.. 1 THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES This book is DUE on the last date stamped below. JAN 301926 & 1826 7 19J2* JAN ? . T927 JtlL_ 1 5 1929 ocr 7 I? ^929 OfcT 1 9 1911 17 6 190 HAR 5 J7 1932) WAR 3 MAY10J93 OCT 2 6 1950 JUN 2 3 i960 JUN191961 X' ELEMENTS OF SYNTHETIC SOLID GEOMETRY, ELEMENTS SYNTHETIC SOLID GEOMETRY BY N. F. DUPUIS, M.A., F.R.S.C. PROFESSOR OF PORE MATHEMATICS IN THE UNIVERSITY OF QUEEN'S COLLEGE, KINGSTON, CANADA gorfe MACMILLAN AND CO. AND LONDON 1893 All rights reserved COPYRIGHT, 1893, BY MACMILLAN AND CO. Norfoooto J S. Gushing & Co. Berwick & Smith. Boston, Mass., U.S.A. Library PREFACE. THE matter of the present work has, with some varia- tions, been in manuscript for a number of years, and has formed the subject of an annual course of lectures to mathematical students by whom the subject has been well received as one of the most interesting in the earlier part of a mathematical course. I have been induced to present the work to the public, partly, by receiving from a number of Educationists inquiries as to what work on Solid Geometry I would recommend as a sequel to my Plane Geometry, and partly, from the high estimate that I have formed of the value of the study of synthetic solid geometry as a means of mental discipline. To me it seems to exercise not only the purely intel- lectual powers in the development of its theorems, but also the imagination in the mental building-up of the necessary spatial figures, and the eye and the hand in their representations. In this work the subject is carried somewhat farther than is customary in those works in which the subject of solid geometry is appended to that of plane geometry, VI PREFACE. but the extensions thus made are fairly within the scope of an elementary w>rk, and are highly interesting and important in themselves as forming valuable aids to the right understanding of the more transcendental methods. It appears to me that it is a prevalent custom to lay too little stress on synthetic methods as soon as plane geometry is passed, and to hurry the student too rapidly into the analytic methods. If mathematical knowledge is all that is required, this may possibly be an advan- tageous course ; but if mental culture is, as it should be, the chief end in a university education, this custom- ary usage is not the best one. I have found it convenient to divide the work into four parts, each of which is further divided into sec- tions. The first part deals with a consideration of the descrip- tive properties of lines and planes in space, of the poly- hedra, and of the cone, the cylinder, and the sphere. Here I would feel like apologising for the introduction of a new term, were it not that I believe that its intro- duction will be fully justified by a careful perusal of the work. Legendre, in his notes to his geometry, proposed to use the word ' corner ' (coin) for the figure formed by the meeting of two planes, and he considered that the different polyhedral angles should receive special names as being geometrical figures of different species. Without PREFACE. vii discussing this idea, I have employed the word 'corner' to denote a solid or polyhedral angle of not less than three faces, while I have retained the expression 'dihedral angle' in its usual sense. If a dihedral angle be cut by a plane, this cutting plane necessarily cuts through both faces, and the figure of intersection is a plane angle. Whereas, if any polyhedral angle be cut by a plane which intersects all its faces, the figure of section is not a simple angle, but a polygon. Thus the plane angle and the dihedral have this in common, that they can both be measured by the same kind of angular unit, while the affinities of the polyhedral angle are with the polygon. Moreover, the trihedral angle is a geometrical func- tion of three plane angles and three dihedral angles, neither of which exists without the other, and every polyhedral angle is a geometrical function or combina- tion of plane and dihedral angles, and these form its elements. Hence I have used the term 'three-faced corner' for 'trihedral angle,' and generally 'w-faced corner ' for ' ?i-hedral angle.' This nomenclature is very convenient ; but if any Teacher prefers the older forms, he can readily make the necessary change in language. The rectangular parallelepiped should certainly be supplied with some convenient name. I have adopted the term ' cuboid,' as proposed by Mr. Hayward, as being both convenient and suggestive. Vlll PREFACE. The second part of the work deals with areal rela- tions, that is, the relations among the areas of squares and rectangles on characteristic line-segments of the prominent spatial figures. The majority of the results, besides being highly inter- esting in themselves, form data for subsequent higher work. The third part is devoted to stereometry and planimetry. In this are developed the principal rules and formulae for the measurement of volumes and surfaces of the more prominent spatial figures which admit of such measurement, and a special section is given to the con- sideration of volumes and surfaces generated by moving areas and lines, and to the development of the theorems of Pappus or Gruldinus. The fourth and last part begins with an explanation. of the principles of conical or perspective projection. By the application of these principles in projecting a circle into a cone and cutting the cone by a plane, the student is introduced to the conic, and is led to under- stand its meaning, and the relations of the various conies to one another. The more common properties of the conies are then easily obtained through a study of the curve as a plane section of a circular cone. The latter half of this part is given to spheric geometry. The spheric figure (tri- angle and polygon) is considered as the section of a PREFACE. IX corner by a sphere whose centre is at the apex of the corner. The study of spheric figures is thus brought into line with the study of the corner or solid angle, and the leading properties of the spheric triangle are thus most easily and directly obtained. The whole work is presented to the younger mathe- matical reader in the hope that it may prove worthy of his careful attention. At the close of the work there is a large collection of miscellaneous exercises, many of which, being connected with the subjects of inversion and of polar reciproca- tion in space, are highly suggestive. I have to acknowledge my indebtedness to Mr. W. R. Sills for assistance in reading the proof-sheets. N. F. D. QUEEN'S COLLEGE, Oct. 1, 1893. CONTENTS. PAST I. DESCRIPTIVE GEOMETRY. SECT. PAGE 1. The Line and the Plane. Intersection. Axial Pencil. Generation of Plane. Normal. Cone-circle. Con- structions . 4 2. Two Planes. Dihedral Angle. Planar and Non-planar Figures. Skew Quadrilateral 17 3. Sheaf of Lines or Planes. Solid Angle or Corner. Properties of Three-faced Corners or Trihedral An- gles. Certain Loci 29 4. Polyhedra. Euler's Theorem C'+ F = E + 2. Tetra- hedron. Parallelepiped. Pyramid. Prism. Regular Polyhedra. Constructions. Nets 45 6. Cone, Cylinder, and Sphere. Tangent Lines and Planes. Conditioned Spheres 61 PA.ET II. AREAL RELATIONS INVOLVING LINE-SEGMENTS OF SPATIAL FIGURES. 1. Tetrahedron. Parallelepiped.- Cuboid. Regular Poly- hedra 78 2. Sphere. Radical Plane. Radical Line. Tangent Cone. Polar 93 Xll CONTENTS. PART III. STEREOMETRY AND PLANIMETRY. SECT. PAGE 1. Polyhedra. Theory of Laminae. Frustum. Prisma- toid. Prismoidal Formula. Applications .... 100 2. Closed Cone. Frustum. Cylinder. Sphere. Zone and Segment 124 3. Special Processes. A. Spatial Figures generated by Translation of a Plane Figure. Pyramid. Cone. Sphere. Groin, etc. B. Figures of Revolution. C. Mean Centre of Area. Theorems. Guldinus' Theorem for Volumes 132 4. Areas of Surfaces. Developable Surfaces. Cone. Cylinder. Sphere. Mean Centre of Figure. Guldi- nus' Theorem for Surfaces 155 PAET IV. PROJECTIONS AND SECTIONS. 1. Perspective Projection. Various Projections. Vanishing Point and Line. Anharrnonic Relations. General Theorem 165 2. Plane Sections of the Cone. Classification of Conies. Degraded Forms. Common Properties. Foci and Directrices. Special Study of Ellipse. Theorems of Apollonius. The Parabola 175 3. Spheric Sections. General Ideas of Spheric Geometry. Spheric Line. Pole and Equator. Lune. Spheric Triangle. Polar Triangle. Properties of Spheric Triangles. Spherical Excess. Superposability and Symmetry. Cases of Ambiguity 197 SOLID OE SPATIAL GEOMETET, PART I. DESCRIPTIVE GEOMETRY. 1. Solid or Spatial Geometry, or the Geometry of Space, deals with the properties and relations of figures not confined to one plane (P. Art. 19) .* The elements of spatial figures are the point, the line, the curve, the plane, and the curved surface. The first four of these are defined in plane geometry (P. Arts. 12, 14, 17) ; but we repeat here the definition of the plane, as upon that definition several corollaries and other definitions depend. Def. A plane is a surface such that the join of any two arbitrary points in it lies wholly in the surface and coincides with it. Cor. 1. A line cannot lie partly within a plane and partly without it. For the part within the plane must have at least two points in the plane, and must there- fore coincide with the plane throughout its whole extent. 1 References marked P. are to the Author's ' Geometry of the point, line, and circle in the plane.' 1 2 SOLID OR SPATIAL GEOMETRY. Cor. 2. A line not coincident with a given plane meets the plane at only one point. 2. A plane is not necessarily limited in extent ; or, in other words, a plane extends to infinity in all its direc- tions. For the plane must be coextensive with every coincident line. Every plane thus theoretically divides all space into two parts, one lying upon each side of the plane. The use of planes thus considered is common in spherical astronomy. 3. In plane geometry the geometric figure is drawn upon the plane of the paper, which properly represents the plane upon which the figure is supposed to lie. In spatial geometry, however, we have only one plane, that of the paper, to stand for and represent all the planes which may be involved in any spatial figure. This is an unavoidable source of confusion to beginners, as the pictured figures in spatial geometry are not representa- tions of the real figures in the same sense as in plane geometry. Thus equal line-segments and equal angles in a spatial figure will not, in general, appear as equal segments or equal angles in the pictured representation. So, also, squares and circles in space will not, in general, appear as squares and circles on our single available plane, that of the paper. Properly constructed models simplify matters to a very great extent, and should be employed whenever available. The construction of proper models is, however, always difficult, and often impracticable, and for several reasons they cannot serve all the pur- poses of a diagram. And hence beginners should ac- DESCRIPTIVE GEOMETRY. 3 custom themselves to reading and interpreting spatial diagrams. These diagrams can be considered only as an aid to building up the figure in the imagination, and facility in reasoning from such diagrams will depend very largely upon the readiness with which the reasoner can make this imaginary construction. The student is accordingly advised to give some care and patience to the constructing of spatial diagrams. To represent a plane we usually represent a rectangular segment of the plane, and this generally appears in the diagram as some form of parallelogram. SECTION 1. THE LINE AND THE PLANE. 4. Theorem. Two planes which coincide in part coin- cide altogether. Proof. The part throughout which the planes coincide must be part of a plane, and must therefore admit of an indefinite number of arbitrary points being taken within it, of which no three are in line. These points taken two and two determine an indefinite number of arbitrary lines which coincide in part with both planes. And the planes thus coinciding (Art. 1. Cor. 1) along an indefinite number of arbitrary lines, coincide altogether, and form virtually but one plane. Def. An indefinite number of lines can lie in one plane. The totality of these is called a plane of lines, although the lines, having only one dimension, do not make up any portion of the plane in which they lie. 5. Theorem. The figure of intersection of two planes is a line. Proof. Let U and V be two \ planes, and let A and B be any \ A two points in their figure of in- / tersection. Join A, B by a line. * \ Then, since A and B are two points in U, the join AB lies wholly in U (Art. 1. Def.). 4 THE LINE AND THE PLANE. 5 For a similar reason the join AB lies wholly in V. Hence it is common to the planes, and is their figure of intersection; and thus the figure of intersection is a line. Cor. 1. Any number of planes may have one com- mon line. For if they pass through the same two points, A and B, they have the join of A and B as a common line. Def. A group of planes having one common line is an axial pencil, and the line is the axis. In contra- distinction to this the pencil of lines in a common plane (P. Art. 203. Def.) is called a flat pencil Cor. 2. As the line of section of two planes cannot return into itself and form a closed plane figure, so two planes cannot form a closed spatial figure. 6. Theorem. Through any three points not in line, 1. One plane can pass. 2. Only one plane can pass. A, B, C are any three points not in line. 1. One plane can pass through A, -B, and C. Proof. Let the plane contain- ing A and B be rotated about the join of A and B. In a complete revolution this plane passes through every point in space, and therefore in some position, V, it passes through C. b SOLID OE SPATIAL GEOMETRY. 2. Only one plane can pass through A, B, and C. Proof. Take D, E, any points in the joins AC and BC respectively. Then D and E and their join lie in U, and in every plane through A, B, and C. Therefore every plane through A, B, and C coincides with U, and forms with U virtually but one plane. Cor. 1. Any three points not in line determine a single plane. Def. Any number of elements so disposed as to lie in one and the same plane are said to be complanar or coplanar. Thus all the parts of a figure in plane geom- etry are complanar. Cor. 2. Two intersecting lines are complanar and determine one plane. For, taking a point in each line, and the point of inter- section, we have three points not in line, and the plane through these is the plane of the lines. Cor. 3. Parallel lines are complanar. For they have a common point at infinity (P. Art. 220. Def.). 7. Generation of a plane. L and M are any two lines intersecting in C, and N is a third line intersecting L in B, and M in A. Then L, M, N are complanar. 1. When L and M are fixed and N is variable, N generates a plane. Therefore, a plane is generated by a variable line which is guided by two intersecting fixed lines. THE LINE AND THE PLANE. Def. The variable line N is called the generator, and the fixed guiding lines are directors. 2. Let C go to infinity, and L and M become parallel. Therefore, a plane is generated by a variable line guided by two fixed parallel lines. 3. Let the point A remain fixed, while B moves along L. Then, a plane is generated by a variable line which passes through a fixed point and is guided by a fixed line. 4. Let the point A go to infinity ; i.e. let the genera- tor N, fixed in direction only, be guided by the fixed line L. Then, a plane is generated by a variable line having a fixed direction and guided by a fixed line. 8. Theorem. At the point of intersection of any two lines a third line can be perpendicular to both. AB and CD are lines intersecting in 0. Then some line OP is perpendicular to both AB and CD. Proof. Let OP be _L to AB, and let it revolve about AB as an axis, being fixed, until it _ comes into the plane of AB and CD at OE and at OF. Then AB, CD, EF are complanar. (hyp.) Similarly and ZDOFis >a~|. 8 SOLID OR SPATIAL GEOMETRY. Therefore, in revolving OP from the position OE to the position OF the Z DOP changes from less than a right angle at DOE to greater than a right angle at DOF; and hence at some intermediate position OP is _L OD. Cor. If AB is _L to CD, and OP is J_ to both, we have three lines mutually perpendicular to each other. Def. 1. Three concurrent lines mutually perpendicu- lar to one another are called the three rectangular axes of space, and their planes are the rectangular co-ordi- nate planes of space. These three lines admit of length measures in three directions, each perpendicular to the other two. Hence, space is said to be of three dimen- sions, or to contain three dimensions, and it is frequently spoken of as tri-dimensional space, in contradistinction to the two-dimensional space of a single plane, or of plane geometry. Def. 2. A line lying in a particular plane is a planar line of that plane ; and when only one plane is under consideration, a planar line will mean a line in that plane. Def. 3. When OP is perpendicular to both AB and CD, it is perpendicular to the plane which these lines determine (Art. 6. Cor. 2). OP is then a normal to the plane, and is the foot of the normal. Also, the plane is a normal plane to the line OP. 9. Theorem. A normal to a plane is perpendicular to every planar line through the foot of the normal. THE LINE AND THE PLANE. 9 OP is _L to OA and OB, and OC is any line through complanar with OA and OB. Then OP is _L to OC. Proo/. Take OA=OB= any con- venient length. Join .4.B, cutting OC in <7, and join P^, P, P<7. The right-angled triangles POA and PO.B are congruent, and there- fore PA = PB. Hence the A APE and -40J3 are each isosceles, and PC and OC are lines from the vertices to the common base AB. ... PB 2 - PC 2 = BC - CA = OB 2 - OC 2 ; (P. Art. 174.) and . . PB 2 - OB 2 = PC 2 - OC 2 . But POB being a ~l, (hyp.) PB 2 -OB 2 = OP 2 = PC 2 - OC 2 . .: ZPOCisal. Cor. 1. If is fixed while OA revolves about OP as an axis, OA generates a plane to which OP is a normal. Def. A line is perpendicular to a line which it does not meet when a plane containing one of the lines can have the other as a normal. Cor. 2. A normal to a plane is perpendicular to every line in the plane, and all normals to the same plane are parallel to one another. Cor. 3. From any point without or within a plane, only one normal can be drawn to the plane. 10. Theorem. Of the line-segments from a point with- out a plane to the plane : 1. The shortest is along the normal through the point. 10 SOLID OR SPATIAL GEOMETRY. 2. The feet of equal segments are equally distant from the foot of the normal, and conversely. 3. Of unequal segments, the longer lies further from the normal than the shorter does, and conversely. P is any point, and PO is normal to the plane U, not passing through P. A, B, O are points in U. 1. PO is < PA, A being any point in U other than 0. Proof. Z POA is a 1 ; (Art. 9. Cor. 2.) .-. Z. PAO is acute, and PO < PA ; (P. Art. 62.) and the normal segment PO is the shortest segment from P to the plane U. 2. PA = PB ; then OA = OB. Proof. The right-angled triangles POA and FOB have their hypothenuses equal, and the side PO in common. They are therefore congruent (P. Art. 65), and OA = OB. Conversely, if OA = OB } the congruence of the same triangles gives PA = PB. 3. PC is >PA; then OC is > OA. For the two triangles POA, POG, being each right- angled, give PC 2 =P0 2 +OC 2 ; and PA* = PO 2 + OA 2 ; .-. PC 2 - PA 2 = OC 2 - OA\ But PC>PA; .'.OC>OA. And conversely, if 0(7 > OA, then PC> PA. THE LINE AND THE PLANE. 11 Cor. When PA = PB, OA = OB. Therefore, if PA is of constant length and variable in position, the foot A describes a circle having as centre and OA as radius. The generation of this circle from a fixed point, P, by a line segment, PA, of constant length, is similar to that of the circle in plane geometry (P. Art. 92), except that in the present case the fixed point is not in the plane of the circle. Def. 1. The circle described on ?7with the vector PA, and from the fixed point P, has a relation to the cone, to be considered hereafter, and we shall accordingly call it a cone circle to the vertex P. Evidently any circle may be considered as a cone circle, and when so considered, it has an indefinite num- ber of vertices, all lying upon the line which passes through its centre and is normal to its plane. Def. 2. The distance of a point from a plane is the length of normal intercepted between the point and the plane. 11. Def. 1. The projection of a point on a plane is the foot of the normal from the point to the plane, and the projection of a line-segment on a plane is the join of the projections of its end-points upon the plane. It follows, then, that the projection of a line upon a plane which it meets is the planar line which passes through the point where the given line meets the plane, and through the foot of the normal, drawn from any point on the given line to the plane. Def. 2. The angle between a given line and its pro- jection upon a plane is taken to be the angle between the given line and the plane. 12 SOLID OR SPATIAL GEOMETRY. Def. 3. The angle between two non-complanar lines is the angle between two intersecting lines respectively parallel to the given lines. 12. Theorem. The angle between a line and its pro- jection on a plane is less than the angle between the given line and any planar line not parallel to the pro- jection. The line PO meets the plane U in 0; ON is the projection of OP on U\ OA is a line through 0, parallel to the planar line L, which is not parallel to the pro- jection ON. Then Z PON is < Z POA. Proof. From P draw PN perpendicular to ON PN is normal to the plane U (Art. 11. Def. 1). Take OA = 0-ZVand join PA and AN. Since Z PNA = 1, PA is > PN. And in the triangles POA and PON, PO is common, OA = ON, and PA > PN; .-. Z POA is > Z PON. (P. Art. 67.) And as L is any planar line not parallel to ON, the Z PON, between PO and its projection on U, is less than that between PO and any line in the plane, not parallel to ON. Cor. 1. Since two intersecting lines make with one another two angles which are supplementary (P. Art. SPATIAL CONSTRUCTION. 13 39), we may say more accurately that the angles, between a line and its projection upon a plane, are the least and the greatest of all the angles made by the given line with lines lying in the plane. Cor. 2. Since 0, P, N are complanar (Art. 6), and Z PNO is a 1, the Z OPN is the complement of the Z PON. Therefore the angle between a line and a plane is the complement of the angle between the line and a normal to the plane. Cor. 3. Let OB be a planar line _L to OP. Since PN is normal to U, OB is J_ to PN (Art. 9. Cor. 2) ; and hence OB, being _L to OP and PN, is J_ to ON. Therefore planar lines which are perpendicular to any line that meets their plane are also perpendicular to the projection of that line upon the plane. 13. Def. A line is parallel to a plane when it meets that plane at infinity. Cor. Any plane through one of two parallel lines is * parallel to the other line. For if L and M be two parallel lines, and the plane U contains L and not M, it can meet M only where L meets M. But L and M meet at infinity (P. Art. 220) ; there- fore M meets U at infinity, or is parallel to U. SPATIAL CONSTRUCTION. 14. In making constructions in space we assume the ability : 1. To draw through any given point a line parallel to a given line. 14 SOLID OK SPATIAL GEOMETKY. 2. To pass a plane through any given point or line. 3. To make a plane construction, according to the principles of plane geometry, upon any assumed or deter- mined plane. Ex. 1. Problem. From a given point without a plane to draw a normal to the plane. Let P be the point, and U be the plane. Con. Draw any line OB in U, and from P draw PO _L to OB (P. Art. 120). In t/draw ON _L to OB; and from P draw PN _L to ON. PN is the normal required. For OB is, by construction, to both OP and ON, and therefore to the plane of these lines, and hence to PN, which lies in this plane (Art. 9. Cor. 2). Therefore PN is _L to OB and to ON, and is conse- quently normal to U. Ex. 2. Problem. To draw a common perpendicular to two non-com planar lines. Let L, M be the two non- complanar lines. Con. In M take any point, A, and through A draw the line ^/"parallel to L (Art. 14. 1). M and N determine a plane, M / r U, which is parallel to L. From any point B in L draw BC normal to U (Ex. 1). Then, as L is parallel to U, BC is _L to L. SPATIAL CONSTRUCTION. 15 Draw CD parallel to L to meet M in D, and from D draw DE _L to L. Then DE is _L to both L and Jf, or is their common perpendicular. For DE is _L to L by construction, and being thus parallel to CB, EC is a rectangle, and ED is normal to U, and therefore _L to jlf. Cor. Since CD can meet Jf in only one point, only one common perpendicular can be drawn to two noii- complanar lines. EXERCISES A. 1. How many planes at least determine one line ? 2. How many lines at most are determined by 3 planes ? by 6 planes ? by n planes ? 3. How many planes at most are determined by 4 points ? by 8 points ? by n points ? 4. Draw a normal to a plane from a point in the plane. 5. Through one of two non-complanar lines, to pass a plane to be parallel with the other line. 6. Show that the common perpendicular to two non-complanar lines is the shortest segment from one line to the other. 7. From a given point in one of two non-complanar lines, to draw a segment of given length to meet the other. The solutions are two, one, or none. Distinguish these cases. 8. Given two non-complanar lines, to draw a segment from one to the other so as to be perpendicular to one of them. 9. Given two non-complanar lines, to draw a segment from one to the other so as to make equal angles with each. Show that this angle may vary from a right angle to the complement of one- half the angle between the given lines. 16 SOLID OR SPATIAL GEOMETRY. 10. PO meets the plane U (Fig. of Art. 12) at an angle of 30, and PN is normal to U. OA is a planar line making the angle POA = 60. Show that cos AON = \ V3. 11. PO meets U at an angle a, and ON is the projection of OP on U. OA is a planar line making the angle POA = p. Show that cos A ON = cos a 12. Through the point, where a given line meets a plane, to draw a planar line to make a given angle with the given line. Examine the limits of possibility. SECTION 2. Two PLANES DIHEDRAL ANGLE SECTIONS. PLANE 15. Def. Parallel planes are such as meet only at infinity, i.e. which do not meet at any finite point. Cor. 1. Planes which have a common normal are parallel. For if the planes meet at any finite point, two perpendiculars can be drawn from that point to the same common normal, one in each plane. But this is impossible (P. Art. 61). Cor. 2. Planes which are not parallel intersect in a line not at infinity. This line is common to the two planes, and is the common line of the planes. When two planes are parallel, their common line is at infinity. 16. U and V are two planes having AB as their common line. From any point, P, in AB draw PC in U and PD in V, each perpendic- ular to AB. The angle CPD is defined as the angle between the planes U and V. Therefore : 17 18 SOLID OR SPATIAL GEOMETRY. Def. 1. The angle between two planes is the angle between two lines, one in each plane, and both perpendic- ular to the common line of the planes. AB is normal to the plane of PC and PD, and is therefore perpendicular to (71Tand DX (Art. 9. Cor. 2). Hence, if CY be _L to PD, and DX to PC, CY is normal to V, and DX is normal to U. And these normals, being complanar, intersect in some point, E, and the angle CED is the supplement of the angle CPD. Hence, if we consider CY and DX in the same sense, i.e. from distal extremity to foot, or vice versa, the angle CED is the angle between the normals to the planes, and therefore : Def. 2. The angle between two planes is the supple- ment of the angle between normals to the planes. When CP is perpendicular to PD, the planes are per- pendicular to one another, and CP is normal to V, and DP to U. Hence : Def. 3. Two planes are perpendicular to one another when one of them contains a normal to the other. 17. Def. When PC is perpendicular to PD, and each is perpendicular to AB, the three planes U, V, and the plane of PCD are mutually perpendicular to one another. These planes are then called the rectangular co-ordinate planes of space, and the common point, P, is the origin. If we assume the positions of these three planes, and therefore the position of the origin, the position of any point in space can be determined by giving its distances from these planes, each distance being affected with a DIHEDRAL ANGLE PLANE SECTIONS. 19 proper algebraic sign. This is the fundamental principle in analytic geometry of three dimensions. 18. If PQ be any line in U, and PR be any line in V, meeting the common line AB, in the same point, P, PQ and PR are complanar (Art. 6. Cor. 2) ; and if W denote their plane, PQ is the common line of U and W, PR is the common line of TFand V, and AB is the common line of "Fand U, and these three lines are concurrent at P. Therefore, three planes, no two of which are parallel, and which do not form an axial pencil, determine one point, and this point is the point of concurrence of the three common lines of the planes taken in twos. This point is at infinity when the three common lines are all parallel. Cor. Three planes cannot form a closed figure. For the planes determine, at most, three concurrent lines, which, meeting in one common point, can never meet in any other points. 19. Def. When a spatial figure, S, is cut by a plane, U, the combination of elements common to /Sand J7form upon U a plane figure, which is called the plane section of S by U, or simply the section of S by U. This definition suggests to us a relation existing be- tween plane and spatial geometry. Plane geometry may be aptly described as a plane section of spatial geometry. The plane upon which the figures of plane geometry lie (P. Art. 11) is the plane of section, and the figures themselves may be considered as sections of spatial figures. 20 SOLID OR SPATIAL GEOMETRY. From this connection we may be led to expect that relations existing among plane figures are only particular cases of more general relations existing among spatial figures. And hence we naturally look for many analogies amongst the results of plane and of spatial geometry. Some of these have appeared already, and others will present themselves in the sequel. And it is worthy of note how often the number two of plane geometry be- comes three in spatial geometry. Thus two points deter- mine one line, while three points determine one plane ; two lines in the plane determine one point, while it requires three planes to determine one point. 20. The following theorems are self-evident : 1. The section of a line is a point. 2. The section of a plane is a line. Hence spatial figures composed of lines and planes give, in section, plane figures composed of points and lines. Def. Sections made by parallel planes are parallel sections. 21. Theorem. Parallel sections of a plane are parallel lines. Proof. If ?7and U' be parallel planes which cut the plane W, the common lines CHFand U'Wboth lie in W, and as J7and U' meet only at infinity (Art. 15), these common lines meet only at infinity and are parallel. Cor. 1. The section of a system of parallel planes is a system of parallel lines. PLANE SECTIONS. 21 Cor. 2. The section of an axial pencil is a set of parallel lines when the section-plane is parallel to the axis ; in other cases it is a flat pencil. 22. Theorem. Parallel sec- tions of two intersecting planes contain the same angle. U and V are intersecting planes, and W and X are two parallel planes of section, the sections being the lines BA, BO, ED, and EF. A D Then and AB is parallel to DE, BC is parallel to EF. (Art. 21.) Def. If W be normal to the common line of the planes U and V, the section is called a right section. Hence, the angle between two planes is the angle be- tween the two lines which form the right section of the planes. A system of any number of planes admits of a right section when all the common lines of the planes are parallel. In every case, the term " right section " must have reference to some particular line or set of parallel lines. 22 SOLID OR SPATIAL GEOMETRY. DIHEDRAL ANGLE. 23. When we cut two intersecting planes, U and F, by a third plane, X, we get (Fig. of Art. 22), 1. A point E, the vertex of the angle DEF; 2. The lines ED and EF, forming the arms of the angle DEF. Now, 1st, to the plane angle DEF corresponds the dihedral angle between the planes U and V; and, 2d, to the vertex E of the plane angle corresponds the com- mon line, BE, of the two planes, this line being called the edge of the dihedral angle ; and, 3d, to the arms ED and EF of the plane angle correspond the planes U and V, called the faces of the dihedral angle. Thus in section a dihedral angle becomes a plane angle, the faces become arms, and the edge becomes the vertex. If the section be a right section, the plane angle and the dihedral angle have the same measure. And as a plane angle is generated by rotating a line about a point in the line taken as a pole (P. Art. 32), so a dihedral angle is generated by the revolution of a plane about any line in the plane, taken as an axis. The angular measurements are thus the same for plane and dihedral angles. 24. Def. The plane which is normal (Art. 9. Def. 3) to the join of two given points at its middle point, is the right-bisector plane of the join of the points. Cor. Since a line-segment has only one middle point, and a plane has only one normal at any given point, it DIHEDRAL ANGLE. 23 \ follows that a given line-segment has only one right- bisector plane. A section through the segment gives the segment and its right-bisector, of plane geometry. 25. Theorem. Every point upon the right-bisector plane of a segment is equally distant from the end points of the segment. Let AB be a given segment, and let U be the right-bisector plane of the segment, passing through its middle point C, and let P be any point on U. Then P is equidistant from A and B. Proof. Since A, B, and P are complanar, let the plane W pass through these points. In the section by W we have the segment AB and its right-bisector CP; and hence PA = PB (P. Art. 53). It will be here noticed that the proof is obtained immediately by reducing the theorem to depend upon the corresponding one in plane geometry. In like manner we readily prove the converse : Every point equidistant from the end points of a given line-segment is upon the right-bisector plane of the seg- ment. Cor. From this it appears that the locus of a point which is equidistant from two fixed points is the right- bisector plane of the join of the points. 26. Def. The planes which pass through the edge of a dihedral angle and make equal angles with its faces are the bisectors of the dihedral angle. 24 SOLID OR SPATIAL GEOMETRY. The proofs of the following theorems may be obtained at once by making them to depend upon the correspond- ing theorems in plane geometry. 1. The two bisectors of a dihedral angle are perpen- dicular to one another. For proof, make a right section of the dihedral angle and apply (P. Art. 45). 2. Any point upon a bisector of a dihedral angle is equally distant from the faces of the angle. For proof, make a right section through the point and apply (P. Art. 68). 3. Any point equidistant from the faces of a dihedral angle is on one of the bisectors of the angle. Proof as in 2. 27. Theorem. Any two lines are divided similarly (P. Art. 201. Def.) by a system of parallel planes. L and M are two lines cut / w _j. t s by the parallel planes U, V, and Xj A" / W. Then L and M are similarly divided. Proof. A, B, C and A 1 , B', /. C" are corresponding points of / \ 1 section of the two lines. Through / ^-*\o. _J C ' / A draw the line N parallel to ^- w L, and let it meet the planes at A, P, and Q. Then L and N being complanar (Art. 6. Cor. 4), AA is II to PB' is II to QC'; .-. A'B' = AP, and B'C' = PQ. DIHEDRAL ANGLE. 25 But ACQ is a triangle, and BP is parallel to CQ; .: AB:BC=AP:PQ = A'B' : B'C 1 . Or the lines L and M are similarly divided. Cor. 1. The parallel planes of a system divide all lines similarly. Cor. 2. The segments of parallel lines intercepted between the same two parallel planes are equal. 28. Theorem. If three concurrent non-complanar lines be divided similarly in relation to the point of concur- rence, the triplets of corresponding points determine a system of parallel planes. L) M, N are three non-com- planar lines concurrent at O, and are divided at A, B, C, , A', B', C", ..-, and A", B", C" , so that OA : AB : BC= OA : A'B' : B'C' = OA" : A"B" : B"C". Then the planes determined by AA'A", BB'B", CC'C", etc., are parallel. Proof. AA is II to BB' is II to CC', and A A " is II to BB" is II to CC". (P. Art. 202. Cor. ) Let OP be normal to the plane AA'A". Then OP is J_ to AA 1 and AA" (Art. 9. Cor. 2), and therefore to BB' and BB", and to CC' and CC". 26 SOLID OR SPATIAL GEOMETRY. Hence OP is normal to the planes BB'B" and CC'C", and the three planes AA'A", BB'B", CC'C" are accord- ingly parallel (Art. 15. Cor. 1). Cor. 1. Since AA' is II to BB', and AA" is II to BB", etc., the A AA'A", BB'B", and CC'C" are similar. But the concurrent lines L, M, N determine three planes whose common point is 0; therefore parallel sections of three non-parallel planes are similar triangles. Cor. 2. Since any polygon may be divided into triangles, and similar polygons into similar triangles similarly placed (P. Art. 206), it follows that : Parallel sections of any number of planes having a common point are similar polygons. 29. Def. Four non-complanar lines which intersect two and two in four points, form a skew-, or a gauche-, or a spatial quadrilateral. The sides of the skew quadrilateral and its two diag- onals are six lines connecting four points in space, and form the six edges of a figure, to be described hereafter, called the Tetrahedron. The skew quadrilateral is a plane quadrilateral with one vertex, and the sides forming it raised out of the plane. 30. Theorem. The joins of the middle points of the opposite sides of a skew quadrilateral bisect one another. ABCD is a skew quadrilateral, AB and BC lying in a plane different from the plane of CD and DA. AC and BD are the diagonals. SKEW QUADRILATERAL. 27 E, F, G, H are middle points of the sides upon which they lie. Then EG and FH bisect one another. Proof. EF and GH are both parallel to AC, and equal to half AC (P. Art. 202); they are therefore equal and parallel to one another. Therefore EFGH is a par- allelogram, and its diagonals EG and FH bisect ,one an- other (P. Art. 81. 3). Cor. 1. Let / and J be the middle points of the diagonals AC and BD. Then ACBD is a skew quadrilateral, and the joins of middle points of opposite sides are FH and IJ. Therefore FH and IJ bisect one another ; and hence FH, IJ, and EG mutually bisect each other. Cor. 2. A, B, C, D are four points in space, and AB, AC, AD, BC, BD, and CD are their six connectors. Therefore if four points in space be connected two and two by six line-segments, the joins of the middle points of these connectors taken in opposite pairs are concurrent, and mutually bisect one another. EXERCISES B. 1. Draw a line equally inclined to two intersecting planes. Is the problem definite or indefinite ? 2. If U and F be two planes, and U contains a normal to V, show that F contains a normal to U. 28 SOLID OR SPATIAL GEOMETRY. 3. Two lines may be drawn, one on each of two intersecting planes, so as to make an angle with one another of any magnitude from zero to a straight angle. 4. If three concurrent non-complanar parallel lines be divided homographically, the planes determined by the triplets of corre- sponding points, all pass through a common line. When is this line at infinity ? 5. If the sides of a skew quadrilateral are equal, the diagonals are perpendicular to one another. 6. What theorem is obtained from 30 by bringing D to the plane of ABC? 7. Draw the shortest path from one point to another so as to touch a given plane in its course, both points being upon the same side of the plane. 8. Show that a skew quadrilateral cannot have four right angles. How many can it have ? 9. A, B, C, D are four non-complanar points. Show that the locus of a point which is equidistant from A and B, and also equi- distant from C and Z>, is a line perpendicular to both AB and CD. 10. If A, B, C, D, E, F be any 6 points in space, a point can be found which is equidistant from A and B, equidistant from C and D, and equidistant from E and F. SECTION 3. SHEAF OF LINES AND PLANES SOLID ANGLE OK COKNEK. 31. Def. Three or more non-complanar lines meeting in a point form a sheaf of lines, and three or more planes passing through a common point form a sheaf of planes. The common point is in each case called the centre of the sheaf. The lines and planes which form a sheaf pass through the centre and extend indefinitely outwards from it, but usually we have to consider only those portions which lie upon one side of the centre, and the centre is then commonly called the vertex or apex of the figure. In a sheaf of lines the determined planes form a sheaf of planes, and in a sheaf of planes the determined lines f rm a sheaf of lines. So that practically a sheaf of lines and a sheaf of planes are only the same figure differently viewed. Cor. From Article 27 it follows that the lines of a sheaf are similarly divided by a system of parallel planes. And from Article 28 it follows that if a sheaf of three lines has its lines similarly divided with reference to the centre, the triplets of corresponding points deter- mine a set of parallel planes. 32. A non-central section of a sheaf of lines and the determined planes is a set of points with their determined 29 30 SOLID OR SPATIAL GEOMETRY. lines ; and the non-central section of a sheaf of planes and the determined lines is a set of lines with their determined points. Thus the reciprocity between a sheaf of lines and a sheaf of planes is analogous to that between a set of points and a set of lines in plane geometry. 33. Def. If the points in the section of a sheaf of lines be so disposed as to form the vertices of a polygon without re-entrant angles, and only those planes of the sheaf be considered, which, in the section, form the sides of the polygon, the combination of lines and planes in the sheaf forms a solid angle, or a polyhedral angle, or a corner. L, M, N, K is a sheaf of four lines with centre 0. Let the sheaf be cut by the plane V, giving in section the points A, B, C, D correspond- ing to L, M, N, K, respectively. If the polygon ABCD is with- out re-entrant angles, the figure formed by the lines L, M, N, K, and the portions of deter- mined planes, LOM, MON, NOK, KOL, intercepted be- tween these lines, is a solid angle, or a corner. is the vertex of the corner, L, M, N, K, forming the edges or axes of the dihedral angles are its edges ; the planes LOM, MON, NOK, and KOL are its faces ; and the angles LOM, MON, NOK, and KOL are its face-angles. The term corner or solid angle does not involve any SOLID ANGLES. 31 particular length of. line, or extent of plane, or magni- tude of angle. It involves the existence of a number of lines forming edges, with the same number of planes limited by these lines and forming faces, and all meeting at a common point to form a vertex. 34. A corner may have any number of faces greater than three, and the same number of edges. The one figured in the preceding article is a four-faced corner, or a tetrahedral angle. A section of a three-faced corner is a triangle; and as the triangle is the most important of all polygons, so the three-faced corner, or trihedral angle, is the most important of all corners. A corner will be indicated by writing its vertex followed by a point, and then the letters indicating points upon its several edges. Thus the symbol ABCD denotes the four-faced corner as figured in the preceding article. 32 SOLID OR SPATIAL GEOMETRY. PROPERTIES OF TRIHEDRAL ANGLES, OR THREE- FACED CORNERS. 35. Theorem. In any three-faced corner the sum of any two face angles is greater than the third. ABO is the three-faced corner. Proof. If the face angles are all equal to one another, the truth of the theorem is evident. If they are une- qual, let the angle LON be > than LOM. In the plane of L and N draw OK, making the angle LOK= LOM, and on M and K take OB = OD any conven- ient length, and let A be any point on L, other than 0. Let the plane of ABD cut N in C. Then AAOB = AAOD. (P. Art. 52. ) .-. AD = AB, and Z ADB = Z ABD. .-. Z CDB is > Z CBD, and CB is > CD. (P. Art. 62. 2.) But in the &BOC and DOC, BO = DO by construc- tion, OC is common, and BC > (7Z>. .-. Z JBO<7 is > Z DOC, (P. Art. 67.) and v Z AOB = Z J.OZ) by construction, .-. Z ,405 + Z 50C is > Z .40D + Z /)OC. Or Z ^10 + Z 0(7 is > TRIHEDRAL ANGLE OR THREE-FACED CORNER. 33 36. Problem. To find the locus of a point equidistant from the three edges of a three-faced corner. O is the vertex, and L, M, N the edges of the three- faced corner. Let P be a point on the required locus, and PA, PB, PC be perpendic- ulars upon the edges L, M, and JV respectively. In the right-angled triangles POA, POB, POC, PO is a common hy- pothenuse, and PA = PB = PC by hypothesis. Therefore the triangles are congruent, and OA = OB = OC. And the circle through A, B, C is a cone circle with and P as two vertices. Therefore OP passes through the centre of this circle and is normal to its plane. Hence the construction : take OA = OB= OC and join with the centre of the circle through A, B, and C; this join is the locus required. Def. The locus just found is a line equally inclined to the three edges, and is an isoclinal line to the edges. A plane normal to this line is also equally inclined to the edges and is an isoclinal plane to the edges. Cor. Since the edges may be considered as indefinite lines extending through the vertex and forming a sheaf of three, the three measures OA, OB, OC may each be taken in two opposite directions, or we can have eight variations of sign in all. But four of these are the other four reversed. Therefore three lines forming a sheaf have four iso- clinal lines and four isoclinal planes through the centre. 34 SOLID OR SPATIAL GEOMETRY. 37. Def. Corners are equal when they can be super- imposed so as to form virtually but one corner. In this superposition the vertices coincide, and the edges coincide in pairs, one from each corner. 38. Theorem. Two three-faced corners are equal when the face angles of the one are respectively equal to the face angles of the other, and they are disposed in the same order about the vertices. LMN and 0' L'M'N' are two three-faced corners having Z. LOM = Z L'O'M', Z MON= Z. M'O'N', Z NOL = Z N'O'L', and having these disposed in the same order about the vertices ; i.e. so that the order of magnitude of the angles is according to the same species of rotation for each. Then the corners are equal. Proof. Take OA = OB=OC = O'A' = O'B' = O'C 1 , A and A' being on corresponding edges, etc. The A A OB = A A' O'B', and AB = A'B 1 . Similarly, BC = B'C', and CA = C'A', and the A ABC = A A'B' C'. Therefore when A 1 B'C' is superimposed on ABC, the centres of their circumcircles coincide, and the normals to the planes of these circles at their centres coincide, and hence the vertices of the corners, lying on these nor- mals, coincide (Art. 36), and the two corners, coinciding in all their parts, form virtually but one corner. c' TRIHEDRAL ANGLE OR THREE-FACED CORNER. 35 39. Two triangles may be congruent and yet not be superposable until one of them is turned over in the plane. This operation, which is possible and allowable in plane geometry, is not always practicable in spatial geometry. Suppose the two three-faced corners of the previous article to be so placed that the triangles ABC and A'B'C' lie in one plane, and and 0' are upon the same side of this plane. Then the triangles are directly superposable and the corners are superposable and equal. But if the triangles ABC and A'B'C' be in the same plane and be directly superposable while and 0' are upon opposite sides of the plane, or if and 0' be upon the same side of the plane while the triangles are not superposable until one of them is turned over in the plane, then the two corners, although having correspond- ing parts respectively equal, are not superposable, and are not, therefore, equal according to definition. A little consideration will show that in the non-super- posable case, the face angles are disposed in opposite orders about the vertices of the two corners. Def. Two three-faced corners having corresponding parts respectively equal but not being superposable are said to be equal by symmetry, or to be symmetrical 1 to each other. Symmetrical figures are related to each other in the same manner as an object and its image in a plane mirror, or as the right and the left hand; and they 1 The term conjugate and opposable have both been employed to express the condition here described. But it is obvious that the well- known term symmetrical expresses exactly what is meant, and can- not therefore be profitably superseded by any other word. 36 SOLID OK SPATIAL GEOMETRY. might be called right-handed and left-handed figures if there were any means of distinguishing between which should be called right-handed, and which left-handed. In certain parts of crystallography the means of distin- guishing is apparent, and this terminology is employed. Two superposable figures can be in perspective with respect to a centre at infinity, while two symmetrical fig- ures can be in perspective with respect to a centre which is the middle point of the joins of corresponding parts. Cor. It is readily seen that two n-faced corners may be superposable and equal, and also that they may be symmetrical and not superposable. But where there are more than three faces, new possi- bilities arise, for the face angles may be equal in number and respectively equal in magnitude, and yet the corners may be neither equal nor symmetrical. 40. Theorem. Of two dihedral angles of a three- faced corner and the opposite face angles, 1. The greater face angle is opposite the greater dihedral angle ; 2. The greater dihedral angle is opposite the greater face angle. LMN is a three-faced cor- ner having as vertex, and L, M, .ZVas edges. From A, any point in L, draw AB _L to M and AC J_ to N, and from B and C draw, in the plane MN, perpendiculars to M and N respectively, and let these perpendiculars meet in D. Join OD. TRIHEDRAL ANGLE OK THREE-FACED CORNER. 37 The angles ABD and ACD are respectively the meas- ures of the dihedral angles whose edges are M and N (Art. 16. Def. 1) . The A ADB and ADC are right-angled at D and have AD as a common side, the triangles ABO and ACO are right-angled at B and C, and have AO as a common hypothenuse, and the A DOB and DOC are right-angled at B and C, and have OD as a common hypothenuse. 1. LetZ^BZ)be>Z^lC'Z); then Z .40(7 is > ZAOB. Proof. Since Z ABD is > Z ^<7A therefore, Z jB^lD is < Z CL4D, and BD is < GO ; and .-. BO is > CO, and .40 is > AB, and .-. Z .40(7 is > Z AOB. 2. This, which is the converse of 1, follows from the law of Identity (P. Art. 7). Cor 1. If a three-faced corner has two dihedral angles equal, it has two face angles equal ; and conversely, if it has two face angles equal, it has two dihedral angles equal. Cor. 2. A three-faced corner with three equal dihe- dral angles has three equal face angles, and conversely. Cor. 3. If A, B, C denote the dihedral angles, and a, b, c denote the opposite face angles, the order of mag- nitude is the same for A, B, C, and a, &, c. It will be noticed that in this theorem and its corol- laries the relations between the dihedral angles and face angles are analogous to those between the angles and sides of a plane triangle. 38 SOLID OE SPATIAL GEOMETRY. Def. A three-faced corner with its edges mutually perpendicular to one another is a rectangular corner or a right corner. It has all its dihedral angles right angles, and all its face angles right angles. 41. Problem. Being given the face angles of a three- faced corner, to construct plane angles which shall have the same measures as the dihedral angles. LMN is the given three-faced corner. To draw a plane angle which shall have the same measure as the dihedral angle whose edge is L. Constr. Through A, any point in Z,, draw a plane normal to L, and cutting M and N in B and C. (i) (2) In (2) take O'A 1 = OA, and through A' draw a line, K, perpendicular to O'A'. Draw O'B', making the Z A'0'B' = ZAOB, and O'C", making the angle A'O'C' = /. AOC. Also, draw O'B" = O'B' and making the Z C'O'B" = COB. Join C'B". The A DA'C' constructed with B'A 1 , A'C', and C'B" as sides, has the angle C'A'D equal in measure to the dihedral angle whose edge is L. TRIHEDRAL ANGLE OR THREE-FACED CORNER. 39 Proof. Since L is normal to the plane of AB and AC, the Z BAC measures the dihedral angle at L (Art. 16. Def. 1). And in the construction O'B' = OB and O'C' = 00, and hence A'B' = AB and A'C' = AC; and also we have made 0'B"C' congruent with OBC. Hence the A DA'C' is congruent with BAC, and the Z DA'C' measures the dihedral angle at L. Similarly, the other dihedral angle may be found. Cor. 1. Since in the foregoing construction only one triangle is possible with the given elements, the dihedral angles of a three-faced corner are completely given when the face angles are given; and hence the measures of the dihedral angles are expressible in terms of those of the face angles. Cor. 2. A three-faced corner is given when its face angles and their order with respect to the vertex are given. In w-faced corners where n is greater than 3, the giving of the face angles does not determine the dihedral angles, and does not therefore determine the form of the corner. We have the analogue of this" in plane geometry, where the giving of the sides of a polygon, with more than three sides, does not determine the form of the polygon. In general, corners of more than three faces are not of much importance unless they are regular. Def. A regular corner has all its face angles equal and all its dihedral angles equal. 42. Theorem. In any corner the sum of the face angles is less than a circumangle. 40 SOLID OR SPATIAL GEOMETRY. Proof. Let the corner have n faces. Cut it by a plane, and we have, as section, a polygon of n sides, the sum of whose internal angles is 2 (n 2) Is. Denote, in general, a basal angle of one of the result- ing triangular faces by B, and a face angle by F. At each vertex of the polygonal section, three faces meet to form a three-faced corner, viz. the section itself and two faces of the original corner. .-. 2-B is > the sum of the internal angles of the section, i.e. > 2(n- 2) Is. (Art. 35.) But 2.B -f 2F-. Or the sum of the face angles is less than a circumangle. 43. Let ABC be a three-faced corner, and let PS be normal to the plane A OB, PR normal to the plane COA, and PQ normal to the plane BOG. The angle QPR is the sup- plement of the dihedral angle at OC, RPS is the supple- ment of the dihedral angle at OA, and SPQ is the supple- ment of the dihedral angle at OB (Art. 16. Def. 2). Therefore P QRS is a three-faced corner in which the face angles are supplementary to the dihedral angles of ABC. Also, since OB is normal to the plane SPQ, etc., the face angles of ABC are supplementary to the dihe- dral angles of P-QRS. RECIPROCAL CORNERS. 41 Similar reasoning will apply to a corner of any number of faces. Def. Corners so related that the dihedral angles in the one are supplementary to the face angles in the other are called reciprocal corners. Cor. 1. Employing the notation of Art. 40. Cor. 3, for one of the corners, and the letters accented for the other, we have A + a' = B + b'= C+c'= . = A' + a = B'+b = C'+c=--'= 21s. Now, in any corner a'-f&'+c'-j is<4~|s; (Art. 42) and A + B+ C -\ \-a' + b' + c'-\ =2n~\s, where n denotes the number of faces. .-. A + B+C+-- is >(2n 4)1*. That is, the sum of the dihedral angles of any corner is greater than the difference between twice as many right angles as the figure has faces, and a circumangle. Cor. 2. Making n = 3, we see that the sum of the dihedral angles of a three-faced corner is greater than two right angles and less than six right angles. 44. Problem. Given the dihedral angles of a three- faced corner, to construct the face angles. Constr. Take the supplements of the given dihedral angles, and considering these as face angles, construct the corresponding dihedral angles by Art. 41. The sup- plements of these latter angles are the required face angles. This construction is evident from the preceding article. 42 SOLID OR SPATIAL GEOMETRY. Cor. 1. It is readily seen that only one set of face angles can be obtained when a set of dihedral angles is given ; so that when the dihedral angles of a three-faced corner are given, the face angles are given also ; and the face angles can be expressed in terms of the dihedral angles. Cor. 2. A three-faced corner is given when the dihe- dral angles, and their order, are given. To construct a three-faced corner when its face angles are given is analogous to constructing a triangle when its sides are given ; and to construct the corner when its dihedral angles are given is analogous to constructing the triangle when its angles are given. And this latter is a definite problem with respect to the corner, but an indefinite one with respect to the triangle. 45. Problem. To find the locus of a point equidistant from three given points not in line. Let A, B, C be the points, and let U be the right- bisector plane of AB, and V be the right-bisector plane of AC (Art. 24. Def.). Every point equidistant from A and B is on U (Art. 25. conv.), and every point equidistant from A and C is on V. And the required locus is the common line of U and V. But this line evidently passes through the cir- cumcentre of the triangle ABC and is normal to its plane. Hence the locus of a point equidistant from three given points, not in line, is the axis of vertices of the circum- circle of the three points considered as a cone-circle. Cor. The three right-bisector planes, of the joins of three points, taken two and two, form an axial pencil. LOCI. 43 46. Problem. To find a point equidistant from four given points which are not complanar, and no three of which are in line. Let A, B, C, D be the four points, and let PO be the locus of a point equidistant from A, B, and (7. Join D, the fourth point, to any one of the other three, as C, and draw the right- . bisector plane, X, of CD. N \ As D is not complanar with A, B, and C, the plane X is not parallel to PO, and therefore * A meets PO at some point 0. But . is equidistant from A, B, and (7, and it is also equidistant from C and D. Therefore is equidistant from A, B, C, and D. Cor. 1. The line OP is the common line to three bisector planes, namely, those of AB, BC, and CA (Art. 45. Cor.), and X is a fourth plane which goes through the point 0. The two remaining bisector planes, those of AD and CD, must pass through the same point 0. Therefore the six right-bisector planes of the joins of four non-complanar points, of which no three are in line, pass through a common point and form a sheaf of planes. Cor. 2. The four points can be combined to form four different triangles, and the lines, such as PO, which pass through their circumcentres and are normal to their planes, all pass through and form a sheaf of lines. Cor. 3. As the line PO can meet the plane X in only one point, there can be only one point equidistant from A, B, C, and D. 44 SOLID OR SPATIAL GEOMETRY. EXERCISES C. 1. Any face angle of a three-faced corner is greater than the difference between the other two. 2. Show how to construct a corner symmetrical with a given corner. 3. Show that the three bisectors of the dihedral angles of a three-faced corner have a common line, and that this line is an isoclinal to the three faces. 4. There are four isoclinal lines through the vertex to the three faces of a three-faced corner. 5. In the figure (2) of Art. 41 denote O'A' by p. Then A'D = A' B> = p tan c ; O'B" = O'B' -psec c; A'C' = ptanb; O'C' = psecb; and C'D 2 = C'B"' 2 = DA 12 + AC" 2 - 2 DA A'C' cos A (P. Art. 217.) = O'C' 2 + O'B" 2 -2 O'C'- O'B" cos a. . . substituting, and dividing by p 2 , tan 2 c + tan 2 6 2 tan c tan b cos A = sec 2 c + sec 2 b 2 sec c sec b cos a, whence by reduction and dividing by cos b cos c, cos a = cos b cos c + sin b sin c cos A ; or, cos A = (cos a cos b cos c) /sin b sin c ; which expresses a dihedral angle in terms of the face angles. 6. Express a face angle in terms of the dihedral angles. (Employ the property of the reciprocal corner.) 7. If the face angles of a three-faced corner are each 60, show that the cosine of a dihedral angle is |. 8. In 46 where is the locus if A, B, C are in line ? 9. In 47 where is the point if the four points be complanar ? where if three points be in line ? SECTION 4. POLYHEDEA. 47. Def. A. spatial figure formed of four or more planes so disposed as to completely enclose a portion of space is a polyhedron. It is analogous to the polygon in plane geometry, and its plane section is always some form of polygon. The faces of the polyhedron are those portions of planes which are concerned in forming the closed figure, but for generality the term is sometimes extended to outlying parts of these planes. The adjacent faces meet by twos to form edges, and the edges are concurrent in groups of three or more to form corners. When a polyhedron is such that no line can meet more than two of its faces, it is convex. 48. Theorem. In any polyhedron the sum of the num- ber of faces and the number of corners is greater by two than the number of edges. Proof. Any polyhedron may be supposed to be built up by beginning with one face, and to it adding a second face, and then a third, and so on until the figure is completed. Denote, in general, the number of corners by C, the number of faces by F, and the number of edges by E. 45 46- SOLID OR SPATIAL GEOMETRY. 1. Let us start with a single face, U. The number of edges is the same as the number of corners, and we have one face. Therefore the equation C+F = E+\. is satisfied. 2. To U add the face F. In so doing V loses one of its edges, BC, and two of its corners, B and C, by union with similar parts of U. So that in adding V we increase F by 1, and we increase E by one more than the increase of (7; and hence the equation C + F=E-t-l is still satisfied. 3. To U and V add W. This new face loses two of its edges, DC and CG, and three of its corners, D, C, and G. Here again we add one face and one more edge than corner, so that C + F=E-{-Hs still satisfied. 4. It is readily seen that in adding any face whatever, that face loses one more corner than edge by union with other faces, until we come to the last face necessary to complete the polyhedron. This face loses all its edges and all its corners, so that by adding this face we increase the number of faces by 1 without interfering with the numbers of edges or cor- ners. And hence in the completed polyhedron we have This beautiful theorem is usually attributed to Euler, and is known as Euler's theorem on Polyhedra, but it appears to have been known before his time. THE TETRAHEDRON. 47 CLASSIFICATION OF POLYHEDRA. 49. Polyhedra may be classified as follows : 1. Tetrahedron. 2. Parallelepiped, Cuboid, Cube. 3. Pyramid, Frustum of Pyramid. 4. Prism, Truncated Prism. 5. Prismatoid, Prismoid. 6. The five Regular Polyhedra. 7. A number of Semi-regular Derived Polyhedra. This classification is not exhaustive, and its divisions are not mutually exclusive. It includes, however, all the polyhedra usually met with. Polyhedra are not equally important in any sense, and only a few can be said to be important in a descriptive sense. THE TETRAHEDRON. 50. The three planes which form a three-faced corner, and any fourth plane, not through the vertex, which cuts them all, form the closed figure called a Tetrahedron. The tetrahedron A BCD has four triangular faces, four three-faced corners, and hence four vertices and six edges, i.e. the six joins of four non- complanar points no three of which are in line. Def. Any face of the tetra- hedron may be taken as the base of the figure. The 48 SOLID OR SPATIAL GEOMETRY. three edges which bound the base are then called basal edges, and the other three are lateral edges. The joins of the middle points of opposite edges are diameters. There are thus three diameters, EG, FH, and IJ. 51. Theorem. The diameters of a tetrahedron bisect one another. Proof. ABCD is a skew quadrilateral, and BD and AC are its diagonals. But the joins of the middle points of opposite sides of a skew quadrilateral bisect one another (30). Therefore EG, FH, and IJ bisect one"another. Def. 1. The point of concurrence of the diameters is the centre of the tetrahedron. And a section through the centre parallel to a pair of opposite edges is a middle section, as EFGH. Cor. There are three middle sections, and these pass through the middle points of the six edges taken in groups of four. The middle sections are evidently parallelograms, and they intersect by twos along the three diameters. Def. 2. A median of a tetrahedron is the join of a vertex with the centroid (P. Art. 85. Def. 2) of the opposite face. There are thus four medians, one to each face. 52. Theorem. The medians of a tetrahedron pass through the centre, and are divided at that point so that the part lying between the centre and a face is one- fourth of the whole median to that face. THE TETRAHEDRON. 49 In the tetrahedron ABCD, I is the middle point of 'BC, and J of AD, and IJ is thus a diameter. IP is one-third of I A, and P is thus the centroid of the face ABC (P. Art. 85), and DP is the median to the face ABC. Evidently DP and IJ are corn- planar, and intersect in some point 0. Then is the centre. Proof. Draw JQ II to DP to meet TA in Q. Then, as J is the middle point of AD, so Q is the middle point of AP (P. Art. 84. Cor. 2). And v QP=PI and OP is II to JQ, is the middle point of JI, and is therefore the centre (P. Art. 84. Cor. 2). Hence the medians pass through the centre. PO = \ QJ, and QJ= % PD. Again, NOTE. A face of a polyhedron is a segment of a plane, and is in form triangular, rectangular, etc. But in order to avoid such uncouth words as parallelogramic we shall speak of these faces as being triangles, rectangles, parallelograms, etc., although not using these terms strictly as defined in plane geometry. This usage will shorten language and cannot possibly lead to confusion. 50 SOLID OR SPATIAL GEOMETRY. THE PARALLELEPIPED. 53. Def. The parallelepiped has six faces, of which each pair of opposite ones are parallel planes. The contraction ppd. will be frequently used for the word 'parallelepiped.' . Since parallel planes cut any other plane in parallel lines (Art. 21), and since the planes AC and A'C' are parallel and cut the paral- lel planes AD' and A'D, it follows that AB, CD, A'B 1 , and C'D' are all parallel. Similarly, AD, BC, A'D', and B'C' are parallel, and AC', A'C, BD 1 , and B'D are parallel. Thus the faces of a ppd. are parallelograms congruent in opposite pairs, and the twelve edges are in parallel sets of four in each set. The corners, which are eight in number, are each three-faced, and the three edges which meet at any one vertex give the directions of all the edges, and these are therefore called direction edges. Cor. 1. As the Z.BAD = Z. B'A'D', the Z DAC' = Z D'A'C, and the Z BAG' = Z B'A'C, the corners having their vertices at A and A' contain face angles which are respectively equal, but these are disposed in opposite orders about the vertices. The same is true for any other pair of opposite cor- ners. THE PAEALLELEPIPED. 51 Therefore, opposite corners of a parallelepiped are symmetrical. Cor. Considering three-faced corners composed of the same face angles as being of the same variety, there are at most only four varieties of corner in any parallelepiped. These will be called representative corners. 54. By considering the forms of representative corners, all ppds. may be divided into two classes, the acute and the obtuse. A denoting any angle, let A' denote its supplement. Let A, B, (7, all acute or all obtuse, be the three face angles at one corner of a ppd. Then the representative cor- ners are easily seen to be ABC, AB'C', A'BC', and A'B'C. (1) If A, B, C are acute, A', B', C' are obtuse. Therefore, if a parallelepiped has one corner formed of acute face angles, the other rep- resentative corners contain one acute and two obtuse face angles, each. This is an acute parallelepiped. (2) If A, B, C are obtuse, A', B', C' are acute. Therefore, if a parallelepiped has one representative corner composed of obtuse face angles, the other repre- sentative corners have, each, one obtuse and two acute face angles. This is an obtuse parallelepiped. It thus appears that no one ppd. can contain all the kinds of corners belonging to ppds. 52 SOLID OR SPATIAL GEOMETRY. 55. Def. The join of opposite vertices in a ppd. is a diagonal. These are four in number, viz. AA', BB 1 , CC', and DD' (Fig. of 53). Since AD is II to JB(7, is II to D'A' and equal to it, AD'A'D is a parallelogram, and its diagonals bisect one another. Hence AA' and DD' bisect one another ; and similarly, AA' and BB' bisect one another, etc. Therefore, all the diagonals of a ppd. pass through a common point, and are bisected at that point. The common point of the diagonals is the centre. 56. Theorem. Every line-segment passing through the centre of a parallelepiped, and having its end-points upon the figure, is bisected at the centre. Proof. PQ (Fig. 53) is a line-segment passing through the centre, 0, and having its end-points P, Q in the face AC and A'C' respectively. Join AP and A'Q. Then AP and A'Q are complanar, since PQ passes through 0; and the plane of APand A'Q cuts the parallel faces AC and A'C' in parallel lines (Art. 21. Cor. 1). .-. APis II to A'Q. Also, AO = A'0, and and ZOAP=^OA'Q. .-. AAOP=AA'OQ, and OP=OQ. Cor. The centre of a ppd. is the centre of every cen- tral section. THE PARALLELEPIPED. 53 57. As a parallelepiped has three direction edges, three sections may be made normal to each of these edges respectively. These sections will be forms of the paral- lelogram. Def. 1. If none of the sections are rectangles, the ppd. is tridinic, and none of its angles, whether face or dihedral, are right angles. 2. If one section is a rectangle, the ppd. is diclinic, and four dihedral angles, whose edges are parallel, are right angles. 3. If two sections are rectangles, the ppd. is mono- clinic, and two sets of four dihedral angles are right angles. 4. If the three sections are rectangles, all the faces are rectangles, and all the dihedral angles are right angles, and all the corners are right corners (Art. 40. Def.). The figure is then a cuboid. 1 Cor. In the cuboid all the diagonals are equal, and the direction lines are mutually perpendicular to one another. Def. 2. A cuboid with its edges equal is a cube. The faces of the cube are squares. The analogues of the ppd., the cuboid, and the cube, are in plane geometry the parallelogram, the rectangle, and the square. 1 This term was proposed by Mr. Hayward. Before the appearance of Mr. Hayward's work I used the term orthopiped for a rectangular parallelepiped. But cuboid is evidently a better and a more convenient term. 54 SOLID OR SPATIAL GEOMETRY. THE PYRAMID. 58. Def. 1. When a corner of any number of faces is cut by a plane which cuts all the faces, the closed figure so formed is called a pyramid. The cutting plane is the base, and the planes which form the corner are faces of the pyramid. The edges which bound the base are basal edges, and those which belong to the corner are lateral edges. The vertex of the corner is the vertex or apex of the pyramid. Def. 2. Pyramids are classified into triangular, square, etc., according to the character of the base. A triangular pyramid is a tetrahedron. 59. Def. If a pyramid be cut by a plane parallel to its base, the portion lying between the base and this cut- ting plane is called a frustum of a pyramid. The frustum has thus two bases, a lower and an upper, or a major base and a minor base. From Art. 28. Cor. 2, it follows that the two bases of the frustum of a pyramid are similar polygons. THE PRISM. 60. When the vertex of a pyramid goes to infinity in a direction normal to the base, the lateral edges become parallel lines, and the resulting figure is not a closed figure. But under like circumstances the frustum be- comes a closed figure with two congruent bases, and is called a prism. If one edge of a prism is normal to a base, all the edges are normal, and the lateral faces are rectangles. This is called a right prism. THE REGULAR POLYHEDRA. 55 And if one of the lateral edges is inclined to the base, they are all inclined at the same angle. This is an oblique prism. Prisms are usually named from the character of the right section. Thus a right rectangular prism is a cuboid, and a parallelepiped may be a right prism or an oblique prism, depending upon its kind (Art. 57). THE REGULAR POLYHEDRA. 61. Def. A regular polyhedron is one in which all the faces are regular polygons of the same number of sides, and all the corners are formed by the same number of faces. This implies that all the edges are equal, that all the face-angles are equal, and that all the dihedral angles are equal. On account of the perfect symmetry of the figure, it must have a definite centre equally distant from each face and equally distant from each vertex. The normal at the centre of each face passes through the centre of the figure, and the line from a vertex to the centre is an isoclinal to the edges of that vertex and to the faces of that vertex. One of the regular polyhedra is familiarly known as the cube. 62. Theorem. There cannot be more than five regular polyhedra. Proof. The least number of faces which can form a corner is three, and these must not be complanar. There- fore the three face-angles must together be less than a 56 SOLID OK SPATIAL GEOMETRY. circumangle, or a face-angle must be less than four- thirds of a right angle (Art. 42). The only regular polygons having their internal angles less than of a right angle are (P. Art. 133. Cor.) the equilateral triangle, the square, and the regular penta- gon; and these alone can form the face of a regular polyhedron. Equilateral Triangle. A corner may be formed of 3, 4, or 5 equilateral tri- angles, and may therefore be three-, four-, or five-faced. 1. The three-faced corner gives the regular tetrahedron, with 4 faces, 4 corners, and 6 edges. 2. The four-faced corner gives the regular octahedron, with 8 faces, 6 corners, and 12 edges. 3. The five-faced corner gives the regular icosahedron, with 20 faces, 12 corners, and 30 edges. Square. Only one corner, a three-faced, can be formed by squares. 4. This gives the cube, with 6 faces, 8 corners, and 12 edges. Regular Pentagon. Only one corner, a three-faced one, can be formed. 5. This gives the regular dodecahedron, with 12 faces, 20 corners, and 30 edges. These are the five regular polyhedra. 63. Euler's theorem, Art. 48, gives F+ C=E + 2. Now the numbers denoted by F and C are evidently interchangeable, while E remains the same. That is, THE REGULAR POLYHEDRA. 57 if we have a given polyhedron, we can form another polyhedron in which the number of corners is the same as the number of faces in the given polyhedron, and the number of faces is the same as that of the corners in the first polyhedron, while the number of edges remains the same in both. These polyhedra may be called reciprocals of each other, as either may be formed from the other by a sort of reciprocation, the changing of points into planes, and planes into points. If a point be taken in each face of any polyhedron, preferably the centre where there is one, and these points be joined in every way, provided we join only points which lie on adjacent faces, the joins form the edges of a polyhedron which is reciprocal to the original polyhedron. If the new polyhedron be treated in the same way, we obtain a third polyhedron, which is reciprocal to the second, and is accordingly of the same species as the first. 64. Applying the principles of the preceding article to the regular polyhedra, we readily see that the octa- hedron is the reciprocal of the cube, and the dodeca- hedron is the reciprocal of the icosahedron. The tetrahedron, having the number of its faces and vertices the same, gives another tetrahedron by recipro- cation ; or the tetrahedron is self-reciprocal. 65. Interesting models of all the polyhedra may be made by drawing proper figures on cardboard; then cutting out the entire piece, and cutting half-way through 58 SOLID OK SPATIAL GEOMETRY. the remaining lines. The piece of cardboard may now be folded along these lines to form the intended figure, and the edges be fastened together with glue. The figure drawn on the cardboard is called a net. The net for an obtuse parallelepiped is given in the diagram. The faces are denoted by V, V, and W, those having the same letter being opposite, and therefore con- gruent parallelograms. The edges which come together are denoted by the same small letter. Those having the same letter attached must, of course, be the same in length. The three obtuse angles concerned are denoted by A, B, and (7. All the other angles are then known. If the angle C were acute, as indicated by the dotted lines, the ppd. would be acute. And the same results would be obtained by making either A or B acute. As the net is drawn, the ppd. will be triclinic. If the U faces be rectangles, the ppd. will be diclinic ; if both C/"and Fare rectangles, it will be monoclinic; and if all NETS. 59 the faces be rectangles, the figure will be the cuboid ; and if all squares, the cube. The accompanying diagrams give nets for the regular polyhedra other than the cube. Nets for prisms and pyramids and frusta need no description. 60 SOLID OB, SPATIAL GEOMETRY. EXERCISES D. 1. The faces of a polyhedron are 3 squares and 2 triangles. Find the number of edges and of corners and classify the figure. 2. If an n-hedron has all its faces triangles, the number of its corners is J (n + 4). 3. If P, Q, .7?, S be the centroids of a tetrahedron, the recipro- cal having P, Q, E, S as vertices has the same centre as the original. Also the diameters and medians of the two tetrahedra coincide, except in length. 4. In the regular tetrahedron the diameters are perpendicular to one another. 5. If the diameters of a tetrahedron terminate in the centres of the faces of a cube, then the edges are diagonals of the faces. Thence show how the cube may be transformed into a regular tetrahedron. 6. If AA', SB', CO', and DD' are diagonals of a cuboid, show that the middle points of AB, BC, CA>, A'B', B'C', and C'A are complanar. Find the form of the section through these points. 7. The join of A 1 with the middle point of AB, and the join of C" with the middle point of BC, divide each other into parts which are as 2 to 1 (Ex. 6). 8. The centres of the adjacent faces of a ppd. are joined. What closed figure is formed ? Describe its characteristics. SECTION 5. THE CONE, THE CYLINDER, AND THE SPHERE. 66. The three figures here mentioned are the simplest spatial figures having curved surfaces, and they are fre- quently spoken of as the three round bodies. The cone and the cylinder can be generated by the motion of a straight line, and they are consequently called ruled surfaces. The sphere is not a ruled surface, but a surface of double curvature. Def. A surface which can be generated by the revolu- tion of a plane figure about an axial line in its plane, is a surface of revolution. The sphere is a surface of revolution. The cone and the cylinder may or may not be surfaces of revolution. Solid geometry furnishes other interesting examples of ruled surfaces besides the cone and cylinder, and of surfaces of revolution besides the sphere. As examples of the first we have the common conoid, the hyperboloid of one sheet, and the elliptic paraboloid; and of the second, the oblate spheroid, the prolate spheroid, and the anchor ring. THE CONE. 67. Def. 1. In general, a variable line which passes through a fixed point and is guided by a fixed plane 61 62 SOLID OK SPATIAL GEOMETRY. curve, not complanar with the point, generates a cone, or has a cone as its locus. is a fixed point, and APE is a fixed curve not com- planar with the point. The variable line L passes through 0, and meets the curve APB. Then L generates a cone. Cor. Since L is unlimited in length, the cone extends in- definitely outwards upon both sides of 0, and is not a closed figure. Def. 2. is the centre of the cone, and the two parts into which it divides the cone are called the two nappes or sheets of the cone. The fixed curve APB is the director, and the line L is the generator of the cone. Any line which coincides with the generator in any of its positions is called a generating line. Thus every line passing through and lying on the conical surface is a generating line. 68. The director may be any form of curve. If it becomes a line, the cone degrades into a plane (Art. 7. 3) ; and if the director becomes a point, the cone becomes the line through that point and the centre. Thus the line and the plane may be looked upon as limiting forms of the cone. THE CONE. 63 69. When the director is a circle, and the centre is a vertex to that circle as a cone-circle (Art. 10. Def. 1), the cone is a right circular cone, and the line through the centre of the circle and the centre of the cone is the axis of the cone. The circular cone is a figure of revolution, and is the most important of all cones. The word 'cone' as hereafter employed will mean a right circular cone, unless otherwise qualified. 70. Let C (Fig. of 67) be the centre of the circular director APE. Then CP is constant, and CO is con- stant, and OCP is a 1 . Therefore the Z POC is constant. This angle is the semi-vertical angle of the cone. Hence a circular cone is generated by a line which revolves about a fixed axial line while meeting the latter in a fixed point and at a fixed angle. Cor. 1. Every section of a circular cone, normal to the axis, is a circle. Cor. 2. Every section of a circular cone, through the axis, is two lines intersecting at a fixed angle the vertical angle of the cone. Cor. 3. Every section of a circular cone through the centre is two lines ; for the plane meets the cone along two generating lines. Cor. 4. Any point on the axis of a circular cone is equidistant from the surface on all sides, and the axis is thus an isoclinal line to the surface. 71. Theorem. Only two generating lines of a cone are complauar. 64 SOLID OR SPATIAL GEOMETRY. Proof. Since the generating lines all pass through 0, any two of them are complanar. Let any two particular generating lines meet the director circle in A and P. The plane of these lines meets the plane of the circle in a line (5), and as a line can meet a circle in only two points (P. Art. 94), the plane of OA and OP has only two points coincident with the circle, and therefore only two generating lines lie in this plane. 72. Theorem. A line which is not a generating line can meet a cone in only two points. Proof. Let M be the line, not passing through 0; and let the plane U pass through 0, and contain M. If U cuts the cone, it contains two generating lines ; and since it contains M, the two generating lines are complanar with M, and meet it in two points, and in only two points ; and these points are common to M and to the cone. Therefore, the line M can meet the cone in two, and in only two, points. 73. If the two points in which a line M, which is not a generating line, meets a cone become coincident, the line becomes a tangent line to the cone, and has one point only, a double point (P. Art. 109. Def. 2) in common with the cone. The plane determined by a tangent line and the generat- ing line through its point of contact is a tangent plane to the cone, and touches the cone along this generating line, which, as it represents the union of two lines, is a double line. THE CYLINDER. 65 Cor. 1. Evidently all tangent planes to a cone pass through the centre and form a sheaf of planes. Cor. 2. All tangent planes to a cone intersect one another in lines which pass through the centre and form a sheaf of lines. 74. Def. A line through the centre perpendicular to a generating line of a cone generates a second cone, which is the reciprocal of the first. When the vertical angle of the cone is a right angle, these two cones become coincident, and form but one cone. THE CYLINDER. 75. When the centre of a cone goes to infinity in the direction of the axis, and the director curve remains finite, the cone becomes a cylinder, and the axis of the cone becomes the axis of the cylinder. Hence : Def. 1. A cylinder is the locus of a line which keeps a fixed direction and meets a fixed plane curve which is not complanar with the line. Def. 2. A circular cylinder is generated by one of a pair of parallel lines while revolving at a fixed distance about the other parallel as a fixed axial line. The fixed line is the axis of the cylinder. Cor. 1. The cylinder, as defined, is not a closed figure. Cor. 2. A line can meet a circular cylinder twice, and only twice. Cor. 3. Sections of a circular cylinder normal to the axis are equal circles. SOLID OR SPATIAL GEOMETRY. THE SPHERE. 76. Def. A sphere is the locus of a semicircle which revolves about its limiting centre line as an axial line. BAD is a semicircle, and AB is its limiting diameter. When ADB revolves about AB as an axis, the semicircle generates a sphere of which OD is a radius. Gor. 1. All the radii of a sphere are equal to one another. Therefore, Def. A sphere is a surface every point on which is equi- distant from a fixed point within called the centre. Cor. 2. The sphere is a closed figure, so that to pass from without the sphere to within, or from within to without, it is necessary to cross the surface. Cor. 3. A point is within a sphere, on the sphere, or without it, according as its distance from the centre is less than, equal to, or greater than the radius of the sphere. Cor. 4. Two spheres which have the same centre and the same radius coincide in all their parts and form virtually but one sphere. 77. Theorem. Every plane section of a sphere is a circle. Let DEP be the plane section and P be any point on it (Fig. of 76). THE SPHERE. 67 Then, being the centre of the sphere, OP is constant, and P lies in the plane of section. Therefore (Art. 10. Cor.) the section is a circle. Def. The section by a plane through the centre of the sphere is the largest circle producible, and is called a great circle of the sphere. All other sections are small circles. Cor. A great circle of a sphere has its centre coinci- dent with that of the sphere ; and the generating semi- circle of the sphere is one-half of one of its great circles. 78. Theorem. A line can meet a sphere in two, and in only two, points. Proof. If a line meets a sphere, any plane containing the line gives in section a circle cutting the line ; and as the circle cuts the line twice, and twice only, so a line can meet the sphere in two, and in only two, points. Def. A line which meets a sphere is a secant line, and the part within the sphere is a chord. A secant through the centre is a centre line, and its chord is a diameter. A plane which cuts a sphere is a secant plane, and when it passes through the centre it is a diametral plane. 79. Theorem. The join of the centre of a sphere with the middle point of a chord is perpendicular to the chord. Let DE be a chord whose middle point is C (Fig. of 76) ; then OC is 68 SOLID OE SPATIAL GEOMETRY. The plane of and DE gives in section a circle with DE as chord, and as centre. And C being the middle point of the chord, OC is _L DE (P. Art. 96. 4). Cor. 1. Diameters of the same small circle bisect one another, and being chords of the sphere, the join of the centre of a small circle with the centre of the sphere is normal to the plane of the small circle, i.e. to the plane of section. The converse of this is evidently true. Cor. 2. Lines through the centres of small circles and respectively normal to their planes meet at the centre of the sphere. Cor. 3. The plane normal to any chord at its middle point contains the centre of the sphere. For this plane is the right-bisector plane of the chord, and therefore contains every point equidistant from the end points of the chord. But the centre of the sphere is equidistant from the end points of the chord. 80. Problem. To find the centre of a given sphere. 1st Solution. Draw, on the sphere, two small circles whose planes are not parallel, and draw normals to the planes of these circles at their centres. These normals meet at the centre of the sphere (Art. 79. Cor. 2). 2d Solution. Draw any three non-parallel chords and their right-bisector planes. These planes have the centre as their common point (Art. 79. Cor. 3). Cor. 1. In the first solution, if the planes of the circles are parallel, the normals also are parallel ; and THE SPHERE. 69 as they pass through the same point, the centre of the sphere, they are coincident (P. Art. 70. Ax.). Therefore the centres of parallel sections of a sphere lie upon a centre line normal to the planes of section, and are therefore collinear. Cor. 2. In the second solution, if the chords are parallel, so also are their right-bisector planes ; and as these planes are concurrent, they are also coincident. Therefore, the middle points of parallel chords in a sphere are complanar, and lie upon a diametral plane normal to the chords. 81. When the two points in which a line meets a sphere become coincident, the line becomes a tangent line to the sphere and touches the sphere in a double point. Hence, for a sphere to touch a given line at a given point is equivalent to two conditions. 82. Theorem. A tangent line to a sphere is perpen- dicular to the radius to the point of contact, and con- versely. Proof. The plane determined by the tangent line and the radius to the point of contact gives in section a circle with its tangent line and radius, and as the same angle is involved, the truth of the theorem follows (P. Art. 110). Def. An indefinite number of perpendiculars may be drawn to a radius at its extremity ; these are all tangent lines, and they all lie in a plane to which the radius is normal. This plane is a tangent plane to the sphere. 70 SOLID OR SPATIAL GEOMETRY. Cor. A tangent plane is normal to the radius to the point of contact. 83. Theorem. Through any four non-complanar points, of which no three are in line, one, and only one, sphere can pass. Proof. It is shown in Art. 46. Cor. 3, that one, and only one, point is equidistant from four given non-com- planar points, no three of which are in line. If this point be taken as centre, and its distance from any one of the given points be taken as radius, the sphere so determined passes through the four given points. Cor. 1. Four non-complanar points, no three of which are in line, determine one sphere. Cor. 2. Spheres which coincide in four non-coniplauar points coincide altogether. Def. Four or more points so situated that a sphere can pass through them are conspheric, and when these points form the vertices of a figure, the figure is inscribed in the sphere, and the sphere circumscribes the figure. When a sphere has all the sides of a skew polygon as tangent lines, the sphere is inscribed to the polygon, and the polygon is circumscribed to the sphere. With a polyhedron it is different. For a sphere may have the edges as tangent lines, or the faces as tangent planes, but not both. The sphere having the edges as tangent lines is the tangent sphere to tlie edges, and the one having the faces as tangent planes is the tangent sphere to the faces. THE SPHERE. 71 Cor. 3. A regular polyhedron, on account of its com- plete symmetry, has all its vertices conspheric, all its edges tangent lines to a sphere, and all its faces tangent planes to a sphere, and these three spheres have the same centre. 84. The vertices of a skew quadrilateral are neces- sarily conspheric ; for from the definition (29) they are four non-complanar points, no three of which are in line. . Let A, B, C, D be the vertices taken in order, and let the sides AB, BC, CD, and DA be considered as lines of indefinite length. Cor. 1. Let A and B become coincident. Then AB becomes a tangent line to the sphere. Therefore, a sphere can touch a given line at a given point, and pass through any two other points whose join is not complanar with the given line. Cor. 2. Let A, B, and C become coincident. Then the lines AB and BC become two tangent lines inter- secting on the sphere at B, and these determine a tan- gent plane. Therefore, a sphere can touch a plane at a given point and pass through any one other point which does not lie in the plane. As four points properly situated are necessary to determine a sphere, touching a plane at a given point is equal to three conditions, and the point of contact is thus a triple point. Cor. 3. Let A and B become coincident at one point, and C and D become coincident at another. Then the 72 SOLID OR SPATIAL GEOMETRY. line AB becomes a tangent line at one point, and the line CD a tangent line at another, and these two tangents are not complanar. Therefore, a sphere may touch two non-complanar lines at any two given points, one in each line. 85. Theorem. The figure of intersection of two spheres is a circle, and the common centre line of the spheres passes through the centre of the circle and is normal to its plane. Proof. Let and 0' be the centres of the spheres, and P be a point on their figure of intersection PQR. Then OP, and O'P, and 00' are constant for all positions of P. Therefore, P lies on a cone-circle to which and 0' are vertices, and hence 00' passes through the centre, C, of the circle, and is normal to its plane. Cor. 1. OP and O'P being given, CP decreases as 00' increases, and vice versa. When OPO' is a right angle, the tangent planes to the two spheres are perpendicular to one another, and the spheres intersect orthogonally. Cor. 2. When P comes to C, the circle PQR becomes a point upon the line 00'. Therefore, when two spheres touch, they do so at a single point, and the common centre line passes through the point of contact. 86. APBR is a sphere with as centre and 0' any point without the sphere. O'P is a tangent line from THE SPHERE. 73 0', touching the sphere at P. PQR is the small circle through P, whose plane is normal to 00'. 1. 00' and OP are constants, and Z OPO' is a right angle, since O'P is a tangent (Art. 82). Therefore O'P is constant, and P always lies on the small circle PQR, which is a cone-circle to and 0' as vertices. Therefore all tangent lines from a given point to a sphere are equal. 2. O'P is the generator of a cir- cular cone which touches the sphere along the small circle PQR, and 0' is the centre or ver- tex of the cone. Def. The cone of which O'P is the generator is the tangent cone for the point 0'. The circle PQR is the circle of contact, and its plane is the polar plane of the point 0' with respect to the sphere ; and the point 0' is the pole of the plane. 3. When 0' comes to A, the tangent cone and the polar plane of 0' unite to form the tangent plane at A ; hence a tangent plane is a double plane representing the limiting form of the tangent cone, and the limiting posi- tion of the polar plane as the pole comes to the sphere. Evidently, then, a tangent plane is the polar plane to its point of contact. 87. Problem. To find the locus of a point equidistant from three planes, no two of which are parallel, and which do not form an axial pencil ; i.e. from three planes which form a sheaf. 74 SOLID OE SPATIAL GEOMETBY. Let ABC, ACD, ADB be the planes having A as their common point. Let the internal and external bisecting planes of the dihedral angle whose edge is AB be denoted by ab and AB respectively, and similarly for the other dihedral angles. Also, let Af denote the com- mon line of ab and ac. Then, as every point on ab is equidis- tant from the planes ABC and ABD, and every point on ac is equidistant from the planes ACB and ACD, every point on the intersection of ab and ac, that is, on A i} is equi- distant from the three given planes. The line A { is thus inclined to all the planes at the same angle, and it will be called the internal isoclinal line to the planes. Again, every point on AB is equidistant from the planes ABC and ABD, and every point on AC is equi- distant from the planes ACB and ACD. Therefore, every point on the common line of AB and AC, that is on the line A d , is equidistant from the three planes, and A d is an external isoclinal line to the planes. Similarly, A b and A c are external isoclinals to the same three planes. Therefore, the required locus consists of the four isoclinal lines to the planes. These isoclinals pass through A, the common point, and form the centre locus of a sphere which touches the three concurrent planes. THE SPHERE. 75 Cor. Each isoclinal line is the common line to three bisector planes which form an axial pencil, viz. : A { of ab, ac, ad; A b of ab, AC, AD ; A c of ac, AB, AD; and A d of ad, AB, AC. 88. Problem. To find the centre of a sphere which shall touch four planes so situated as to form a tetra- hedron. Employing the notation of Art. 87, we have four iso- clinal lines to three of the planes, at each vertex of the tetrahedron, or 16 in all. These are A { , B { , C { , D { as internal ones, and A t , A c , A d , B a , B c , B d , C a , C b , C d , D a , D b , D c , as external ones. Denote the planes opposite A, B, C, D by a, ft, y, 8, respectively. Then A { is the locus of a point equidistant from (3, y, 8 ; and B ( from a, y, 8. Therefore, a point equidis- tant from y and 8 lies upon both A t and B,, and hence these lines intersect, and C t and D t pass through the point of intersection. Hence (1) A a B C D t , meet to give one point required. Similarly, each of the following groups of four lines gives a point equidistant from the four planes: (2)A a B a ,C a , D a ; (3) B e A b , C b , D b ; (4) G e A c , B c , D c ; and (5) D A d , B d , C d . Again, A b is the locus of a point equidistant from /3, y, 8, and B a from a, y, 8. Therefore, A b and B a intersect in a point equidistant from a, (3, y, 8, and C d and D c pass through this point. 76 SOLID OR SPATIAL GEOMETRY. Hence these lines meet in groups of four to give three points equidistant from the four planes ; namely, (6) A b , B a , C d , A; (7) A c , C a , B d , A; (8) A d , D a , B c , C b . Thus eight spheres, in all, can be found, each of which shall touch four planes so situated as to form a tetra- hedron. EXERCISES E. 1. If the director, figure in the generation of a cone (61) is a polygon, what figure is formed ? 2. Show that the cone is a limiting case of an n-faced corner, and explain how. 3. If the radius of a sphere is the generator of a circular cone, the figure of intersection of the sphere and cone is a circle. 4. The centre locus of a sphere which touches a plane at a given point is a normal to the plane at the given point. 5. What is the centre locus of a sphere which touches a line at a given point ? which touches two parallel lines ? which touches two intersecting lines ? which touches two intersecting planes ? PART II. AREAL RELATIONS INVOLVING LINE-SEGMENTS ABOUT SPATIAL FIGURES. 89. The theorem in plane geometry that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the sides, the theorem that the rectangle on the parts of a secant line between a point and a circle is equal to the square on the tangent from the point to the circle, and others of this nature, express areal relations, involving line-segments of plane figures. Many important relations of a similar nature exist among the line-segments connected with spatial figures. These we propose to consider in this part of the work. 77 SECTION 1. THE SKEW QUADRILATERAL AND THE POLYHEDRON. 90. Theorem. In a skew quadrilateral the sum of the squares on the sides is greater than the sum of the squares on the diagonals, by four times the square on the join of the middle points of the diagonals. Proof. ABCD is a skew quadrilateral, and AC and BD are its diagonals, having / and J as their middle points. D I is median to A CD A, and BI is median to A CBA. .-. CD*+DA 2 + CB 2 + BA 2 =2 (CP+DP+ CP+BI 2 ); or But IJ is median to the A DIB ; or, = CA 2 + BD 2 + 4 IJ 2 . Q. E. D. This important theorem is true of all quadrilaterals, whether plain or skew (P. Art. 173). 78 THE SKEW QUADRILATERAL. 79 91. For the tetrahedron let us adopt the following notation : Taking ABC as the base and D as the vertex, denote the lateral edges DA, DB, DC by a, b, and c respectively, and the basal edges BC, CA, AB by a 1} &!, and c : respectively. Then a and a t are opposite edges, etc. Theorem. In any tetrahedron four times the sum of the squares on the diameters is equal to the sum of the squares on the edges. Proof. The skew quadrilateral with its diagonals forms the tetrahedron. The results of Art. 90 give : 4 IJ* = a? + c 2 + a! 2 + C/ - & 2 - b?, (Fig. of 90. ) 4 FH 2 = b 2 + c 2 + b? + d 2 - a 2 - a?, 4 EG 2 = a 2 + 6 2 + ai 2 + &i 2 - c 2 - Cj 2 . Therefore, by addition, 4 (/J 2 + l^ff 8 + EG 2 ) = a? + 6 2 + c 2 + i 2 + &i 2 + c/ ; or, denoting, in general, a diameter by d and an edge by e, Cor. In the regular tetrahedron, all the diameters being equal, and all the edges being equal, gives, So that if the diameter is equal to the side of a square, the edge is equal to the diagonal of the square (P. Art. 180. Cor.). 80 SOLID OR SPATIAL GEOMETRY. 92. Theorem. In any tetrahedron, nine times the square on a median is equal to the dif- ference between three times the sum of the squares on the con- terminous edges and the sum of the squares on the remaining edges. Proof. In the tetrahedron D ABC, P is the centroid of ABC. Then DP is the median to the face ABC. Bisect AP in Q. And A Q = QP = PI = i AL DQ is median to the A ADP; .: AD 2 + DP 2 = 2AQ 2 + 2DQ 2 . (P. Art. 173.) Also, DP is median in the A QDI; And eliminating DQ 2 between these relations, we obtain 3 DP 2 = AD 2 + 2DI 2 - \AI\ But v .41 is median in the A ABC, and DI is median in the A BCD, .-. 2 AP = AB 2 + AC 2 -2 BI 2 = c, 2 + b 2 - < and 2DI 2 = DB 2 + DC 2 -2 BI 2 = b 2 + c 2 -^af; whence 3 DP 2 = a 2 + b 2 + c 2 - 1 (a, 2 + &/ + q 2 ) , or 9 DP 2 = 3 2a 2 - Sa^. Q. E. D. 93. Theorem. In any tetrahedron, nine times the sum of the squares on the medians is equal to four times the sum of the squares on the edges. THE SKEW QUADRILATERAL. 81 Proof. Let m lf m 2 , m s , m 4 denote the medians. Then from 86, D as vertex, 9 m? = 3 a 2 + 3 b 2 + 3 c 2 - a? - b? - c? ; A as vertex, 9m 2 2 = 3 a 8 + 3 V + Stf - a, 2 -b 2 -c 2 ; B as vertex, 9 m 3 2 = 3 a* -f 3 6 2 + 3 Cj 2 a 2 &/ c 2 ; C as vertex, 9 m 4 2 = 3 a a 2 + 3 &/ -f 3 c 2 - a 2 - o 2 - c, 2 . Whence by addition, and denoting a median in general by m and an edge in general by e, we have 92m 2 = 42e 2 . Q.B.D. (7o?*. 1. In the regular tetrahedron 9 2m 2 = 36 m 2 , and Cor. 2. The median in a regular tetrahedron is the same as the perpendicular from the vertex to the base, and denoting it by p, we have Cor. 3. Denoting a dihedral angle of the regular tetrahedron by E, sin E = DP/DI= e V6 -*- $e ^3 = | V 2 - And cos E = \. 94. In the regular tetrahedron we have the circum- scribed sphere, the tangent sphere to the edges, and the inscribed sphere. Denoting the radii of these by R, p, and r respectively, R = OD = f p = {e V6, (Art. 52.) p = 01 = | d = \e V2, (Art. 91. Cor.) 82 SOLID OR SPATIAL GEOMETRY. THE PARALLELEPIPED. 95. Denote the direction edges by a, 6, c, and an edge in general by e, and a diagonal in general by d. Theorem. The sum of the squares on the diagonals is equal to the sum of the A< squares on the edges. Proof. Since the faces are all parallelograms, and AB is II to B'A, AC to C'A', etc., A A 12 + BB 12 = AB 2 + BA' 2 + A'B' 2 + B'A 2 . Similarly, CC' 2 + DD' 2 = CD 2 -f DC' 2 + C'D' 2 + D'C*. Whence, by addition, Sd 2 = AB 2 + CD 2 + A'B' 2 + C'D' 2 + BA' 2 + D'C 2 + B'A 2 + DC 12 . And BA' 2 + CD' 2 = BC 2 + CA' 2 + A'D' 2 + D'B 2 ; and B'A 2 + C'D 2 = DA 2 + AC 12 + C'B 12 + B'D 2 . .: 2d 2 = 2e 2 . Q.E. D. Cor. 1. As the edges are separable into three groups of four equal edges each (Art. 53), Cor. 2. In the cuboid the diagonals are all equal, and d 2 = a 2 + b 2 + c 2 . Cor. 3. In the cube a = b = c ; THE PARALLELEPIPED. 83 96. Problem. To find a diagonal of a given parallele- piped by a plane construction. In Fig. (1), let AO'BC be the given ppd., and AA' be the diagonal required. Analysis. Draw A'P, BQ, CR, perpendiculars on AO, produced if necessary. P, Q, R are the projections, on AO, of A', B, and C. The projection of the middle point of OA' is the same point as the projection of the middle of BC, i.e. it is the middle point of RQ. .;. OP=OR+OQ. Construction. In Fig. (2) construct the faces AB, AC, and OA', disposed as in the figure. Draw CR, BQ _L on AO, produced if necessary. 84 SOLID OR SPATIAL GEOMETRY. Take OP=OR+ OQ, and draw PT _L to AO. With as centre and OA' as radius, describe a circle cutting PT in T. Join AT. AT is the required diagonal. Proof. OR is the same for both figures, and so also is OQ, and therefore OP; and AO being the same in both, AP is the same in both. Also, 0!F of Fig. (2) is made equal to OA' of (1). .-. A OPT of (2) ==AOPA'of (l),and PT= P^'. Hence A APT of (2) = A APA! of (1), and AT of (2)= Ad' of (1). In like manner any other diagonal can be constructed. Cor. Let the face angles about the vertex A' be all acute, and the figure is an acute ppd. (Art. 54). Denote Z BA'C by X, Z CA'O' by v, and Z OM'5 by M . Then, Fig. (2), Z 50(7! = A, Z COP = /*, Z BOP=v. Now, + PT 2 = (.40 + OP) 2 + O^t' 2 - OP 2 = a 2 + 2 a (OQ + 072) THE PARALLELEPIPED. 85 But, OA' 2 = b* + c 2 + 2 be cos A, (P. Art. 217. ) OQ = b cos v, OR = c cos /*. .-. ^IT 72 = AA' 2 = a 2 + 6 2 + c 2 + 2 6c cos X + 2 ca cos /* + 2a& cos v. And this expresses the square on the longest diagonal, i.e. the one extending between the vertices having three acute face angles. The other diagonals are given by making two angles obtuse in every possible way. They are : a 2 + b 2 + c 2 + 2 be cos X 2 ca cos p. 2ab cos v, a 2 + 6 2 + c 2 2 6c cos X + 2 ca cos /A 2 a& cos v, a 2 + 6 2 + c 2 2 &c cos A 2 ca cos p. + 2 a& cos v. For the diagonals of an obtuse ppd. it is only neces- sary to change throughout the algebraic sign of every cosine term. REMARK. In making constructions like the foregoing care must be exercised that every measured segment is taken in its proper sense, or with its proper sign. By taking the face angles about A' all acute, the per- pendiculars A'P, BQ, CR, all fall to the right of 0. Under a different arrangement of angles, some or all of these might have fallen to the left of 0. In any case if M is the middle point of QR, OP is to be taken equal to 2 OM, whatever the sign may be. 97. Let 0-ABC be a cuboid, and OP be a diagonal. Then OP 2 = OA 2 + OB 2 + OC- (Art. 95. Cor. 2). 86 SOLID OR SPATIAL GEOMETRY. But if OX, OY, OZ, the direction lines of the cuboid, meeting in 0, be taken as the three rectangular axes of space ; (Art. 8. Def. 1), OA is the pro- 1 jection of OP on OX, OB is the projection of OP on OY, and OC, of OP on OZ. Therefore, the square on any line-segment is equal to the sum of the squares of the projections of the segment on any three mutually perpendicular lines. 98. Denoting OA by a, OB by b, and OC by c ; also Z.POA by a, Z POB by ft, Z POC by y, we have OA 2 a 2 B ---,. OP 2 a 2 +6 2 +c 2 ' with the symmetrical expressions for cos 2 ft and cos 2 y. .-. cos 2 a + cos 2 /S -f cos 2 y = 1. Def. The angles a, ft, y are direction angles of the line OP, and determine the direction of OP relatively to the three axes. The cosines of these angles are the direction cosines of OP. These angles are interdependent, and the result of this theorem shows that the sum of the squares of their cosines is unity. Cor. The position of a point, P, in space is known relatively to the origin 0, and the axes OX, OY, OZ, when we are given OP, and the angles which OP makes THE OCTAHEDRON. 87 with the axes ; or when we are given the length of the projections of OP upon the axes. For the projections are the direction edges of a cuboid of which OP is the diagonal. This is the fundamental principle entering into analytic spatial geometry. THE OCTAHEDRON. 99. The octahedron may assume a variety of forms, but we shall confine ourselves to those in which the point of intersection of the axis is the middle point of each axis, or the centre of the figure. In general the octahedron is the reciprocal of the parallelepiped, formed by joining the centres of adjacent faces. The three joins of the centres of opposite faces of the ppd. in pairs are the axes of the octahedron, and hence from the nature of a ppd. (Art. 37) the octahedron may be triclinic, diclinic, monoclinic, right or regular; the right octahedron coming from the cuboid, and the regular from the cube. 100. Theorem. In any oc- tahedron, the sum of the squares on the twelve edges is equal to twice the sum of the squares on the three diameters AA' } BE', and CO. 88 SOLID OR SPATIAL GEOMETRY. Proof. The section along any two diameters, being a parallelogram, gives AA' 2 + BB' 2 = AB 2 + BA' 2 + A'B 12 + B'A 2 , BB 12 + CC' 2 = BC' 2 + C'B 12 +B'C 2 + CB 2 , CC' 2 + AA' 2 = AC 2 + CA' 2 + A'C 12 + C'A 2 . Whence, by addition, 2(AA' 2 + BB' 2 + CC' 2 ) = 2 2d 2 = 2e 2 . Cor. If the octahedron is regular, all the edges are equal and all the diameters are equal, and therefore and the section ACA'C' is a square. THE REGULAR DODECAHEDRON. 101. Let AE, AB, and AG be the three edges which meet to form the corner of a regular dodecahedron. Let Q be the centre of the face ADB, and be the centre of the circumscribed sphere. THE REGULAR DODECAHEDRON. 89 Since all the faces are congruent, BEG is an equi- lateral triangle, and OA passes through its centroid P, and is normal to the plane of the triangle (Art. 79. Cor. 1). Z ABE = 36, and BE = 2 BH= 2 AB cos 36 =2 e cos 36. Then, BP= BE - V 3 = |e V 3 cos 36 - Also, . BPA = 1 , AP 2 = AB 2 -BP 2 , or ^P 2 =e 2 (l-|cos 2 36). But if AA' be a diameter of the circumsphere, ABA' is a ~1, since B is on the sphere. .-. AA'-AP=AB 2 (Art. P. 169) ; or 2R-AP=e 2 . Whence, R = % e ^/ 3 V|3-4cos 2 36} Or R= gV 3 =6 y 1.401258... 4V sin 6 - sin 66 Again, we have, .-. /3 = e VI 1.401258 -0.25} = exl.309016..., and r = ^/\R 2 -AQ 2 l. But AQ = ABsecBAQ 1 e .-. r = e* / j 1.401258 2 - j \ 1 4 sin 2 36 j = 6x1.113516-.-. 90 SOLID OR SPATIAL GEOMETRY. EXERCISES F. 1. Two opposite edges of a tetrahedron are perpendicular to one another when of the remaining edges the sums of the squares upon opposite edges, taken in pairs, are equal. 2. What does the theorem of Art. 91 become when the four vertices of the tetrahedron become complanar ? 3. What does the theorem of Art. 92 become when D comes to the centroid of the triangle ABC? 4. Show that the tangent of the angle made by an edge of a regular tetrahedron with one of the faces is ^/'2. 5. In the cube, P is the middle point of AB, and 8 is the middle point of A'B' ; show that the acute angle of the section through Q, Z>, Sis c 6. In the cube, DK is JL from D upon the diagonal BB' ; show that DK= | e \/6 ; and that CK = e. 7. In the cube, the join of the middle point of AB with _B', and the join of the middle point of AD with D', divide each other into parts which are as 2 : 1. 8. The angle between two diagonals of a cube is cos- 1 . 9. In the cube, the angle between a diagonal and a face is cos- 1 -- V3 10. In the cuboid, the angle subtended at the centre by the middle points of two conterminous edges is cos-i a 2 / V(a 2 + 6 2 ) (a 2 + c 2 ), with variations in the letters for the different cases. 11. In the cuboid, the angle between diagonals is cos- 1 (a 2 - 6 2 - c 2 )/(a 2 + 6 2 + c 2 ), with symmetrical variations. THE KEGULAR DODECAHEDRON. 91 12. In the cuboid, the from a vertex upon a diagonal is a V& 2 + c 2 / \/a 2 + 6 2 + c 2 , with symmetrical variations. 13. In an octahedron, there may be, at most, six different lengths of edges. 14. If the semi-diameters of an octahedron be a, b, c, and Z (6c) = X, Z (ca) = M, and Z (a&) = v, then the squares of the edges are a 2 + 6 2 2 aft cos v, 6 2 + c 2 2 6ccos X, c 2 + a 2 2 ca cos /x. 15. In a right octahedron, the cosines of the dihedral angles are 6 2 c 2 + c 2 a 2 - 2 6 2 , c 2 a 2 + a 2 & 2 - 6 2 c 2 , a 2 6 2 + 6 2 c 2 - c 2 a, each divided by a 2 6 2 + 6 2 c 2 + c 2 a 2 . 16. In a regular octahedron, the perpendicular from the centre upon a face is eV&. 17. In a regular otcahedron, the cosine of a dihedral angle is - x. 18. The section through the middle points of AC', AB', B'C, CA', A'B, and B' C is a hexagon with opposite sides parallel, and is regular if the octahedron is regular. 19. A section of an octahedron parallel to any face is a hexagon. 20. The radius of the tangent sphere to the edges of a regular octahedron is \ e. 21. The squares of the radii of the three spheres of a regular octahedron are in harmonic proportion. 22. In a regular dodecahedron, jR^ 4 92 SOLID OR SPATIAL GEOMETRY. 23. In a regular dodecahedron, and / , = | 24. In a regular dodecahedron, if D be the dihedral angle, sin D = |V 5 - 25. In the regular dodecahedron show that 11.R 2 exceeds 15r 2 by 3e 2 . 26. In the icosahedron, .R = e-J( +v/ -j ; 27. In the icosahedron if D be the dihedral angle, cosD = y-v/5> or sinZ) = f. 28. A sphere touches one face of a regular tetrahedron exter- nally, and the three others internally. Show that its radius is %p ; and that the distance from the further vertex at which it a touches the three faces is - -^3. u 29. If a regular cube and octahedron be circumscribed to the same sphere, their vertices are conspheric. 30. If a regular dodecahedron and icosahedron be circumscribed to the same, sphere, their vertices are conspheric. SECTION 2. THE SPHERE. 102. Def. If P be any point, and a line through P meets a given sphere in A and B, the rectangle PA PB is called the power of the point P with respect to the given sphere. Cor. A point is without a sphere, on the sphere, or within it, according as the power of the point with respect to the sphere is positive, zero, or negative. 103. The power of a fixed point with respect to a given sphere is independent of the direction of the line whose segments form the rectangle which measures the power. Proof. Let the line through P meet the sphere in A and B. Since P, A, B are in line, P, A, B, are complanar, being the centre of the sphere. The section by this plane is a great circle, with a secant line through P cutting the circle in A and B. And PA ' PB is constant in value for this circle (P. Art. 176). And since all great circles have the same centre and equal radii, PA PB is constant for every great circle, and therefore for the sphere. Cor. If A and B become coincident, the secant line becomes a tangent, and the rectangle PA PB becomes the square on the tangent. 93 94 SOLID OR SPATIAL GEOMETRY. Therefore, the power of an external point with respect to a given sphere is the square on the tangent from the point to the sphere; and all tangents from the same point to the same sphere are equal. 104. S and S' are two circles with centres A and B and radical axis L (P. Art. 178). Let the whole system revolve about the com- mon centre-line AB as an axis, while retaining the fixed relations of the several parts. The circles describe spheres, and the radical axis, L, describes a plane normal to AB. Also PE PD = PE' PD' remains true for the spheres. And since P may be any point on the plane described by L, the. power of P with respect to each sphere is the same. Def. The locus of a point of which the power is the same with respect to two given spheres is the radical plane of the spheres. Cor. 1. Evidently, the radical plane of two spheres is normal to the join of their centres, and divides the distance between the centres so that the difference of the squares on the two parts is equal to the difference of the squares on the conterminous radii. THE SPHERE. 95 Cor. 2. The tangents to two spheres, from any point on their radical plane, are equal. Cor. 3. The plane of the circle of intersection of two spheres is their radical plane. 105. Let Su S 2 , $ 3 , 4 be four spheres, and let U i2 denote the radical plane of Si and S 2 , etc. The four spheres have the six radical planes, C7" 12 , C/i 3 , Ui*> UK, U 24 , and U M . A point whose power with respect to Si and S 2 is the same is on the plane C7" 12 , and a point whose power with respect to Si and S s is the same is on the plane [Tig. Therefore, a point whose power with respect to Si, S 2 , and S 3 is the same is on the common line of Uu and C/i 3 , and is evidently on the plane U^. Therefore, the radical planes of three spheres have a common line, and from any point on this line tangents to the spheres are equal. We shall call this line the radical line of the three spheres. In a section through the centres of the spheres, this line gives the radical centre of the three resulting great circles. Cor. 1. The radical line of three spheres is normal to the plane through their centres. Cor. 2. The six radical planes to four spheres inter- sect by threes to form four axial pencils. The axes of these pencils may be denoted by L m , i m , L m , and L^ L m being the common line to U i2 , U&, and %. Cor. 3. The line L m meets the plane Uu in one point only, and it evidently meets U 2i and U& in the same point. 96 SOLID OR SPATIAL GEOMETRY. Therefore, there is, in general, one point from which tangents to four given spheres are equal ; or of which the power is the same with respect to four given spheres. This is the radical centre of the four spheres. EXERCISES G. 1. Two secants are drawn through the same point, P, within a sphere, and meet the sphere in A, B, and C, D respectively. Then PA PB = PC PD. 2. If a, b be the parts into which the plane of a small circle divides the diameter through its centre, the area of the small circle is vab. 3. If three spheres intersect two and two, the planes of the small circles of intersection form an axial pencil. 4. If four spheres intersect two and two, the planes of the circles of intersection pass through a common point. 5. Where is the radical centre of four spheres whose centres are complanar ? 6. Under what condition will four spheres have a line of radical centres ? (The spheres are then coaxal.) 7. The tangent cones, common to three spheres taken two and two, have their vertices collinear. 8. The tangent cones, common to four spheres taken two and two, have their vertices complanar. 9. If P and Q be two points in the line L, and U and V inter- secting in M be the polar planes of P and Q with respect to a sphere, then every plane through M is polar to some point in L ; and L and M are perpendicular to each other. 10. Any rectilinear figure has a corresponding rectilinear figure such that every side in the first figure has a side perpendicular to it in the second. PART III. STEREOMETRY AND PLANIMETRY; OR THE MEAS- UREMENT OF VOLUMES AND SURFACES. 106. A closed spatial figure includes within its boun- daries a portion of space separated from all other parts of space. This portion of space considered with respect to extent, and not with respect to form, is called the volume of the closed figure. As our primary ideas of a spatial figure were probably derived from concrete objects such as blocks of wood or stone, the volume of a spatial figure is also called its solid contents, and the figure itself is called a solid. Hence the name Solid Geometry. Also considering a closed spatial figure as a surface, which after the manner of a closed vessel might be filled with a liquid, the volume is sometimes called the capacity of the figure. The measuring of volumes, or solid contents, or capaci- ties is called Stereometry. Def. Equal spatial figures are those which have equal volumes, and therefore congruent figures, when having volumes, are necessarily equal. 97 SECTION 1. POLYHEDEA. 107. Theorem. Two cuboids with congruent bases have their volumes proportional to their altitudes. B-ACD and F-EGH are two cuboids having their bases congruent. Then vol. B ACD : vol. F EGH = BD : FH. Proof. If this propor- tion is not true, let vol. B-ACD: vol. F EGH = BD : FI, where Flis different in length from FH; and first let FI be less than FH. As a general case let BD and FH be incommensurable. Take some u.l (P. Art. 150. 3) less than IH which will measure BD, and divide BD and FH into parts equal to this u.l. One point of division, at least, must fall at some point, J, between / and H. Through all the points of division pass planes parallel to the bases. These divide the cuboids B ACD and F EGJ into congruent and therefore equal cuboids. /. vol. B - ACD : vol. F . EG J = BD : FJ and vol.B-ACD:vol.F-EGH=BD:FI (hyp.). .. vol. F- EGJ: vol. F- EGH= FJ: FI. POLYHEDRA. 99 But vol. F-EGJ< vol. F EGH-, .: FJis. The area of the trapezoid evidently lies between the sum of the external rectan- gles and the sum of the internal rectangles. Now, any external rectangle as Cc is congruent with an internal rectangle below it, CQ ; except that the lowest external rectangle has no corresponding and con- gruent internal one, and the uppermost internal rectan- gle has no congruent external one. Let E denote the sum of the external rectangles, and / denote the sum of the internal ones. Then .: the difference between the sum of the external rectangles and the sum of the internal rectangles is less OF LAMINAE. 107 than the lowermost external rectangle ; and this is true however many rectangles be formed. But the lowermost rectangle can be made as small as we please, by making its altitude sufficiently small ; i.e. by making the number of parts into which we divide AE sufficiently great. And hence the area of the trap- ezoid is the limit of the sum of either series of rectan- gles as the number of rectangles is indefinitely increased. 118. Now, let AETN be a vertical section of a frus- tum of a pyramid (Art. 59), in which AN and ET are sections of the bases. Divide AE into any number of equal parts, and through the points of division pass planes parallel to the bases. On the figures of section construct a series of inscribed prisms, BP, CQ, DR, ES , and a series of circum- scribed prisms, Aa, Bb, Cc, Dd---. The volume of the frustum lies between the sum of the internal prisms and the sum of the external prisms. But any external prism, except the lowermost, has a congruent internal prism below it, and any internal prism, except the uppermost, has a congruent external prism above it. Hence if E denotes the sum of the external prisms, and / of the internal prisms, E 1= prism Aa prism ES = vol. of lowermost external prism vol. of the uppermost internal prism. And this is true, however many equal parts AE is divided into. 108 SOLID OB SPATIAL GEOMETRY. Therefore, the volume of the frustum differs from the sum of either series of prisms, by less than the volumes of the series of prisms differ from each other ; that is, by a quantity less than the lowermost external prism ; and this difference may be made as small as we please by dividing AE into a sufficiently large number of parts. Hence, the volume of the frustum is the limit of that of either series of prisms, when the number of prisms is indefinitely increased. Cor. This theorem is exceedingly important, for the least consideration will show that nothing in the inves- tigation requires that AE, or any edge, should be a straight line, and hence that the theorem holds true when the boundary of the figure, between the parallel bases, is composed partly or wholly of curved surfaces ; also that the theorem is true when one or both bases reduce to lines or points. 119. Def. When a spatial figure is cut by two indef- initely near parallel planes, the prism, having one of the sections as base, and the distance between the planes as altitude, is called a lamina of the spatial figure. When two figures are confined between the same two parallel planes, the laminae determined by two indefinitely near planes, parallel to the confining planes, are corre- sponding lamince. Usually the planes which determine a lamina are supposed to be infinitely near, so that a lamina is one of the prisms of the preceding article, taken at its limit. Cor. 1. From Art. 118, it appears that two figures which have all corresponding laminae equal are them- THE PYRAMID. 109 selves equal; and two figures which have all corre- sponding laminae in the same ratio are themselves in that ratio, the one to the other. Cor. 2. Since corresponding laminae have the same altitude, their volumes are proportional to their bases ; and hence corresponding laminae are equal when corre- sponding sections are equal ; and corresponding laminae are in the same proportion to one another as are the cor- responding sections. THE PYRAMID. 120. Theorem. Pyramids are equal whose bases are equal and whose altitudes are equal. Proof. Let the trian- gular pyramids, D ABC and H-EFG, have their bases equal, and also their altitudes equal, and let them be so placed that their bases are compla- nar, and their vertices are upon the same side of this plane. Then D and H lie in a plane parallel to the plane of the bases. Let abc and efg be corresponding sections. Then (Art. 28. Cor. 2) A abc ^ A ABO, and A efg ~ A EFG. But (P. Art. 218. 2), A abc : A ABC = a& 2 : AB 2 . 110 SOLID OR SPATIAL GEOMETRY. And since DA and DB are cut by parallel planes, AB is II <(h. .-. ab 2 : AB 2 = Da 2 : DA 2 . And (Art. 27) Da: DA = He: HE; .: ab 2 : AB 2 = Da 2 : DA 2 = #e 2 : HE 2 = e/ 2 : EF 2 , or A abc : A ABC = A efg : A EFG. But A ABC = A .EF& ; (hyp. ) . . A abc = A efg. And as corresponding laminae are equal, the volumes of the pyramids are equal (Art. 119. Cor. 1). And since all pyramids may be divided into triangu- lar pyramids, Therefore, any two pyramids are equal whose bases are equal and whose altitudes are equal. Cor. Two frustums of pyramids which have their two bases respectively equal and their altitudes equal are themselves equal. 121. Theorem. A triangular prism can be divided into three equal pyramids. Proof. A BCD is a triangular prism. Pass a plane through the points A, C, and E, and another plane through (7, D, and E. The prism is divided into three equal pyramids. For C FDE and E CAB have their bases DEF and ABC equal, and their altitudes the same as that of the prism. These pyramids are therefore equal (Art-. 120). Also the pyramids C ADE and C ABE have their THE PYRAMID. Ill bases ADE and ABE equal (P. Art. 141. Cor. 1), and have their vertices coincident. Therefore they have the same altitude and are equal. And the prism is thus divided into three equal pyramids. Cor. 1. As each triangular pyr- amid is one-third of the corre- sponding triangular prism, and as every prism can be divided into triangular prisms ; Therefore every pyramid is one-third of the prism having the same base and altitude as the pyramid. Cor. 2. If B denotes the area of the base of a pyramid, and h denotes its altitude, vol. of pyramid = ^ hB. Cor. 3. Pyramids with equal bases are to one another as their altitudes, and pyramids with equal altitudes are as their bases. 122. Theorem. The frustum of a triangular pyramid may be divided into three pyramids, two of which have o. the bases of the frustum as their bases, and the altitude of the frustum as their alti- tude, and the third of which is a mean proportional between the first two. ABCDEF is a triangular frustum. 112 SOLID OB SPATIAL GEOMETRY. The plane through A, E, F cuts off the pyramid A DEF, whose base DEF is the upper base of the frustum. From the remaining figure the plane through A, E, C cuts off the pyramid E ABC, whose base ABC is the lower base of the frustum. We have left the pyramid E AFC. Join BF and CD. The pyramids B AEG and F- AEG having the com- mon base AEC are as their altitudes, and the altitudes are as PB to PF, or BC to EF. .-. B-AEC:F.AEC=BC:EF. Again, the pyramids C-AEF (which is the same as F-AEC) and D-AEF having the common base AEF are to one another as CQ is to QD, or AC to DF. But, since the bases are similar (Art. 28. Cor. 2), BC : EF = AC : DF. Or the pyramid F- AEC, or C- AEF, is a mean propor- tional between the pyramids E ABC and A DEF. Cor. If B and B' denote the bases of the frustum, and li the altitude, vol. of E-ABC=^liB, vol. of A-DEF= \hB', and .-. vol. of F- AEC = The volume of the frustum is accordingly : vol. = \li {B 4- B' + Vi^B'f . 123. The volume of the frustum may also be found as follows : Let be the vertex of the pyramid from which the THE PYRAMID. 113 frustum is formed, and let OP be the altitude of the pyramid. Also let OP' be the altitude of the pyramid DEF, which is removed in forming the frustum. Then, The frustum = pyr. 0-ABC-pyr. 0-DEF, and OP-OF = h. Since any area may be expressed as a square, let 6 2 = B or the base ABC, and b 12 = B' or the base DEF. Then OP: OP = OA : OD = AB : DE = b : b', and OP-OP':OP=b-b': b. .-. OP(b-b')=bh. bh .-. OP = OP' = b'h and b-V 6-6'' frust. = | OP- 6 2 - 1 OP - b 12 Cor. The volume of a frustum of a pyramid is the sum of the two bases and a mean proportional between the bases, multiplied by one-third of the altitude. 124. Def. A triangular prism with non-parallel bases is called a truncated triangular prism, or a wedge. Let ABC, DEF be the bases of the wedge, of which ABC is nor- mal to the lateral edges. Through D pass a plane II to ABC and draw AP _L to BC. 114 SOLID OR SPATIAL GEOMETRY. The wedge = the prism A BCD + the pyramid D EFGH. But the prism = A ABC x AD ; and the = i A ABC(BE+ CF - 2 AD) ; .-. vol. of wedge = 1 A ABC (AD + BE + CF). Or if e,, e 2 , e 3 denote the edges, and B the area of a right section, vol. =i(e 1 + e 2 + e 3 ). 125. Theorem. If a tetrahedron be cut by a plane which bisects two edges and passes through an opposite vertex, the volume of the tetra- hedron is equal to four-thirds of the prism having the section as base, and the perpendicular from any other vertex on the plane of section as altitude. A BCD is a tetrahedron, and E and F are middle points of AB and AC respectively. AP is perpendicular upon the plane EFD. Then : EF is II to BC, and bisects AB, EFis one-half BC, and AAEF= \AABC (P. Art. 218. 2). The pyramids having these triangles as bases have D as a common vertex; .. tetrahedron A BCD = 4 tetr. A - DEF Q. E. D. PRISMATOID AND ALLIED FORMS. 115 PRISMATOID AND ALLIED FORMS. 126. Def. A polyhedron with two parallel polygonal bases, and all its lateral faces plane rectilinear figures, and all its lateral edges the joins of vertices of opposite bases, is a prismatoid. This definition includes the prism, pyramid, and frus- tum of a pyramid as special cases, and is more general than any of these. When none of the faces are triangles, the figure is the frustum of a pyramid, or a prismoid, according as the lateral edges are, or are not, concurrent when produced. 127. ABCD and EFG are parallel bases of a prisma- toid, and AEB, EBF, FBC, CFG, etc., are triangular faces, which, in the figure given, are seven in number. G B . A If n denotes the number of sides in one base, and n' in the other, it is readily seen that the number of faces cannot be greater than n + n'. 116 SOLID OR SPATIAL GEOMETRY. But if an edge of one base be connected by lateral edges with a parallel edge of the other base, two trian- gular faces become a quadrangular face, and the whole number of faces is reduced by one. Thus, if EF were parallel to AB, the edges AE, EB, and BF would be complanar, and the two triangular faces AEB and BEF would become one quadrangular face, AEFB. If two other edges of the bases become parallel, a like reduction may take place, and the whole number of faces be reduced by two. And finally, if the bases have the same number of sides, and each edge in one base be connected with a parallel edge in the other, all the faces become quadran- gular, and the figure becomes a frustum of a pyramid or a prismoid, according as the edges, when produced, are or are not concurrent. Even with the same bases, however, the general appear- ance of the figure will vary with the different ways of connecting the vertices of the bases by the lateral edges. 128. Def. Take H, I, J, etc., middle points of the lateral edges, AE, BE, BF, etc., respectively. Since HI is parallel to AB (P. Art. 84. Cor. 2. 2) and His parallel to EF, and JKio BC, etc., it follows that H, I, J, etc., lie in a plane which bisects all the lateral edges, and is parallel to the bases. The section by this plane is called the middle section. The middle section contains, at most, n -f- n' sides, there being always as many sides as there are faces in the prismatoid. The middle section may contain re-entrant angles, although no such angles are found in either base; and PRISM ATOID AND ALLIED FORMS. 117 it will frequently have such angles when the bases are polygons of different species, or when their vertices are connected in some particular order. Cor. The middle section bisects the altitude. 129. Volume of the prismatoid. Take P, any point in the plane of the middle section, and join it to A, D, E, H, and N. Denote the altitude of the prismatoid by h. G A Then, P ADE is a tetrahedron, and PNH is a section through a vertex, P, and the middle points, H and N, of two ppposite edges. .-. vol. of P ADE = | h x A PNH. (Art. 125.) Similarly, by joining P to all the remaining vertices, B, C, F, etc., and to the remaining middle points, /, J, K, etc., we have, Sum of all the tetrahedra of which P - ADE is the type = | h x (the sum of the A of which PNH is the type), or 2 (P ADE) = | hi (A PNH). 118 SOLID OE SPATIAL GEOMETRY. But 2 (A PNH) = the area of the middle section, and denoting the area of the middle section by M } Now, after removing all these tetrahedra, we have left two pyramids having P as a common vertex, and the bases of the prismatoid as their respective bases. The altitude of these prisms being \h (Art. 128. Cor.), their volumes are ^hB and ^hB', where B and B' are the areas of the bases of the prismatoid. .-. vol. of prismatoid = - (B + B' + 4 M ). 6 Cor, The prismatoid is equal to four pyramids, two having the bases of the prismatoid as their bases and half the altitude of the prismatoid as their altitude, and two having the middle section as their bases and the altitude of the prismatoid as their altitude. Cor. The formula of the present article is known as the prismoidal formula. On account of its extremely wide range of applicability it is the most important of all formulae connected with the determination of the volumes of the more prominent spatial figures. The following examples are some illustrations of its application. (a) Prism. Here the two bases and the middle sec- tion are all congruent. Hence, vol. = - (B+B + 4 B) = hB. (Art. 115. Cor. 2.) (b) Pyramid. The upper base vanishes, and the mid- dle section is one-fourth the lower base. REGULAR POLYHEDRA. 119 * . . vol. = I ( B + + B) = $ hB. (Art. 121. Cor. 2. ) (c) Frustum of a pyramid. Let B, B' be the bases, and M be the middle section. And since any area may be expressed as a square, let B = 6 2 , B' = b' 2 , and M= m 2 . Then 2 m =& + &'. = - (B + B' + V.B'). (Art. 122. Cor.) o (d) Tetrahedron, in terms of a middle section (Art. 51. Def. 1) and the length of the common perpendicular to the edges parallel to the section. In this case, which has an important subsequent appli- cation, both bases vanish, and we have vol. = REGULAR POLYHEDRA. 130. A regular polyhedron of n faces is divisible into n congruent pyramids whose bases are the several faces of the polyhedron, and whose altitude is the radius of the in-sphere to the polyhedron. 120 SOLID OR SPATIAL GEOMETRY. i Hence, if n be the number of faces, B be the area of a face, and r be the radius of the in-sphere, we have vol. = -.Br. o (a) Kegular Tetrahedron. Cor. As the expression for the volume may be writ- 1 / e \ 3 ten - ( - I , therefore the cube on the side of a square whose diagonal is the edge of a regular tetrahedron is three times the tetrahedron. (6) Kegular Octahedron. Cor. This volume may be written Therefore, the cube on the diagonal of a square whose side is the edge of a regular octahedron is six times the octahedron. (c) Regular Dodecahedron. By the methods of plane geometry, we find the area of a regular pentagon with side e to be -B=4e ! REGULAR POLYHEDRA. (d) Kegular Icosaliedron. EXERCISES H. 1. If a plane parallel to the bases, and midway between them, be passed through the prism of Art. 121, compare the areas of the sections of the three pyramids. 2. Apply the conditions of Ex. 1 to Art. 122. 3. A plane of section passes through the middle points of the parallel edges of a wedge, one of whose bases is a right section (Art. 124). Find the area of the section. 4. If e v e. 2 , e 3 be the three parallel edges of a wedge, show that K e i + e z + e a) is tne distance between the centroids of the bases. 5. Apply the prismoidal formula to find the volume of a wedge. 6. A prismoid has both bases parallelograms with angle 6, and the sides are a, b for the one, and a', V for the other. Find its volume, its altitude being h. 7. Show that the cube on the side of a square whose diagonal is the edge of a regular octahedron is three- fourths of the octahedron. 122 SOLID OR SPATIAL GEOMETRY. 8. If a regular tetrahedron and a regular octahedron have the same edge, the octahedron is four times the tetrahedron. 9. AA', BB', CC', DD', being diagonals of a cube, show that the plane through DBC' cu|s off a pyramid whose volume is one- sixth that of the cube. 10. The direction edges of a cuboid are a, b, c, and a plane passes through the three distal extremities of these. Show that the area of the section is jVa 2 6 2 4- 6 2 c 2 + c 2 a 2 . 11. AA', BB 1 , etc., are the diagonals of a ppd. Show that a plane through DBC' cuts off a pyramid which is one-sixth the ppd. 12. The direction edges of a ppd. are a, b, c, and the angles between them are Z (be) = X, Z (ca) = /u, Z (aft) v. Then the vol. is abc Y/ { 1 cos 2 X cos 2 /j. cos 2 v + 2 cos X cos p cos v}. OA, OB, OC are the direction edges; Z COB = X, Z COA = M, ZAOB = v. Let CP be normal to the plane of A OB, and PQ, PE be J_s upon OA and OB. vol. of ppd. = OA- OB sin v CP. (P. Art. 215.) ot OQPR are concyclic, and OP is a diameter of the circumcircle ; (P. Art. 228.) and But and CT=V(L~ \ sin 2 v) sm v ' vol. = ab -v/( c2 s i n2 * ~ QR 2 )- QR 2 = OO, 2 + OR 2 - '20Q OR cos v ; OQ 2 = c 2 cos 2 fi, and O.R 2 = c 2 cos 2 X. REGULAR POLYHEDKA. 123 Whence, by substitution, vol. = abc - x /{ 1 cos 2 X cos 2 fj. cos 2 v + 2 cos X cos p cos v}. 1 13. Show from the character of the result in Ex. 12 that if in fcny ppd. X, /*, v are all acute, all vertices except the one opposite have one acute and two obtuse angles, etc. 14. With the vertices of a ppd. as centres, equal spheres are described to cut the ppd. Then the volume removed by all the spheres is equal to that of one of the spheres. 15. Show that space may be wholly divided up into regular octahedrons and tetrahedrons, and that there will be twice as many of the latter as of the former. 1 The area of a parallelogram whose sides are a, b and angle 6 is ab sinff, or ab-^/(l cos 2 0), and the volume of the ppd. is 6c v /(l cos 2 A cos 2 /x etc.). On account of the analogy in form, the expression 1 cos 2 A cos 2 /u etc. is sometimes called the square of the sine of the solid angle O- AB (7, and it usually appears in the matrix form 1 cos A cos yu COS A. . 1 COS v COS fi COS V 1 The analogy, however, is one of form only, as there are no func- tions of solid angles really corresponding to the sine, cosine, tan- gent, etc., of plane angles. SECTION 2. CONE, CYLINDER, SPHERE. THE CONE. 131. The cone of Art. 67 is not a closed figure, and consequently does not admit of measurement for volume. But if the cone be cut by a plane which does not pass through the centre, and which makes, with the axis, an angle greater than the vertical angle, a closed figure is formed by the conical surface and the plane. It is this closed figure -that is called a cone in relation to stere- ometry. The centre of the cone is, in this relation, called the apex or vertex, and that portion of the section plane which forms a part of the enclosing figure is the base of the cone. The word 'cone,' whenever having reference to stereo- metrical relations, will mean this figure. 132. As the director curve may be of any form, and as the plane of section may assume different relative directions, the variations in the cone are unlimited. If the cone be circular, and the plane of section be perpendicular to the axis, the figure is the right circular cone ; and this is the most important of all the cones. The base is a circle, and the axis of the cone passes through the centre of the circle. 124 THE CYLINDER. 125 A right circular cone is generated by a right-angled triangle while revolving about one of the sides as an axis. The other side then generates the base (Art. 9. Cor. 1), and the hypothenuse generates the convex surface. 133. The cone may be looked upon as the limiting form of a right regular pyramid, when the number of sides in the base is indefinitely increased, and the length of each side is correspondingly diminished. But the volume of any pyramid is one-third of its altitude multiplied by the area of its base ; Therefore, the volume of a cone is one-third of its altitude multiplied by the area of its base. Cor. If the base be circular and its radius be r, its area is Trr 2 . And if h be the altitude of the cone, the VOl. = ^ 7!T 2 7i. 134. The frustum of a cone is the limit of the frus- tum of a pyramid, and its volume is therefore But if r and r' be the radii of the bases, B = Trr 2 , B' = Trr' 2 , and .-. vol. = i 7 r THE CYLINDER. 135. When the cylinder of Art. 75 is cut by two parallel planes which cut completely through the surface, a closed figure is formed, which is the cylinder of stereometry. 126 SOLID OK SPATIAL GEOMETRY. When the planes are perpendicular to the axis of the cylinder, the figure is a right cylinder. Otherwise it is an oblique cylinder. 136. It is obvious, from the definitions, that the cylinder is the limiting form of the prism, when the number of sides in the base is indefinitely increased and the lengths of each side correspondingly diminished. Hence the measure of a cylinder is the area of the base multiplied by the altitude (Art. 115. Cor. 2). Cor. If the cylinder be circular and right, and r be the diameter of the base, vol. = where h is the altitude. THE SPHERE. 137. ABCD is a tetrahedron in which the edge AB is equal and perpendicular to the edge CD, and KJ, joining the middle points of these edges, is the common perpendicular to them. THE SPHERE. 127 Also, K'RQTis a sphere having its diameter K 'J' = KJ. We shall prove that corresponding laminae of the tet- rahedron and of the sphere are in a constant ratio, by proving that corresponding sections are in a constant ratio. Proof. Let parallel planes pass through AB and CD. Then KJ is a common normal to these planes, "and if the sphere be placed between the planes with K'J' parallel to KJ, the planes will touch the sphere at K' and at J'. Let the sphere and tetrahedron be relatively so placed, and let EG and RQT be corresponding sections of the figures (Art. 119). Then EFGH is a rectangle, and QRT is. a circle, and KP = K'P'. KP AE EH Now and KJ AD DC' PJ- DF EF KJ DB AB' .*. by multiplication KP PJ = EF - EH = a EG KJ 2 AB* ~~ AB 2 ' Also, denoting the radius of the sphere by r, K'F P'J' = P'R 2 = 1 TT P'R 2 = 1 OQRT K'J' 2 K'J' 2 IT' KJ 2 TT' 4^ Therefore . KP - PJ = K'P ' - P'J', 128 SOLID OR SPATIAL GEOMETRY. Hence the corresponding sections of the tetrahedron and of the sphere are in a constant ratio ; and the vol- umes of the tetrahedron and the sphere are in the same ratio. Cor. 1. The tetr. : the sphere = AE 2 : 4?rr 2 . But the tetr. =%KJx mid. sec. (Art. 130. d.) .-. vol. of sphere =%irr\ Cor. 2. The expression for the volume of a sphere may be written f 2r irr 2 . But 2 r is a diameter of the sphere, and -n-r 2 is the area of a great circle. Therefore, 2r Ti-r 2 is the volume of the right circular cylinder which circumscribes the sphere. Hence a sphere is two-thirds of its right circumscribing cylinder. 138. As the prismoidal formula applies to any portion of the tetrahedron confined between planes, each parallel to AB and CD, and since laminae of the sphere hold a constant relation to corresponding laminae of the tetra- hedron, it follows that the prismoidal formula applies to any portion of the sphere limited between parallel planes. Thus, applying the formula to the whole sphere, we have B = 0, B' = 0, M= Trr 2 , and h = 2r. vol. = (0 + + 4 Trr 2 ) = f Trr 3 . THE SPHERE. 129 139. Def. A portion of a sphere enclosed between two parallel planes is usually called a zone of the sphere ; but if one of the planes is a tangent plane, the zone be- comes a segment of the sphere. L 140. Volume of a zone. Let a sphere be cut by par- allel planes, given in section, in the diagram, by AB and CD-, and let XY denote in section the plane which is parallel to the cutting planes, and half way between them. The data usually furnished from which to find the volume of the zone, are the radii CD and AB of the two bases, and the length of their common normal AC, or the altitude of the zone. Hence we suppose AB, AC, and CD to be the known quantities. We have, being the centre of the sphere, OA 2 + AB 2 = OC 2 + CD* = OX 2 + XT 2 , since each expression is the square on the radius of the sphere. CD 2 Hence and hence AB 1 = 4 AX 2 + CD 2 + 40A'AX = AX 2 + XY 2 + 20A- AX. .: 4 OA . AX=AB 2 - CD 2 - 4 AX 2 = 2AB 2 -2XY 2 -2AX 2 ; 2 XY 2 = AB 2 + CD 2 + 2 AX 2 . 130 SOLID OR SPATIAL GEOMETRY. Now, TT XY 2 is the area of the middle section, and TT AB 2 and TT CD* are the areas of the bases, and 2 AX is the altitude of the zone ; .-. vol. = T Or, denoting the radii of the bases by r and r' } and the altitude by h, Cor. If r = 0, the zone becomes a segment, and its volume is irh Sr^ h 2 . 141. The expression for the volume of the zone may be transformed as follows: Draw DE _L to AB. 3AB 2 + 3 CD 2 + AC 2 = 2(AB 2 + CD 2 + AB CD) and TT AC (AB 2 + CD 2 + AB . CD) is the volume of the frustum of the cone which has the same bases and altitude as the zone. And AC being the projection of BD on OC, if we denote the angle between BD and AC by /?, AC=BDcos/3. = sphere on BD as diameter x cos ft. Therefore, the zone exceeds the inscribed conical frus- tum by the sphere on the slant height as diameter multiplied by the cosine of the semi-vertical angle of the cone. THE SPHERE. 131 EXERCISES I. 1. Compare the volume of a sphere (1) with that of the circum- scribed cube ; (2) with that of the inscribed cube. 2. Compare the volume of the sphere with that of the circum- scribed regular tetrahedron. 3. A cone circumscribes a sphere and has its slant height equal to the diameter of its base. Show that vol. of cone : vol. of sphere = 9:4. 4. If in Ex. 3 a plane passes through the circle of contact, the vol. of cone removed is f the vol. of sphere removed. 5. A cylinder of radius a passes centrically through a sphere of radius r. Show that the volume removed from the sphere is | irr 3 (1 cos 3 0), where sin 6 = - . 6. A circular cone with semi-vertical angle a has its vertex at the centre of a sphere of radius r. Show that the volume common to the cone and sphere is f Trr 3 (1 cos a). 7. A right circular cone has its vertex lengthened out into a linear edge equal and parallel to a diameter of the base. Show that the volume is one-half that of the circumscribing cylinder. (The resulting figure is known as the common conoid.) 8. A cone whose semi-vertical angle is 45 has the diameter of a sphere as its axis, and its vertex on the sphere. Show that one- fourth of the sphere lies without the cone. 9. The cone of Ex. 8 has its semi-vertical angle equal to a; then the part of the sphere lying without the cone is irr 3 (1 + cos 2 a) 2 . SECTION 3. 142. In this section we propose, under three heads, A, B, and C, to explain and illustrate some special methods of measuring volumes, by applying these methods to the cone, cylinder, sphere, and some other spatial figures. A. SPATIAL FIGURES GENERATED BY THE MOTION OF A PLANE FIGURE. 143. When a variable plane figure moves so that a fixed point lying in its plane describes a line or curve not complanar with it, the plane figure describes or generates a spatial figure. The plane figure is then the generator, and the line or curve is the path of the particular point which de- scribes it. The case as here stated is too general for use, especially in elementary geometry or by elementary methods. We therefore subject the elements of the description to certain conditions, usually as follows. (1) The generator is a closed plane curve, being in- variable in form, while being either variable or constant in dimensions. (2) The path is a line normal to the plane of the generator. This line will be called the axis. (3) The generator preserves its orientation, i.e. any fixed line of the generator is invariable in direction ; or 132 GENERATED FIGURES. 133 any fixed point in the generator describes a line or curve complanar with the axis. This line or curve, whose form depends upon the nature of the variation of the generator, is a guide to the motion of the generator, and forms the director. Thus if the nature of the variation of the generator is given, the director is also given; and if the director is given, the nature of the variation is given. 144. Let PQR be a variable circle, whose centre, C, moves along the fixed line AB normal to the plane of the circle. AB is the axis. O F (1) Let P, any point on the circle, be guided by the fixed director line L, which meets AB in D. Then, evidently the generating circle describes a cone having D as vertex and AB as axis. The radius CP is in a constant ratio to DC. Hence a variable circle, whose centre moves on a fixed line normal to its plane, and whose radius varies as the distance of the centre from a fixed point in the line, describes a cone. 134 SOLID OR SPATIAL GEOMETRY. (2) If the generating figure in case (1) were a polygon, the figure generated would be a right pyramid. (3) Let P move on the line M parallel to AB. The circle describes a cylinder, and a polygonal genera- tor describes a prism. (4) Let P be guided by the circle, Z, to' which AB is a centre line, and EF a diameter. The circle PQR then generates a sphere whose diam- eter is EF. If be the centre of the director circle, it is evident that OP 2 + CO 3 = OP 2 = constant. Therefore, a variable circle, whose centre moves on a line normal to its plane, and whose radius so varies that the sum of the squares on the radius and on the distance of the centre of the circle from a fixed point in the line is constant, generates a sphere. (5) If the generator in case (4) were a polygon, the figure generated would be a polygonal groin; the most common groin is the square one. In a similar manner many other figures may be gener- ated, such as the oblate spheroid, the prolate spheroid, the hyperboloid, the paraboloid, the ellipsoid, etc. 145. Consider a number of equidistant points along the axis. Let the generator at these points be taken as bases of prisms or cylinders whose altitudes are the dis- tances between consecutive points. We have then a series of prisms or cylinders, of equal altitude, inscribed in or circumscribed about the spatial figure, as the case may be. GENERATED FIGURES. 135 But (Art. 118) the volume of the spatial figure is the limit of either series of prisms or cylinders, when their number is indefinitely increased and their altitudes cor- respondingly diminished. Hence if we can obtain an expression for the total volume of any number of such elementary prisms or cylinders, we can deduce the expression for the volume of the spatial figure, by imposing the condition that the number of elementary prisms or cylinders shall be infinite. In carrying out this operation we assume the two fol- lowing relations, which are proved in almost any work on algebra : (A) 1 + 2 +3 +...+M =|n 2 + in; (B) l 2 + 2 2 + 3 2 +...+n 2 = -^ 3 + ^ 2 + |n, where n denotes any positive integer, and the series extends from 1 to n. 146. Let X be a closed plane figure, which remains invariable in form while varying its dimensions. Let a given point P be guided by the line AH, and let a point Q move on AC. Then X describes a spatial figure, a cone or pyramid, having some position of the generator at B, as above. Let X denote the area of the variable figure, X, at any stage in its variation, and let B denote the area of CDE, the final stage of X. 136 SOLID OR SPATIAL GEOMETRY. Then X: B = PQ 2 : HC 2 . (P. Art. 218. 5.) And from similar triangles, APQ and AHC, PQ 2 :HC 2 =AP 2 :AH 2 . Ap2 Denote AH by h, and let AH be divided into n equal parts, and let AP be ra of these parts. Vfi Then AP = --AH, n and .-. X=-B. n 2 But the elemental cylinder, or prism, on X as base has - AH, or -, as its altitude, and therefore its volume is n n a* This expresses the volume of any element, a particular one being got by giving a particular value to m. ra = 1 gives the first element, lying next A ; m = 2 gives the second, etc., and m = n gives the last, lying next H. The sum of tnese elements is This holds true for all integral values of n. When we go to the limit by making n infinite, the fractions GENERATED FIGURES. 137 and become zero, and the sum of the elements 2 n 6n 2 becomes the volume of the spatial figure (Art. 145). . . volume = J Bh. As B may have any closed form whatever, this ex- presses the volume of any species of cone or pyramid which forms a closed spatial figure. 147. Let the generating figure, X, of constant form, but variable in dimensions, be guided by the axis OA, and by the circular quadrant CQA as a director, being the centre of the quadrant. Let CDE be the generator in the position in which lies in its plane, and let S denote the area of CED, and X denote the area of the generator in any position. Then, since PQ is _L to OA, PQ 2 = OQ 2 - OP 2 = OC 2 - OP 2 . X: S = PQ 2 : OC 2 ; (P. Art. 218. 5.) But X ~ OC 2 S S Now, denote OA by r and divide it into n equal parts, and let OP be ra of these parts. Then OP=r, and 00 =r. 138 SOLID OR SPATIAL GEOMETRY. The elemental cylinder having X as base has - for n altitude, and its volume is therefore The sum of these is 1 + 1 + 1 + - n terms efi 11 1 } = 7"O < 1 ------ > I 3 2n 6nM at the limit, when n becomes infinite. And the volume generated while moving over the whole diameter is vol. = | rS. The value of this expression for volume depends upon the value of S. 1. If S is a circle, its area is Trr 2 , and the figure gener- ated is the sphere. .-. vol. of a sphere = f 7T?- 3 . 2. If S is a square, and the middle point of its side is at (7, the area is 4^, and the figure is the common groin, and its vol. = f r 3 ; since the groin extends only from to A. 3. If S is a regular hexagon with a vertex at (7, we have a hexagonal groin, and its volume is SPATIAL FIGURES. 139 148. By varying the form of the generator, and also of the director curve, a great variety of spatial figures may be described. 1. With a circle as director and an ellipse as generator, QR being the major axis, we get the oblate spheroid; and with QR as minor axis, the prolate spheroid. 2. With an ellipse as director, and major axis as axis, and an ellipse as generator, we get the ellipsoid; with circle as generator we get the prolate spheroid. 3. With parabola as director, and a circle as generator, we get the paraboloid of revolution ; and with ellipse as generator we get the elliptic paraboloid. EXERCISES J. The axes of an ellipse being a and 6, its area is irab. 1. Show that the volume of a prolate spheroid is irab 2 , where a>b. 2. Show that the volume of an oblate spheroid is ira 2 6, where a>b. 3. In the figure of Art. 147, if CQA were a quadrant of an ellipse, and OA = a and OC = b, then . + -- = 1. Hence OC 2 OA 2 find the volume of an ellipsoid when the axes of the generating ellipse are b and c at the position S. 4. In Art. 146, if PQ 2 = c AP, where c is a constant, show that the volume described is one-half that of the circumscribing cylinder. 5. OC is an axial line cut by a curve in and C, and PM is a perpendicular from a point P on the curve to the axis OC. If PM = a(OM- OC - OiTf 2 ), show that the volume described by the curve in a revolution about the axis is T 8 5 of that of the circum- scribing cylinder between and (7. 140 SOLID OR SPATIAL GEOMETRY. B. FIGURES OF REVOLUTION. 149. When a plane figure revolves about an axial line lying in its plane, the plane figure generates a spatial figure bounded wholly or partly by curved surfaces, and called, from its mode of generation, a figure of revolution. Under the same circumstances the area of the plane figure generates a volume of revolution, i.e. the volume of the figure of revolution. The area of a plane figure may be considered as the limit of the sum of a set of elements, composed of inscribed rectangles with equal but indefinitely small altitudes. In revolution ? these elements of area describe or generate elements of volume, whose sum has for its limit the volume of the gen- erated spatial figure. 150. Let AC be a rectangle, and let it revolve about the axial line PR, parallel to AD. The volume generated by the rectangle AC is the difference between the volumes generated by PC and by PD. But the vol. by P<7 = TT P.B 2 EC, and the vol. by PD = TT - PA 2 BC; .: the vol. by AC= IT BC(PB 2 - PA*) = TT BC(PA + PB) (PB - PA). FIGUEES OF REVOLUTION. 141 If Q be the middle point of AB, PQ is the distance of the centre of the rectangle from the axis of revolution, = 2PQ; .: vol. by AC = 2* PQ - BO AB, = area of AC x the circumference of the circle traced by the centre of AC. Therefore, the volume described by a rectangle in one revolution about an axial line parallel to its side, and which does not cross the rectangle, is the area of the rectangle multiplied by the length of the path of its centre. Cor. 1. When the axial line passes through the cen- tre of the rectangle, the length of path described by that centre is zero, and hence the volume described is zero. From this it appears that if a revolving plane figure is crossed by the axis of revolution, the parts of the figure lying upon opposite sides of the axis generate volumes which must be taken in opposite senses, or with opposite signs. Cor. 2. From the figure we have 2PQ = 2PA + AB ; and hence 2 TT PQ = 2-n- PA + v AB. But when AC is an elemental rectangle, and we go to the limit by indefinitely diminishing AB, PQ has for its limit either PA or PB, these being finally the same. Hence, if the elemental rectangle AC is to be taken at the limit, PA may be taken for PQ. 151. Volume of a cone of revolution. The rectangle AC revolves about AB as an axis. 142 SOLID OR SPATIAL GEOMETRY. The triangle ACB generates a cone of revolution ; the rectangle generates a cyl- inder; and the triangle ACD describes that part of the cylinder which remains after the cone is removed. On -4Ctake P, Q, any near points, which at the limit become coincident, and draw PR, QS, perpendicular to AB, and PT, Q F, perpendicular to AD. Then PS, being an elemental rectangle of the triangle ACB, generates an element of the cone; and PF, in like manner, generates an element of the portion of the cylinder which remains after removal of the cone. But vol. of element by PS = TT PR 2 PE. And vol. of element by PF= ir(PR + FR) PT PF. And from similar triangles PR A and QEP, PJl = QE Qr PR = PF RA ~ EP' r PT ~ PE element by PS = PR element by PR + FR And this relation being true for any, and therefore for every, pair of corresponding elements, is true for their sums. But at the limit, when Q comes to P, PR and FR become the same. ... the limit of ^(elements by PS) = 1. 2 (elements by PF) 2 Or, cone by ABC : figure by ACD = 1:2. FIGURES OF REVOLUTION. 143 Whence it follows that the cone is one-third of the cylinder. REMARK. In the foregoing investigation we might, according to Cor. 2 of Art. 150, have taken the element described by PFas being TT 2 PR PT PF, since the element is finally to be taken at its limit. The quadrant DPA, r v 152. Volume of a sphere, and its circumscribed square DBAC, revolve about CA as an axis. The quadrant generates a semisphere, and the square generates the right circum- scribed cylinder. On the arc DA take P and Q, any near points which at the limit approach to coinci- dence. Draw PR, QS, perpendicu- lars to CD, and PT, QV, per- pendiculars to DB. Produce DC, making CG = DC. The rectangle PS, being an element of the circle, describes an element of the sphere, and the rectangle PV for similar reasons describes an element of that part of the cylinder which lies without the sphere. The volume of the element described by PS is, at its limit when Q comes to P, 2ir CR PR RS ; and the volume of the element described by PV is, at its limit, 144 SOLID OR SPATIAL GEOMETRY. at It element by PS _ 2 PR Rtf CR " element by PV PT '' But the &DPR and PGR being similar, CD PT RD ~ PR ~ PR And the A PEQ and PRO being similar at the limit when Q approaches P, RS = PE = PE = PR PF~ PF~ EQ~ CR .: at It., element by PS = 2 x element by PV. And this being true for each, and therefore every pair of corresponding elements, is true for their sums. Therefore the volume generated by the quadrant is twice the volume generated by the figure DPAB. Or the volume of a sphere is two-thirds that of the circumscribing right cylinder. Cor. 1. If r be the radius of the sphere, the volume of the circumscribing cylinder is -n-r 2 -2r- ) and hence the volume of the sphere is f irr 3 . Cor. 2. From the foregoing investigation it follows that wherever Q is taken on the arc, with CA as axis, the volume generated by the segment of the circle, DSQP, is two-thirds the volume generated by the rec- tangle DSQV. 153. Volume generated by an isosceles triangle revolv- ing about an axis which passes through the vertex but does not cross the triangle. FIGURES OF REVOLUTION. 145 The isosceles A OPQ, with PQ as base, revolves about the axis OD passing through the vertex 0. Let PQ meet OD'mD, and draw the altitude OR, and project P, R, Q, on OD at A, C, and B. Also draw QE parallel to OD. and hence, A PEQ ^ A ORD. Therefore, OD - PE = OR PQ ; also, APEQ^AOCR, and PQ-CR=OR-EQ; .: OD-CR-PE = OR 2 EQ = OR 2 - AB. Now, the vol. described by A OPQ = vol. of cone by OP A + vol. of cone by DP A vol. of cone by OQB vol. of cone by DQB, = |TT OD (PA 2 - QB 2 ) = ^7T'OD'2CR'PE = |TT. OR 2 >AB. Therefore, the volume described, in one revolution, by an isosceles triangle revolving about a line through its vertex, and lying without it, is the continued product of the projection of the base of the triangle upon the axis, the area of the square on the altitude, and the constant f IT. 154. Let equidistant points A, B, C, etc., be taken in the arc of a circle of which is the centre and OL is a centre line not crossing the arc. The A A OB, BOO, ..., are all isosceles and congruent. 146 SOLID OR SPATIAL GEOMETRY. The volume described by these triangles in revolving about OL as axis, p being the common apothem, is |^ 2 7r(pr. of AB on OL + pr. of BC on OL + ...). But at the limit when the number of points A, B, C ... is indefinitely increased, and the distance between them is correspondingly diminished, the generating figure be- comes the sector of a circle, p becomes the radius, and the sum of the projections of the bases of the triangles is the projection of the arc, and the figure generated is a sector of a sphere. Therefore, the volume of a sector of a sphere = f Ti-r 2 x pr. of the generating arc on the axis. Cor. If the generating arc forms a semicircle, its projection on the axis is 2r, and the figure generated is a sphere. A yoL of a gphere = 4 ^ EXERCISES K. 1. Solve Ex. 6 of Set I., by the principle of 153. 2. AX is an axial line, and PM is a perpendicular to this line from a point P on a curve which starts from A. If PM 2 = cAM, where c is a constant, show that the volume described by one revolution about AX, is one-half that of the circumscribing cylinder. 3. The volumes of the circumscribing cylinder, the sphere, and the cone with the same base and altitude as the cylinder, are as the numbers 3, 2, and 1. 4. The volumes of the cylinder circumscribing a semisphere, the semisphere, and the cone with base and altitude of the cylin- der, are as the numbers 3, 2, 1. 5. A plane cuts a sphere, and its circumscribed cylinder parallel to the base ; then twice the volume of the segment is equal to the intercepted volume of the cylinder and twice the volume of the sphere on the altitude of the segment as diameter. THEOREM OF PAPPUS. 147 C. THEOREM OP PAPPUS OR GULDINUS FOR VOLUMES. 155. The mean centre of a system of complanar points for a system of multiples is defined (P. Art. 240) as the point of intersection of two lines, L and M, for which 2(a-AL)=0, and 2(a-AM) = Q; where A is a representative point, AL and AM represen- tative perpendiculars from A to L and M respectively, and a a representative weight or number. Also (P. Art. 241), if be the mean centre of the system, and L be any line complanar with the system, We have to deal here with the mean centre of the area of a figure, and later on with the mean centre of the perimeter of a figure. 156. When a plane figure has an axis of symmetry, the mean centre of the figure lies on this axis. For every point in the area upon one side of the axis of symmetry there is a point upon the other side exactly corresponding in every respect. So that if L be the axis of symmetry and A lt A 2 be corresponding points, we have A V L + AL = 0. And since the whole area is represented by pairs of such corresponding elements, 2(a AL) = 0, or L passes through the mean centre. Cor. 1. When a figure has two axes of symmetry, the mean centre of area is the point of intersection of the axes. 148 SOLID OR SPATIAL GEOMETRY. This is the case with the square, the rectangle, the rhombus, all regular polygons, the circle, and some other figures. 157. If the area of a figure be supposed to be made up of elemental squares, the centres of these squares, being their mean centre of area, will represent the points, A, B, C, etc., in a system of points, and'the areas of the several squares will represent the weights. But since the squares are all equal, the weights are all equal, and may be left out of consideration. With this understanding we have for the mean centre of area, = 0, where L passes through this centre; and = n- OL, where is the mean centre, L is any line not passing through 0, and n is the number of elements under consideration. 158. Theorem. The join of the mean centres of two systems passes through the mean centre of the system composed of the two taken together. This theorem is almost self-evident. For if 2(a f A'L) = and 2(a" A"L) = denote the two systems, and L is the join of their mean centres, we have at once which is of the type 2 (a AL) = 0. Cor. If any number of systems have their mean centres collinear, the mean centre of the system com- posed of all taken together lies on the line of collinearity. 159. Theorem. The mean centre of a parallelogram is its geometric centre, i.e. the intersection of its diagonals. THEOREM OF PAPPUS. 149 Let ABCD and EFGH be two congruent parallelo- grams, superposable with E on A, F on B, G on C, and H on D. Their mean centres of area are then coinci- dent. But the parallelograms are also superposable with E on C, F on D, G on A, and H on B ; and their mean centres of area are again coincident. Hence the mean centre of each is the geometric centre. Cor. In like manner it may be shown that when any figure has a geometric centre, that centre is also the mean centre of its area. 160. Mean centre of the area of a triangle. Let BD be a median to the triangle ABC. Draw EF and GH two near lines each parallel to AC, and draw El, FJ parallel to BD. The parallelogram EJis an element of the area of the triangle, and the sum of the areas of these elements, when taken at the limit, is the area of the triangle. But as BD bisects EF and JJ, it passes through the mean centres of all the elements of which EJ is a type. Therefore ( Art. 158), the centre of area of the triangle lies on BD ; and as it lies on both of the other medians, the centre of area of a triangle is its centroid. Def. On account of the foregoing, we shall call the mean centre of area of any figure its centroid. 161. Theorem. The orthogonal projection of the mean centre of any complanar system is the mean centre of the projection of the system for the same multiples. 150 SOLID OR SPATIAL GEOMETRY. Let A, B, (7, ' be the elements in the plane U, and A', B', C', be their projections on the plane F Take L, any line in U, through the mean centre 0, and let L' and 0' be the projection of L and on F. Then 2 (a AL) = 0. (P. Art. 240.) But AL, BL, CL, etc., are all parallel, and A'L', B'L', C'L', etc., are all parallel. Therefore, the /. (AL A'L') = Z (BL B'L') = etc., and hence ___ _____ say . A'L'~B'L'~ or and L' passes through the mean centre of the projected system. And as this is true for all directions of L and L' in their respective planes, 0' is the mean centre of the projected system ; or the projection of the mean centre of the system in U is the mean centre of the pro- jected system in F. 162. Def. Let us call, in general, a figure of the type of the cylinder or prism, but with non-parallel bases, a cylindroid. Suppose a system of near equidistant planes parallel to the axis to cut the cylindroid. These divide it into laminae parallel to the axis. Now suppose a second set of planes, parallel to the axis, to cut the first system at right angles, and to have the distance between consecu- tive planes the same as in the first system. THEOREM OF PAPPUS. 151 These planes divide the cylindroid into elementary prisms on square bases. These form the prismatic ele- ments of the figure, and the sum of their volumes, at the limit, as their bases are indefinitely diminished and their number is correspondingly increased, is the volume of the cylindroid. 163. AGB is a cylindroid having the base ABH nor- mal to the axis, and the base COD oblique to the axis. Let PQ be a prismatic element, the area of whose base is ft, and let the line CD be taken parallel to the common line of the planes of the bases, and let AB be the orthogonal projection of CD on the lower base. Draw QFA. to AB, and FE nor- mal to the base ABH. Then FE H is parallel to QP, and meets CD in some point E. Draw ER to PQ. Join EP. Then EP is _L to CD, and ERQF is a rect- angle, and EF= QR. The volume of the prismatic element PQ is And since the bases of all the prismatic elements have the same area, and EF is constant, the sum of the volumes is But ^((3)EFis, at the limit, the volume of the cylin- der HK, whose base is ABH, and altitude EF. In order that this may be equal to the cylindroid, we must have 2(/3 PR)= ; and as every element PR is in 152 SOLID OR SPATIAL GEOMETKY. a constant ratio to the corresponding element EP, and ft is constant, we must have ^,(EP) = 0. Or CD must pass through the centroid of the upper base, CGD, of the cylindroid; and (Art. 161) AB passes through the centroid of the lower base. Hence, however the directions of the planes of section which give the bases may vary, provided they do not meet within the limits of the cylindroid, the volume remains unchanged, while the distance between the centroids of the bases remains the same. Cor. The volume of a cylindroid is the area of a right section multiplied by the distance between the centroids of the bases. 164. Let the plane figure X, invariable in form and dimensions, move from a position AB to another posi- tion CD, in such a manner that its direction of motion, whether follow- B ing a line or a curve, is always C normal to its plane. Take two near positions of X as at GH and JK, and consider these as bases of a cylindroid forming an element of the figure generated by the motion of X. If P and Q be the centroids of the bases, the volume of the elementary cylindroid, GK, is X PQ, where X is the area of the generating figure. And at the limit, when P approaches indefinitely to Q, the sum of the cylindroids is the generated spatial figure, and the sum of the elements PQ is the path of the centroid of X. THEOREM OF PAPPUS. 153 Therefore, when a plane figure, invariable in form and dimensions, moves in a path which is at every instant normal to the plane of the figure, the whole volume described is the area of the figure multiplied by the length of path moved over by the centroid of the figure. This is the statement of the theorem as first given by Pappus (about 300), and afterwards reproduced by Guldinus (1577-1643), and usually called after his name. Cor. When a plane figure revolves about a complanar axis, the direction of motion of the centroid is at all times necessarily normal to the plane of the figure, and the volume described in one revolution is the area of the plane figure multiplied by the circumference traced by its centroid. Ex. A circle revolves about a complanar line lying without it ; the figure generated is called an anchor ring. To find its volume. Let r be the radius of the generating circle, and R be the distance of its centre from the axis. Then EXERCISES L. 1. Find the position of the centroid of a semicircle. 2. The circle which generates an anchor ring is divided by a diameter parallel to the axis ; compare the volume described by the outer and the inner half of the circle. 3. A semicircle revolves about its limiting diameter. Any segment whose chord is parallel to the axial line describes a volume equal to that of a sphere on the chord of the segment as diameter. 154 SOLID OR SPATIAL GEOMETRY. 4. The distance from the centre of a circle to the centroid of 2 c 3 any segment, is - , where c is the half chord of the segment, and 3 S S is its area. 5. The distance of the centroid of a segment from its chord is 2 c c' 2 . 3S 2v + where c' is the chord of half the arc, and v is the versed sine of the arc (P. Art. 176. Cor. 1). 6. An arc of a circle revolves about its chord ; the volume generated is {4 C 3fl-3(c 2 -?; 2 )}. The figure generated is called a circular spindle. 4r 7. The centroid of a semicircle is at the distance from the centre of the circle. 8. A semicircle revolves about a tangent at its middle point. The volume described is 9. A square with side s revolves about a line through one vertex, making an angle with a side, and not crossing the square. The volume described is 7rs 3 (sm -f cos 0*), 10. A plane cuts through a right circular cylinder so as to cut one base only. The volume of the portion removed is where h is the height of the convex part, r is the radius of the cylinder, and v, c, and S denote the versed sine, semichord, and area of the segment of the base. This figure is called an ungula of a right circular cylinder. SECTION 4. PLANIMETRY THE MEASUREMENT OF THE AREAS OF SURFACES, OR SUPERFICIES. 165. When a spatial figure is bounded by plane faces only, the area of its surface is the sum of the areas of its faces. For such figures no special method is required outside of the processes of plane geometry. The area of a curved surface is usually derived from that of a polyhedron by going to the limit, and suppos- ing the number of polyhedral faces to be indefinitely increased while the size of each face is correspondingly diminished. In some curved surfaces, however, we may suppose the surface to be brought to coincide with a plane by a sort of unrolling of the surface without stretching or distorting it in any of its parts. Such surfaces are said to be developable ; and when the surface is brought to coincide with a plane, it is said to be developed on the plane. Thus a sheet of paper may be rolled into a cone or a cylinder, but it cannot be bent into a sphere. The cylinder and the cone are accordingly developable surfaces, while the sphere is not. It is readily seen that none but ruled surfaces can be developable. Kuled surfaces are not, however, all devel- 155 156 SOLID OR SPATIAL GEOMETRY. opable, and those which are not so are called skew sur- faces. 166. Development of the conical surface. Let be the centre of a circular cone, and L be a generating line. On L take any point, P, and through P draw the cone-circle APB with as vertex. With any point, Q, as centre, and QD = OP as radius, describe an arc, DE, equal in length to the circumfer- ence of the circle APB. The figure QDE, a sector of a circle, is the develop- ment of the conical surface lying between the centre and the cone-circle APB. It must be remarked that the construction here given is theoretical only, since we have no method in elemen- tary geometry of constructing an arc of one circle equal in length to a given arc of another circle, when the circles have different and incommensurable radii. This difficulty will not, however, vitiate any application to be made of this principle. PLANIMETRY. 157 167. Area of surface of right circular cone. For the closed cone APB it is evident that the area of the curved surface is equal to the area of its development, i.e. of the circular sector QDE, and this is one-half the length of the arc DE multiplied by the radius QD. But the arc DE is equal in length to the circle APB, and the radius QD is equal to OP. Therefore, denoting the circumference of the base by C, and the slant height, AO or BO, by S, curved surface = CS. 168. Frustum of a right circular cone. Drawing a second cone-circle, apb, to the vertex 0, and the development Qde, we have for the frustum, area = sector QDE sector Qde. Or, denoting the circumference of apb by c, 2 area = OP -C- Op- c. But OP = Pp + Op ; and = -; Op c . Pp _ C-^c , Op c and 2 area = Pp - C+ Op (C c), or area of surface = \ Pp (C + c). 169. In the cylinder, 0, and therefore Q, goes to infin- ity, and QD and QE become parallel. Hence DE and de become equal and parallel lines, and the development DdeE is a rectangle. 158 SOLID OR SPATIAL GEOMETRY. Or, if h be the height of the cylinder, and r be the radius of the base, convex surface = 2 -n-rh. 170. Area of the surface of a sphere. Let AB be a quadrant of a circle which generates a semisphere by revolving about OB as an axis. Take two near points on the curve, P and Q, which at the limit come into coincidence, and draw the chord PQ. This chord describes the convex surface of a frustum of a cone, and the area of the surface is P r q (Art. 168.) where Pp and Qq are J_s upon OB. Take R, the middle point of PQ, and draw Rr _L to OB and join RO, and also draw QS _L to Pp. Then the surface described by PQ is 2 TT PQ . Rr. And on account of the similar A PQS and ORr, the surface described by PQ is 27T- OR-pq. And the convex surface described by a system of chords, forming the sides of a regular polygon^ is 27T- OR-^(pq). PLANIMETRY. 159 But at the limit when P comes to Q, the apothem, OR, becomes the radius, and the polygon becomes the circle ; and the surface described by any arc, PQ, is 2 Trt* x proj. of the arc on the axis. Now, 2 TTT is the circumference traced by D, a point on the circumscribed rectangle ACBO, and the projection of the arc is equal to DE ; Therefore, the convex surface described by the arc PQ, is equal to the convex surface described by DE. Hence, if a sphere and its circumscribed right cylinder be cut by two planes parallel to the bases of the cylinder, the area of the curved surface intercepted between the planes is the same for the sphere as for the cylinder. Cor. The area of the surface of a sphere is equal to that of the curved surface of its circumscribed right cylinder. Therefore, the area of the surface of a sphere is 471-r 2 , or four times the area of a great circle. 171. We may consider a sphere as circumscribing a conspheric polyhedron with an indefinite number of very small faces. Considering these faces as bases of pyra- mids having their vertices in common at the centre of the sphere, the sum of these pyramids at the limit, when the number is indefinitely increased, and the size of each base is correspondingly diminished, is the volume of the sphere, and the sum of their bases is the surface of the sphere. 160 SOLID OR SPATIAL GEOMETRY. But each pyramid is ^ Bh, and their sum is ^ S ( Bli) . Or writing S for 2-B, and r for h, where V is the volume of the sphere, and S is the area of the surface. THEOREM OF PAPPUS OR GULDINUS FOR SURFACES. 172. For convenience we shall call the mean centre of the perimeter of a plane figure its centre of figure. The general theorems respecting the mean centre as developed in Arts. 158 and 160 apply to the centre of figure in the same manner as to the centre of area. 173. Two equal line-segments, AB and CD, are con- gruent whether A is placed on C, and B on D, or A on D, and B on C, and hence the centre of figure of a line- segment is its middle point. Then, in any rectilinear figure which has an axis of symmetry, the sides exist in congruent pairs which are symmetrically disposed upon opposite sides of the axis of symmetry. And hence if A and A' denote the middle points of two sides forming a symmetrical pair, AL = A'L, or AL + A'L = ; where L is the axis of symmetry. Therefore, 2 (AL) = ; or L passes through the cen- tre of figure. Hence, when a rectilinear figure has an axis of sym- metry, the centre of figure lies upon that axis. THEOREM OP PAPPUS. 161 Cor. When a rectilinear figure has two axes of sym- metry, their point of intersection is the centre of figure. 174. Let AGE be a cylindroid having the base ABH normal to the axis, and the base CGD oblique to it. Suppose the convex surface to be divided in very narrow strips of equal width throughout, and paral- lel to the axis of the cylindroid and equal in width to one another ; and let b denote the breadth of one of these elements of surface, and let ST denote the line along the middle H of the strips. Then ST is normal to the base ABH. Let CD be parallel to the common line of the planes of the bases, and let AB be the projection of CD on the lower base. Draw TF _L to AB and FE _L to CD. Also draw EVA- to ST, and join ES. Then, EVTFis a rectangle, and VT = EF. The area of the element represented by ST is b-ST,orb- EF+ b - 8V. And the sum of these elements, of which the one represented by S T is the type, is, But EFis constant, and 2(&) is the circumference of the base ABH. Therefore 2(b'EF) is the convex surface of the cylinder, or prism, whose base is ABH, and whose alti- 162 SOLID OK SPATIAL GEOMETRY. tude is EF. And that this may be equal to the whole convex surface we must have 2(& SV) = 0. But as all the elements have the same width, b is constant, and SV: SE is constant; or, CD passes through the centre of figure of the upper base ; and hence AB passes through the centre of figure of the lower base. Therefore, the area of the convex surface of a cylin- droid is the circumference of a right section multiplied by the distance between the centres of figure of the bases. 175. Let the plane figure X, invariable in form and dimensions, move with centre of figure on the path OPQR, and so that the direction of the path is at all points normal B to the plane of the figure, and let GH and JK be two near positions, which at the limit come into coin- cidence. The surface of the elementary cylindroid QK is the circumference of X x PQ ; and the area of the surface of the figure generated by the motion of X is S(PQ) X circum. of X. But 2(PQ) is the path OPQR---, and circumference of X is constant. Therefore, the area of the surface described by a plane figure, invariable in form and . magnitude, which moves so that its direction of motion is at each point normal to THEOREM OF PAPPUS. 163 its plane, is the circumference of the generating figure multiplied by the length of path described by its centre of figure. Cor. When a plane figure revolves about a complanar line as axis, the direction of motion is necessarily normal to the plane of the figure, and the surface described has for its area the circumference of the figure multiplied by the circumference traced by its centre of figure. Ex. To find the surface of an anchor ring. The centre of figure is the centre of the generating circle, and the circumference traced is 2irR. .-. area of surface = 2irr 2-TrR = ki^Rr. 176. The two theorems which go under the name of Guldin's theorems, but which were discovered by Pappus, express relations of the highest importance in mathe- matics both pure and applied. They enable us to find the centroid of a generating figure when the volume of the generated figure is known, or the centre of figure of a generating figure when the area of the surface of the generated figure is known, and vice versa. Thus knowing the volume of a sphere, we can readily find the centroid of a semicircle, and knowing the sur- face of a sphere enables us to find the centre of figure of a semicircular arc. EXERCISES M. 1. The circle describing an anchor ring is divided by a diameter parallel to the axis. Show that the difference between the sur- faces described by the outer and the inner part is eight times the area of the generating circle. 164 SOLID OR SPATIAL GEOMETRY. 2. The convex surface of a cone is irrs ; and the entire surface is irr (r + s) ; where s is the slant height, and r is the radius of the base. 3. The entire surface of a conical frustum is 4. The areas of the surfaces of the regular polyhedra are as follows : Tetrahedron, e 2 ^/3 ; Cube, 6e 2 ; Octahedron, e 2 2^/3; Dodeca. hedron, e 2 15 v /(l + f ^5) ; Icosahedron, e 2 5^/3. 5. The distance between the centre of a circle and the centre of figure of any arc of the circle is > where I is the length of I the arc, and c and r denote as usual. 6. The area of the surface of a circular spindle is -{2cc' 2 -(c 2 -" bisects the segment A"B". In this projection we notice that no part of the pro- jected segment lies between A" and B" in the finite, but that the projection of the line-segment AB extends from A" upwards to C" at infinity, and thence returns, from below, to B". We have thus reversed the segments of the original line, so that the finite part ACB extends through infin- PERSPECTIVE PROJECTION. 173 ity, and the infinite part BDA becomes finite ; and thus D" represents the external point of bisection of AB, that is, the point at infinity, projected into the finite. We have thus a graphic illustration of the theorem that the point at infinity on any line-segment is the external point of bisection of the segment. Cor. 1. A circle may be so projected that any point within it may become the centre of the curve of pro- jection. Let G be the given point in the circle, and let ACB be a diameter, and ECF a chord perpendicular to this diameter. On any plane parallel to ECF project the segment ACB so that C" may be the centre of A'B', by 1. In this case no part of the circle goes to infinity, and the projection is a closed curve. Cor. 2. A circle may be so projected that any point without it may become the centre of the figure of pro- jection. Apply the principle involved in 2. In this case the circle becomes two curves which extend to infinity in opposite directions, and which do not meet one another. The diameter AB projects into a common axis to the two curves, and the projection of the given point bisects the part of the common axis intercepted between the curves. EXERCISES N. 1. Show that a tetragram may be projected into a rectangle. 2. A given line may be projected to infinity, and two given angles be projected into required angles. 174 SOLID OR SPATIAL GEOMETRY. 3. ABC is a triangle, and DE is parallel to AC, D being on 5(7, and E on AB. CE and AD intersect in 0. Then BO is a median ; and if BO meets DE in P and AC in Q, BPOQ is a harmonic range. 4. ABC is a triangle, and AD, BE and CF are parallel, Z> being on BC, F on AB, and on ^4(7. Show that AE : EC = AF ED : BF CD. 5. A chord of a circle is projected to infinity. Show that the pole of the chord becomes the point of intersection of tangents which touch the projection at infinity. SECTION 2. PLANE SECTIONS. 188. The definition of a plane section, and some general facts with regard to it, are given in an earlier portion of this work (see Arts. 19 et seq.). Evidently the plane section of any polyhedron is a polygon, and the plane section of a sphere is a circle. Such sections offer no distinctive features other than what belong in general to polygons and circles. But when we make plane sections of the cone or cylinder, we are introduced to curved figures which are not circles, and with which we have not hitherto become acquainted. These we propose to consider. PLANE SECTIONS OF THE CIRCULAR CONE. 189. The spatial projection of a circle, from a centre of projection for which the circle is a cone-circle, is a cir- cular cone. The section of this cone by a plane, variable in direction, is a variable curve, which, in passing through several distinctive phases in its variation, constitutes a class of plane curves which are known as conic sections, or simply conies. Hence a circle may be projected into any conic ; and conversely, any conic can be projected into a circle. And thus any conic may be projected into any other conic. 175 176 SOLID OR SPATIAL GEOMETRY. CLASSIFICATION OF CONICS. 190. Let be the centre of the circular cone AKB, L being a generating line, and let V denote a plane of section passing through any point Q. Then, 1. When V is normal to the axis of the cone, the sec- tion is a circle, DCQ, or C. Now, let a tangent to the circle, C, at Q be drawn in the plane V, and let the plane revolve about this tan- gent line as an axis. Then, 2. When V makes with the axis of the cone an angle less than a right angle, and greater than the semivertical angle of the cone, the section is an Ellipse, E. In this case the section- plane, V, cuts completely through one nappe of the cone, and does not meet the other nappe. Hence the ellipse consists of a single closed curve, as represented by the figure E. 3. When V makes with the axis of the cone an angle equal to the semivertical angle of the cone, the section is a Parabola, P. In this case V is parallel to a single generating line, and cuts only one nappe of the cone, but does not cut through it, and thus the curve extends indefinitely out- wards in one direction. CLASSIFICATION OF CONICS. 177 The parabola is consequently a single curve, but not a closed curve. 4. When V makes with the axis an angle less than the semivertical angle of the cone, the section is a Hyperbola. In this case V is parallel to two generating lines which make a finite angle with one another, and cuts into both nappes of the cone, but does not cut through either. Therefore, the hyperbola, H, consists of two curves, or rather two branches extending infinitely outwards in opposite directions, and separated from one another by a finite interspace, QQ'. 191. Degraded forms. All the conies may take what are called degraded forms ; that is, forms which they assume as limiting forms, under the sequence of varia- tion, but which are not visible curves. 1. Suppose that while the different directions of the section plane, which give the several conies, remain the same, the section plane moves up to 0. Then, (a) The circle and ellipse reduce to a point called a point-circle and a point-ellipse respectively. Of course there is no final distinction between a point-circle and a point-ellipse, but their names indicate their origin. (6) The plane of the parabola becomes a tangent plane to the cone, and touches the cone along two coinci- dent lines ; and thus the parabola degrades into two coincident lines. (c) The plane of the hyperbola gives in section two generating lines which make a finite angle with one 178 SOLID OR SPATIAL GEOMETKY. another, and the hyperbola thus degrades into a pair of intersecting lines. Hence a pair of intersecting lines is frequently called a rectilinear hyperbola. Let V be the plane which gives the hyperbola H, H (Fig. 190), and let V be parallel to Fand pass through 0. V gives in section the rectilinear hyperbola which cor- responds to H, H; and if we draw OT to the middle point of QQ', and by parallel projection in the direction TO, we project the hyperbola H, H upon the plane F 7 , we have, on that plane, a hyperbola and its correspond- ing rectilinear hyperbola. The two lines which form the latter are then called the asymptotes of the former. 2. If G remains fixed while Q moves up to 0, the ellipse becomes a double line-segment, and is called a line-ellipse. 192. From the generation of the various conies, as now explained, we deduce 1. That the circle is a special form of the ellipse, and that properties of the circle are special cases of more general properties belonging to the ellipse. And as only one direction of the plane of section, relatively to the axis of the cone, can give the circle, the circle has only one form, or all circles are similar to one another. 2. That the parabola stands intermediate between the ellipse and the hyperbola, and is the form through which one of these curves passes into the other. Also, since only one direction of the plane of section relatively to the axis of the cone can give the parabola, the curve has only one form, and all parabolas are similar to one another. COMMON PROPERTIES OF CONICS. 179 3. That both the ellipse and the hyperbola are varia- ble in form, the ellipse varying from the circle at the one limit to the parabola at the other, and the hyperbola varying from the parabola at the one limit to the form of two coincident lines at the other. 4. That all the conies have many properties in common. COMMON PROPERTIES OF CONICS. 193. As a line can meet a circular cone but twice (72), so a line can meet a conic section in two and only two points. When these two points become coincident, the line becomes a tangent line, and the point of contact is a double point. The conies constitute a distinct class of curves. Being the simplest curves that it is possible to have, they are called curves of the first order. All curves not conies belong to a higher order, and cannot be obtained as sections of a circular cone by a plane, nor as sections of any cone, one of whose sections is a conic. Curves are classified according to the number of times they may be met by a line under the most favourable cir- cumstances, and all curves other than conies can be met by a line in more than two points, either real or imag- inary. Cor. A tangent to a conic lies completely without the conic, except at the point of contact. 194. Z is a sphere, and PFG is a tangent cone touching the sphere in the small circle BEG. 180 SOLID OR SPATIAL GEOMETRY. Denote the plane of BEC by U, and let V be a plane of section of the cone, touching the sphere in S, and meeting U in the line DH. Let TFdenote the plane containing the axis of the cone and being normal to DH. Then W is per- pendicular to both U and F AQP is the conic formed by F, and P is any point on this conic. BD is a centre line of the circle BEC, and is the common line of U and W, and SA is the common line of W and F This latter line, SA, is the principal axis of the conic, or simply the axis, and it is perpendicular to DH. The conic is evidently symmetrical about the axis SA, or the principal axis. Draw PH J_ to ZXffand PM parallel to DH. Then, P, M, D, H are complanar, all lying in V, and MHis a rectangle, and therefore PH= MD. Join PS and PO, and^ pass a plane through P, par- allel to U, giving the circular section GPF, and meeting W along the line GF. Then BEC and GPF being cone-circles with as vertex, OP =OF, and OE=OC; .-. CF=EP=SP. (Art. 86. 1.) COMMON PROPERTIES OF CONICS. 181 Similarly, SA = CA. But, from similar triangles CAD and FAM, CF:MD=CA:AD, or SP:PH = SA:AD. But the ratio SA : AD is independent of the position of P on the conic, and remains constant while P moves along the conic. Therefore, denoting this constant by e, we have SP = e = a constant. PH Hence a conic, considered as the locus of a variable point, may be denned as follows : A conic is the locus of a point which, being confined to one plane, so moves that its distance from a fixed point ($) is in a constant ratio (e) to its distance from a fixed line (DH), all being complanar. This definition is usually adopted in analytical conies, and it is sufficiently general to include every conic. Def. The point S is the focus, and the line DH is the directrix. A is the vertex of the conic, and the constant e is the eccentricity. PM is an ordinate to the principal axis, and PS is the focal distance of the point P. 195. Let the accompanying figure represent the sec- tion by the plane W. A second sphere, Z', may be drawn, to touch the cone and the plane V at S'. Then BC and B'C' representing the sections of the circles of contact, the planes of these 182 SOLID OR SPATIAL GEOMETRY. circles are parallel, and they cut the plane V in parallel lines represented in section at D and D'. Hence the conic has two foci, 8 and S', two vertices, A and A', and two parallel directrices represented in section by D and D'. The figure as here drawn applies particularly to the ellipse, but it may serve as a type for all the other conies. T IT In the parabola one vertex, focus, and directrix are at infinity. In the hyperbola the second focus is given by the point of contact of the sphere Z". Thus in the ellipse the curve lies between the direc- trices, while in the hyperbola the directrices lie between the vertices of the two branches of the curve. COMMON PROPERTIES OF CONICS. 183 196. In the figure of Art. 195, considered as a plane figure, Z is the incircle, and Z* is an excircle of the triangle AOA'. Therefore, A8=A'S' (P. Art. 135. Ex. 1) ; so that both foci are similarly situated with respect to the corresponding vertices. Also, drawing A'T parallel to CA, A'S' = A'B' = A'T=AS = AC. Therefore, the triangles D'A'T and DAG are con- gruent, and A'D' = AD ; or the directrices are similarly situated with respect to the corresponding foci and vertices. Hence it follows that the same curve may be drawn from either vertex with the corresponding focus and directrix, and therefore that a line drawn at right angles to the principal axis, and bisecting the distance between the foci, is an axis of symmetry of the curve. And as the principal axis is also an axis of symmetry, a conic has, in general, two axes of symmetry bisecting one another at right angles. These are called the axes of the conic. In the parabola one axis is at infinity. 197. The character of the conic is determined by the value of its eccentricity. In the figure to Art. 194, e = SA:AD=CA:AD. 1. Let AD be infinite. Then e = 0, and the plane V is parallel to the plane V, and the conic is a circle. Therefore, the eccentricity of a circle is zero. In this case both spheres touch Fat the centre of the 184 SOLID OR SPATIAL GEOMETRY. circular section, and the foci of the circle become coinci- dent at the centre of the circle. 2. Let AD=CA. Then e = 1 ; and the triangle CAD being isosceles, Z ADC= Z ACD = Z 5(70 = Z OBC. Therefore, AD is parallel to OB, and the conic is a parabola. Hence the eccentricity of the parabola is unity. In this case a second sphere cannot be drawn in any finite position so as to satisfy the conditions for a focus. 3. When AD is < GO and > CA, the value of e lies between zero and unity. The Z.ACD is> the /.ADC, and the plane V, being inclined to the axis of the cone at an angle greater than the semivertical angle of the cone, cuts through one nappe and gives the ellipse. 4. When AD is < AC and>0, e lies between unity and infinity ; the angle ADC is greater than ACD, and the plane V, being inclined to the axis of the cone at an angle less than the semivertical angle of the cone, cuts into both nappes and gives the hyperbola. 198. When the centre of a circular cone goes to infinity, the cone becomes a cylinder, and the only possi- ble plane sections are the circle, the ellipse, and two parallel lines representing the parabola and hyperbola. The ellipse, including the circle as a particular case, is the most- important of the conic sections, and we propose to develop some of its THE ELLIPSE. 185 more prominent properties, through its relationship to the circular cylinder. Let ADB be one-half of the right section of a circular cylinder, and let AEB be the corresponding half of an oblique section of the same cylinder. Then ADB is a semicircle with AB as diameter, and AEB is one-half of an ellipse. From (7, the centre of both the circle and the ellipse, draw CD and CE perpendicular to AB, the former in the plane of the circle, and the latter in that of the ellipse. Then ED is a segment of a generating line of the cylinder. On the ellipse take any point P, and draw PQ parallel to ED, and QG parallel to DC. The&ECD and PGQ are similar, and CD=CQ = the radius of the cylinder, and CD is > GQ. Therefore, ED is > PQ. But CE 2 =CD 2 +DE 2 =CQ 2 +DE 2 , and this is >CQ-+ Whence CE is > CP. Or, CE is the longest segment from C to the ellipse, and is the semiaxis-major of the ellipse. In like manner it may be shown that CP is > CA ; or that CA is the shortest segment from the centre to the ellipse, and is the semiaxis-minor of the ellipse. These axes are perpendicular to one another. 199. On account of the similar triangles, PGQ and EDC (Fig. of 198), PG:GQ=EC:CD. But EC : CD is constant for a constant direction of the plane of oblique section. .-. PG : GQ = a constant. 186 SOLID Oil SPATIAL GEOMETRY. Hence the following construction for an ellipse ; is a quadrant of a circle with cen- tre (7, and GQ is a chord _L to AC. Take GP, a constant multiple of GQ. The locus of P is an ellipse whose semiaxis-minor is AC. Again, draw PH _L to CE and let CQ meet HP in R. Then, from similar triangles and CGQ, RP^PQ. GC GQ' RH PG . = = constant. AQD And Pjff is a constant part of RH ; hence the the- orem If on a chord of a circle, perpendicular to a fixed diameter, a point be taken so as to divide the chord in a constant ratio, the locus of the point is an ellipse, and the fixed diameter is the major or the minor axis of the ellipse, according as the point divides the chord inter- nally or externally. Def. The circles which have the major and minor axes of the ellipse as their diameters are the major and minor auxiliary circles to the ellipse. 200. In the figure of Art. 198, let AEB be a semicircle with AB as diameter and C as centre, and let CE be perpendicular to AB. Project orthogonally on any plane, F, passing through AB. Then GQ = GP cos PGQ, and as PGQ is a con- THE ELLIPSE. 187 stant angle, G Q is in a constant ratio to GP and is less than GP. Hence Q lies on an ellipse of which AB is the major axis. Therefore, the orthogonal projection of the circle on any plane not parallel to its own is an ellipse. This result also follows directly from the statements of Art. 190, since, in general, the perspective projection of a circle on any plane which cuts through one nappe is an ellipse. But in the case of perspective projection the same diameter of the circle does not project into an axis of the ellipse. 201. Conjugate diameters. Let ADBE be a right section of a circular cylinder, by the plane U, and let adbe be an oblique section by the plane V. Also let AB and DE be perpendicular diam- eters of the circle, and FG be a chord parallel to AB. Now let the whole figure on ?7be projected ant-orthogonally (Art. 177) on V. Then c is the centre of the ellipse and ab and de, the pro- jections of AB and DE, are a pair of conjugate diameters of the ellipse, And FG is bisected at H. Whence, from the nature of parallel projection, fg is bisected at 7i, and is parallel to ab. Therefore, de bisects all chords parallel to ab ; and in like manner ab bisects all chords parallel to de. 188 SOLID OE SPATIAL GEOMETRY. Hence, when two diameters are conjugate, each bisects all chords parallel to the other. Cor. 1. Conjugate diameters are such that each is parallel to the tangents at the extremities of the other. Cor. 2. Conjugate diameters in the ellipse are the parallel projections of orthogonal diameters in that circle whose projection gives the ellipse. Manifestly an indefinite number of pairs of conjugate diameters may be found, and each diameter has one, and only one, conjugate. Cor. 3. When the plane F is parallel to AB or DE, ab and de are perpendicular to one another, and become the principal diameters of the ellipse. 202. Since Aa, Bb, etc., are generating lines of the cylinder, they are lines of contact of tangent planes (Art. 73). Let four tangent planes to the cylinder touch it at Aa, Bb, Dd, and Ee. The section of these by ?7is a square, whose sides are parallel to AB and DE; and this section remains constant in area however AB and DE may be drawn, provided they are diameters which intersect orthogonally. The section of the tangent planes by V is a parallelo- gram whose sides are parallel to ab and de. Now (Art. 116), the area of the square is equal to that of the parallelogram multiplied by the cosine of the angle between U and V. Therefore, the area of the parallelogram is unaffected by any change in the position of F", which does not change the inclination of that plane to the axis of the THE ELLIPSE. 189 cylinder ; that is, which does not change the form of the ellipse. Hence, in any given ellipse the parallelogram formed by tangents at the extremities of conjugate diameters, or the parallelogram on a pair of conjugate diameters taken in both length and direction, is constant. Cor. If a and b denote the semiaxes, and a 1 and b' denote a pair of conjugate semidiameters, and be the angle between them, a'b' sinO=ab. 203. Let APQB be a semicircle on a plane, U, and let CP and CQ be radii at right angles to one another. Let the whole be projected orthogonally upon a plane, V, passing through AB, and in- clined to U at a fixed angle, 0. And let CP 1 , CQ' be the pro- A" jections of CP and CQ respec- tively. Then (Art. 201. Cor. 2), AP'Q'B is a semiellipse, and CP 1 and CQ' are a pair of semi-conjugate diameters. Draw PE and QF _L to AB, and join PE and Q'F. Then P'E and Q'F are _L to AB ; and since PCQ is a right angle, CF= EP, and EC = QF. Also, PP = EP sin 0, and QQ' = FQ sin = CE sin 0. And CP 12 + CQ' 2 = CE 2 + EP 2 - PP' 2 + CF Z + FQ? - QQ* = 2 CP 2 - (EP 2 + FQ 2 ) sin 2 = <7P 2 (2-siu 2 0) = a constant. 190 SOLID OR SPATIAL GEOMETRY. Therefore, the sum of the squares on a pair of con- jugate diameters of an ellipse is constant. Cor. Denoting the parts as in Art. 202. Cor., a' 2 + b' 2 = a- + 6 2 . The results of Arts. 202 and 203 are known as the theorems of Apollonius. 204. Let F be a plane, cutting a circular cone so as to give an ellipse, and let P be any point on the ellipse. P is a point on the cone. Let Z be the sphere which touches V at the focus S, and Z' be the sphere which touches Fat the focus $'; also let K denote the circle of contact of Z with the cone, and K' denote the circle of contact of Z'. Draw through P a generating line of the cone. This line cuts Kin k, and K' in k' } and at these points touches the spheres Z and Z'. But K and K' being cone circles to the vertex 0, kJc' is constant for all positions of the generating line. But PA; = PS and PA;' = PS 1 , being tangents to the spheres. ^ sp+ ps , = a constant . Therefore, in any given ellipse, the sum of the dis- tances of any point on the curve from the two foci is constant. 205. The result of the preceding article furnishes a convenient practical method of drawing an ellipse. Over two pins placed at S and S' put a loop of inex- tensible thread, and keep it stretched by a pencil at P. The locus of P is an ellipse of which S and S' are the foci. THE ELLIPSE. 191 For the whole length of thread being constant, and the part SS' being constant, it follows that SP + PS' is constant. Cor. By considering the phase when P comes to A or to B, we readily see that SB+BS'=AB, A s c and hence that the whole length of thread is 2 AS', or 2BS. 206. Let PT be a tangent to the ellipse at P, and let Q be any point, other than P, on this tangent. Then, since the tangent has only one point in common with the curve (Art. 193. Cor.), QS' cuts the curve in some point B. Then, SQ + QS 1 is > SR + ES' ; or SQ + QS 1 is>SP + PS 1 . Hence SP+PS' is the shortest route from S to S' by way of the line PT, and hence SP and S'P are equally inclined to PT. Therefore, in any ellipse, the lines from the foci to the point of contact of any tangent are equally inclined to the tangent. 207. GQH is a circular cone ; APQB is an elliptic section, and CPD and EQF are circular sections cutting the elliptic section in the lines PM and QN. CD and EF are diameters of the circles, and AB is the axis of the ellipse, all lying in the plane W (Art. 194). PM and QN are perpendicular to AB and to the lines CD and EF respectively. 192 SOLID OR SPATIAL GEOMETRY. Therefore CM MD = PM 2 , and EN- NF= QN 2 . But A AMD ~ A ANF, and A ENB ~ A CMB. .-. AM:MD = AN:NF; and MB : CM= NB : EN. Therefore, by multiplication, AM . MB : MD - CM= AN- NB-.NF- EN; or AM MB : PM 2 = AN- NB : QN 2 . In a similar manner it is proved that a like rela- tion holds true for the hyperbola. Therefore, In the ellipse and the hyperbola, if perpendicu- lars be drawn from points on the figure to the prin- cipal axis, the squares on these perpendiculars are proportional to the rectangles on the parts into which each perpen- ' dicular divides the prin- cipal axis. Again, let A'P'N' be a parabolic section. Then A'N' is parallel to OG, and EM' = GN'. But from the similar A A'M'F and A'N'H, A'M 1 :M'F= A'N' : N'H. And 1 : EM 1 = 1 : GN'. .-. A'M' : EM 1 - M'F= A'N' : GN' N'H', or A'M' : P'M 12 = A'N' : Q'N 12 . THE PARABOLA. 193 Hence, in the parabola, the squares on perpendiculars from points on the figure to the axis are proportional to the parts of the axis intercepted between the vertex of the parabola and the foot of each perpendicular. Def. The perpendiculars of this article are called ordinates to the principal axis, and the segments into which they divide the axis are called abscisses. So that in the ellipse and hyperbola the square on any ordinate is in a constant ratio to the rectangle on its abscissae. In the parabola, however, one abscissa is infinite, and we have only the finite one to consider. Then, the square on an ordinate is in a constant ratio to its abscissa. EXERCISES O. 1. Show that the area of an ellipse is vab. 2. A right cylinder has its base an ellipse with axes a and b. Show how to cut it by a plane so that the section may be a circle. 3. If P be a point on a hyperbola, of which S and F are foci, then SP - PF = constant. 208. As the parabola is a limiting form of the ellipse (Art. 192. 2), the fundamental properties of the parabola may be obtained from those of the ellipse by supposing that one focus of the ellipse goes to infinity, while the other focus remains at a finite distance from the vertex. We shall, however, obtain these relations by means of the perspective projection of the circle. In the accompanying diagram, the right-hand figure is the projection of the left-hand one, and for the sake of convenience in comparison, a point and its projection are denoted by the same letter in both figures. 194 SOLID OR SPATIAL GEOMETRY. In (I), BE' is a tangent to the circle APB, touching it at B, and BAK is a centre line. B'PQ is any secant line from B', and PT and Q T are tangents meeting at T. B' V is a tangent from B' touching the circle at V, and TK is perpendicular to BA. The circle is projected into a circular cone, and the plane which gives the parabola in section touches the circle at A and is parallel to the tangent BB ! , and BB' is projected to infinity. (i) (ID In the projection (II), A becomes the vertex A of the parabola, and BB' goes to infinity (Art. 181. Cor. 2), and hence the lines through B in (I) become a system of parallels in the projection (II), and thus in (II) KAG is the axis of the parabola, and TVHcc is parallel to the axis. So, also, the lines through B' in (I) project into a system of parallels in (II); that is, QPB' and VB' COMMON PROPERTIES OF CONICS. 195 become the parallels QPao and Foo, or the tangent at V is parallel to PQ. Now in (I) B' is pole to VB, and therefore QHPB' is a harmonic range (P. Art. 311. 2). Therefore, also, in (II), QHPao is a harmonic range, and H is the middle point of QP. Hence 1. In the parabola, any line parallel to the axis is a diameter, and bisects all chords parallel to the tangent ( Foo ) at its vertex. Thus PQ is bisected by TVH. The direction of the diameter conjugate to VH co is given by PQ, but its position is at infinity. Again, in (I), T is pole to PQ, and TVHB is a har- monic range (P. Art. 311. 2). Therefore, in (II), TVH so is harmonic, and F bisects TH, Hence: 2. In the parabola, tangents at the end-points of any chord (PQ) meet upon the diameter to that chord (TH); and the part of the diameter intercepted between the chord and the point of meeting of the tangents is bisected by the curve . TH is bisected at F Again, in (I), TK\s the polar of G (P. Art. 267), and hence KAGB is a harmonic range. Therefore, in (II), KAGcc is harmonic, and A bisects KG. Hence : 3. In the parabola, if two tangents be drawn from any point to the curve, and a perpendicular be drawn from the same point to the axis, the part of the axis intercepted between the foot of the perpendicular and the chord of contact (PQ) is bisected by the vertex of the curve. KG is bisected at A, 196 SOLID OR SPATIAL GEOMETRY. 209. Since the circle can be projected into any conic, and auharmoiiic properties are projected without change, all the properties of the circle which depend upon anharmonic or harmonic relations are equally true for all the conies. Thus the theorems in plane geometry, given in Arts. 311, 312, 313, 314, and many of the following ones, are true when we read conic for circle. To enter any further into this subject is beyond the scope of this work. The student who desires to pursue this most interesting subject will find it fully developed in Salmon's 'Conies,' or Cremona's 'Projective Geometry,' or in Poncelet's great work, the 'Traite des proprietes projective des figures.' EXERCISES P. 1. In the parabola, prove that the tangent at any point on the curve makes equal angles with the axis and the line joining that point to the focus. 2. P being a point on a parabola, if PM be drawn perpendicu- lar to the axis and PN perpendicular to the tangent at P, the part MN intercepted on the axis is constant. 3. The tangent at the vertex of a parabola bisects the part of any other tangent lying between the point of contact and the axis. 4. The tangent at the vertex of a parabola, and the perpendicu- lar from the focus upon any other tangent, meet the latter tangent at the same point. 5. In the projection of Art. 208, if TK (1) projects into the directrix, then Gr will project into the focus. SECTION 3. SPHERIC GEOMETRY. 210. When a spatial figure is cut by a sphere, the elements common to the. spatial figure and the sphere form a figure which lies on the sphere in the same manner as a plane figure lies upon its plane. Such a figure is a spheric figure, as being confined to a spherical surface, and the geometry of such figures is called spheric geometry or spherical surface geometry, On account of the uniform curvature of a sphere, a spheric figure may be moved about upon the spherical surface upon which it lies, without undergoing any neces- sary change in the relations of its parts, just as a plane figure may be moved about over the plane surface upon which it lies. There are many analogies between spheric geometry and plane geometry, and many of the theorems and of the methods employed are more or less alike. But there are also fundamental differences. The prominent analogies will be exhibited in the sequel. 211. It must be understood in the beginning that spheric geometry does not deal with a comparison of the properties or relations of figures drawn upon differ- ent spheres ; its purpose is not this, but to investigate 197 198 SOLID OR SPATIAL GEOMETRY. the properties which belong to a figure in consequence of its lying upon a sphere. Hence all spheric figures are supposed to lie on one and the same sphere, just as all the figures in plane geometry are supposed to lie on one and the same plane. The radius of the particular sphere is altogether arbi- trary, and, except in the case of metrical theorems or problems, the radius may be left out of the consideration. The centre of the sphere will be referred to as the centre. 212. Every section of a sphere by a plane is a circle, and when the plane contains the centre of the sphere, the section is the largest circle in this way obtainable, and is called a great circle of the sphere. It will appear hereafter that when a plane figure involving the line has an analogue in spheric geometry, the line is represented by the great circle. And as there can be no straight line in connection with any spheric figure, we shall, for the sake of the analogy, commonly speak of a great circle as a spheric line. Then all other circles are spheric circles. Any limited part of a spheric line is a spheric arc, and parts of other circles are circular arcs. 213. The spheric line, unlike a planar line, returns into itself without passing to infinity. Evidently the spheric line divides the whole spherical surface into two congruent parts, just as the planar line may be said to divide the whole plane into two congruent parts. The parts of the plane, however, extend to infin- ity, while those of the spherical surface do not. SPHERIC GEOMETKY. 199 We have here a fundamental distinction between plane and spheric geometry, namely, that spheric geometry has no infinity. 214. As any plane which gives in section a spheric line must pass through the centre, any two points on the sphere, not in line with the centre, determine, with the centre, one plane, and therefore one spheric line. Thus, like a planar line, a spheric line is determined by any two points, provided they are not collinear with the centre. And through any two points not collinear with the centre, one, and only one, spheric line can pass. When two points are in line with the centre, the three points determine only one line, a diameter of the sphere, and through this any number of planes can pass, giving in section the same number of spheric lines. Now in plane geometry, any two points determine one line, unless the points be at infinity. For in this latter case, since all parallels of a system meet at infinity, two points at infinity, in opposite directions, determine a system of parallels. We see, then, that for two points to be collinear with the centre, in spheric geometry, is analogous to two points at infinity in opposite directions, in plane geometry. Thus spheric lines passing through a pair of opposite points are analogous to parallel lines in plane geometry ; and hence there is no theory of parallels in spheric geometry as in plane geometry. Cor. Any number of points on a sphere, no two of which are collinear with the centre, and no three of 200 SOLID OK SPATIAL GEOMETRY. which are complanar with the centre, determine as many spheric lines as there are groups of the points taken two and two. The corresponding theorem in plane geometry is, that any number of points, no two of which are at infinity, and no three of which are in line, determine as many lines as there are groups of the points taken two and two. 215. The normal, through the centre, to the plane of a great circle, meets the sphere in two opposite points which are end-points of a diam- eter. These points are poles of the great circle ; and in relation to the poles the circle is called the equator. Thus every spheric line has two poles, and any point on the sphere, considered as a pole, has an opposite pole, and an equator. Thus AB, in the figure, is normal to the plane EGFH, and passes through the centre 0, and meets the sphere at A and at B. Then A and B are poles of the spheric line EGFH, and reciprocally EGFH is the equator to the poles A and B. Evidently the angle AOE, subtended at the centre between a pole and any point on its equator, is a right angle, and the spheric arc AE is one-fourth of a whole circumference. If a quadrant of a great circle has one extremity fixed at A while the other moves over the surface of the SECTION OF TWO PLANES. 201 sphere, the moving extremity will describe the great circle or spheric line EGFH, which is the equator to the point A as pole. Hence the quadrant AE is the spheric radius of the great circle described ; or the spheric radius of a great circle is a quadrant. Cor. If any point P be taken in the quadrant when the quadrant moves over the surface having A fixed, P will describe a circle PRQupon the sphere. As the arc AP, and therefore the chord AP, is constant, PRQ is a cone-circle to A as vertex, and its plane is normal to AB, and therefore parallel to the plane of EGFH. And as A and B are points on the sphere from which the spheric circle PRQ can be described by means of constant arcs, AP or BP, these points are poles to the circle PRQ, and the arc AP, as also BP, is its spheric radius. Thus every circle on a sphere has two spheric radii which are supplementary to one another, and the poles of any great circle are poles to all spheric circles whose planes are parallel to that of the great circle. SECTION OF Two PLANES. 216. The sphere which has its centre on the common line of two intersecting planes has, in section by these planes, two spheric lines which intersect in opposite points, or at the end-points of a diameter. Thus the two planes which have in common the line AB (Fig. of 215) give by their intersection with the 202 SOLID OR SPATIAL GEOMETRY. sphere the two spheric lines AEBF and AGBH, inter- secting in A and B, and mutually bisecting one another. The angle between these spheric lines is equal in measure to the dihedral angle between the planes which give rise to the lines. But if EGFH be the equator to A, EO and GO are each perpendicular to AB, and there- fore EOG measures the dihedral angle between the planes, and hence also the angle between the spheric lines. But the angle EOG = (arc EG) -~ EO; and since EO is supposed to be constant, our investigations being con- fined to one and the same sphere, Therefore, the angle between two spheric lines is proportional to the arc which they intercept upon the equator to their points of intersection as poles. Cor. 1. If, through A, tangents to the spheric lines be drawn, AT to AEBF and in its plane, and AS to AGBH and in its plane, the angle TAS is equal to EOG, and is the angle between the spheric lines. Therefore, the angle between two tangents drawn to two spheric lines at their point of intersection is the angle between the spheric lines. Cor. 2. When the angle EOG is a right angle, OG is normal to the plane of AEBF, and G is a pole to the circle AEBF. Therefore, two spheric lines are perpendicular to one another when one of them passes through a pole of the other ; and in this case each passes through both poles of the other. 217. The spheric figure AEBGA, formed by two spheric lines between their points of intersection, is a THREE PLANES. 203 lune. The points A and B are vertices of the lune, and the angle between the spheric lines forming its sides is the angle of the luue. Evidently every lune is accompanied by an opposite congruent lune, as AEBGA and AFBHA; and any two spheric lines divide the whole spheric surface into four lunes which are congruent in opposite pairs. We have the analogous case in plane geometry, where any two intersecting lines divide the whole plane into four sections, which, although extending to infinity, may properly be said to be congruent in opposite pairs. It will be seen from this and other cases that the analogy between plane and spheric geometry is descrip- tive rather than metrical in kind. THREE PLANES SECTION OF THREE-FACED CORNER OR TRIHEDRAL ANGLE SPHERIC TRI- ANGLE. 218. Just as a plane section of any corner is a plane polygon with a side corresponding to and given by each face of the corner, so the section of a corner by a sphere with its centre at the vertex of the corner is a spheric polygon, whose sides are parts of spheric lines, and the number of whose sides is the same as that of the faces forming the corner. In spheric as in plane geometry the most important polygon is the triangle. In plane geometry three given lines can form but one triangle, since they determine at most but three points. But, as every spheric line meets every other spheric line 204 SOLID OR SPATIAL GEOMETRY. in two points, three spheric lines, which are not concur- rent, determine six points, and these may combine in threes to form eight spheric triangles. Therefore, any three non-concurrent spheric lines divide the surface of the sphere into eight triangles. 219. Let AHA'G, BIB'J, and CFC'E be three non- concurrent spheric lines. They meet in the six points A, B, C, A', B', C', of which A is opposite A', B opposite B', and C opposite C'. The eight determined triangles are ABC, ABC', AB'C, A'BC, A'B'C', A'B'C, A'BC', AB'C'. Since A is opposite A, etc., the arc AB = the arc A'B'. Similarly, arc BC = arc B'C' and arc CA = arc C'A'. Also, as ACC'A' and ABA'B' determine two planes, the angle at A is equal to the angle at A'; and so also the angle at H f B is equal to the angle at B', and the angle at C to the angle at C'. And thus the opposite triangles ABC and A'B'C' have all their parts in the one respec- tively equal to the corre- sponding parts in the other. But the triangles are not superposable. For taking the centre as a point of refer- ence, ABC and A'B'C' are in opposite orders of rotation. SPHERIC TRIANGLE. 205 In the case of plane triangles we could invert one of them, or turn it over in the plane, and then superimpose them ; but this operation is clearly impossible in a spheric figure. Spheric triangles related in this manner are symmetrical, or conjugate, to one another, and they are evidently produced by the sections of two symmetri- cal three-faced corners having a common vertex at the centre (Art. 39). The eight triangles are symmetrical or conjugate in opposite pairs as follows : ABC and A'B'C', ABC' and A'B'C, AB'C and A'BC 1 , ABC and AB'C'. 220. Let ABC (Fig. of 219) be a spheric triangle formed by section of the three-faced corner ABC. The angles at A, B, and C, whose measures are respec- tively those of the three dihedral angles of the corner, are called the angles of the triangle, and are usually denoted by A, B, (7; and the arcs BC, CA, and AB are called the sides of the triangle, and are denoted by a, b, c. Here two views confront us. If linear units are to be introduced, and arcs are to be considered with respect to length, the length of the radius of the sphere is involved, and our investigations are con- fined to some one sphere whose radius is known or deter- minable with reference to the unit of measure employed. On the other hand, if our operations and results are to have no reference to the length of the radius, we must take for the side of the spheric triangle, not the arc itself, but its ratio to the radius. This ratio is an angle, a face angle of the corner which in section gives the spheric triangle. 206 SOLID OR SPATIAL GEOMETRY. Spheric geometry then ceases to have any distinct relation to the sphere, except in name, involves no rela- tion of length, and becomes a geometry of direction only. In what follows we shall not confine ourselves exclu- sively to either view, but shall adopt that which serves our purpose at the time. In the majority of applications, however, the second one is the only view that can be adopted. Thus in applying the results of spheric geometry to the visible surface of the heavens, anything like a linear unit is out of the question. According to the second view, a spheric triangle con- sists of six parts, all angles. Three alternate parts, called the angles of the triangle, are respectively equal in measure to the dihedral angles of a three-faced corner, and the remaining three, called the sides of the triangle, are respectively equal to the face angles of the same corner. And thus all the relations which exist between the dihedral angles and face angles of a three-faced corner, exist also between the angles and sides of a spheric triangle. The adaptation to a triangle described on a given sphere is easily effected; for it is only necessary to express the sides in radians and then multiply the result by the radius of the sphere. 221. In any three-faced corner the sum of two face angles is greater than the third (Art. 35). Therefore, the sum of two sides of a spheric trian- gle is greater than the third side ; and the difference SPHERIC TRIANGLE. 207 between any two sides of a spheric triangle is less than the third side of the triangle. 222. Theorem. The shortest path from one point to another, on the surface of a sphere, is along the spheric line through the points. Let A and B be the points, and C be any point on the spheric join AB. With A and B as poles describe circles PCD and QCE to pass through C. Take P, any point on the circle PCD, and draw the spheric arcs AP, and PB cutting the circle QCE in Q. Then APB is a spheric triangle, and AP+PB>AB. (221) Therefore, P lies without the circle QCE, and the circles PCD and QCE touch at C. Now let ADEB be any path on the sphere from A to B. Then the path from A to D may be brought to extend from A to C by turning the circle PCD about its centre A, until D comes to C. In a similar way, the path from B to E may be brought to extend from B to C. But by this change the whole path is shortened by the distance DE. That is, the path is shortened by making it pass through C, a point on the- spheric line through A and B. In like manner, each part of the path is shortened by making it pass through some arbitrary point on the spheric line AB. 208 SOLID OB SPATIAL GEOMETRY. Hence the path is shortest when every point lies on the spheric line from A to B. Cor. 1. When two spheric circles touch, the spheric line through their poles passes through their point of contact. Cor. 2. When two points on a sphere are in line with the centre, an indefinite number of equal shortest paths may be drawn from one point to the other. 223. In any three-faced corner, considering two face angles and the opposite dihedral angles, the greater face angle is opposite the greater dihedral angle ; and con- versely, the greater dihedral angle is opposite the greater face angle (Art. 40). Hence in any spheric triangle, considering two sides and the opposite angles, the greater side is opposite the greater angle; and conversely, the greater angle is oppo- site the greater side. Cor. 1. If a spheric triangle has two equal sides, it has two equal angles ; and conversely, if it has two equal angles, it has two equal sides. Cor. 2. The order of magnitude of A, B, C, the angles of a spheric triangle, is the same as that of a, b, c, the sides of the triangle. Hence spheric triangles, like plane ones, are equi- lateral, and isosceles, and scalene. 224. When the three face angles of a three-faced cor- ner are given, the dihedral angles also are given (Art. 41. Cor. 2); and conversely, when the dihedral angles SPHERIC TRIANGLE. 209 are given, the face angle also are given (Art. 44. Cor. 2). Therefore, 1. When the sides of a spheric triangle are given, the angles also are given, and all the parts are known. This case holds also for plane triangles. 2. When the three angles of a spheric triangle are given, the sides also are given, and all the parts are known. This does not hold for plane triangles, and this funda- mental difference between spheric and plane geometry is due to the fact that spheric geometry has no theory of similar figures, a theory which plays so important a part in plane geometry. Similarity requires an equality of tensors, and therefore involves the consideration of linear extension. But in a spheric triangle, where all the parts are angles, there is no place for linear extension, and hence no similarity exists beyond absolute equality. Similar spheric triangles might be drawn upon spheres of different radii, but the comparison of these, although belonging to spatial geometry, does not belong to spheric geometry (Art. 211). 225. In any corner the sum of the face angles is less than a circumangle (Art. 42). Hence the sum of the sides of a spheric triangle is less than a circumangle; or if we introduce the radius, is less than a circumference. Cor. 1. When two sides of a spheric triangle become straight angles, the third side vanishes and the figure becomes a lune. Cor. 2. When each side becomes a right angle, the 210 SOLID OR SPATIAL GEOMETRY. planes forming the faces of the corner become the rec- tangular co-ordinate planes of space (Art. 8. Cor.), and each angle becomes a right angle. Hence a spheric triangle which has each side a right angle has also each angle a right angle. Such a triangle is a quadrantal triangle. Cor. 3. If two sides of a spheric triangle be produced to meet, a second triangle is formed which is said to be co-lunar with the first. These two triangles have an angle and the opposite side respectively equal, while the remaining two sides in the one are supplementary to the corresponding sides in the other, and the remaining angles in the one are supplementary to the remaining angles in the other. Cor. 4. Any spheric triangle has three colunar triangles. 226. The polar triangle. When the vertices of one spheric triangle are poles to the sides of another spheric triangle, the first triangle is said to be polar to the second. And the second triangle is also polar to the first. Let A', B', C' be poles of a, b, c respectively, and let a', &', c' be the sides of the spheric tri- angle having A', B', C' as ver- tices. Since C' is pole to AB, C'O is _L to OB (Art. 215) ; and since A is pole to BC, A'O is _L to OB. Therefore, OB is normal to the plane of OA and 0(7, SPHERIC TRIANGLE. 211 and B is therefore pole to the spheric join of A'C 1 ; that is, to b'. Similarly, A is pole to a' and C to c'; and the spheric triangle ABC is polar to A'B'C'. Hence when one spheric triangle has its vertices poles to the sides of a second triangle, the vertices of the second triangle are poles to the sides of the first, and the two triangles are polar to each other. Cor. 1. A quadrantal triangle is polar to itself. Cor. 2. The points A', B', C' determine eight triangles (Art. 218), every one of which might be said to be polar to ABC. Or more generally, A, B, C determine one set of eight triangles, and A', B', C' determine a second set; and every triangle of one set has every triangle of the other set as a polar triangle. It is easily seen, however, that two triangles in either set are conjugate (Art. 219), and the remaining six are co-lunars of these. So that if we agree that the spheric triangle formed from three given spheric lines is to be considered as being the triangle (triangle or its conjugate), each of whose sides are less than a straight angle or a semi-circumference, then each spheric triangle has but one polar triangle, (a triangle or its conjugate). 227. The spheric triangles ABC and A'B'C' being polar to one another, A'O is normal to the plane of a, and B'O is normal to the plane of 6. But the angle between two planes is the supplement of the angle between normals to the planes (Art. 16. Def. 2); and the angle between the planes is the angle C, and the angle between the normals is the side c'. 212 SOLID OR SPATIAL, GEOMETRY. Or the sum of an angle of a spheric triangle and the corresponding opposite side of the polar triangle is a straight angle. Cor. 228. From the preceding article, A + B + C + a' + V + c' = Sir. But (Art. 225) a' + V + c' > and < 2 TT ; .-. ^4 + B + (7 is < Sir and > TT. That is, the sum of the angles of a spheric triangle is variable, lying between the limits of two right angles and six right angles. And hence, in every spheric tri- angle the sum of the angles exceeds two right angles. The amount by which the sum of the three angles exceeds a straight angle is called the spherical excess of the triangle. If we denote it by E, we have E=A+B+ CTT. 229. It has been shown (Art. 219) that conjugate spheric triangles, although having all their corresponding parts respectively equal, are not superposable, but corre- spond to one another after the manner of the right and the left hand. Hence in the determination of a spheric triangle from its parts, there is always the kind of ambiguity which results from not knowing whether a particular triangle or its conjugate is the one required. This ambiguity disappears in the case of an isosceles spheric triangle, for this triangle is conjugate to itself. SPHERIC TRIANGLE. 213 For the sake of simplicity, then, we shall agree in what follows that a spheric triangle is given or deter- mined when we know the triangle or its conjugate. 230. A spheric triangle is given when the three sides are given, or when the three angles are given. This follows from Art. 41. Cor. 2, and Art. 224. 231. A spheric triangle is given when two sides and the included angle are given. Let ABC, A'B'C' be two spheric triangles in which Z C" = Z C, a' = a, and b' = b ; and let the parts be dis- posed in the same order. Place C" on C and C'A' along CA. Since 6' = b, A' coincides with A. Also, since Z C" = Z (7, the side a' will lie along the side a, and as a' = a, B' will coincide with B. And as A and B determine only one spheric line, A'B> coincides with AB, and the triangles coincide in all their parts. Therefore, the triangle ABC is given when two sides and the included angle are given (see P. Art. 66). 232. A spheric triangle is given when two angles and the included side are given. Let A and B and the side c be given. Then the sides a' and b' and the included angle C' is given for the polar triangle, and therefore the polar triangle is given (Art. 231). Hence the original triangle is given. 214 SOLID OR SPATIAL GEOMETRY. 233. Let ABC be an isosceles spheric triangle with CA = CB, and hence the Z A = Z B. Draw the spheric line D'CD to the middle point, D, of the base AB. Then CD is median to the base. The triangles ACD and BCD having AC and AD respectively equal to BC and BD, and the included angles A and B equal, are conju- gate. Therefore, Z CD A = Z CM = 1, and Z ^ICD = Z BCD. Hence the median to the base of an isosceles spheric triangle is the right bisector of the base, and the bisector of the vertical angle. Hence, also, every point on the spheric line D'CD is equidistant from A and B, distance being measured along a spheric line. In the same manner as in plane geometry (P. Art. 54) it is shown that every point equidistant from A and B, and lying on the spheric surface, is on the right bisector of the base AB, i.e. on the spheric line D'CD. Cor. In spheric geometry the join of A and B is either ADB or AD'B; i.e. there are two joins whose sum makes up the whole spheric line. The bisector of one of these joins evidently bisects the other also ; as at D and D'. SPHERIC TRIANGLE. 215 234. Let P (Fig. of 233) be the pole of AB, and let C be any point, lying between P and the arc AB, on the spheric line PD. Then PA = PD, and PA is < PC + CA, (Art. 221.) and .-. CD is < CA. And ZPAD = ^PDA = ~\, and Z CAD is < T Therefore, the least distance from C to the spheric line AB is along the perpendicular CD. Also, two equal spheric lines, CA and CB, can be drawn from C to AB, and these lie upon opposite sides of CD, and are equally inclined to it, and meet the line AB at equal distances from the foot of the perpendicular. 235. From C draw any spheric line, CE, to meet AB, and to cut PA at some point F between P and A. Then, PE is, at which position CE has its minimum value. Cor. 1. Since all spheric lines are of the same length, or contain the same angle, the greatest and least spheric arcs from a point on the sphere to any spheric line, AB, are the parts of the spheric line perpendicular to AB, which are intercepted between C and AB. Cor. 2. If two equal spheric arcs be drawn from a point on the sphere to a spheric line which is not its equator, they are equally inclined to the longest spheric arc from the given point to the given line, and lie upon opposite sides of it. 236. As in a plane triangle, so in a spheric one, when two sides and an angle opposite one of them are given, the triangle may be ambiguous. Owing to the facts, however, that any two spheric lines intersect in two points, and that the sum of the angles of a spheric tri- angle is not a fixed quantity, the condition for ambiguity is much more complex than in plane geometry. Also, unlike a plane triangle, a spheric triangle may be ambiguous when two angles and a side opposite one of them are given. THE AMBIGUOUS CASE. 237. In the following examination we assume that the relative magnitudes of the parts given are such as to determine a real triangle, so that we shall not be concerned with conditions which lead to impossible or THE AMBIGUOUS CASE. 217 vanishing triangles, although such conditions are easily obtained. For the given parts let us take the sides a and b and the angle A, and let us consider the subject under three cases, according as A is less, equal to, or greater than, a right angle. For a general diagram let A CD and AED be two spheric lines at right angles, and let C be a pole of AED. Through A and D draw the spheric line APD, making the angle CAP less than a right angle, and the spheric line AQD, making the angle CA Q greater than a right angle. Take any point C" between A and C, and any point C" between C and D. Let the side 6 be measured from A along the arc ACD, A i and let the given angle A be the angle at A. Then a is drawn from some point on the arc ACD to the arc APD or AED or AQD, according as A is less than a right angle, equal to a right angle, or greater than a right angle. One such triangle is represented at AC'P, where a denotes the side C'P. CASE I. Let A = Z. CAP, < f . Then the shortest distance from a point on ACD to the arc APD is perpendicular to APD (Art. 234), and is therefore not along ACD. 1. Let b = AC', < |, and let P' be the foot of the perpendicular from C". 218 SOLID OR SPATIAL GEOMETRY. Two equal arcs can be drawn from C' to APD, one on each side of C'P' (Art. 234), and the triangle may be ambiguous. For a case of ambiguity, however, a must be < C'A, i.e. < b. .. A < f , b < f , a < b is ambiguous. 2. Let 6 = .4(7=1, and let P be the foot of the per- pendicular from C. Then evidently the case is ambiguous if a is less than AC and greater than CP, and CP, being on the equator to A, measures the angle A (Art. 216). "> A .-. A<\, b = f, a^, , is ambiguous. 3. Let & = AC", > f . Then the case will be ambiguous if a < C"D. .-. -4z, a<(7r 6) is ambiguous. Cor. In all the foregoing cases it is readily seen that the ambiguity disappears when the triangle becomes right angled by a being drawn perpendicular to the arc APD. CASE II. Let A be the angle CAE = f . Since C is a pole of AED, from any point on ACD, two equal spheric arcs can be drawn to AED, one lying on each side of AC and equally inclined to it (Art. 234). The two triangles thus formed would be conju- gate and not ambiguous (Art. 229). But if the point C be taken, all spheric arcs from C to AED are equal, and the triangle is indeterminate. Therefore, with A = f there is no real ambiguity, but when b < \ and a > 6, and also when b > \ and THE AMBIGUOUS CASE. 219 a>(7r 6), two triangles are obtained which are conju- gate ; and when b = f , the triangle is indeterminate. CASE III. Let A be the angle (Z4Q>|. Since the angle CAQ is greater than a right angle, the longest spheric arc from any point on ACD to AQD is perpen- dicular to AQD, and is therefore not along ACD. 1. Let b = AC', (TT 6). .-. A > f , 6(7r 6) is ambiguous. 2. Let 6 = AC '= \ ; then if Q be the foot of the per- pendicular from C, it is readily seen that the case will be ambiguous if a is less than CQ and greater than CD. But CQ being on the equator to A measures the angle A (216). %, 6=f, a ^ , is ambiguous. 3. b = AC", > \. Then, if Q" be the foot of the perpendicular from C", the triangle will be ambiguous if a lies between C"Q" and C"A. .-. A>%, b > f, a > 6 is ambiguous. 220 SOLID OR SPATIAL GEOMETRY. 238. The results of the preceding article are collected in the following table : &<| 6 = 1 6<| A<\ a< b ambiguity. a > A b < O - 6) conjugate. a b indeterminate. a < b > (TT - b) conjugate. A>\ a >(T - 6) ambiguity. < A>b ambiguity. a>b ambiguity. We see from the table that ambiguity occurs only when A is not a right angle. 239. By making use of the polar triangle we can readily investigate the cases of ambiguity when two angles and a side opposite one of them are given. For when a triangle is ambiguous, its polar is ambiguous, and vice versa. The table corresponding to that of the last article is here given : -B>f B=I B<1 > f A>B ambiguity. AB ambiguity. A > (ir - B) ambiguity. a = f A(v-B) conjugate. A = B indeterminate. A>B<(K-B) conjugate. 0a< B ambiguity. A f. But, also, Afis < 1, and Ae is > f . And a and c are both less, or equal to, or greater than, f. 3. Let b = AC". Then b > f ; and c"e is < f, and C"fis > f. But Ae is > f, and Afis < f. Therefore, when b < f , both a and c are less than, equal to, or greater than, f . When 6 = f, both a and 6 are equal to f . When 6 > f , b and c are > f , or a and c are = f , or b and a are > f . And the theorem is proved. 241. Theorem. The sum of two angles of a spheric triangle, and the sum of the sides opposite these angles, are both less than, equal to, or greater than, a straight angle. Let CDF be a lune, and let EF be equator to C and D. Through G, the middle point of EF, draw the spheric line AB, meeting the sides of the lune in A and B. Then CAB is a spheric triangle. It is evident that the triangles CAB and DBA are congruent, and therefore that Z CAB == Z DBA, and Z CBA = Z DAB, etc. 222 SOLID OR SPATIAL GEOMETRY. Then Z CAB + Z CBA = Z CAB + Z BAD = -, and CA + CB= CA + AD = ir. Therefore, if the sum of two angles is a straight angle, so also is the sum of the two opposite sides. Now take B 1 , any point between B and D, and draw the spheric arc AB'. Then Z ABB' + Z .R4.B' + ^BB'A > TT ; (Art. 48. Cor. 2.) and Also, CL4 + CB' > CA + CB>Tr. Therefore, when the sum of two angles is greater than a straight angle, the sum of the opposite sides is also greater than a straight angle. And since these sums decrease and increase together, the theorem follows. Hence (A + B) and |(a + &) are both > f , both = f, or both < f. This relation is commonly expressed by saying that %(A-\-B) and ^(a + &) are of the same affection. EXERCISES Q. 1. The area of a spheric triangle is Er 2 , where E is the spherical excess. 2. The area of a spheric polygon is (S A (n 2)7r}r 2 , where SJ. is the sum of the angles, and n is the number of sides. 3. The area of an equilateral spheric triangle is one-fourth that of the surface of the sphere. Show that its angle is 120, and find its side. EXERCISES. 223 4. AB is a spheric arc, and C is its middle point. The locus of P, such that Z APC = Z BPC, is two spheric lines perpendicular to one another. Prove this, and state its analogue in plane geometry. 5. If the direction from A to B, two places on the earth, is estimated along a spheric line, and in terms of the angle which this line makes with the meridian of the first place, show that if A and B have different latitudes and longitudes the direction from A to B is not the opposite of the direction from B to A. 6. If a spheric triangle be formed by cutting a three-faced corner by a sphere, the centre of the sphere being the vertex of the corner, show (i.) that the isoclinal line to the edges of the corner gives the centre of the circle circumscribing the spherical triangle ; (ii.) That the isoclinal line to the faces gives the centre of the inscribed circle of the triangle. 7. What are given by the external isoclinal lines to the corner ? 8. A spheric line is described by a quadrant which has one extremity fixed (215) ; what is the analogue in plane geometry ? 224 SOLID OK SPATIAL GEOMETRY. MISCELLANEOUS EXERCISES. 1. Non-parallel lines do not necessarily intersect. 2. Two circles in space may pass within one another and have two, one, or no points in common. 3. From the definition of a tangent (P. Art. 109) show that the tangent to a circle lies in the plane of the circle. 4. A plane which is normal to the common line of two planes is perpendicular to both planes. 5. If any number of planes meet in parallel lines, the normals to these planes, from the same point, are complanar. 6. The sum of the normals from a point A to the planes U and V is the same as that of the normals from B to the same planes. Show that if P be any point in the line AB, the sum of the normals from Pto U and Vis constant. 7. Show that Ex. 6 holds good for any number of planes, U, V, W, etc. 8. If the sum of the normals to the planes U and V be the same for any three points, A, B and C, it is the same for every point in the plane of ABC. 9. The right-bisector plane of the common perpendicular to two lines bisects the join of any two points, one on each line. 10. A perpendicular is drawn to the base of a regular pyramid and meets the faces, produced where necessary. Then, the sum of the distances of the points of intersection from the base is constant. 11. Find in a given plane, a point equidistant from three given points. 12. 'Determine on a given line the point which is equidistant from any two given points. 13. The bisecting plane of a dihedral angle of a tetrahedron divides the opposite edge into segments which are proportional to the areas of adjacent faces. 14. The shortest chord through any point within a sphere is normal to the diametral plane containing the point. MISCELLANEOUS EXERCISES. 225 15. If one intersection of a sphere by a cylinder is a circle, so also is the other intersection. 16. If one intersection of a sphere by a cone is a circle, so also is the other intersection. 17. The perpendiculars to the faces of a tetrahedron, at their centroids, are concurrent. 18. The vertices of a cuboid are conspheric. 19. The edges of a given three-faced corner pass through three fixed points. Show that its vertex is fixed. 20. The medians of a tetrahedron, taken in both length and direction, form a quadrilateral. 21. In a three-faced corner the planes through the edges bisect- ing the dihedral angles form an axial pencil. 22. In a three-faced corner the planes through the bisectors of the face-angles and perpendicular to the faces form an axial pencil. How many such pencils in all ? 23. What is the locus of a point equidistant from two given points ? 24. What is the locus of a point equidistant from two complanar lines? 25. What is the locus of a point equidistant from two given planes ? 26. What is the locus of a point equidistant from three parallel lines ? 27. A point is equidistant from a fixed line and a fixed plane ; show that its locus is a ruled surface. 28. Through a given line pass a plane perpendicular to a given plane. 29. Through a given point pass a plane normal to a given line. 29J. The locus of a point whose joins with two given points is in a constant ratio is a sphere whose centre-line passes through the given points. 226 SOLID OR SPATIAL GEOMETRY. 30. Given a plane and any three points, show that a point may be found in the plane such that its joins with the given points shall make equal angles with the plane. 31. Through a given point draw a line to intersect two given non-complanar lines. 32. Through a given point, P, in a plane draw a planar line which shall be at a given distance from a given point, Q. What are the limits of possible solution ? 33. Draw a line from a point, P, to a plane, M, which shall be parallel to the plane N, and of given length. 34. Given Z, M, N, three non-complanar lines, draw a line to intersect L and M and be perpendicular to N. To be parallel to N. 35. Through a given point to draw a line which shall meet a given line and a given circle not complanar with the line. 36. Two points are upon opposite sides of a plane. Find the point in the plane for which the difference of its distances from the given points shall be a maximum. 37. In Ex. 36 find a point in the plane which shall be equidis- tant from the given points. 38. Cut a given four-faced corner by a plane so that the section shall be a parallelogram. 39. L and M, two non-complanar lines, meet their common per- pendicular in A and B. If P be any point on L, and Q on M, P#2 = AB 2 + Ar 2 + BQ 2 -2AP- BQ cos 0, where is the angle between the lines L and M. 40. is the centre, e an edge, and A a vertex of a ppd., and P is any point. Then, SP4 2 = 8 PO 2 + \ Se 2 . 41. O is the centroid, and a is a side of any triangle, and P is any point in space. Then, SPvl 2 = 3 PO 2 + 1 Za 2 . 42. is the centre, A is a vertex, and e is an edge of a tetrahe- dron, and P is any point. Then, SP^l 2 = 4 PO 2 + J Se 2 . MISCELLANEOUS EXERCISES. 227 43. From a fixed point three mutually perpendicular lines are drawn to a fixed plane. Show that the sum of the squares of the reciprocals of these lines is constant, being the square of the recip- rocal of the perpendicular from the point to the plane. (Compare P. Ex. 32, p. 131.) 44. In any four-sided prism the sum of the squares on the twelve edges is greater than the sum of the squares on the four diagonals, by eight times the square on the join of the common mid-points of the diagonals, taken in pairs. 45. What are the axes of symmetry of a cube? Of a cuboid? Of a regular tetrahedron? 46. Two spheres may have two, one, or no common tangent cones. Distinguish the cases, and explain those where the spheres have contact of the same and of opposite kinds. 47. Of four spheres, each one touches three others. Show that their tangent planes, at the points of contact, form a sheaf of planes. 48. The common tangent cones to three spheres, taken in twos, have their centres in four collinear rows of three. 49. The common tangent cones to four spheres, taken in twos, have their centres lying by sixes upon four planes. 50. Two circles in parallel planes are the intersections of the planes by two different cones. Show that the centres of the cones and the centres of the circles form a harmonic range. 51. The difference between two faces of a tetrahedron is less than the sum of the other two. 52. Show that to bisect a pyramid by a plane parallel to the base requires the solution of a cubic equation. 53. The cube having the diagonal of another cube for its edge has 3\/3 times the volume of the other. 54. One cube has its face equal to the surface of another cube. Compare their volumes, and also their edges. 228 SOLID OR SPATIAL GEOMETRY. 55. If P be any point within a parallelepiped whose diagonals are AA 1 , BB', etc., the pyramids P ABCD + P A'B'C'D' = $ the ppd. What if P be without ? 56. If a, 6, c, d be the four altitudes of a tetrahedron, and a', &', c', d' be the corresponding perpendiculars from any point to the faces, show that a'b'c'd' , h H (-=!. abed 57. ABCD is a tetrahedron, and P is any point. If AP, J5P, etc., meet the faces in o, 6, etc., then Pa . Pb . PC . Pd =l Aa Bb Cc Dd~ 58. Three mutually perpendicular lines pass through a fixed point in a sphere. Show that the sum of the squares of the three determined chords is constant. 59. In Ex. 58, the sum of the squares on the six segments into which the chords are divided by the point, is constant. 60. A spherical shell six inches in diameter has the interior cavity one-half the volume of the sphere. Find the thickness of the shell. 61. Three equal spheres touching each other lie upon a table, and a fourth equal sphere rests upon the three. How far is the centre of the fourth from the table ? 62. In Ex. 61, the radius of the fourth sphere is n times that of the others. What is the case when n = 1 f V3 ? 63. A sphere touches each of three mutually perpendicular con- current lines. Find the distance from the centre of the sphere to the point of concurrence. A cylinder of revolution whose section through the axis is a square is an equilateral cylinder; and the cone of revolution whose section through the axis is an equi- lateral triangle is an equilateral cone. MISCELLANEOUS EXERCISES. 229 64. If an equilateral cone and an equilateral cylinder be in- scribed in the same sphere, (1) The surface of the cylinder is a mean proportional between the surfaces of the sphere and cone ; (2) The volume of the cylinder is a mean proportional between the volumes of the sphere and cone. 65. If an equilateral cone and an equilateral cylinder be circum- scribed about the sphere, (1) The surface of the cylinder is a mean proportional between the surfaces of the sphere and cone ; (2) The volume of the cylinder is a mean proportional between the volumes of the sphere and cone. If P, Q be points on a centre line of a sphere, such that OP OQ jR 2 , where is the centre of the sphere and R is the radius, the points P and Q are inverse points with respect to the sphere. And when two figures are such that every point in the one is the in- verse of a corresponding point in the other, the figures are inverse to one another (P. Art. 260) . 66. The inverse of a line is a complanar circle through the centre of inversion. 67. The inverse of a circle is a complauar circle, unless the first circle passes through the centre of inversion. 68. The inverse of a sphere is a sphere, unless the first sphere passes through the centre of inversion, when its inverse is a plane. 69. The inverse of a plane is a sphere through the centre of inversion. 70. A sphere which passes through a pair of inverse points with respect to another sphere cuts the other orthogonally. 71. A sphere which cuts two spheres orthogonally has its centre on the radical plane of the two. 230 SOLID OR SPATIAL GEOMETRY. 72. A cube circumscribed to a sphere is inverted with respect to the sphere. Show that the spheres produced pass by threes through common points and cut one another orthogonally. 73. The locus of a point with respect to which two spheres can be inverted into equal spheres is a sphere having a common radi- cal plane with the two. 74. The locus of a point with respect to which three spheres can be inverted into equal spheres is a circle. 75. There are two points, real or imaginary, with respect to which four spheres can be inverted into equal spheres. 76. What is the locus of a point from which two given spheres subtend the same angle ? 77. What is the locus of a point from which three spheres sub- tend the same angle ? 78. The joins of the foci to any point on a hyperbola are equally inclined to the tangent at that point. 79. If an ellipse and a hyperbola have the same foci, the curves intersect orthogonally. 80. A sector of a circle revolves about a diameter parallel to the chord of the sector. The volume described is f irr 5 sin 0, where 2 9 is the angle of the sector. 81. The volume of a segment of a sphere is I wr s (2 - 3 cos + cos 2 0}, where 2 is the angle subtended by the segment. 82. A plane figure, invariable in form and dimensions, moves with its centre on a path which is inclined to its plane at a constant angle, a. Show that the volume described is the area of the figure x the length of path x sin o. 83. The generator of Art. 143 does not preserve its orientation, but revolves about the path. Show that this does not affect the volume described, if the centroid is confined to the path. MISCELLANEOUS EXERCISES. 231 84. A square, side s, moves with its centre on a circle, and its plane perpendicular to the path, but revolves about the path as an axis. Show that the volume described is 2 wrs 2 , if r > \ s V2. 85. A spheric line is described by a quadrant rotating about one of its end-points as a centre (Art. 215). What is the analogue in plane geometry ? A plane through one of two inverse points, normal to the join of the points, is the polar plane to the other point, and this latter point is the pole of the plane, with respect to the sphere of inversion. 86. If the point P lies on the polar plane of Q, then Q lies on the polar plane of P. 87. The polar of a line is a line at right angles to the given line. 88. Explain how the process of Art. 63 is one of polar recipro- cation. 89. Show that the tetrahedron may be a polar reciprocal to itself. 90. A sphere touches the twelve edges of a cube. What is the polar reciprocal of the cube with reference to the sphere, and how is it situated ? 91. The distances of any two points from a polar centre are pro- portional to the distances of each point from the polar plane of the other. 92. The centre locus of a sphere which cuts two given spheres orthogonally is their radical plane. 93. The centre locus of a sphere which cuts three spheres orthogonally is their radical line. 94. All the spheres which cut two spheres orthogonally pass through two fixed points. 95. All the spheres which cut three given spheres orthogonally pass through three fixed points. 232 SOLID OR SPATIAL GEOMETRY. 96. A sphere which cuts four spheres orthogonally is fixed. What exception ? 97. All the spheres, which have contact of like kind (P. Art. 291) with two given spheres, are cut orthogonally by one and the same sphere. 98. All the spheres which have contact of like kind with three given spheres are cut orthogonally by the same three spheres. 99. A line is cut harmonically by a point, a conic, and the polar of the point with respect to the conic. 100. A line is cut harmonically by a point, a sphere, and the polar plane of the point with respect to the sphere. In crystals, whether formed in the laboratory or by slow geological processes, we have examples of natural polyhedra. These are forms derived from prisms or parallelepipeds by transformations closely allied to polar reciprocation, the replacement of corners or points by planes. In crystallography the relative direction of the plane which forms a face of the crystal is of primary importance ; its distance from the centre is only a sec- ondary consideration. Through the centre of the cube let the three rectan- gular axes of space be drawn parallel to the direction edges of the cube, and let them be denoted by X, Y and Z. Every plane cuts these axes either at finite points or at infinity, and hence every plane makes on these axes three intercepts, which may be finite or infinite. Denote the intercepts by x, y, z, where these letters denote measures on the respective axes, but may be equal or unequal in value. The giving of these inter- cepts determines the relative direction of the plane. If a plane which forms a face of a crystal is parallel to the face of the original cube, it is looked upon as a MISCELLANEOUS EXERCISES. 233 face of the cube, and if parallel to a face of the regular derived octahedron, it is considered to be a face of the octahedron, etc. 101. Show that the plane (x, y, z) is parallel to the plane (mx, my, mz). 102. Show that the plane (x, y, 2) is parallel to the plane , ^, 1]. What inference do you draw as to the absolute values z z ) of the intercepts ? 103. Show that the plane (x, y, z) is parallel to the plane (-x, -y,z). 104. The planes (2, 1, 1) and ( J, 1, 1) are perpendicular to one another. 105. The planes (a, a, 6) and ( a, a, ] are perpendicular to one another. * ' 106. Show that (1, GO , co ) is a cubic face, and write the remain- ing faces of the cube. In representing planes in this way, by three quantities taken in one order of rotation, it is usual to employ the reciprocals of the intercepts, as some of the final results are simplified by this means. These will be called the three parameters of the plane, and will be denoted by h, Jc, I in particular, and by any letter or quantity in general. 107. Write the faces of the cube in the parametric notation. 108. Show that (1, 1, 1), or, in general (a, a, a), is a face of the regular octahedron. 109. How does the plane (1, 1,1) cut the cube ? And how does (1, 0, 0) cut the octahedron ? 110. A plane with three equal parameters truncates a corner of the cube. Describe what is meant by truncating a corner of a cube, and compare the dihedral angles formed. 234 SOLID OR SPATIAL GEOMETRY. 111. A plane with two parameters equal, and the third zero, truncates an edge of the cube. What is the character of this truncation ? Compare the dihedral angles formed ; what is their value ? 112. Show that the plane (1, 0, 0) truncates the corner of the octahedron ; and that (1, 1, 0) truncates an edge of the octahedron. 113. A plane with three unequal parameters bevels a corner of the cube. Describe the operation of beveling a corner of the cube. 114. The plane having two parameters equal and the third un- equal, all being finite, cuts the corner of the cube in a way which will be called trunco-bevelment. Define this term. 115. By what change does a trunco-beveling plane of a corner of a cube become a truncating plane of the edge ? 116. The cube admits of eight truncating planes to the corners. Describe the figure formed, on the supposition that all these planes are equidistant from the centre, and the faces of the cube are com- pletely cut away. 117. The cube admits of twelve truncating planes to the edges. Describe the figure formed by these planes. (This is the rhombic dodecahedron.) 118. To what figure does the plane (1, 1, 0) belong? The plane (1, 0, 1) ? 119. How does the plane (a, a, 0) cut the octahedron ? 120. The cube admits of two beveling planes at each of its twelve edges. Explain how, and describe the figure to which these faces belong. (This is the tetra-hexahedron, or four-faced cube.) 121. To what figure do the planes (1, 2, 0) and (2, 1, 0) belong? 122. a. The cube admits of three trunco-beveling planes at each corner. How many faces has the figure to which these planes belong ? 6. Show that these planes may be disposed in two different ways, and describe the difference in the resulting modification of the original corner. What relation does it hold to the octahedron ? (This is the triakis-octahedron, or three-faced octahedron.) MISCELLANEOUS EXERCISES. 235 123. The cube admits of six beveling planes at each corner. Write these planes, and give the character of the figure formed. (This is the hexakis-octahedron, or six-faced octahedron.) 124. To what figures do the planes (1, 1, 2), (1, 1, 0), (1, 2, 3), (1, 2, 2) belong? Figures formed from the cube by putting in all the possible planes given by varying the order of the param- eters in any one symbol, as (1, 2, 3), (2, 3, 1), ( 2, 1, 3), etc., are called holohedral figures. Those formed by putting in one-half the possible planes, in alternate posi- tions, are hemihedral figures. 125. One beveling plane is put in at each edge of a cube so as to alternate the positions of these planes. Show that the resulting figure will have pentagonal faces. (This is the pentagonal dodeca- hedron.) 126. Show that the tetrahedron is a hemihedral form derived from the cube, and give its mode of derivation. 127. The cube admits of three beveling planes at each corner, applied in alternate positions. Write these planes and show how they are applied. (The figure is the pentagonal icosi-tetrahedron.) 128. The cube admits of six beveling planes at four corners alter- nate in position. Write these planes. (The figure is the hexa- tetrahedron. 129. If p be the length of normal, from the origin, on the plane, and a, /3, 7 be the direction angles of p (Art. 98), show that cos a = hp, cos /3 = kp, and cos 7 = Ip. 130. Show that p = 1 / V/i 2 + k 2 + I 2 . 131. Show that cos a = h/ Vfi 2 + k 2 + I' 2 , with symmetrical ex- pressions for cos /3 and cos 7. 132. If be the angle between the normals to two planes, cos6=pp'(hh' + kk'+ II'), where the accented letters refer to the second plane. 236 SOLID OK SPATIAL GEOMETRY. 133. Show that cos e = (hh 1 + kk' + H')/V{(W + k 2 + I 2 ) (h 12 + k' 2 + Z' 2 )}. 134. The angle between the planes (1, 1, 1) and (1, 1, 1) is cos" 1 ^. 135. The angle between the planes (1, 0, 0) and (1, 1, 1) is cos' 1 ^ V3. 136. Find the cosine of a dihedral angle of the regular tetra- hedron. 137. Find the cosine of the dihedral angle of a regular octahe- dron. 138. Find the cosine of the angle between a face of the cube and that of the octahedron. 139. The type plane (1, 2, 3) cuts the cube. Find the angle between two adjacent planes, and also between one of these planes and an adjacent face of the cube. 140. Find the angle between (1, 2, 3; and (1, 1, 1). 141. The edge made by the planes (a, b, 0) and (6, a, 0) is truncated by the plane (6 + a, b a, 0). 142. Determine the ratios of the intercepts of any plane which bevels the edge of the rhombic dodecahedron. 143. The face of a pentagonal dodecahedron being (0, 1, a) with necessary variations, show that for the regular figure a=2(-\A>l), and thence show that the cosine of a dihedral angle of this figure is INDEX OF TERMS, ETC. The numbers refer to the articles. Abscissa 207 Acute ppd 54 Anchor ring ..... 164 Angle of obliquity . . . 116 Ant-orthogonal projection 177 Asymptotes 191 Axes of space .... 8 Axial pencil 5 Axis of conic 194 Basal edges 50 Capacity 106 Centre of figure .... 172 Centroid 160 Circle of contact ... 86 Col-unar triangle . . . 225 Common line 15 Complanar 6 Cone 67, 131 Cone-circle 10 Conic 189 Conjugate diameters . . 201 Conjugate triangles . . 219 Conspheric 83 Co-ordinate planes ... 8 Corner . 33 ART. Cube 57 Cuboid 57 Cylinder 75, 136 Cylindroid. . . . . . 162 Developable surface . . 165 Diametral plane .... 78 Diclinic ppd 57 Dihedral angle .... 23 Direction angles .... 98 Direction cosines ... 98 Direction edges .... 53 Director 7 Directrix 196 Eccentricity 194 Ellipse 190 Equator 215 Euler's theorem .... 48 Figure of revolution . . 149 Focal distance . . . . 194 Focus 194 Frustum 59 Generator 7 Great circle 77 Groin 144 Hyperbola 190 237 238 INDEX. Isoclinal ART. 36 . Prismoidal formula. ART. 130 Lamina 119 Projection 11 Lateral edges 50 Pyramid 58 Lune 217 Quadrantal triangle 225 Median 51 Radical centre .... 105 Middle section . . . 61, 128 Radical line 105 Monoclinic ppd Nappes of cone .... Net 57 67 65 Radical plane .... Reciprocal corners . . . Rectilinear hyperbola . 104 43 191 Normal 8 Representative corners 53 Normal plane .... 8 Right corner 40 Oblique prism .... Obtuse ppd 60 54 Right-bisector plane . . Right-circular cone . 25 69 Ordinate 194 Right prism . 60 Orthogonal ororthographic rroi . 177 Right section Ruled surface 22 66 Parabola . . ... 190 Secant line 78 Parallel section .... 20 Secant plane 78 Parallelepiped .... Perspective projection . . Planar 53 177 8 Segment of sphere . . . Semiaxis major .... Semiaxis minor .... 139 198 198 Plane 1 Semivertical angle . 70 Plane of lines .... Plane section 4 19 Sheaf of lines and planes . Sheets of a cone 31 67 Point circle ..... 191 Skew quadrilateral . 29 Point ellipse 191 Skew surface 165 Polar plane 86 Small circle 77 Polar triangle .... 226 Solid angle ..... 33 Pole 215 Solid contents . . . 106 Polyhedral angle 33 Sphere 76 Polyhedron ... 47 Spheric arc . . 212 Power of a point . . . Prism 102 60 Spheric figure .... Spheric line 210 212 Prismatic element . . . Prismatoid 162 126 Spheric radius .... Spheric triangle . 215 218 Prismoid . 126 Soherical excess . Wfl INDEX. 239 Surface of revolution Symmetrical ... Tangent cone Tangent sphere .. Triclinic ppd Trihedral angle .. ART. . 66 39, 219 . 86 . 83 . 57 34 Unit volume 109 Vanishing line .... 181 Vanishing point .... 181 Volume 106 Wedge 124 Zone of sphere . . . . 139 D92 oc UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is nirr ^ *u_ ...... 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