841 
is supposed to stand erect, as AD and CD, 
fig. 16. 
39. Altitude of. a figure is its perpendicular 
height from the base to the highest part, as 
EF, fig. 16. 
f 40. Area of a plane figure, or other sur- 
face, means the quantity of space contained 
within its boundaries, expressed in square 
feet, yards, or any other superficial mea- 
sure. 
41. Similar figures are such as have the 
same angles, and whose sides are in the same 
proportion, as fig. 17. 
] 42. Equal figures are such as have the 
[ same area or contents. 
43. A circle is a plane figure, bounded by 
j a curve line returning into itself, called its 
j circumference, ABCD (fig. 18), every where 
equally distant from a point E within the cir- 
j cle, which is called the centre. 
44. The radius of a circle is a straight line 
drawn from the centre to the circumference, 
j as EF (fig. 18). The radius is the opening 
of the compass when a circle is described ; 
j and consequently all the radii ot a circle 
must be equal to each other. 
45. A diameter of a circle is a straight line 
. drawn from one side of the circumference to 
the other through the centre, as CB (fig. 18). 
j Every diameter divides the circle into two 
i equal parts. 
46. A segment of a circle is a part of a 
j circle cut off by a straight line drawn across 
it. This straight line is called the chord. A 
i segment may be either equal to, greater, or 
i less than, a semicircle, which is a segment 
| formed by the diameter of the circle, as 
1 CEB, and is equal to half the circle. 
47. A tangent is a straight line drawn so as 
j just to touch a circle without cutting it, as 
GH (fig. 18). The point A, where it touches 
i the circle, is called the point of contact. And 
a tangent cannot touch acircle in more points 
I than one. 
48. A sector of a circle is a space com- 
j prehended between two radii and an arc, as 
IK, fig. 19. 
49. lire circumference of every circle, 
| whether great or small, is supposed to be di- 
vided into 360 equal parts, called degrees; 
and every degree into 60 parts, called mi- 
j mites ; and every minute into 60 seconds 
To measure the inclination of lines to each 
other, or angles, a circle is described round 
the angular point as a centre, as IK, fig. 19 ; 
J and according to the number of degrees, mi- 
| notes, and seconds, cut off by the sides of the 
| angle, so many degrees, minutes, and se- 
1 conds, it is said to contain. Degrees are 
f marked by °, minutes by and seconds by " ; 
thus an angle of 48 degrees, 15 minutes, and 
7 seconds, is written in this manner, 48° 15' 
7". 
50. A solid is any body that has length, 
breadth, and thickness : a book, for instance, 
is solid, so is a sheet of paper ; for though its 
thickness is very small, yet it has some thick- 
ness. The boundaries of a solid are sur- 
faces. 
51. Similar solids are such as are bounded 
I by an equal number of similar planes. 
52. A prism is a solid, of which the sides 
[ are parallelograms, and the two ends or bases 
are similar polygons, parallel to each other. 
Prisms are denominated according to the 
number of angles in the base, triangular 
prisms, quadrangular, heptangular, and so 
Vol. I. 
GEOMETRY, 
on, as figs. 20, 21, 22, 23. If the sides are 
perpendicular to the plane of the base, it is 
called an upright prism ; if they are inclined, 
it is called an oblique prism. 
53. When the base of a prism is a paral- 
lelogram, it is called a parallelopipedon, as 
figs. 22 and 23. Hence a parallelopipedon 
is a solid terminated by six parallelograms. 
54. When all the sides of a parallelopipe- 
don are squares, the solid is called a cube, as 
fig. 23. 
55. A rhomboid is an oblique prism, whose 
bases are parallelograms (fig. 24). 
56. A pyramid (figs. 25 and 26) is a 
solid bounded by, or contained within, a 
number of planes, whose base may be any 
polygon, and whose faces are terminated in 
one point, B, commonly called the vertex of 
the pyramid. 
57. When the figure of the base is a trian- 
gle, it is called a triangular pyramid ; when 
the figure of the base is a quadrilateral, it is 
called a quadrilateral pyramid, &c. 
58. A pyramid is either regular or irregu- 
lar, according as the base is regular or irre- 
gular. 
59. A pyramid is also right or upright, or 
it is oblique. It is right, when a line drawn 
from the vertex to the centre of the base, is 
perpendicular to it, as fig. 25 ; and oblique, 
when this line inclines, as fig. 26. 
60. A cylinder is a solid (figs. 27 and 28), 
generated or formed by the rotation of a 
rectangle about one of its sides, supposed to 
be at rest : this quiescent side is called the 
axis of the cylinder. Or it may be conceived 
to be generated by the motion of a circle, in 
a direction perpendicular to its surface, and 
always parallel to itself. 
61. A cylinder is either right or oblique, 
as the axis is perpendicular to the base or in- 
clined. 
62. Every section of a right cylinder taken 
at right angles to its axis, is a circle ; and 
every section taken across the cylinder, but 
oblique to the axis, is an ellipsis. 
63. A circle being a polygon of an infinite 
number of sides, it follows, that the cylinder 
may be conceived as a prism, having such 
a polygon for bases. 
64" A cone is a solid (figs. 29 and 30), hav- 
ing a circle for its base, and its sides a con- 
vex surface, terminating in a point A, called 
the vertex, or apex of the cone. It may be 
conceived to be generated by the revolution 
of a right-angled triangle about its perpendi- 
cular. 
65. A line drawn from the vertex to the 
centre of the base is the axis of the cone. 
66. When this line is perpendicular to the 
base, the cone is called an upright, or right 
cone ; but when it is inclined it is called an 
oblique cone. 
67. If it be cut through the axis from the 
vertex to the base, the section will be a tri- 
angle. 
68. If a right cone be cut by a plane at 
right angles to the axis, the section will be a 
circle. 
69. If it be cut oblique to the axis, and 
quite across from ofie side to the other, the 
section will be an ellipsis, as fig. 31. A sec- 
tion of a cylinder jnade in the same manner 
is also an ellipsis ; and that is easily conceiv- 
ed : but it does not appear so readily to most 
people, that the oblique section ot a cone is 
an ellipsis : they frequently imag'ne that it 
will be wider at one end than the other, or 
what is called an oval, which is the shape of 
an egg. But that this is a mistake, any one 
may convince himself by making a cone, 
and cutting it across obliquely ; it will be 
then seen that the section, in whatever di- 
rection it is taken, is a regular ellipsis ; and 
this is the case, whether the cone be right or 
oblique, except only in one case in the ob- 
lique cone; w hich is when the section is taken 
in a particular direction, which is called sub- 
contrary to its base. 
70. W hen the section is made parallel to 
one of the sides of the cone, as tig. 32, the 
curve ABC, which bounds the section, is 
called a parabola. 
71. When the section is taken parallel to 
the axis, as fig. 33, the curve is called an hy- 
perbola. 
These curves, which are formed by cutting 
a cone in different directions, have various 
properties, which are of great importance in 
astronomy, gunnery, perspective, and many 
other sciences. 
72. A sphere is a solid, terminated by a 
convex surface, every point of which is at an 
equal distance from a point within, called the 
centre, fig. 34. 
73. It may be conceived to be formed by 
making a semicircle revolve round its diame- 
ter. This maybe illustrated by the process 
of forming a ball of clay by the potter’s 
wheel, a semicircular mould being used for 
the purpose. The diameter of the semicircle 
round which it revolves, is called the axis of 
the sphere. 
74. The ends of the axis are called poles. 
75. Any line passing through the centre of 
the sphere, and terminated by the circumfer- 
ence, is a diameter of the sphere. 
76. Every section of a sphere is a circle ; 
every section taken through the centre of the 
sphere is called a great circle, as AB, fig. 34 ; 
every other is a lesser circle, as CD. 
77. Any portion of a sphere cut off by a 
plane is called a segment ; and when the plane 
passes through the cent.e, it divides the 
sphere into two equal parts, each of which is 
called a hemisphere. 
78. A spheroid is a solid (fig. 35), gene- 
rated by the rotation of a semi-ellipsis about 
the transverse or conjugate axis; and the 
centre of the ellipsis is the centre of the sphe- 
roid. 
79. The line about which the ellipsis re- 
volves is called the axis. If the spheroid be 
generated about the conjugate axis of the 
semi-ellipsis, it is called a prolate spheroid. 
SO. It the spheroid be generated by the 
semi-ellipsis by revolving about the trans- 
verse axis, it is called an oblong spheroid. 
8 1 . Every section of a spheroid is an el- 
lipsis, except when it is perpendicular to that 
axis about which it is generated ; in which 
cases it is a circle. 
82. All sections of a spheroid parallel to 
each other, are similar figures. 
A frustum of a solid, means a piece cut 
off from the solid by a plane passed through 
it, usually parallel to the base of tin solid, as 
the frustum of a cone, a pyramid, &c. 
There are a lower and an upper frustum, 
according as the piece spoken of does or does 
not contain the base ot the solid. 
83. Ratio is the proportion which one mag- 
nitude bears to another of the same kind, 
