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 ar.34 (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 Excube, extetrahedron, exoctahedron, exdode- Extreme and mean ratio, a straight line said last terms Foot of a perpendicular, is the point in which def. 33 Geometrical Mean, def. 33. See "Arithme- be in 34 (a) The differences of magnitudes, which same common ratio cor. 42 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.] 1 Geometry, its subject 48 127 See also "Cone," "Cylinder," "Plane," "Sphere." Great Circle of a sphere See "Spherical Geometry." def. 184 ,,, 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 68 be 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 conic sections. Those in the Appendix are moreover (b) Magnitudes A, B, C, &c., which are by perspective projection. See "Conic Sections." 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. See "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. Homologous sides of similar figures def. 57 Homologous terms of a proportion def.33 Hyperbola, one of the conic sections def. 215 See" Conic Section." 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-dodecahedron, 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 Lacommensurable, 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 nx n + 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 (f) The parts of a line divided in medial ratio are incommensurables of this class, and their ratio may be approxi mated 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 46 (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. 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.") def. 1 See "Curve," "Straight Line," and Projection." Locus (Lat., place), (B) def. 106 (C) [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 [(e) 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 conault the errata, square 109 109 (e) 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. 110 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.] [(b) 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, straight lines produced in a given ratio, lie in another parallel plane cor. 135 (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 D E be divided into any two arcs, A B C, CDE, and if upon the chords of these arcs semicircles are described, as in the adjoined figure, the lunes ABCb, CDEd shall be together equal to the triangle A CE. B D E 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.] [(b) The lune which is included by a semicircumference Cd E and a quadrant C DE, 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, CO A, COE are also equal; therefore, since 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. > antecedents, and for its consequent the product of their consequents. sch. 45 167 Oblique, a term applied to angles (whether rectilineal, dihedral, or spherical) which are not right angles. Oblique cone, def. (See "Cone.") Oblique cylinder, def. (See "Cylinder.") 166 Obtuse (Lat. blunted), a term applied to angles (whether rectilineal, dihedral, or spherical) which are greater than right angles. Octagon, (figure of eight sides) regular. (See "Regular Polygon"). Octahedron (solid contained by eight planes) regular. See "Regular Polyhedron." Opposite cone, def. (also "opposite conical surface") 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." 66 sch. 121 Original point, line or plane, in projection 214 186 180 Orthographic projection. See "Projection." Parallel planes, def. (See "Plane.") def. 220 def. 208 Parallel circles, of the sphere See "Sphere." def. 184 125 Parallel to a plane, a straight line said to be 125 Parallel to a straight line, a plane said to be 125 def. 2 Parallel ruler, an instrument for drawing parallel lines Parallel straight lines. (See "Straight line.") def. 2 note, 24 Parallelogram (a) Its opposite sides and angles are equal to one another 15 (b) Its diagonals bisect one another 15 (c) It is bisected by each of its diago nals cor. 16 (d) The squares of its diagonals are, together, equal to the squares of its four sides cor. 24 (e) If a quadrilateral figure has its opposite sides equal to one another, or its opposite angles equal, or if its diagonals bisect one another, or if it is bisected by each of its diagonals, or if the squares of its diagonals are, together, equal to the squares of the four sides, the quadrilateral is a parallelogram 15, cor. 17, and cor. 24 (f) The complements of parallelograms 139 (d) The squares of its four diagonals are together equal to the squares of the twelve edges sch, 140 (e) The complements of the parallelopipeds 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 adjoining faces.] (9) 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 The prisms, into which a parallelopiped is divided by its diagonal plane, are symmetrical; and, therefore, except in the case of the rectangular parallelopiped, their equality cannot be established by coincidence; a circumstance which has been overlooked in Book XI. prop. 28, of Simson's Euclid. See "Polyhedron." |