121

121

two-thirds of a cylinder upon the same base and of the same altitude, cor. 179]. Hendecagon (figure of 11 sides) regular, to inscribe in a circle, very nearly Heptagon (figure of 7 sides) regular, to inscribe in a circle, very nearly Hexagon (figure of 6 sides) regular. "Regular Polygon" and "Circle." Hexahedron (solid contained by 6 planes) regular. See "Regular Polyhedrons." Homologous edges of similar polyhedrons, are the homologous sides of their similar faces.

See

Hypotenuse of a right-angled triangle def. 3 See "Triangle."

[The side opposite to the right angle is called the Hypotenuse also in rightangled spherical triangles.]

Icosa-lodecahedron, a solid derivable either from the icosahedron, or from the dodecahedron

163

Icosahedron (solid contained by 20 planes) regular. See "Regular Polyhedrons." Inclination of a straight line to a straight line, is the acute angle which the former makes with the latter.

of a plane to a plane, is the acute dihedral angle, which the former makes with the latter.

of a straight line to a plane

def. 125 Incommensurable, magnitudes said to be 32 (a) If one magnitude is incommensurable with another, it is incommensurable with every magnitude which is commensurable with that other

cor. 37 (b) Although the ratio of two incommensurable magnitudes can never be exactly expressed by numbers, yet two numbers may be, obtained which shall express it to any required degree of

exactness

48

(c) Magnitudes are incommensurable, when the process for finding the greatest common measure leads to no conclusion, but has an unlimited number of steps cor.35 (d) If P and Q are two magnitudes of the same kind, and if Q be contained in P any number of times with a remainder R, which is to Q as Q to P, P and Q are incommensurable

sch. 73 (e) The same being supposed, if n be the number of times that Q is contained in P, the ratio of Q to P shall lie between the ratios of any two successive terms of the series 1, n, a, b, c, &c., where a, the third term, is equal to nxn+1, b, the fourth term = n a +n, c, the fifth term nba, and so on, every successive term being

equal to n times the last, together with the last but one * sch. 73 (f) The parts of a line divided in medial ratio are incommensurables of this class, and their ratio may be approximated to by the series 1, 1, 2, 3, 5, 8, &c.: the side and the sum of the side and diagonal of a square are incommensurables of the same class, and their ratio may be approximated to by the series 1, 2, 5, 12, 29, &c. sch. 73 Infinite arc is an arc of unlimited extent; such as, for example, occurs in perspective projection, when any point of the original curve lies in the vertical plane. 211 The hyperbola affords an example of infinite arcs with asymptotes, the parabola of infinite arcs without asymptotes. Infinite extent. Straight lines and arcs of infinite extent may have finite projections, and vice versa 209, 210 Inscribed in a circle, a rectilineal figure said to be

79

(a) Two fixed magnitudes A, B, are said to be limits of two others P, Q, when P and Q, by increasing together, or by diminishing together, may be made to approach more nearly to A and B respectively, than by any the same given difference, but can never become equal to, much less pass A and B e.g. two circles are the limits of the similar inscribed or circumscribed regular polygons (b) If two magnitudes A and B are the limits of two others P and Q, and if P is always to Q in the same constant ratio, A is to B in the same ratio* 46 For examples of this theorem see 56, 96, 146, 171, 173.

46

Limits of a geometrical problem, are of frequent occurrence, and are commonly indicated by the construction: loci are useful in determining them 27, 106, 124 Line, (also straight line" "curved line.")

6.

def. 1

See "Curve," "Straight Line," and Projection."

Locus (Lat., place),

def. 106

[More generally, a locus is any part of space, every point of which, and none else, satisfies certain conditions.]

When said to be a simple locus, when

a plane locus, when of higher dimensions

107

(a) All points which are equidistant from two given points

107

107

(b) All points which are equidistant from two given straight lines [(c) The extremities of all equal parallels whose other extremities lie in the same given straight line, I. 16. cor.] (d) All points which divide lines falling from a given point to a given straight line in the same given ratio 108 (e) The vertices of all triangles upon the same base, which have the side terminated in one extremity of the base greater than the side terminated or between two parallel planes, or such

There are some very important errors in the latter part of the demonstration of this theorem, for the correction of which the reader is desired to consult the errata.

(c) The vertices of all triangles upon the same base, which have the side terminated in one extremity greater than the side terminated in the other extremity, and the sides (or which is the same thing, the squares of the sides) in a given ratio (d) The vertices of all triangles upon the same base, which have the square of one side in a given ratio to the square of the other side diminished by a given square sch. 110 (e) The vertices of all triangles upon the same base, which have the square of one side diminished by a given square in a given ratio to the square of the other side diminished by another given square sch. i10 Examples of Loci satisfying conditions in Solid Geometry.

[(a) All the points in space which are equidistant from two given points lie in the plane which bisects the distance between them at right angles.] [(6) All the points in a given plane, which are equidistant from two given points without the plane, lie in the common section with the given plane of the plane which bisects at right angles the distance between the two points.] (c) All the points which are equidistant from three given points, not lying in the same straight line, lie in the straight line which is drawn perpendicular to their plane from the centre of the circle which passes through them. cor. 151

[(d) All the points equidistant from a straight line and plane, or from two given planes, lie in the straight line or plane which bisects their angle of inclination.]

(e) The extremities of all equal parallels whose other extremities lie in one and the same plane, lie in a plane parallel to it. cor. 134

(f) The points which divide all straight lines drawn from a point to a plane,

(D) Examples of Loci satisfying conditions in Spherical Geometry.

[(a) All points upon the surface of a
sphere, which are equidistant from
two given points of that surface, lie
in the circumference which bisects
their distance at right angles.]
[(b) All points which are equidistant
from the circumferences of two given
great circles, lie in the circumference
which bisects the angle between them.]
(c) The vertices of all equal spherical
triangles upon the same base and
upon the same side of it, lie in the
circumference of a small circle, such
that one equal and parallel to it may
be drawn through the extremities of
the base
cor. 199

(d) The vertices of all spherical triangles upon the same base, which have the vertical angle equal to the sum of the other two, lie in the circumference of a small circle, whose pole is the middle point of the base, and its polar distance half the base sch. 201 Lowest terms of the ratio of two magnitudes, See "Numerical ratio." Lune, (Lat., moon) spherical . def. 180 See "Sphere" and "Spherical Geometry." Lunes (contained by circular arcs in the same plane) quadrature of.

[(a) If a semicircumference A B C DE be divided into any two arcs, ABC, CDE, and if upon the chords of these arcs semicircles are described, as in the adjoined figure, the lunes ABCb, CDE d shall be together equal to the triangle ACE.

For, semicircles (III.33.) being as the squares of their diameters, the semicircles upon A C and CE are together equal to the semicircle upon AE; therefore, taking away the segments ABC and C DE, the lunes which remain are together equal to the triangle ACE.]

[(6) The lune which is included by a semicircumference C d E and a quadrant C D E, is equal to the triangle COE, whose vertex O is the centre of the quadrant.

For if the arcs A C, C E, in the former figure, are equal to one another, each of them will be a quadrant, and the two lunes will be equal to one another; and the two triangles, COA, COE are also equal; therefore, since

E

the halves of equals are equal, the lune CDEd is equal to the triangle COE]

Maximum (Lat., greatest) is a name given to the greatest among all magnitudes of the same kind which are subject to the same given conditions: as minimum (Lat., least) is, on the other hand, the name given to the least.

For examples of maxima and minima
on a plane See III. § 5, p. 99.
on a spherical surface, See 199 and
sch. 201.

def. 32

cor. 36

cor. 36

(a) The lowest terms are those which are determined by the greatest common measure; and are, therefore, prime to one another (b) Any other terms are equimultiples of the lowest terms (c) The lowest terms may be found from any terms by dividing them by their greatest common factor: they serve when found to determine whether two numerical ratios are different, or only different forms of the same ratio cor. 36 (d) A numerical ratio, which is com pounded of any number of ratios, has for its antecedent the product of their

antecedents, and for its consequent
the product of their consequents.

sch. 45

Oblique, a term applied to angles (whether
rectilineal, dihedral, or spherical) which are
not right angles.
Oblique cone, def. (See "Cone.") 167
Oblique cylinder, def. (See "Cylinder.") 166
Obtuse (Lat. blunted), a term applied to an-
gles (whether rectilineal, dihedral, or sphe-
rical) which are greater than right angles.
Octagon, (figure of eight sides) regular. (See
Regular Polygon")
sch. 121
Octahedron (solid contained by eight planes)
regular. See "Regular Polyhedron."
Opposite cone, def. (also "opposite conical
surface")

214

186

Opposite points, on the surface of a sphere,
def. (See "Spherical Geometry.")
Orb, spherical, def. (See " Sphere.")
Ordinate, in a conic section

See "Conic Section."

180

def. 220

(d) The squares of its diagonals are, to-
gether, equal to the squares of its four
sides
cor. 24
(e) If a quadrilateral figure has its op-
posite sides equal to one another, or
its opposite angles equal, or if its dia-
gonals bisect one another, or if it is
bisected by each of its diagonals, or if
the squares of its diagonals are, to-
gether, equal to the squares of the
four sides, the quadrilateral is a paral-
lelogram 15, cor. 17, and cor. 24
(f) The complements of parallelograms
about the diagonal of a parallelogram
are equal to one another
(g) If one angle of a parallelogram is
a right angle, all its angles are right
angles
cor. 16
(h) Every parallelogram is equal to a
rectangle having the same base and
the same altitude, i. e. to the product

of its base and altitude

16

cor. 16

From two Greek words, signifying" along one
another."

Parallelopiped
def. 127
(a) Its opposite faces are similar and
equal parallelograms, and its opposite
edges are [equal and] parallel. 139
(b) Its four diagonals pass through the
same point, and bisect one another
sch. 140
(c) It is bisected by each of the diago-
nal planes *

139

sch. 140

(d) The squares of its four diagonals
are together equal to the squares of
the twelve edges
(e) The complements of the parallelo-
pipeds about the diagonal plane of a
parallelopiped are equal to one another
cor. 140

[(f) If one face is at right angles to
each of its adjoining faces, every face
is at right angles to each of its adjoin-
ing faces.]

(g) Every parallelopiped is equal to a
rectangular parallelopiped, having the
same base and the same altitude; i. e.
to the product of its base and altitude
cor. 142

(h) Parallelopipeds upon the same, or
upon equal bases, and between the

(1) Parallelopipeds which are equiangular (so that a solid angle of the one may be made to coincide with a solid angle of the other) are to one another in the ratio which is compounded of the ratios of their edges cor. 144 Rectangular. See "Rectangular Parallelopiped."

Part, or measure def. 31 Pentagon (figure of five sides) regular. See "Regular polygon" and " Circle." Pentedecagon (figure of 15 sides) regular. See "Regular polygon" and "Circle." Perimeter of a plane figure

Minimum of. See "Circle."

def. 2

When said to be parallel or perpendicular to a straight line

(A) (a) A plane, and one only, may be made to pass through a straight line and a point without it, or three given points not in the same straight line, or the sides of a given rectilineal angle, or two given parallels 127 (b) Any number of parallels through which the same straight line passes, are in one and the same plane cor. 128 (c) Any number of planes may be made to pass through the same straight line cor. 128

(d) The common section of two planes is a straight line 128 (e) If there be three planes, and if the common section of two of the planes be not parallel to the third plane, the three planes shall pass through the same point; viz., the point in which the common section of two of the planes meets the third plane sch. 156 (f) A plane surface is less than any other surface having the same contour lem. 167

128

(b) Any number of straight lines, which are drawn at right angles to the same straight line from the same point of it, lie all of them in the plane which is perpendicular to the straight line at that point cor. 129 (c) If the plane of a right angle be made to revolve about one of its legs, the other leg will describe a plane at right angles to the first leg cor. 129 (d) If a straight line be perpendicular to a plane, and if from its foot a perpendicular be drawn to a straight line taken in the plane, any straight line, which is drawn from a point in the former perpendicular to meet the foot of the latter perpendicular, shall likewise be perpendicular to the straight line taken in the plane 129

(e) If a straight line be perpendicular to a plane, and if from any point of it a perpendicular be drawn to a straight line taken in the plane, the straight line which joins the feet of the perpendiculars shall likewise be perpendicular to the straight line taken in the plane 129

129

(ƒ) Straight lines which are perpendicular to the same plane are parallel ; and conversely, if there be two parallel straight lines, and if one of them be perpendicular to a plane, the other shall be perpendicular to the same plane (9) Perpendiculars to the same plane, which are drawn to it from points of the same straight line, lie in one and the same plane cor. 130 (h) A straight line may be drawn perpendicular to a plane of indefinite extent, from any given point, whether the given point be without or in the plane; but from the same point there cannot be drawn more than one perpendicular to the same plane (From a point to a plane the perpendicular is the shortest distance; and of other straight lines which are drawn from the point to the plane, such as are equal to one another, cut the plane at equal distances from the foot of the perpendicular; and such as are unequal, cut the plane at unequal distances from the foot, the greater being always further from the perpendicu lar, and conversely (k) If from any point taken without a plane, a sphere be described with a radius less than the perpendicular, it

130

131