by the figure, shall be to one another in the same ratio. Of such figures it may be demonstrated

1. That if any rectilineal figure whatever be inscribed in, or circumscribed about the one, a similar rectilineal figure may be inscribed in, or circumscribed about the other. (II.32.) 2. That their areas are in the duplicate ratios (or as the squares) of any two homologous lines, and their perimeters as those lines. (II. 28.) Again, according to this definition, solid figures will be similar, when a point may be found in each, and twe straight lines drawn from each of those points, in the same direction from one another, such, that any two straight lines which are drawn from the same points at equal angles to these, and terminated by the figure, shall be to one another in the same ratio; and of such figures it may be demonstrated

1. That if any polyhedron whatever is inscribed in, or circumscribed about the one, a similar polyhedron may be inscribed in, or circumscribed about the other.

2. That their solid contents are in the triplicate ratio (or as the cubes) of any two homologous lines, and their convex surfaces in the duplicate ratio (or as the squares) of those lines. (II. 28.)]

Similarly divided, straight lines said to be 57 Similarly placed, straight lines said to be, in similar figures

57

Simple locus. (See "Locus.") def. 107 Small are, in spherical geometry, is the arc of a small circle.

Small circle, of the sphere, def. (See "Spherical Geometry.")

184

def. 1

Solid Solid Geometry, is that part of geometry which treats of solid figures and lines in different planes. See "Polyhedron," &c. Solid angle, def. (also its "edges")

125

When said to be a right angle sch. 203 (a) If a solid angle is contained by three plane angles, any two of them are together greater than the third; and any one is greater than the difference of the other two 137

137

(b) The plane angles, which contain any solid angle, are together less than four right angles (c) If two solid angles are each of them contained by three plane angles, and if two of these and the included dihedral angle in the one, be equal to two and the included dihedral angle in the other, each to each; their other dihedral angles are equal, each to each, and the third plane angle of the one is equal to the third plane angle of the other cor. 157 (d) If two solid angles are each of them contained by three plane angles, and

(f) Given two of three plane angles con-
taining a solid angle, aud the dihedral
angle which they contain, to find the
third plane angle, and the other two
dihedral angles by a plane construc-
tion
cor. 157

(g) Every solid angle is measured by
the spherical surface described about
its angular point with a given radius,
and included between its planes

sch. 203 (h) The properties of solid angles are analogous to those of spherical triangles and polygons sch. 203

See "Right Solid Angle." Solid content. See "Content." Solid of revolution, is any solid which is generated by the revolution of a plane figure about an axis in the same plane.

(a) If an isosceles triangle revolve about
an axis, which passes through its ver-
tex, the convex surface generated by
the base is equal to the product of that
portion of the axis which is inter-
cepted by perpendiculars drawn to it
from the extremities of the base, and
the circumference which has for its
radius the perpendicular drawn from
the vertex to the base; and the solid
generated by the triangle is equal to
one-third of the product of this surface
by the same perpendicular
175
(b) If any triangle revolve about an
axis, which passes through its vertex,
the solid generated by the triangle is
equal to one-third of the product of
the surface generated by the base, and
the perpendicular drawn from the
vertex to the base
cor. 176
(c) If the half of any regular polygon
of an even number of sides revolve
about the diagonal, the whole convex
surface is equal to the product of the
diagonal and the circumference of a
circle which has the apothem for its
radius; and the whole solid content
is equal to one-third of the product of
this surface and the apothem 176
See also "Cylinder," Cone,"
"Sphere."

Sphere, def. (also its "radius," and "diameter") Is generated by the revolution of a semicircle about its diameter 127

four times the area of the generating circle cor. 178 =D, if D is the diameter of the sphere

cor. 178 the convex surface (or two-thirds of the whole surface) of the circumscribed cylinder cor. 178

(d) The whole solid content

one-third of the product of the radius and surface 178 one-third of the product of the radius and four times the area of the generating circle cor. 179 =D, if D is the diameter of the sphere cor. 179

two-thirds of the content of the circumscribed cylinder. cor. 179 (e) If a polyhedron be circumscribed about a sphere, the contents of the sphere and polyhedron will be to one another as their convex surfaces

cor. 179 The surfaces of any two spheres are as the squares of the radii, and their solid contents are as the cubes of the radii

179

(B) Of certain portions of the Sphere. Spherical segment, sector, orb, ungula, zone, lune def. 179, 180 These portions are generated by the revolution of certain portions of a circle 180 (a) If a semicircle revolve about its dia

meter, the zone, which is generated by any are of the semicircle, shall be greater than the surface generated by the chord of that arc, and less than the surface generated by the tangent of the same are which is drawn parallel to the chord, and terminated by the radii passing through its extremities (b) Every spherical zone is equal to the product of the circumference of the generating circle and that portion of the axis which is intercepted between its convex surface and base, or (if it be double-based) between its two

bases

180

181

(c) If a cylinder having the same axis be circumscribed about the sphere, any zone generated about that axis is equal to that part of the convex surface of the cylinder which is inter

cor. 181

181

(e) Every spherical sector is equal to one-third of the product of its base and the radius of the sphere; and the same may be said of every spherical pyramid [() In the same, or in equal spheres, any two sectors are to one another as their bases; and the same may be said of any two spherical pyramids] (g) Every spherical segment upon a single base is equal to the half of a cylin der having the same base and the same altitude, together with a sphere of which that altitude is the diameter

(0) Similar zones and lunes, and also the triangular or polygonal surfaces which are the bases of similar pyramids, are as the squares of the radii; and similar sectors, pyramids, segments, orbs, and ungulas, are as the cubes of the radii sch. 184, 207 Spheres, intersection and contact of sch. 151 Spherical Geometry, is that part of geometry which treats of figures and lines upon the surface of a sphere.

Great and small circles, axis, poles, and
polar distance of a circle, parallel cir-
cles, opposite points
def. 184
Spherical arc, spherical angle (right,
acute, and obtuse), spherical triangle,
polar triangle, spherical polygon def.
184, 185, 186

When two points are said to be joined

on the surface of a sphere def. 184 When two triangles are said to be symmetrical. def. 185 When two triangles are said to be similar sch. 207

(A) Of great and small Circles.

(a) Every plane section of a sphere is a circle, the centre of which is either the centre of the sphere, or the foot of the perpendicular which is drawn to its plane from the centre of the sphere

186

(b) The radius of a great circle is the same with the radius of the sphere; and the radius-square of a small circle is less than the radius-square of the sphere by the square of the perpendicular which is drawn to its plane from the centre of the sphere cor. 186 (c) Either pole of a circle of the sphere is equally distant from all points in the circumference of that circle, whe ther the direct or the spherical distance be understood 186 (d) Any circle of a sphere may be conceived to be described round either of its poles as a centre, with the spherical distance of that pole as a radius cor. 186 (e) The distances of any circle from its two poles are together equal to a semicircumference cor. 187 (f) Equal circles have equal polar distances; and conversely (g) Circles whose polar distances are together equal to a semicircumference are equal to one another cor. 187 (h) If a point be taken within a circle which is not its pole, of all the spherical arcs which can be drawn from it to the circumference, the greatest is that in which the pole is, and the other part of that are produced is the least; and, of any others, that which is nearer to the greatest is always greater than the more remote; and from the same point there can be drawn only two arcs that are equal to

187

(B) Of Spherical Angles.
(a) Any two spherical arcs may be pro-
duced to meet one another in two
points which are opposite to one
another
cor. 187

(b) If any number of angles are formed
at the same point, their sides, being
produced, shall pass through the op-
posite point, [and there form as many
angles equal to the former respectively,
each to each]
cor. 187
(c) A spherical angle has the same
measure with the dihedral angle of its
planes, and with the plane angle of
the tangents at the angular point
def. 185
(d) Every spherical angle is measured
by the spherical arc which is described
about the angular point as a pole, and
included between its legs; or, which is
equal to it, by the distance between
the poles of its legs
188
(e) The spherical angles which one arc
makes with another upon one side of
it, are either two right angles or are
together equal to two right angles 189
(f) If two spherical arcs cut one an-
other, the vertical or opposite angles
are equal
cor. 189
(g) All the spherical angles about any
the same point are together equal to
four right angles

(C) Of Spherical Triangles.

cor. 189

(a) If one triangle be the polar triangle

(e) The three sides of a spherical triangle (and, generally, all the sides of any spherical polygon) are together less than the circumference of a great circle cor. 189 (d) In every spherical triangle, the sum of the angles is greater than two and less than six right angles 189 (e) A spherical triangle may have two or three right angles, or two or three obtuse angles cor. 189 (f) If a spherical triangle has two of its sides equal to one another, the opposite angles are likewise equal, and conversely 190 (9) If one side of a spherical triangle be greater than another, the opposite angle is greater than the angle opposite to that other; and conversely 190 (h) If one side of a spherical triangle is produced, the exterior angle is less than the sum of the two interior and opposite angles; and the exterior angle is equal to or greater than, or less than either of the interior and opposite angles, according as the sum of the two sides not common to them is equal to, or less than, or greater than a semicircumference cor. 190 (1) Two spherical triangles have all their sides and angles equal, each to each, when they have,

1. Two sides, and the included angle equal:

or 2. Two angles and the interja cent side:

or 3. The three sides:

(e) Every spherical polygon is measured by the excess of the sum of its angles together with four right angles above twice as many right angles as the figure has sides cor. 196 (f) Spherical triangles which stand upon the same base, and between the same equal and parallel small circles, are equal to one another; and conversely, equal spherical triangles which stand upon the same base, and upon the same side of it, are between the same equal and parallel small circles (g) Of equal spherical triangles upon the same base, the isosceles has the least perimeter; and of spherical triangles upon the same base, and having the same perimeter, the isosceles has the greatest area 199, 200 (h) Of all spherical triangles which have the same two sides, the greatest is that in which the angle included by them is equal to the sum of the other two angles 200

198

Of which triangle it may be remarked:

1. The two sides are by less than a semicircumference, and the angle included by them is greater than a right angle.

2. The difference between the lune which has the same angle, and the triangle, has always the same surface, viz., a fourth of the surface of the sphere sch. 201 (1) A spherical polygon, which has all its sides equal, and all its angles equal, is greater than any other which has the same number of sides and the same perimeter sch. 201 (k) A spherical polygon, which has all its angles lying in the circumference of a circle, is greater than any other which has the same sides sch, 201