. 64). 
11. To divide a parallelogram into a given number of equal parts, by lines 
parallel to one of its sides. 12. From an acute angled parallelogram to cut 
off a given part by a line perpendicular to two of the sides (pl. 3, fig. 65). 
13. To divide a parallelogram, abcd, into a certain number of equal parts, 
by lines proceeding from a given point in one of the sides (fig. 66). 
14. From a trapezoid, abcd, to cut off a given part, for instance the half, 
by a line parallel to its parallel sides (fig. 67). 15. To cut off from any 
quadrilateral, abcd, a given part, by a line proceeding from a corner, a, or 
from a given point, e, in one of its sides (figs. 68, 69). 
6. OF THE CIRCLE AND ITS MEASUREMENT. 
A circular line cannot have more than two points in common with a 
straight line (fig. 70). A straight line intersects or touches the circle, 
according as it has two points in common with the circumference, or only 
one; in either case we must consider the line as indefinitely produced in 
either direction. We obtain a tangent, when we draw a perpendicular to 
the extremity of a radius or diameter (fig.71). On the other hand, a 
radius drawn to the point of tangency of a tangent, will be perpendicular 
to it; whence it follows, that to any point of a circumference only one 
tangent can be drawn. Lines drawn from the same point, tangent to a 
circumference, are equal to each other, e. g. su = sv in fig. 72. 
Equal angles at the centre of the same circle, or of equal circles, have 
equal chords and areas, and the reverse. An angle at the centre is mea- 
sured by the number of degrees contained by its arc. An inscribed angle 
is half the angle at the centre of the same arc, and is therefore measured 
by the half of its are. An angle formed by a tangent and a chord is mea- 
sured by half the are included between the tangent and the chord (jig. 78). 
Inscribed angles resting upon the same or upon similar ares are equal 
( fig. 75). When two chords intersect each other, either within the circle, 
or when produced, without it, the angle thus formed is measured in the 
first case by half the sum, and in the second by half the difference of the 
two arcs included between the chords (fig. 74). Every angle inscribed in 
a semicircle is a right angle (fig. 77). If at-any given point of a diameter 
a perpendicular be drawn to the circumference, it will be a mean propor- 
tional to the two segments of the diameter (fig. 78). 
From the preceding propositions may be obtained the solution of the 
following problems: 1. To find the centre of a circle or of a circular are 
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