2 MATHEMATICS. 
solids. Geometry 1s again divided into the lower and higher, of which the 
former treats of rectilineal figures and the circle, of bodies bounded by 
planes, and finally of the cylinder, cone, and sphere ; the latter of curved 
lines, of surfaces inclosed by them, and of the solids and curved surfaces 
which they generate. 
I. PLANIMETRY OR PLANE GEOMETRY. 
1. GENERAL IDEAS. 
Lines, as already mentioned, are divided into straight (pl. 1, fig. 1) and 
curved (fig. 2). A broken line (fig. 3) cannot be said to be a particular 
species of line, but only a combination of several straight lines. A mixed 
line (fig. 4) is the union of straight and curved. The idea of horizontal 
and vertical lines (figs. 5, 6) is essentially foreign to pure Geometry. In 
applied or practical Geometry we call a line horizontal, when it runs in the 
same direction with the plane of the horizon, or the surface of still water, 
sometimes termed level ; and vertical or perpendicular, when it cerresponds 
to the direction of the plumb-line, or that of a string to which hangs a 
freely suspended weight ; every other straight line is called slanting or 
oblique (fig. 7). Two straight lines in the same plane are said to be 
parallel (fig. 8) when they never meet however far they may be produced, 
or when they are everywhere equally distant from each other. Curved 
lines also, if they possess the latter property, are sometimes called parallel, 
although this extension of the idea is hardly allowable. 
Two straight lines, when they meet in a point (fig. 9), form an angle 
with each other: this is the name which is given to their inclination or 
separation. The lines are called the sides, and the point where they meet 
the vertex. Any angle, even if it has both sides curved (jig. 10), or one 
side curved and the other straight (fig. 11), may still be reduced to a recti- 
lineal angle. In elementary Geometry, all angles are rectilineal. If one of 
the sides of an angle is extended beyond the vertex, a second angle is 
formed, which is called the adjacent angle of the first. If the two adjacent 
angles are equal (fig. 7), each is called a right angle (also fig. 12). All 
right angles are equal; and on that account they are used as a standard by 
which to measure other angles. Every angle smaller than a right angle is 
called acute (fig. 18), and every angle that is larger is called obtuse 
(fig. 14). Two or more angles that have a common vertex, and lie on the 
same side of a straight line in such a manner that this line constitutes one 
side of the first angle and one of the last, are altogether equal to two 
right angles. The angles about a point are together equal to four right 
angles. 
A flat space bounded by lines is called a figure. A figure is called recti- 
lineal if it is bounded by straight lines ; if by curved lines, curvilineal ; and 
it is a mixed figure when it is inclosed by both straight and curved lines. 
Plane Geometry deals only with plane figures (figures that lie in a plane 
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