A straight line when said to be perpendicular to a plane

125 (a) If a straight line stand at right angles to each of two other straight lines at their point of intersection, it shall be at right angles to the plane in which they are

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 (f) 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 per pendicular 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 131 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

(C) Of a plane and inclined straight line. Angle of inclination of a straight line to a plane def. 125 (a) If a straight line be inclined to a plane; of all the angles which it makes with straight lines meeting it in that plane, the least is the angle of inclination; and, with respect to every other of these angles, a second angle may always be drawn which shall be equal to it viz., upon the other side of the angle of inclination; but there cannot be drawn in the plane more than two straight lines with which the inclined straight line shall make equal angles, one upon each side of the angle of inclination

132

[(b) If a straight line cuts a plane, every straight line which is parallel to it shall cut, and be equally inclined to the same plane.]

(c) Through a given point in a given plane, to draw a straight line at right angles to a straight line which is inclined to the plane at that point 153

(D) Of a plane and parallel straight line. (a) If one straight line is parallel to

another, it is parallel to every plane which passes through that other 132 (b) If a straight line is parallel to a plane, it is parallel to the common section of every plane which passes through it with that plane 132 [(c) If a straight line is parallel to a plane, every straight line, which is parallel to it, is parallel to the same plane, see app. prop. 7.]

cor. 132

(d) If two straight lines are parallel, the common section of any two planes passing through them is parallel to either of them [(e) If there be three planes, and if the common section of two of the planes is parallel to the third plane, the common sections of the three planes are parallel.]

(a) If two straight lines, which cut one another, are parallel each of them to the same plane, the plane of the two straight lines is parallel to that plane cor. 133 (b) Planes, to which the same straight line is perpendicular, are parallel; and, conversely, if two planes are parallel, and if one of them is perpendicular to a straight line, the other is perpendicular to the same straight line 133 (c) Through any given point a plane may be drawn, and one only, which shall be parallel to a given plane

cor.134 (d) Planes, which are parallel to the same plane, are parallel to one another cor. 134

(e) If parallel planes are cut by the same plane, their common sections with it are parallel 134 (f) If two planes, which cut one another, are parallel to other two which cut one another, each to each, the common sections of the first two and of the second two are parallels cor. 134 (g) If two planes are parallel, a straight line which cuts one of them may be produced to cut the other likewise

the perpendiculars to the common section which are drawn in the two planes from any the same point of the common section 136 (0) Dihedral angles, which have the sides of the one parallel, or perpendicular, or equally inclined to the sides of the other, and in the same order, are equal to one another sch. 136 (p) If one plane is at right angles to another, the perpendiculars to the common section, which are drawn in the two planes from any the same point of the common section, are at right angles to one another; and conversely cor. 136 (9) If one plane is perpendicular to another, any straight line, which is drawn in the first plane at right angles to their common section, is perpendicular to the other plane 136 (7) If a straight line is perpendicular to

136

a plane, every plane which passes through it is perpendicular to the same plane (s) If two planes which cut one another are each of them perpendicular to a third plane, their common section is perpendicular to the same plane cor. 137 (1) If through the same point there pass any number of planes perpendicular to the same plane, they all of them pass through the same straight line, viz., the perpendicular which is drawn from the point to the plane cor. 137

(u) If two planes are parallel, a plane which is parallel, or perpendicular, or inclined to one of them, shall be parallel or perpendicular, or equally inclined to the other sch. 136 (v) Through a given point, to draw a plane which shall be parallel to a given plane (x) Through a given point, to draw a plane perpendicular to each of two given planes

154

154

(y) Through a given straight line to draw a plane perpendicular to a given plane

153

Polar triangle. (See "Spherical Geometry.")

Polygon. (See "Rectilineal Figure," "Regular Polygon.")

def. 2 Polygon, spherical. (See "Spherical Geome try.") def. 186 Polyhedron. (Also "Diagonals of a Polyhedron.")

def. 126

126

When said to be regular When two polyhedrons are said to be similar. (See Note upon this def.) 126 [Two polyhedrons are said to be symmetrical, when a face of the one may be made to coincide with a face of the other, and, these being made to coincide, the straight lines which join the solid angles of the one with the corresponding solid angles of the other are perpendicular to, and bisected by, the plane of the faces.]

(a) If S, E, and F represent respectively the number of solid angles, the number of edges, and the number of faces of a polyhedron, SEX F=2 sch. 197

(b) If S represents the number of solid angles, the sum of all the plane angles of the faces is equal to (S - 2) times 4 right angles sch. 197 (c) The solidity of a polyhedron may be obtained by dividing it into pyramids, having for their common vertex one of the vertices of the polyhedron, or some point within it sch. 147 (d) Similar polyhedrons are divided into the same number of similar pyramids, by drawing diagonals from any two corresponding angles, and planes along those diagonals

150

(e) Similar polyhedrons are to one another in the triplicate ratio (or as the cubes) of their homologous edges; and their convex surfaces are in the duplicate ratio (or as the squares) of those edges 150 [(f) If four straight lines are proportionals, any two similar polyhedrons which have the first and second for homologous edges, and any two which have the third and fourth, are proportionals. (IV. 27. cor. 3.]

[(g) Symmetrical polyhedrons may be divided into the same number of

(d) Prisms which have equal altitudes are to one another as their bases; and prisms which have equal bases as their altitudes; also, any two prisms are to one another in the ratio which is compounded of the ratios of their bases and altitudes cor. 145 (e) The solid content of any prism is equal to the product of the principal edge, and the area of a plane section perpendicular to it; and the convex surface is equal to the product of the principal edge, and the perimeter of the same section. In the right prism, this section is the same with the base sch. 145

[(ƒ) Prisms which have for their bases similar polygons, and the principal edges drawn from two corresponding angles of those polygons, making equal angles with each of the homologous adjoining sides, and in the same ratio as those sides, are similar.]

[(9) Similar prisms are as the cubes of their homologous edges. (IV. 35.)]

[(h) Symmetrical prisms are equal to one another.]

(i) If the convex surface of a prism be produced to any extent, the sections of it by parallel planes will be similar and equal polygons sch. 145 Problem (Gr., a thing put forth or proposed) 3 Product, strict meaning of the word when used in geometry sch. 18, 142 And hence it is used synonimously with rectangle or rectangular parallelopiped, these figures being measured by the products of their bases and altitudes, or of their respective dimensions. Progression, arithmetical. tical Progression."

geometrical.

Progression."

harmonical.

Progression."

See "Arithme

See "Geometrical

SeeHarmonical

Of Orthographic Projection. Orthographic projection of a point, line, or figure def. 208 Plane of projection, direction of projection, original def. 208 (a) The orthographic projection of every given point may be found upon the plane of projection

212

(b) The orthographic projection of a straight line (not parallel to the direction of projection) is a straight line-viz., the common section of the plane of projection, and a plane drawn through the original, parallel to the direction of projection 213

(c) If the original straight line be pa rallel to the plane of projection, the orthographic projection of such straight line is a point cor. 213 (d) The orthographic projection of a straight line, which is parallel to the plane of projection, is parallel, and equal to its original cor. 213

213

(e) The orthographic projections of any parallel straight lines are parallels, and have the same ratio to their respective originals (f) The orthographic projection of a curve (the plane of which is not paral lel to the direction of projection) is a curve; and the orthographic projection of a straight line, touching the original curve, is a straight line touching the projection of that curve 214 (g) If the plane of the original curve be parallel to the direction of projection, the projection of a straight line, and the projection of the tangent coincides with it, or, in one case, is a point of it cor. 214 (h) The orthographic projection of a straight line cutting a curve, is a straight line cutting the projection of that curve, except always as in (g) cor. 214

[ The orthographic projection of a

circle is a circle, or an ellipse (app. 25,' 26, 27,)]

(B) Of Perspective Projection. Perspective projection of a point, line, or figure def. 208 Vertex, vertical plane, plane of projection, original def. 208 (a) The perspective projection may be found of every point which is without the vertical plane, but of no point in that plane 208

209

(b) The perspective projection of a straight line (not passing through the vertex) is a straight line-viz. the common section of the plane of projection, and the plane which passes through the vertex and original straight line (c) If the original straight line passes through the vertex, the projection of such straight line is a point cor. 209 (d) The perspective projection of a straight line parallel to the plane of projection, is parallel to its original cor. 209 (e) The perspective projection of a straight line, which is terminated by the vertical plane, is a straight line of unlimited extent in one direction; and the projection of a straight line, which cuts that plane, is the whole of a straight line of unlimited extent, in both directions, excepting only the part which lies between the projections of the extreme points [(f) The perspective projection of a straight line, which is of unlimited extent in one direction, but does not meet the vertical plane (and is not parallel to that plane), is finite.] (g) The perspective projection of straight line, which is not parallel to the plane of projection, passes, if produced, through the point in which a parallel to the original, drawn through the vertex, cuts the plane of projection cor. 208 (h) The perspective projections of parallel straight lines, which are likewise parallel to the plane of projection, are parallels. 209

209

a

The perspective projections of parallel straight lines, which cut the plane of projection, are not parallels, but pass, when produced, all of them through the same point-viz., the point in which a parallel to the originals, drawn through the vertex, cuts the plane of projection 210 (k) The perspective projection of a curve (the plane of which does not pass through the vertex) is a curve; and, if any point of the original curve lies in the vertical plane, the projection shall have an arc of unlimited extent corresponding to the arc of the original curve, which is terminated in that point.

210

(1) The perspective projection of a straight line touching any curve is a straight line touching the projection of that curve, if the original point of contact be without the vertical plane; but if it be in that plane, the projection of the tangent is an asymptote to the projection of the curve 211

(m) If, however, the tangent at the ori ginal point be also in the vertical plane, it has no projection, and the projec tion of the original curve has no asymptote cor. 212

(n) The perspective projection of a straight line, which cuts a curve in a point without the vertical plane, is a straight line which cuts the projection cor. 212

of that curve

cor. 50 (d) If nA > or = or < mB, then nC > or = or <mD; and conversely. cor. 50 (e) If there are four magnitudes A, B, C, D, and if when nA> or = or mB, nC > or = ormD, for all values of n and m, then A, B, C, D are proportionals cor. 51 (f) If four magnitudes are proportionals, the greatest and least together are greater than the other two together

53