172

(g) The solid content of every cone is equal to one-third of the product of its base and altitude; or (if R represents the radius of the base, and A the altitude) R2A 173 and sch. 175 (h) Every cone is equal to the third part of a cylinder which has the same base and the same altitude

cor. 173 and sch. 175

(i) Every cone is equal to a pyramid which has an equal base and an equal altitude cor. 173 and sch. 175 (k) Cones which have equal altitudes are to one another as their bases; and cones which have equal bases, as their altitudes; also any two cones are to one another in that ratio which is compounded of the ratios of their bases and altitudes

cor. 173 and sch. 175

173

(2) The surfaces of similar cones are in the duplicate ratio (or as the squares) of their axes; and their solid contents in the triplicate ratio (or as the cubes) of their axes (m) The convex surface of the frustum of a right cone is equal to the product of the slant side of the frustum by half the sum of the circumferences of its two bases; i. e. of the slant side and the circumference generated by its middle point 174 (n) The solid content of a frustum of a cone, whether it be right or oblique, is equal to the sum of the solid contents of three cones which have the same altitude with the frustum, and, for their bases, its two bases and a mean proportional between them 174 (0) If a cone is cut by a plane which is parallel to its base, the section is a circle, having its centre in the axis of the cone 215 and 229

229

(p) The subcontrary section of an oblique cone is a circle (2) If a cone is cut by a plane which is neither parallel to the base nor subcontrary, nor passes through the ver

tex, the section is a conic section

Conic section

(A)

(B)

vertical plane of - when said to be an ellipse, when a parabola, when an hyperbola Of the three Conic Sections. (a) Every conic section is the perspective projection of a circular section of the cone upon the plane of the conic section by straight lines drawn from the vertex of the cone

216

(b) And in like manner, every circular section may be considered as the perspective projection of the conic section by straight lines drawn from the vertex of the cone cor. 216 (c) The projection of every point in the conic section may be found in the circular section, whether it be an ellipse or a parabola, or an hyperbola cor. 216

(d) The projection of every point in the circular section may be found in the conic section; except in the case of the parabola, the projection of the point in which the vertical plane touches the circular section; and except in the case of the hyperbola, the projections of the two points in which the vertical plane cuts the circular section cor. 216

(e) A conic section cannot be cut by a straight line in more than two points; and, if a straight line touches a conic section, it shall meet it in one point only, viz., the point of contact 218 [() If from any point without a conic section, two straight lines are drawn to touch it, every straight line which is drawn through that point to cut the conic section, shall be harmonically divided by the curve and the chord which joins the points of contact; and the tangents at the points in which every such straight line cuts the curve, shall meet one another in the chord produced.-See lem. p. 220, and the demonstration of App. 20, in p. 227.]

(9) If, through any poin taken within or without a conic section, there are drawn any number of straight lines, each cutting the curve in two points, and if at every such two points tangents be drawn, cutting one another in a point P, the locus of the points P shall be a straight line, and every straight line which is drawn through the point taken to cut the curve, shall be harmonically divided by that straight line and the curve 227 Of the Ellipse and Hyperbola. (a) The curve of the ellipse returns into

itself, and incloses an area: the curve of the hyperbola has four infinite arcs.

aef. 215

217

(b) In both, the curve is symmetrically divided by a certain straight line, which cuts it in two points, and is perpendicular to the tangents at those points 216, 217 Transverse (or major) axis, principal vertices, and centre def. 217 (c) The hyperbola has two asymptotes, which pass through the centre, and make equal angles with the axis upon opposite sides of it, being parallel respectively to the slant sides in which the vertical plane of the hyperbola cuts the surface of the cone (d) If a straight line (not parallel to either asymptote, in the hyperbola) cuts the curve in any point, it may be produced to cut it in a second point; but, in the hyperbola, a parallel to either of the asymptotes cannot meet the curve in more than one point 219 Diameter, vertices, ordinate, abscissæ def. 219, 220 (e) Every diameter bisects its ordinates, and is itself bisected bythe centre 221 (f) If a straight line, which cuts the curve in two points, but does not pass through the centre, be bisected by any diameter, it is an ordinate to that diameter cor. 222

[(g) Two straight lines cannot bisect one another, except they both pass through the centre *.]

(h) Tangents at the extremities of a diameter are parallel cor. 222

222

(i) Tangents at the extremities of any ordinate meet the diameter in the same point, and that in such a manner (fig. p. 221) that C N, CP, CT, are proportionals (k) The squares of any two semiordinates, of the same diameter, are to one another as the rectangles under the abscissæ 223 (2) If two straight lines cut one another, and likewise other two, which are parallel to the two first respectively, and if each of them cuts the curve in two points, or if one or more touch it in a single point, the rectangle under the segments of either of the two first, shall be to the rectangle under the segments of its parallel, as the rectangle under the segments of the remaining one of the first to the rectangle under the segments of its parallel; the square of any of the straight lines being understood, instead of the rectangle under its segments, when it touches the curve instead of cutting it cor. 225

(m) If, in the hyperbola, a parallel to either of the asymptotes be cut by any two parallel straight lines, the seg

* For each would be parallel to the tangent at the extremity of the diameter (f), which is impossible (1.14. Cor. 2.)

and conjugate axis There is no other axis Conjugate diameter of an hyperbola, and conjugate axis sch. 226 (n) The vertices of the conjugate diameters of an hyperbola lie in another hyperbola, which has the same centre and axes as the first sch. 226

Conjugate hyperbolas

sch, 227

The two are mutually conjugate. (0) If two spheres are described, touching the plane of the curve in two points S, S', and the conical surface in two circles, the planes of which being produced, cut the plane of the curve in two straight lines X R, X'R'; then, if PR, PR' are the perpendiculars drawn from any point P in the curve to the lines X R, X' R', SP is to PR, and S'P to P R', in the same constant ratio; SP being less than PR in the ellipse, and greater in the hyperbola 229

(p) The same being supposed, in the ellipse SP+ SPA A'; in the hyperbola, SP ~ S/P = A Á'. cor.

229

The points S, S' are called the foci, and each of the lines X R, X'R', a directrix sch. 229

Of the Parabola.

(a) The curve of the parabola has two infinite arcs def. 215 (b) It is symmetrically divided by a certain straight line, which cuts it in one point only, and is perpendicular to the tangent of that point

216

cor. 218

Axis and principal vertex, def. 217 (c) The infinite arcs of the parabola do not admit of asymptotes (d) The axis of a parabola is parallel to the slant side in which the vertical plane touches the surface of the cone cor. 217

(e) If a straight line (not parallel to the axis) cut the curve in any point, it may be produced to cut it in a second point; but a parallel to the axis cannot meet the curve in more than one point 219

Diameter, vertex, ordinate, abscissa def. 219, 220 (f) Every diameter bisects its ordinates

221

(If two straight lines cut one another, and likewise other two, which are parallel to the two first respectively, &c. (see (1) of the last division) cor. 225

(m) If a diameter be cut by any two parallel straight lines, the segments of the diameter are to one another as the rectangles under the segments of the parallels cor. 225 (n) If a sphere is described, touching the plane of the parabola in a point S, and the conical surface in a circle, the plane of which (being produced) cuts the plane of the parabola in a line RX; then, if PR is drawn perpendicular from any point P in the parabola to the line RX, SP is equal to PR. 228 and cor. 229 The point S is called the focus, and the line R X the directrix sch. 229 Conjugate axis of an ellipse or hyperbola sch. 226

Conjugate diameters of an ellipse or hyper

bola

Conjugate hyperbolas

sch. 226

sch. 227

Consequent of a ratio, is the second term. ·Consequents of a proportion, are the second

and fourth terms.

Construction of a geometrical proposition 3 When said to be a plane construction sch. 26 Contact of two circles, when said to be internal, when external note, 118 point of def. 79 problems of. See " Circle." (I) (m). Continued proportion, magnitudes said to be in (See "Geometrical Progression.") 34 Content of a solid, is the number of times it contains the cubical unit, or unit of solidity sch. 142

The terms" capacity" and "volume" are used in the same sense.

Concave. (See " Convex Side.") def. 1 Convertendo, a rule in proportion. See "Proportion."

Convex, a line or surface is said to be, when its roundness or bulging is everywhere towards the same parts; the test of which, whether it be a line or a surface, is, that it cannot be cut by any the same straight line in more than two points. Convex side of a line or surface, is that side upon which is the roundness or bulging out; and the other side is called the concave side.

def. 1

Convex surface of a prism or pyramid, def.

127 def. 166, 167

of a cylinder or cone, (a) A convex surface is greater than a

(b) Of two surfaces, one of which is convex and is enveloped by the other, the enveloping surface is greatest 168 Corollary of a proposition Cube. (Also "Cube of a straight Line.”) def. 126

3

(a) Cubes are to one another in the triplicate ratio of their edges; i. e. the triplicate ratio of two straight lines is the same with the ratio of their cubes. cor. 144 (b) The difference of two cubes is equal to the sum of three parallelopipeds having the same altitude, viz., the difference of the edges, and for their bases the respective bases of the cubes and a mean proportional between them; i. e., R3 — r3 = (R − r) × (R2 + r2 + Rr) *. lem. 177 (c) If the difference of the edges of two cubes may be made less than any given difference, the difference of the cubes may likewise be made less than any given difference lem. 177

vex and is enveloped by the other, the enveloping curve is greatest sch. 8 Cylinder (also its bases, axis, convex surface) def. 166

When said to be right, when oblique : when two cylinders are said to be similar: when a prism is said to be inscribed in, or circumscribed about a cylinder 166, 167 (a) The convex surface of a cylinder may be supposed to be generated by a straight line which is carried round the circumference of either base so as to be always parallel to the axis 166 (b) A right cylinder is generated by the revolution of a rectangle about one of its sides 167 (c) A cylinder is greater than any inscribed prism, and less than any circumscribed prism; and the convex

surface of the cylinder is greater than that of any inscribed prism, and less than that of any circumscribed prism 168 and sch. 175 (d) A prism may be inscribed in any cylinder (or circumscribed about it) which shall approach nearer to the cylinder, in convex surface or in solid content, than by any given difference 169 and sch. 175

(e) Any two similar cylinders being given, similar prisms may be inscribed (or circumscribed), which shall approach nearer to the cylinders, in convex surface or in solid content, than by any the same given difference cor. 170 and sch. 175 (f) The convex surface of a right cylinder is equal to the product of its altitude and the circumference of its base; or (if R represents the radius of the base, and A the axis) 2 RA. 170

=

(g) The convex surface of any cylinder is equal to the product of its axis, and the perimeter of a section which is perpendicular to the axis sch. 175 (h) The solid content of every cylinder is equal to the product of its base and altitude, =R2 A. 170 and sch. 175 (i) Every cylinder is equal to a prism which has an equal base and an equal altitude cor. 170 and sch. 175. (k) Cylinders which have equal altitudes are to one another as their bases; and cylinders, which have equal bases, as their altitudes; also any two cylinders are to one another in the ratio which is compounded of the ratios of their bases and altitudes cor. 170 and sch. 175

(1) The surfaces of similar cylinders are in the duplicate ratio (or as the squares) of their axes; and their solid contents are in the triplicate ratio (or as the cubes) of their axes 170 and

sch. 175 (m) If a cylinder be cut by a plane which is parallel to its base, the section is a circle, having its centre in the axis of the cylinder, and its radius equal to the radius of the base 231 (n) The subcontrary section of an oblique cylinder is a circle, having its centre in the axis of the cylinder, and its radius equal to the radius of the 232

base

(0) If a cylinder, whether right or oblique, be cut by a plane, which is neither parallel to the axis, nor parallel to the base, nor subcontrary, the section is an ellipse 231 Data (Lat. things given), in geometrical problems, limits of sch. 27, 124 Decagon (figure of ten sides) regular. See "Circle" and "Regular Polygon." Developable, a term applied to the surfaces

def. 125 When said to be right, when acute, when obtuse

125

(a) If a dihedral angle is cut by any two parallel planes, it intercepts equal angles in those planes cor. 135 (b) A dihedral angle is measured by the rectilineal angle of the perpendiculars to the common section which are drawn in its two planes from any the same point of the common section sch, 136

(c) Or by the angle of two perpendicu lars to its planes, which are drawn from any the same point sch. 136 (d) To bisect a given dihedral angle 155 (e) If a dihedral angle is bisected, every point in the bisecting plane is at equal distances from the planes of the dihedral angle

See "Plane"

cor. 155

def. 126

Edge, of a polyhedron See "Polyhedron." Elementary course of Geometry, in the present Treatise P. iv Enneagon (figure of nine sides) regular, to inscribe in a circle, very nearly 121 Enunciation, of a geometrical proposition 3 Equal, circles said to be def.79 ratios said to be def. 32 Equality and inequality, axioms of 4 Equations of two unknown quantities, solved geometrically sch. 124 Equiangular, triangles are said to be, when the angles of the one are equal to the angles of the other, each to each Equilateral triangle

see 59 def. 2 Is equiangular, and conversely. See

ах.34

also "Circle" and "Regular Polygon." Equimultiples of two (or more) magnitudes def. 31 (a) Equimultiples of equal magnitudes are equal; and conversely: also the equimultiple of the greater is greater than the equimultiple of the less; and conversely (b) If two magnitudes, A, A', are equimultiples of other two B, B', which again are equimultiples of other two C, C', the first two shall be equimulples of the last two 34 (c) If two magnitudes A, A' are equimultiples of other two B, B', and again of other two C, C', and if B be a multiple of C, B' shall be the same multiple of C'.

cor. 34

Ex æquali in proportione directâ, or ex æquali, or ex æquo, a rule in proportion. See "Proportion."

Ex æquali in proportione perturbatâ, or ex @quo perturbato, a rule in proportion. See "Proportion."

Excube, extetrahedron, exoctahedron, exdodecahedron, exicosahedron, solids formed from the regular solids see sch. 162 Explementary, an angle said to be note 85 Exterior angle. See "Straight Line," "Tri," "Rectilineal Figure," "Spherical angle," Geometry." Extreme and mean ratio, a straight line said to be divided in. (See "Straight line.") 71 Extremes of a proportion, are the first and last terms def. 33

and in which the first term A' is to A as A to AB. cor. 42 [(c) And the sum of any number of the magnitudes A, B, C, &c. in succession, is equal to the difference of two of the magnitudes A', B', C', D', &c.: thus, A is the difference of A' and B', A+B of A' and C', A + B + C of A' and D', and so on.]

[(d) Hence, if A is greater than B, the sum of the whole series A, B, C, D, &c. continued without end is equal to A'; i. e. the sum of a finite number of terms is less than A', but by the continued addition of new terms may be made to approach to it by less than any given difference.] Geometry, its subject Geometry is distinguished by the epithets of "plane," "solid," and "spherical," according as it treats of plane figures and lines in one plane, or of solid figures and lines in different planes, or of figures and lines upon the surface of a sphere.

1

General theory of proportion 48 General properties of the conic sections 214 Generated, meaning of the word as applied to solids of revolution

127

See also “Cone," "Cylinder," "Plane,” "Sphere."

Great Circle of a sphere

See "Spherical Geometry."

Harmonical mean

(a) The harmonical mean numbers, m and n, is

def. 184

def. 67 between two

2 m n

m + n

,,, will represent the lengths of strings producing with the same thickness and tension the sounds denoted by C, D, E, F, G, A, B, c. note 67 Harmonically divided, a straight line said to be 68 Harmonicals, four straight lines when said to be 68 Harmonical progression, magnitudes said to

be in

68

to what observation the name is owing note, 67 Hemisphere, the half of a sphere, [is equal to

*By "general properties," are here meant such properties as may be declared in the same words, or nearly so, (i. e. with some slight difference arising from their difference of form,) to pertain to the three only such as admit of being derived from the circle conic sections. Those in the Appendix are moreover

by perspective projection. See "Conic Sections."