will not meet the plane; if with a radius equal to the perpendicular, it will meet it in one point only, viz. the foot of the perpendicular; and, if with a radius greater than the perpendicular, its surface will cut the plane in the circumference of a circle, having for its centre the foot of the perpendicular cor. 131 (To draw a plane perpendicular to a given straight line (1) From a given point in the straight line. (2) From a given point without it 153 (m) To draw a straight line perpendicular to a given plane (1) From a given point in the plane. (2) From a given point without it 151 (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.] (d) If two straight lines are parallel, the common section of any two planes passing through them is parallel to either of them cor. 132 [(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.] (f) To draw through a given point1. In a given plane, a straight line which shall be parallel to another given plane 153 154 2. A straight line which shall be parallel to each of two given planes cor. 153 (g) To draw through a given straight line a plane which shall be parallel to a given straight line (E) Of paralle', inclined, and perpendicular planes. (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 perpendi cular to the same straight line (c) Through any given point a plane may be drawn, and one only, which shall be parallel to a given plane 133 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 cor. 209 [(h) If two planes are parallel, a straight line, which is parallel, or perpendicular, or inclined to one of them, shall be parallel or perpendicular, or equally inclined to the other.] (i) If parallel straight lines are cut by parallel planes, the parts of the straight lines, which are intercepted between the planes, are equal to one another 134 (k) Parallel planes are everywhere equidistant cor. 134 (1) If any two straight lines are cut by three parallel planes, the parts of the straight lines which are intercepted by the planes shall be to one another in the same ratio 134 sch. 136 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 (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 (r) If a straight line is perpendicular to a plane, every plane which passes through it is perpendicular to the same plane 136 (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 (t) If through the same point there pass any number of planes perpen dicular 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 154 (x) Through a given point, to draw a plane perpendicular to each of two given planes 154 (y) Through a given straight line to draw a plane perpendicular to a given plane 153 Plane Geometry is that part of Geometry which treats of plane figures and lines in one plane. What constructions are said to be practicable by Plane Geometry sch. 26 def. 107 Plane Locus. (See "Locus.") Plane Section of a solid, is any section made by a plane. of a cone, cylinder, or sphere. See "Cone," "Cylinder," or " Sphere." Planes, angle contained by. See "Dihedral angle." Point Point of contact def. 1 def.79 (a) If a straight line touches a circle, or if a plane touches a sphere, the foot of the perpendicular which is drawn from Polar triangle. (See "Spherical Geometry.") Polygon. (See "Rectilineal Figure," "Regular Polygon.") def. 2 Polygon, spherical. (See "Spherical Geometry.") 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, S — E × 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 propor tionals, 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.] [(9) Symmetrical polyhedrons may be divided into the same number of (b) If the upper part of a triangular prism be cut away by any plane, the remaining solid is equal to the sum of three pyramids, having the same base with the prism, and for their vertices the upper extremities of the diminished edges sch. 147 (c) Every prism is equal to a rectangular parallelopiped, having an equal base and altitude; i. e. to the product of its base and altitude 144 (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 cor. 145 [(h) Symmetrical prisms are equal to one another.] () 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. See "Arithmetical Progression." See "Geometrical geometrical. Progression." (c) If the original straight line be parallel 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 parallel 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 (1) 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 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 a 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 () 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 (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 (2) 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 [(0) The perspective projection of a circle Proportion, and how denoted (also its antecedents and consequents) def. 33 the first and second terms must be of the same kind, and also the third and fourth 33 48 Of commensurable magnitudes 37 general theory of when four magnitudes A, B, C, D, are said to be proportionals; def. [7.] p. 33; and def. 7 p. 49 Proportionals according to def. [7.] are also proportionals according to def. 7. 38, 49 (A) Comparative magnitude of the terms, in a proportion, AB :: C: D*. 52 (a) If A > or = or <B, then C> or <D; and conversely (b) If A > or = or < C, then B > or or <D; and conversely (c) If A > or = or < m m n 53 B, then C or or < or <D; and conversely n 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 = or <mD, for all values of n and m, then A, B, C, D are proportionals cor. 51 (ƒ) If four magnitudes are proportionals, the greatest and least toge ther are greater than the other two together (C) Rules for combining two or more proportions. (a) If A: C:: A': C', and B:C:: B': C', then A‡ B:C:: A′ ± B′ C'; and so, if any number of proportions have the same consequents 54 (b) If A: B:: A': B', and B:C:: B': C', then, ex æquali, A: C :: A': C'; and so, if there are any number of magnitudes A, B, C, D, and A', B', C', D', the ratios of which are the same respectively in the same order. 54 (c) If A B: B': C', and B:C:: A: B', then ex æquo perturbato, ACA: C'; and so, if there are any number of magnitudes A, B, C, D, and A', B', C', D', the ratios of which are the same respectively in a cross order. 55 (e) If four magnitudes, numerically represented by A, B, C, D, are proportionals; and four others, represented by A', B', C', D', are likewise proportionals; then A x A': B x B' :: CX CD x D'; a rule which is called "compounding the proportions," and is applicable to any number of proportions sch. 48 Proportional. See "Third Proportional," "Fourth Proportional," "Mean Proportional." Proportional Straight Lines. See "Straight Line." Pyramid, (also its vertex, base, sides, principal edges, lateral or convex surface, frustum, or truncated pyramid) def. 127 When said to be regular (its axis) 127 When two pyramids are said to be similar When symmetrical. See " Polyhedron." q 126 (a) Triangular pyramids which have equal bases and altitudes are equal to one another. (6) Every pyramid (triangular or other 145 and this will be the case, how great soever the m d numbers m and n may be taken; that is, how great soever the numbers nb and m d may be. But be1 q by less than and nb m d A C = the B D' sch. 47 cause p differs from n b na n b from m d by less than by increasing m and n, This is true, not only when the magnitudes are commensurable proportionals (as is demonstrated in p. 47), but also when four magnitudes are proportionals according to def. 7, and as such, is meant to be asserted by placing it among the properties of proportionals. It is necessary, therefore, to add the following demonstration of the general case: Let A, B, C, D, be proportionals according to def. 7, p. 49; let B be divided into any number, b, of equal parts, and let A contain a of those parts; that is, a certain number exactly together with a fraction, which number and fraction are, together, represented by a; also let D be divided into any number, d, of equal parts, and let C contain c of those parts, c being a whole number and fraction as before; n b m c m d by less than any the same given difference. P nb Therefore, because is always equal to proportionals according to def. 7. For let B and D be divided, each, into any number, n, of equal parts; then, because A contains ths of B, it contains of B as often as the number n is contained in and in like manner C contains th of D as often as n n is contained in ; but, because = -,n is con d α tained in with a remainder, as often as it is con |