SOLID GEOMETRY DEVELOPED BY THE SYLLABUS METHOD BY EUGENE RANDOLPH SMITH, A.M. HEADMASTER, THE PARK SCHOOL, BALTIMORE, MARYLAND (FORMERLY HEAD OF THE DEPARTMENT OF MATHEMATICS, POLYTECHNIC PREPARATORY SCHOOL, BROOKLYN, N.T.) IN CONSULTATION WITH WILLIAM H. METZLER, PH.D. DEAN OF THE GRADUATE SCHOOL, AND HEAD OF THE DEPARTMENT OF MATHEMATICS, SYRACUSE UNIVERSITY NEW YORK : CINCINNATI : CHICAGO AMERICAN BOOK COMPANY COPYRIGHT, 1913, BY EUGENE RANDOLPH SMITH. COPYRIGHT, 1913, IN GREAT BRITAIN. SMITH 8YL. SOLID GEOM. W. P. I PREFACE THE author has always believed that the teaching of mathematics should encourage original thinking on the part of the pupils, and that a maximum of thought is difficult to obtain when the pupil is furnished with a text containing the proofs of the geometrical theorems in synthetic form. He has been using the heuristic method for thirteen years, and has proved to his own satisfac- tion that the average pupil can enter intelligently into the class development of new propositions, and, after a certain amount of training, can analyze propositions of ordinary difficulty with little or no help from the teacher. This book has been written to encourage what is often called the " genetic," " heuristic," or " syllabus " method of teaching, but what is, in effect, simply the development of new work, partly in class, the rest by assignment as orig- inal exercises for preparation before the recitation. It is intended to contain all that a pupil really needs, except that which the teacher, and only the teacher, can best supply. While the proofs are not in the book in full and formal fashion, the author has given as much analysis, suggestion, and guidance as he feels to be wise. In fact, he believes that even this amount of help should often not be used until after the class discussion has covered the topic. The list of propositions is based on the recommendations of the various committees, and especially on that "of the National Geometry Committee of Fifteen. An attempt ill iv PREFACE has been made to reduce the list to a pedagogical minimum, without injuring the subject by over- reduction. The space thereby saved has been devoted to valuable matter not often found in geometries, such as the preliminary chapter, the detailed summaries, the Appendix, and the college examination questions. Solid geometry is an especially promising field for heuristic methods because the pupils studying it are of sufficient maturity to enable them to think with some assurance and originality. If a pupil is ever to obtain the proper training in logical methods of thinking, it ought not to be longer delayed. Besides this, the results in solid geometry in schools and colleges show that former methods have not proved entirely satisfactory. This con- dition is probably due to the fact that the student is often plunged into the subject with little or no understanding of the figures with which he is to deal, the method he should use in studying them, or the relation of this new subject to the mathematics that has gone before. This book attempts to better this condition by introducing a preliminary chapter that shows the relation between plane geometry and solid geometry^ and by building up the solid geometry figures in easy gradation from points, lines, and planes. This arrangement will give the pupil a clear understanding of the formation of the figures with which he is dealing and will lead him, in progressive stages, from the less complicated figures and proofs to those of greater difficulty. Incidentally, many of the difficult parts of the subject have been simplified, and the pupil is given a broad working knowledge of the properties and formulas of solids.* There are three combinations of related parts of the subject into single sections : prisms and cylinders, pyra- PREFACE V mids and cones, and polyhedral angles and spherical poly- gons. The simplification of the subject by these combi- nations is quite marked, and will be evident on inspection of the sections in question. In the last case, it gives op- portunity for some consideration of the way in which each can be made to depend on the other. Some use has been made of two of the most powerful propositions of solid geometry, Cavalieri's Theorem, and the Prismatoid Formula, but only as alternate methods. The National Geometry Committee recommends that these two theorems be used but not proved. The Prismatoid Formula is used for spherical segments. The use of limits and the incommensurable case has been left to the discretion of the teacher. The best present-day thought seems to favor the omission of as much of this work as possible. The exercises are of wide variety in subject and diffi- culty, and include a set of about two hundred and seventy- five college examination questions chosen from recent papers set by various colleges, the New York State Board of Regents, and the College Entrance Examination Board. The exercises are arranged, some after the theorems to which they apply, and the rest in general sets. While the author has made use of all the knowledge of solid geometry that his study and teaching have given him, still, as this text was written chiefly from classroom experiment, it is impossible to give credit specifically to authors from whom the germs of ideas may have been received. He is greatly indebted to Dr. William H. Metzler of Syracuse University, for his painstaking reading of the manuscript and for his valuable criticisms. Mr. Howard F. Hart of the Montclair High School, who has been trying in his classes some of the ideas that appear in vi PREFACE this book, has given the author the benefit of his experi- ence with them, as well as a number of suggestions on other parts of the book. Mr. Clarence P. Scoboria of the Polytechnic Preparatory School, Brooklyn, also has aided in the preparation of the book by using the manuscript in class, and so making a practical test of the ideas represented. Besides those mentioned, the author wishes to express his obligation to many others who have taken an active interest in the preparation of the book. EUGENE RANDOLPH SMITH. CONTENTS PRELIMINARY CHAPTER How TO STUDY SOLID GEOMETRY ...... I. THE USE OF PLANE GEOMETRY IN SOLID GEOM- ETRY II. METHODS OF ATTACK III. THE REPRESENTATION OF SOLID GEOMETRY FIG- URES NOTES . PAGES 193 194-202 203-215 216-231 232 BOOK VI. LINES AND PLANES I. THE PLANE ' . . 233-236 II. RELATIVE POSITION OF LINES AND PLANES IN SPACE; PARALLELS AND INTERSECTIONS . . 237-248 III. LINES PERPENDICULAR TO PLANES .... 249-253 IV. ANGLES BETWEEN PLANES 254-262 V. Locus OF POINTS 263-267 SUMMARY OF PROPOSITIONS 267-272 ORAL AND REVIEW QUESTIONS .... 272-274 GENERAL EXERCISES ....... 274-277 BOOK VII. POLYHEDRONS, CYLINDERS, AND CONES I. THE PRISM AND THE CYLINDER .... 278-292 II. THE PYRAMID AND THE CONE 293-304 SUMMARY OF PROPOSITIONS 305-307 ORAL AND REVIEW QUESTIONS 308 GENERAL EXERCISES 308-313 vii viii CONTENTS BOOK VIII. POLYHEDRAL ANGLES AND THE SPHERE PAGES I. DEFINITIONS; SECANTS AND TANGENTS . . . 314-324 II. SPHERICAL POLYGONS AND POLYHEDRAL ANGLES . 325-343 III. AREA ON A SPHERICAL SURFACE .... 344-349 IV. VOLUME OF A SPHERE AND ITS PARTS . . . 350-357 SUMMARY OF PROPOSITIONS 357-361 ORAL AND REVIEW QUESTIONS 362-364 GENERAL EXERCISES . . . . . . . 364-366 GENERAL SUMMARY OF THE FORMULAS OF SOLID GEOMETRY . 367-369 COLLEGE EXAMINATION QUESTIONS . . . . . 370-392 APPENDIX LOGIC 393 CAVALIERI'S THEOREM . 394 THE PRISMATOID - 396 CIRCULAR CYLINDRICAL AND CONICAL SURFACES . . 397 COMPARISON BETWEEN PLANE GEOMETRY AND SPHERICAL GEOMETRY . . ... . . . . 398 THE GENERAL POLYHEDRON 400 RADIAN MEASURE ... ..... 401 SYMMETRY 403 INDEX Light-faced numbers refer to pages and full-faced numbers to articles. Cones 151, 158 spherical 303 Congruent polyhedral angles 265 Congruent spherical polygons 264 Conical space 147 Conical surfaces 145, 321 Construction 31 Contact, point of 196, 197, 233 Converse of statement 211, 318 Convex figures 102, 238 Coplanar figures 11 Corresponding angles and sides 258 Corresponding sets of prisms 172 Counter-clockwise motion 260 Cubes 128, 253 Cuboids 127 Cylinders 110, 114 Cylindrical space 105 Cylindrical surfaces 104, 321 Decahedron 98 Degree of formulas 311 Descriptive geometry 78 Determined surfaces 4, 9 Diagonals 129 Diameters 189 Dihedral angles 64 Dimensions 129 Directrix 104, 145 Distance on a sphere 248 point to a plane 80 Dodecahedrons x 98, 255 Duplication of the cube 332 Edges 64, 97, 99, 110, 151, 154, 312 Elements 104, 145 Equality of Tunes 287 Equilateral triangles 239 Equivalence 124 Excess, spherical 278 Explementary angles 70 ix Acute angles 70 Addition of lunes 286 Adjacent angles 70 Algebraic analysis 209 Altitude 111, 155, 280 Analysis 203-209 Analytical geometry 266 Angles, between arcs 221 dihedral 64 face 100 line with plane 83 of spherical polygon 237 plane or measuring 65 spherical 222 Area 97, 110, 151 Attack, methods of 203 Axioms 6 , 18, 282 Axis 115, 157, 204 Bases 110, 151, 154 Cavalieri bodies 137 Cavalieri's Theorem 137, 319 Center lines 232 Center sects 232 Center of a sphere 187 Central polyhedral angles 241 Circles, of spheres 203, 207 Circular cones 158 Circular cylinders 114 Circular cylindrical and conical surfaces 321 Circumscribed figures polyhedrons 201 prisms 116 pyramids 159 sets of prisms 172 spheres 195 Classification 203 Clockwise motion 260 Closed surfaces 105, 147 Coincidence 263 Complementary angles 70 INDEX Face angles 100 Faces 64, 97, 98, 99, 110, 151, 154 Figures, representation of 216 Foot, of line 14 Frustums 153 Generation of a conical surface 145 of a cylindrical surface 104 of a prismatic surface 104 of a pyramidal surface 145 of a sphere 188 of a spherical surface 279 of a zone 281 Generatrix 104, 145 Great circles 207 Hexahedrons 98, 253 Icosahedrons 98, 256 Inscribed figures polyhedrons 195 prisms 116 pyramids 159 sets of prisms 172 spheres 201 Intersecting planes 17 Intersection of loci 207 Isosceles triangles 239 Lateral area 110, 151 Lateral faces and edges 110, 154 Lateral surfaces 110, 151 Limit of circumscribed poly- hedron 300 of inscribed polyhedron 282 of prism 116 of pyramid 159 of sets of prisms 175 Loci 88 Logic 210, 318 Limes 259, 285-296 Material spheres 227-230 Measuring angles of dihedral angles 65, 69 of spherical angles 225 Midsections 169 Negative converse of state- ment 210 Negative of statement 210 Non-coplanar figures 11 Normal lines and planes 52 Notation of lunes 288 of spherical polygons 299 Oblique, to a plane 52 Oblique cylinders and prisms 110 Oblique parallelepipeds 127 Oblique sections 106 Obtuse angles 70 Obverse of statement 210 Octahedrons 98, 254 Opposite polyhedral angles 257 Opposite spherical polygons 258 Parallel figures lines and planes 14 sections 106, 148 summary of parallels 41, 43 three planes 39 two planes 21 Parallelepipeds 125, 127 Parts 100 Pentahedrons 98 Perpendiculars lines and planes 52 two planes 70 Plane angles 65 Plane geometry, relation to solid geometry 193, 322 Planes, construction of 31 defined 5 intersecting 18, 34, 35, 38 lines in - 6, 22 lines parallel to 41, 43 lines perpendicular to 50, 53 parallel 21, 39, 41 relative positions of 15, 33 representation of 10 Polar chords 215 Polar distances 215 Polar triangles 273 Poles 204 Polygons, spherical 237 Polyhedral angles 99, 240, 241, 244, 257 Polyhedrons 97, 103, 195, 250, 323 Prismatic space 105 Prismatic surface 104 INDEX XI Prismatoids, defined formula proof Prisms, defined inscribed and circum scribed Projections Pyramidal space Pyramidal surface Pyramids spherical Quadrant Quad ran tal triangles Radian measure Radius of a circular cone of a circular cylinder of a sphere Ratio of solids Reflex angles Relative positions of line and plane of line and sphere of plane and sphere of point and sphere of three planes of two lines of two planes Right angles Right cones Right cylinders Right parallelepipeds Right prisms Right sections Scalene triangles Secants Sections Sectors, spherical Segments, spherical Similar solids Slant height Solid angles Solid geometry Solids, defined Space 181 Sphere, defined 187 182 Spherical excess 278 320 Spherical figures 110, 113 angles 222, 225, 226 cum- cones 303 172 polygons 237, 258 77 pyramids 305 147 sectors 303 145 segments 309, 310 151, 156, 157 surfaces 187 305 triangles 239 wedges 307 217 Squared paper 217 342 Straight angles 70 Subtraction of limes 286 324 Summaries 95, 185, 315, 317 Supplementary angles 70 158 Surfaces 3, 97, 104, 145, 187 der 115 Symmetric figures 325 189 Symmetric polyhedral angles 131 260, 265 70 Symmetric spherical polygons 260, 264 14 Synthesis 203 ! 196 re 197 Tangent re 193 to a cone 162 33 to a cylinder 118 13 to a sphere 196, 197 15 two spheres 233 70, 239 Tetrahedrons 98, 252 158 Transversals 28 110, 114 Triangles, spherical 239 127 Trirectangular triangles 348 110 Truncated figures 152 106 Ungula 307 239 Unit of volume 135 196, 197 101, 106, 148 Vertical dihedral angles 70 303 Vertical polyhedral angles 257 309 Vertices 97, 99, 146, 237 312 Volume 97, 135 157, 158 99 Wedge 312 193, 8, 322 spherical 307 2 1 Zones 280, 281 SUMMARY OF GEOMETRICAL SIGNS -f- plus, sign of addition minus, sign of subtraction X times, sign of multiplication -f- , /, : , divided by, sign of division y/ square root sign = is (or are) equal, or equivalent = is not equal, or equivalent, to = is identical to ^ is congruent to = approaches as a limit ~ is similar to > is greater than > is not reater than < is less than jC is not less than II is parallel to _L is perpendicular to Z. or 2 angle A triangle O parallelogram HH rectangle Q square O circle ^ arc .. therefore . because, since The signs for figures become plural by the addition of s, often within, the sign, as HJ for rectangles. xii SOLID GEOMETRY PRELIMINARY CHAPTER HOW TO STUDY SOLID GEOMETRY The Relation between Solid Geometry and Plane Geom- etry. Solid geometry is not an entirely new subject. It is simply an extension of plane geometry, with which it has much in common. It is true that the conditions are somewhat changed by the use of figures not entirely in one plane, but the large part of solid geometry that treats of single planes of a solid figure introduces very little that is really new, since propositions of plane geom- etry obviously hold for any one plane of a figure. Even in the part of the subject where planes are least used, the definitions, axioms, and propositions of plane geometry are constantly applied, either directly or indirectly. As the methods of attacking a new proposition in solid geom- etry are the same as those used in plane geometry, the solid geometry propositions are made to depend, as far as possible, directly on plane geometry propositions. In order that the pupil may find the transition to figures of three dimensions as easy as possible, this preliminary chapter gives some directions about methods of work. It contains a discussion of the plane geometry propositions that can be used in solid geometry, a review of methods of attack, and a short description of a method for drawing solid geometry figures. The section on drawing figures can best be used as the subject is studied, the different figures being examined as the pupil needs them in his work, 193 194 PRELIMINARY CHAPTER SECTION I. THE USE OF PLANE GEOMETRY IN SOLID GEOMETRY In starting solid geometry, all of plane geometry is at the disposal of the student, but great care must be taken that it is used only for those figures for which the plane geometry proofs will hold. For example, it has been proved in plane geometry that Lines parallel to the same line are parallel to each other ; but as this was proved only for the case when the three lines were in one plane, it can- not be used for the general case in solid geometry. It cannot be used, for instance, to prove that the back edge of the ceiling of a room is parallel to the front edge of the floor because they are both parallel to the front edge of the ceiling. The proofs of some propositions in plane geometry would hold even if the figures used did not lie entirely in one plane ; as, for example, congruence proofs. In a second kind of proposition, as in theorems concerning triangles, the figures, from the conditions of the proposi- tion, necessarily lie in one plane. Such propositions can be used wherever their figures occur in solid geometry. In a third kind of proposition, where there is nothing in the conditions to show whether or not the figure is in one plane, it must be proved to lie in one plane before the plane geometry proof can be assumed to hold. An ex- ample is the proposition Lines perpendicular to the same line are parallel. By looking at the edges meeting at the corner of a room it will be seen that this proposition does not hold for the general case. The propositions of plane geometry that are of most value in solid geometry will be summarized under the three heads indicated in the last paragraph. USE OF PLANE GEOMETRY IN SOLID GEOMETRY 195 A. SOME PARTS OF PLANE GEOMETRY THAT CAN BE USED IN SOLID GEOMETRY WHETHER OR NOT THE ENTIRE FIGURE IS IN ONE PLANE I. (a) All the definitions and axioms. (5) All right angles are equal ; all straight angles are equal. (V) Complements, supplements, or explements of equal angles are equal. II. FIGURES CONGRUENT. (a) The definition : two figures are congruent if they can be made to coincide. (6) Two triangles are congruent if they have three parts of one, of which at least one is a side, equal to the corresponding parts of the other ; unless those three parts are two sides and an acute angle opposite one of them. III. SIMILAR FIGURES. (a) Two polygons are similar (1) if their corre- sponding sides are proportional and their cor- responding angles are equal ; (2) if they are similar to the same potygon. (5) Two triangles are similar if they have (1) two angles equal ; (2) two sides of one propor- tional to two sides of the other, and the included angle equal ; (3) their sides corre- spondingly proportional. (e) Areas of similar figures are proportional to the squares of any two corresponding sects. IV. OTHER COMPARISONS. (a) If two triangles have two sides of one equal to two sides of the other, then the third side of one is correspondingly greater than, equal to, or less than, the third side of the other, ac- 196 PRELIMINARY CHAPTER cording as its opposite angle is greater than, equal to, or less than, the opposite angle of the other, and conversely. (6) Areas of triangles having an angle of one equal to an angle of the other are to each other as the products of the including sides. B. PLANE GEOMETRY PROPOSITIONS THAT CAN BE USED IN SOLID GEOMETRY BECAUSE THE NATURE OF THE FIGURE MAKES IT LIE ENTIRELY IN ONE PLANE See 9. Figures that lie entirely in one plane are a tri- angle, a line and a point outside, two intersecting lines, two parallel lines and therefore a parallelogram, and a circle. I. PROPOSITIONS ABOUT A TRIANGLE. (a) If one side of a triangle is greater than, equal to, or less than, a second side, the angle op- posite the first side is respectively greater than, equal to, or less than, the angle op- posite the second side, and conversely. (b) The sum of two sides of a triangle is greater than the third side; their difference is less than that side. (c) The perpendicular bisectors of the sides of a triangle meet in a point (called the circum- center) equidistant from the vertices. (d) The bisectors of the interior angles of a tri- angle meet in a point (called the incenter) equidistant from the sides. (e) The medians of a triangle meet in a trisection point of each. In an equilateral triangle, the medians are also the altitudes. (/) A line parallel to one side of a triangle cuts the other sides proportionally, and con- versely ; it is in the same ratio to the side to USE OF PLANE GEOMETRY IN SOLID GEOMETRY 197 which it is parallel as the sects it cuts off on the other sides from their common vertex are to those sides. ($r) In a right triangle with the altitude drawn to the hypotenuse, (1) the triangles are simi- lar ; (2) the altitude is the mean propor- tional between the sects of the hypotenuse ; (3) each leg is the mean proportional be- tween the hypotenuse and its own projection upon the hypotenuse. II. PROPOSITIONS ABOUT PARALLELS AND PARAL- LELOGRAMS. (a) Definition : lines in the same plane that never meet are parallel. (5) Through a point there can be but one parallel to a given line. (c) If two parallels are cut by a transversal, two sets, each of four equal angles, are formed, the angles of one set being supplemental to the angles of the other set, and conversely. (t?) In a parallelogram (1) the opposite sides are equal, (2) the opposite angles are equal, (3) either diagonal divides it into congru- ent triangles, (4) the diagonals bisect each other. (0) A quadrilateral is a parallelogram if (1) the opposite sides are parallel, (2) two opposite sides are equal and parallel, (3) the diago- nals bisect each other. (/) Parallelograms having equal bases, and lying between the same parallels, are equivalent. The sect between the midpoints of the legs of a trapezoid is one half the sum of the bases. 198 PRELIMINARY CHAPTER III. PROPOSITIONS IN REGARD TO A LINE AND AN EXTERNAL POINT. (#) There can be but one perpendicular from an external point to a line. (5) From an external point to a line, (1) the per- pendicular is the shortest line; (2) obliques that make equal angles with the perpen- dicular or witli the given line, or that cut off equal distances from the foot of the perpendicular, are equal, and conversely; (3) obliques that make unequal angles with the perpendicular, or with the given line, or that cut off unequal distances from the foot of the perpendicular, are unequal, the one that makes the greater angle with the per- pendicular, or that cuts off the greater distance, being longer, and conversely. IV. PROPOSITIONS ABOUT CIRCLES. (a) Radii, or diameters, of a circle are equal. (5) The diameter is the greatest chord ; it bisects the circle and its circumference. If it is perpendicular to a chord, it bisects the chord, and conversely. (c) One, and but one, circle can be inscribed in, or circumscribed about a given triangle. (d) A central angle is measured by its arc ; an inscribed angle, by one half its arc. (e) Equal chords subtend equal arcs, and are equi- distant from the center, and conversely. The greater chord subtends the greater arc, and is nearer the center, and conversely. (/) A tangent is perpendicular to the radius drawn to the point of contact. USE OF PLANE GEOMETRY IN SOLID GEOMETRY 199 A tangent from a point to a circle is the mean proportional between the sects (from the point to the circumference) of any secant from the same point to the circle. (Ji) A circle can be inscribed in, or circumscribed about, any regular polygon. (i) If the number of sides of an inscribed or a cir- cumscribed regular polygon is increased in- definitely, the perimeter and the area of the polygon approach the circumference and area of the circle as limits. V. PROPOSITIONS ABOUT FIGURES FORMED BY Two INTERSECTING LINES. (a) Any two vertical angles are equal. (6) Bisectors of vertical angles are in a straight line. (c) A quadrilateral is a parallelogram if its diago- nals bisect each other. VI. CONSTRUCTIONS. The given figure is in one plane, and the construction is to be done in that plane, except where it can be equally well done in a different plane, as in (d) and (e) below. Where the given figure can be in more than one plane, as in (a) and part of () fl is not true. But (a) contradicts the condition, so is impossible, and (6) must follow, that is, the negative converse of a true statement is also true. But if C is true, H may be either true or untrue without bringing in a contradiction ; and if H is untrue, C may be true or untrue without bringing in a contradiction, so the truth or untruth of a converse or a negative cannot be es- tablished from a single statement. If the converse or the negative is true, it is because of some one or more other propositions used in connection with the one to which it is related. If either the converse or the negative is true, both are true, for they are negative converses of each other. . See also Law 2. ILLUSTRATION. Suppose it is known that (1) If two lines crossed by a transversal meet, any pair of alternate angles are unequal. Then its negative converse must also be true, that is, (2) If a pair of alternate angles formed by two lines crossed by a transversal are equal, the lines are parallel. Without the use of one or more other geometrical state- ments, it cannot be known whether (8), its converse, or (4), its negative, is true or not. (3) If a pair of alternate angles formed by two lines crossed by a transversal are unequal, the lines meet (are not parallel}. (4) If two lines crossed by a transversal are parallel f any pair of alternate angles are equal. To Prove the Converse of a True Statement. Show that the figure constructed by using the condition and the figure constructed by using the conclusion are identical. For example, to prove (4) from (2) ; 212 PRELIMINARY CHAPTER If a figure is drawn with the angles equal, the lines will be parallel by (2) ; if the lines are drawn parallel, the same figure will result, because Through a point there can be but one line parallel to a second line, which is the other geometrical statement needed in this proof. To Prove the Negative of a True Statement. Show that the figure constructed by using the condition and the figure constructed by using the conclusion cannot be the same. For example, to prove (3) from (2) : If a figure is drawn with the angles equal, the lines will be parallel by (2) ; if another line is drawn through the same point making the angles unequal, it will be a different line, and so cannot be parallel since there can be but one parallel through the point. To prove the conclusion, it is necessary to use the additional geometrical principle that but one parallel to a given line can be drawn through a given point. WARNING. It must not be assumed that the converse and the obverse of a proposition are even likely to be true : if they are true, they can be proved as in the examples just given, or by the method of reasoning from statements covering all possibilities, which will be given later. NOTE. It should be observed in these examples of the methods used in proving a converse or a negative, that a line is added to the figure, and is then proved to coincide or not to coincide with one of the given lines. Nothing is assumed in constructing it as to whether or not it is the same line as the one in the figure, but if it is proved to be the same line, the two are then identical, and all proper- ties of the one are also properties of the other. The use of two lines which may be identical lines should not be confused with the use of two given lines, for all given elements are supposed to be distinct ; as, when two lines are given, they are assumed to be different lines. Were it not for this understanding, the proposi- tion, Lines parallel to the same line are parallel to each other, would need to be worded, Two different lines that are parallel to the same line are parallel to each other. METHODS OF ATTACK 213 The method of reasoning from the negative converse is very common throughout mathematics, as well as wherever reasoning is used in everyday life. The most common method of proving a proposition or other fact is to show : (1) That a certain condition must give a certain con- clusion (which it may require one or more steps of reason- ing to establish) ; (2) That the conclusion is not true. From this it follows by the negative converse law that the condition could not have been true. A few examples outside of mathematics follow : 1. If it were freezing, the puddles would be covered with ice. There is no ice on this puddle, so it is not freezing. 2. If the office boy had been attending to his work, the waste paper basket would have been emptied. It is not emptied, so he has not been attending to his work. 3. If this window had been broken from the inside, the glass would have fallen out. It has fallen inside, so the glass was broken from the outside. Other examples could be given without limit. In mathematics, when several steps are used to reason from the condition to an untrue conclusion, and so to show that the condition must have been untrue, the method of rea- soning is sometimes called reductio ad absurdum. 2. Reasoning from True Statements Covering all Possi- bilities. If statements such that their conditions cover all possi- bilities are true, and no two of their conclusions can be true at once, then their converses are also true. For example, suppose it has been proved that (I) If two sides of a triangle are equal, the opposite angles are equal; and 214 PRELIMINARY CHAPTER (2) If two sides of a triangle are unequal, the oppo- site angles are unequal, the larger angle being opposite the longer side. Then, since the conditions with regard to the sides cover all possibilities, and only one of the conclusions can be true, the converses of these statements must follow. Stated with regard to triangle ABC, Given ; If BC = CA, then correspondingly Z A ^ Z B. It follows that if Z.A $^B, then correspondingly _B<7= CA. If the explosion of the battleship Maine was within the ship, the plates must have been blown outward ; if it was outside the ship, the plates must have been blown inward ; then the converses of these statements are also true, and since it was found that the plates were bent in, the explo- sion must have been outside the ship. The truth of converse statements may also be shown by elimination of possibilities, that is by showing that all but one of the possibilities are untrue, and that the one remain- ing is therefore the true one because it is the only remain- ing possibility. For instance, in the foregoing geometrical illustration, given the same conditions, to find what is true if Z. A > Z B. If BC were equal to CA, then Z A would be equal to /. B, which is not true. If BC were < CA, then Z. A would be CA, because that is the only remaining possibility. Order and Arrangement of a Proof. In writing out a formal proof, it is best to have a definite order of arrange- ment that will show all the important details of the proof. The following arrangement is recommended for all propo- sitions: METHODS OF ATTACK 215 - STATEMENT OF PROPOSITION Diagram Given. The conditions of the proposition. To prove. The conclusion. (Both condition and con- clusion should be in terms of the figure. If it is a con- struction, Required should take the place of To prove.) Proof. 1. The proof should follow in numbered steps. (If it is a construction, the proof may be divided into Construction and Proof.) ILLUSTRATION A line and a point outside determine a plane. Given. Line AB and external point P. To prove. AB and P determine a plane. Proof. 1. Let K and L be two points on AB, then K, L, and P determine a plane (definition of plane, 5, p. 233). 2. But AB is entirely in this plane, for two of its points are in the plane (axiom of line and plane, 6, p. 234). 3. Therefore AB and P determine a plane, for they lie in one, and but one, plane. SECTION III. THE REPRESENTATION OF SOLID GEOMETRY FIGURES Since in solid geometry three-dimensional figures are usually represented on the paper or on the blackboard, that is, on a surface of but two dimensions, the student needs some idea of how to make the figures appear solid. While there is neither space nor time for a study of drawing, it is believed that the following brief directions may be of assistance in giving this idea. Unless the student has given some time to the study of perspective, the parallel line method will be the simplest for him to use. In this, a plane, or rather, since a plane is unlimited in extent, that portion of it shown in the figure, is, in general, represented by a parallelogram. It follows that many solids, since they are bounded by planes, can be drawn by the use of sets of parallel lines. This, of course, is not true of solids the conditions of which require that some portion of a plane shown in the figure shall be a circle, or a polygon other than a parallelogram. Where parallel lines are used, one set is usually drawn horizontally, the other somewhat inclined to the vertical. There are a few general rules that are of assistance in drawing all the figures, the most important being the following : The figures are usually supposed to be viewed from a point a little above, and either directly in front, or a little to one side. 216 REPRESENTATION OF SOLIDS 217 ** Lines nearer to the observer should be somewhat heavier than those at the back of the figure. As planes are supposed to be opaque, all lines appar- ently covered by them should be left out entirely, or, if needed in the figure, should be dotted. Where it seems desirable, lines that should not appear in the figure may be drawn and afterwards erased. A circle, unless viewed from a point directly opposite the center, appears elliptical, the ellipse being narrower as the eye is nearer its plane. It should be noted that two lines that intersect in a drawing do not necessarily intersect in the three-dimen- sional figure, and that the actual length of a line, or the size of an angle, may be very different from its representa- tion, as each part is drawn so as to appear to be of the right size in the three-dimensional figure represented. The Use of Squared Paper. At the beginning, squared paper will be found very convenient for drawing the solid geometry figures, for with it equal lines, parallel lines, and perpendicular lines can be drawn with no instru- ment but the straightedge. As the student gains skill in drawing he should not confine himself to this method, but should draw accurate figures on unruled paper, using compasses and ruler. He should also draw reasonably accurate freehand figures. If models of the solid figures are used in addition to the drawings, the two together will give the best possible idea of the three-dimensional figures. The following explanation shows how to draw parallel or perpendicular lines on squared paper: (1) Parallel Lines. (Fig. 1). Draw the lines so that they connect points the same distance apart, measured 218 PRELIMINARY CHAPTER along the lines of the paper, to the right or left, and up or down. For example, in Fig. 1, to draw a line from C parallel to AB, note that AB extends to the right 5 spaces, and up 15 spaces ; therefore count from C to the right 5 spaces, and up 15 spaces to D, and draw CD. This also makes the lines equal, as can be seen by this illustration, where CD = AB. If the lines are not to be of the same length, the same method is used, and that part of CD that is needed is taken. - (2) Perpendicular Lines. (Fig. 2). Count as before, but draw the new line so that one direction is reversed, that is, up for down, left for right, etc., and so that the two numbers are interchanged. In Fig. 2, to draw a line from P perpendicular to AB, note that AB extends 10 spaces down, and 15 spaces to the right, so draw from P to a point O, 15 spaces down, and 10 spaces to the left, or, 15 spaces up, and 10 spaces to the right. The Plane (Figs. 3, 4). Draw a parallelogram, usually with two edges horizontal. The other positions in which a plane most frequently appears will be taken up in the following figures. If, for any reason, it is desirable to emphasize the fact that the plane is unlimited in extent, one pair of edges can be broken, as in Fig. 4. Two Lines in Space (Figs. 5, 8). Intersecting or par- allel lines do not differ from those in plane geometry. Two lines not in the same plane, called non-coplanar lines, cannot be distinguished from intersecting lines, unless there is more in the figure to help show their relative position. In Fig. 5, the plane M appears to contain CD, and so makes it appear that AB and CD are not in the same plane. In P'ig. 8, AB and ES are not in the same REPRESENTATION OF SOLIDS 219 220 PRELIMINARY CHAPTER PLATE II F/G. 3. PLANES FIG. 4 F/G. 5. TWO NON-COPLAA/AR L/MES F/G. 6. A L/NE AND A PLANE F/G. 7 REPRESENTATION OF SOLIDS 221 plane, as is shown by the points in which they meet the planes M, N, and P. A Line and a Plane (Figs. 5, 6, 7, 8, 9). (1) The Line in the Plane. Draw it within the paral- lelogram (or other outline) that represents the plane, as AC and OB in Fig. 6. (2) The Line Parallel to the Plane. Draw it parallel to a pair of sides of the parallelogram, preferably to the horizontal sides, as ES in Fig. 6. If more than one line is to be drawn through a point parallel to the plane, draw them so that they will be within the outline that repre- sents another plane parallel to the given plane. Thus, in Fig. 8, OK and OL are parallel to plane P. (3) The Line Perpendicular to the Plane. Draw it perpendicular to the pair of horizontal sides, as PO in Fig. 6, AB in Fig. 8, and PO in Fig. 9. If a portion of a plane is represented by an outline other than a parallel- ogram, imagine the parallelogram you would draw to rep- resent that plane, and then draw the line so as to be perpendicular to two sides of that parallelogram. See P'o' in Fig. 7. (4) The Line Oblique to the Plane. Draw it so as not to be perpendicular to the horizontal sides, as PA, PB, PC in Fig. 6, BC and ES in Fig. 8, and PB in Fig. 9. Parallel Planes (Fig. 8). Represent them by paral- lelograms with their corresponding sides parallel. To represent portions of parallel planes, if the conditions of the figure require some other outline, as in Fig. 22, the corresponding sides should still be parallel. Intersecting Planes (Figs. 9, 10). First draw the line of intersection, preferably somewhat oblique to the vertical, 222 PRELIMINARY CHAPTER PLATE III 5 C 8. PARALLEL PLAMES f/G. 9. /MTEPSECr/A/G >LAA/ES * REPRESENTATION OF SOLIDS 223 so that the observer will appear to be looking at one plane from a little above, and at the other from a little to one side. Then fill in the outlines of the planes, keep- ing one pair of sides of the parallelogram representing each plane parallel to the intersection. If the planes are to be perpendicular to each other, make a pair of sides of one parallelogram perpendicular to a pair of sides of the other, as in Fig. 10; otherwise, let the sides not parallel to the intersection be oblique to each other, as in Fig. 9. In Fig. 9, the plane PAO is perpendicular to the plane M and to the plane N, but, on account of its position with reference to those two planes, it cannot be shown in the same way as in Fig. 10. It is made to appear perpen- dicular by being drawn so as to appear parallel to the horizontal sides of the parallelogram representing plane If, and to the corresponding sides of the parallelogram representing plane N. Note that PA and AO are made to appear perpendicular to the intersection by being drawn parallel to a pair of edges. In the same way, in Fig. 10, PO and OA are perpendicular to the intersection, while OB is oblique to it. Three Planes Intersecting in Parallel Lines (Figs. 11, 12). Draw the parallel intersections, dotting what- ever is covered by the planes, as in Fig. 12. The planes can be represented by completing the parallelograms as in these figures, or by using a broken edge, as in Fig. 13. For more than three planes intersecting in parallel lines, see Figs. 17, 18, 19. Three or more Planes Concurrent in a Point (Figs. 13, 14). Choose the point that is to be the vertex, as F, and draw the edges, and fill in the other boundaries 224 PRELIMINARY CHAPTER PLATE W F/G. /O. P&*PN)/C(/LAa FIG. /2 FIG. // THREE PLANES PARALLEL L//VE-S /A/ REPRESENTATION OF SOLIDS 225 /7G. /3 Th 'PEE OA MOPE PZ.A/VES ///A PO//VT A /70./6 T/fAfE OR MORE PLANES MEET/A/G /A/ A COMMO/V L/ME 226 PRELIMINARY CHAPTER afterwards, or draw the other boundary, and connect its vertices to the vertex F. Be careful that the edges behind the planes, as VE and VF in Fig. 14, do not lie so near the other edges as to cause confusion. The termina- tion of the figure can be by a plane, as ABC in Fig. 13, or ABCDEF in Fig. 14, or by a broken boundary, as 'in Fig. 13, when the fact that the figure is unlimited is to be emphasized. Three or more Planes Meeting in a Common Line (Figs. 15, 16). Draw the intersection, then the plane nearest the observer, and so on ; keep one side parallel to the intersection. The hidden lines can be dotted when they are needed in the figure. Parallelepipeds and other Prisms (Figs. 17, 18, 19). Draw one base, then draw parallel equal edges, and fill in the other base last. A pentagonal prism is shown in Fig. 17, an oblique parallelepiped in Fig. 18, and a rectangular parallelepiped in Fig. 19. The Cylinder (Fig. 20). Draw two parallel equal lines to represent two elements, then draw ellipses between their ends. ABCD is a section of the cylinder through an element. The Pyramid (Figs. 13, 14, 21, 22). The figures v- ABC in Fig. 13, and V- ABCDEF in Fig. 14 are pyramids and can be drawn as explained there. In Fig. 21 the pyramid is regular, as is indicated by its regular base, and by the fact that V is on the perpendicular at the center of the base. REPRESENTATION OF SOLIDS 227 PLATE VI Fid 17 f/G. /8 F/G. /9 PARALLELEP/PEOS A/VD Or/Y& A F/G. 20 CYL/MDER A B F/G.2/ F/G. 2 2 PYRAM/DS 228 PRELIMINARY CHAPTER For a frustum, draw a pyramid, with the lateral edges filled in lightly so that they can be erased where they are not needed ; then from some point on one edge, as P in Fig. 22, draw a parallel section by drawing lines parallel to the corresponding edges of the base from each lateral edge to the next lateral edge. For example, PQ is paral- lel to J?<7, etc. The parts of the lateral edges from the upper base to the vertex should then be erased. The Cone (Figs. 23, 24, 25). A cone can be drawn by drawing two lines from the point to be used as its vertex, and then filling in the elliptical base. Figure 23 is an oblique cone with a triangular section ABV ; Fig. 24 is a right cone with its altitude meeting the base at its center. In drawing a frustum, draw two elements between par- allel lines, that is, so that they would be legs of a trapezoid if their ends were connected, then fill in the elliptical bases, as in Fig. 25. The Sphere (Figs. 26, 27, 28, 29). The sphere is rep- resented by a circle, the appearance of solidity being given by the other lines drawn in the figure. Of these, circles on the surface of the sphere are the most frequent. In Fig. 26 a great circle G and a small circle 8 are shown, with their radii, and, since they are parallel, with their common axis PP f . All circles except the one used for the outline of the sphere should appear as ellipses, the ellipse being narrower as it is supposed to be looked at from a point more nearly in its plane, and more like a circle as it is supposed to be looked at from a point more nearly on its axis. To draw the arc of 'a great circle, the arc should be so curved that it will appear to meet the outline circle at the ends of a diameter. In Fig. 27 the spherical triangle REPRESENTATION OF SOLIDS 229 PLATE VII FIG. ^5 230 PRELIMINARY CHAPTER ABC is drawn, and the sides are continued in dotted lines, to show how their circles would appear to pass through the ends of diameters. In this figure, the observer is look- ing toward the center of the sphere from a point such that the line along which he is looking meets the surface in- side the triangle, as is shown by the fact that the sides all curve away from the inside of the triangle. In Fig. 28 the eye is in such a position that the sides of the tri- angle all appear to curve away from a point outside of the triangle. It makes little difference in what position the eye is supposed to be, if the arcs are drawn through the ends of diameters so that they give the appearance of solidity to the sphere. To draw polar triangles, draw the axis of each of the circles that form one triangle, using only that half of the axis that meets the same hemispherical surface on which the opposite vertex lies, and draw great circle arcs through the poles found, using the method shown in Figs. 27 and 28. Other Figures. The diagrams shown are those most fre- quently used in solid geometry, the others of importance being for the most part combinations of these. For ex- ample, polyhedral angles are formed by three or more planes meeting in a point ; a spherical pyramid is a com- bination of the pyramid with great circle arcs, the regular tetrahedron and the regular octahedron are respectively a pyramid, and two pyramids placed base to base, while a regular hexahedron is a rectangular parallelepiped with equal edges. If the pupil has clearly in mind the few principles by which these diagrams have been drawn, the building up of new and more complicated diagrams should not be found difficult. REPRESENTATION OF SOLIDS 231 PLATE Vffl F/o. 28. SPHER/CAL TR/AMGLES F/o.29. POLAR TR/AMGLES 232 PRELIMINARY CHAPTER NOTES Statements marked with an asterisk require proof. Such statements include many simple deductions from the definitions and axioms, not important enough to receive the emphasis given to propositions, as well as other im- portant statements that follow so naturally from the dis- cussion that a more or less informal proof of them seems most satisfactory. Both these classes of statements, which have been italicized, may be considered as corollaries of the definitions and axioms. In order to avoid circumlocution and monotonous repe- tition in the statements concerning mensuration, this book, in speaking of the length of a line, usually omits "the length of," and in speaking of the areas of plane figures usually omits "the area of," wherever there is no doubt as to the meaning. For example, in the state- ment, " The volume of a parallelepiped equals the product of its base and its altitude" "base" manifestly means "the area of the base " and, " altitude," " the length of the alti- tude," and should be so interpreted. BOOK VI. LINES AND PLANES SECTION I. THE PLANE 1. Space. The space in which everything exists is, as far as experience shows, unlimited. It is also divisible, for all the bodies with which we are familiar occupy por- tions of space. While the space studied in solid geom- etry (sometimes called Euclidean space) appears to be the same as that in which we exist, it is assumed only to be a space such that in it the axioms and postulates of geometry are true. Euclidean space is assumed to be unlimited in extent and to be divisible. 2. Solids. Any limited portion of space is called a geometric solid, or simply a solid. The term " solid " must not be confused with " solid body," for the geometric solid is studied as a part of space, and is entirely irrespective of any physical body that might occupy it. 3. Surfaces. That which separates one portion of space from an adjoining portion is called a surface. A surface may be limited or unlimited in extent, and can have limited portions. 4. Determining a Surface. As in plane geometry a straight line is said to be determined by any two of its points, so in solid geometry a surface of any specified kind is said to be determined by given points, lines, etc., if it is the only surface of its kind that contains them. 5. Planes. The surface that is determined by any three of its points that are not in a straight line is called 233 234 LINES AND PLANES a plane surface, or simply a plane. A plane is unlimited in .extent, but limited portions of a plane can be consid- ered. Familiar examples of a portion of a plane are, the surface of a desk, the surface of a blackboard, or any sur- face commonly spoken of as " flat." 6. Axiom of the Straight Line and the Plane. If two points of a straight line are in a plane, the whole line is in the plane. 7. Similarity between the Plane and the Straight Line. The plane occupies the same position among surfaces that the straight line does among lines; as indicated by the axiom, it is straight, that is, a straight line could lie in it, through any two of its points. Another similarity arises from the fact that the straight line is the one line determined by two of its points, while the plane is the one surface determined by three of its points. 8. Solid Geometry. Solid geometry treats of figures that do not lie entirely in one plane. In proving the propositions of solid geometry, any of the definitions, axioms, and propositions of plane geom- etry may be used, as well as the definitions and axioms of solid geometry. It is therefore necessary to find out what planes exist in a figure, and what points and lines are in those planes, in order to be able to apply plane geometry to as great an extent as possible, and to guard against using plane geometry propositions where they do not hold. 9. Determining a Plane. To determine a plane, two facts must be proved: that the determining points or lines are such that they lie wholly in that plane, and that no other plane can contain them. It has already been said that a plane is determined by (1) three points not in a straight line; THE PLANE 235 It is also determined by (2) A. line and a point outside that line ; for two points on the line and the point outside determine the plane, and the line is wholly in the plane since two of its points are in the plane ; (3) two intersecting lines (Prove as in (2)) ; (4) two parallel lines (See the definition of parallel lines). 10. Representation of a Plane. Although a plane is unlimited in extent, only a limited part can be shown in the drawing of a figure, just as in plane geometry only a limited part of an unlimited line is shown. A parallelogram has been found to be, in general, the best representation to use in such a drawing. It is well to make the side sup- posed to be nearer the observer a little heavier than the others. Such a plane is denoted by a single letter, as plane M, by the Betters at opposite vertices of the parallelogram, as plane AB, or by some determining points, or lines, as plane ABC, ifit contains points A, B, and C, or plane , b if it contains lines a and b. For a more extended discussion of figures, see Section III of the Preliminary Chapter. EXERCISES 1. Does a triangle determine a plane ? Why? 2. Why does a three-legged stool stand firmly on the floor, while a four-legged chair or table will sometimes rock ? 3. If four points are given, how many planes can be determined? What is the smallest number of planes in which the sides of a quadri- lateral can lie ? In how many planes can all the sides of a parallelo- gram lie? 236 LINES AND PLANES 4. In a series of parallels such that not more than two lie in any one plane, how many planes would be determined by using three of them ? four ? 5. Show that all transversals of two intersecting lines are in the same plane. 11. Coplanar Figures. Figures in the same plane such as two intersecting lines are said to be coplanar, those not in the same plane are said to be non-coplanar. When figures are said to lie in the same plane, or to be coplanar, it is meant that a plane can be passed through the figures so as to entirely contain them, not that there is necessarily a given plane that does contain them. Similarly, when any figures are said to be non-coplanar, the meaning is that no plane could be passed so as to contain them. The fact that two lines, or other parts, appear in different planes of a given figure does not, therefore, mean that they must be non-coplanar, for there may be some other plane that can be passed through them. SECTION II. RELATIVE POSITIONS OF LINES AND PLANES IN SPACE ; PARALLELS AND INTERSECTIONS 12. Necessity of the Study of Relative Positions. In plane geometry the study of the relative positions of points and lines is very informal, because the possibilities are few ; but in solid geometry, points, lines, and planes admit of many relative positions. In order to give a clear idea of the figures, their more fundamental combi- nations will be studied. ^ 13. Relative Positions of Two Lines. Two lines can be (1) Coplanar, therefore, as in plane geometry ; (a) intersecting. (6) parallel. (2) Non-coplanar, therefore neither intersecting nor parallel. An illustration of non-coplanar lines can be shown by crossing two pencils, and then moving one away from the other without changing the position of either except for this separation. The edge of the side wall and the ceiling of a room, and the edge of the back wall and the floor would also be a pair of such lines. The angle between two non-coplanar lines means the angle between one of them and an intersecting line parallel to the other. All such angles formed for a certain pair of lines are equal, as is proved in 42. When two lines are neither intersecting nor parallel, they are not coplanar. When two lines are given without 237 238 LINES AND PLANES any conditions that show without further proof whether or not they are intersecting or parallel, they cannot be assumed to determine a plane. One of them with a point of the other does, however, determine a plane, and if the second line should also lie in that plane, it may then be possible to prove this fact. 6. Transversals of two non-coplanar lines are neither parallel nor concurrent unless they meet at a point on one of the lines. 14. Relative Positions of a Line and a Plane. A line either (1) lies in a given plane ; (2) intersects the plane ; or (3) is parallel to the plane. A line has already been said to lie in a certain plane if two of its points are in that plane. Other methods of proving ihat it lies in a certain plane will be taken up later. A line intersects a plane if it meets the plane in one point only. The point is called the foot of the line with respect to that plane. A line is parallel to a plane, and the plane is parallel to the line, if they have no point in common. The given line evidently cannot meet any line in the plane, but it is parallel only to such of those lines as are coplanar with it. Why? 15. Relative Positions of Two Planes. Two planes either (1) coincide; (2) intersect ; or (3) are parallel. The study of two planes will also include some use of a line with the two planes. 16. Planes coincide if no point of one is outside the other. Coincidence is proved by showing the two planes to be determined by the same determining points," lines, or points and lines. (See 9.) 17. Two planes intersect when they have at least one common point, but do not coincide* RELATIVE POSITIONS OF LINES AND PLANES 239 18. Axiom on Intersecting Planes. Two intersecting planes have at least two points in common. 19. Theorem I. The intersection of two planes is a straight line. They intersect in two points. What is known of the line through those points? Can the two planes have any other points in common ? 7. Three planes that do not contain the same straight line can have but one point in common. 8. There can be but one line from an external point cutting two non-coplanar lines. 9. If a paper is folded and creased smoothly, why does it form a straight edge along the fold? 20. COR. If a line is parallel to a plane, it is parallel to the intersection of any plane in which it lies, with that plane. 10. If a line in one of two intersecting planes is parallel to the other plane, it is parallel to the intersection. 11. If a line is parallel to a plane, any two parallels drawn from that line to the plane are equal. 21. Parallel Planes. Two planes are parallel when they have no points in common. If two planes are parallel, either plane is, by definition, parallel to all the lines in the other ; also, since any line in one of the planes is parallel to the other plane, it cannot meet any line in that other plane, but is parallel only to such of those lines as are coplanar with it. Why? 12. What is known about the relative positions of two lines that are parallel to the same plane ? 240 LINES AND PLANES 22. Proving a Line to lie in a Plane. It has already been said that a line is in a plane if the plane contains two of its points. There are at present three other ways in which a line can be shown to be in a plane. * 23. A line is in a plane if the plane contains one of its points, and also contains a second line to which the given line is parallel. Compare the plane determined by the parallels, and the given plane. A direct consequence of this statement is * 24. A plane containing one, and but one, of two paral- lels is parallel to the other. For if the second parallel should have a point in the plane, what would follow? When this statement is used to show that a line is parallel to a plane, the following wording is convenient : If a line outside a given plane is parallel to a line in the plane, it is parallel to the plane. * 25. A line is in a given plane if the plane contains one of its points, and if the line and the plane are both paral- lel to a second line. The parallel lines determine a plane ; examine its inter- section with the given plane. * 26- A line is in a given plane if the plane contains one of its points, and if the line and the plane are both paral- lel to a second plane. Use 25. 27. The four methods, thus far considered, of proving a line in a plane, might be summed up as follows : RELATIVE POSITIONS OF LINES AND PLANES 241 A line is in a given plane if one of its points is in that plane, and (a) if a second point of the line is in the plane ; (6) if a second line parallel to the given line is in the plane; () inter- secting, (c) parallel (14) ; of two planes: (a) coinciding, (b) intersecting, (c) parallel (15); of three planes : () intersecting in three lines, (b) intersect- ing in two lines, (c) all meeting in one line, (d) mutually parallel (33). (3) Intersections of a line and a plane, a point (14) ; of two planes, a line (19) ; of three planes : a line (38) ; two parallel lines (35) ; three lines parallel or concurrent (34). SUMMARY OF PROPOSITIONS 271 (4) A line is parallel to a plane : if they have no point in common (14) ; if it is not in the plane and is parallel to a line in the plane (24) ; if it is not in the plane, and the line and the plane are both parallel to the same line or plane (41). (5) A line is perpendicular to a plane : if it is perpendicular to all lines in the plane through its foot (52) ; if it is perpendicular to two intersecting lines in the plane (62) ; if it is perpendicular to a plane that is parallel to that plane (57) ; if it is parallel to a line that is perpendipular to the plane (58) ; if it is in a plane perpendicular to it and is perpendicular to the intersection of the two planes (72) ; if it is the intersection of two planes that are perpendicular to that plane (74). (6) Transversals of parallels : If a line cuts one of two parallel planes, it cuts the other also (28). If a plane cuts one of two parallel lines, it cuts the other also (28). If a plane cuts one of two parallel planes, it cuts the other also (36). IV. ANGLES : (1) Equal plane angles : Angles whose arms are respectively parallel are equal (42). Angles made by equal obliques with a perpendicular^are equal (81). Measuring angles of equal dihedral angles are equal (66). (2) Unequal plane angles: Angles made by unequal obliques with a perpendicular are unequal (81). The angle between an oblique line and its projection upon a plane is the least angle it makes with any line in the plane (82). 272 LINES AND PLANES (3) Dihedral angles having equal plane angles are equal (67) ; are proportional to their plane angles (68) ; are measured by their plane angles (69) . (4) Measuring angles have their plane determined , by two lines in the faces perpendicular to the edge at the same point (65) ; by two intersecting lines perpendicular to the faces of the dihedral angle (75). V. Locus: of points equidistant from two given points, a plane perpendicular to their sect at its midpoint (8ft) ; three given non-collinear points, a line perpendicular to their plane at the circuincenter of the triangle of which they are vertices (90) ; four given non-coplanar points of which no three are collinear, the point of concurrence of the planes perpendicular to their sects at their midpoint (91) ; two intersecting planes, the planes bisecting the dihedral angles formed (92) ; three planes meeting in a point, the lines of intersection of the planes bisecting the dihedral angles (93) ; four planes meeting in four points, the points common to the bisectors of the dihedral angles (94). 96. ORAL AND REVIEW QUESTIONS What are the two most fundamental statements that can be made about a plane ? In order to use plane geometry propositions in proving solid geometry theorems, what must first be done ? By what methods can this be done ? Show that two transversals of the arms of an angle are, in general, coplanar. What is the exception ? In what relative positions can a point and a line lie? a point and a plane ? a line and a plane ? two lines ? two planes ? What is the intersection of a line and a plane? Why? of two planes? Why? State six methods of proving that a line is in a plane. Why is it ever necessary to prove that a line is in a plane ? What is the definition of a line parallel to ORAL AND REVIEW QUESTIONS 273 a plane ? Is there a simpler method of showing that a line is parallel to a plane? What is the definition of parallel planes? State two other methods of proving planes parallel, founded directly on this defi- nition, and state and prove the other theorem used in each method. How many intersecting lines can be parallel to the same line? parallel to the same plane? How many intersecting planes can be parallel to .the same plane? to the same line? State all possibilities for the relative positions of three planes. If three planes intersect in pairs, is it possible for them to have two non-coplanar intersections ? Is it pos- sible with four planes? Are planes necessarily parallel if they con- tain parallel lines ? Are lines necessarily parallel if they are in parallel planes ? How many lines through a point can be perpendicular to the same line ? to the same plane ? How many intersecting planes can be perpendicular to the same line? to the same plane? Which of the following must be parallel (or coincident, including as a special case the line being in the plane) : lines parallel to the same line ; planes parallel to the same plane ; lines parallel to the same plane ; planes parallel to the same line ; lines perpendicular to the same line ; lines perpendicular to the same plane ; planes perpendicular to the same line ; planes perpendicular to the same plane ? How many intersect- ing lines can be perpendicular to the same line? to the same plane? What is the definition of perpendicular planes? How many lines are used to show the planes perpendicular to each other ? Can it be proved with a less number? If two planes are perpendicular to each other, how can a line be drawn in one so as to be perpendicular to the other? If two planes are perpendicular to each other, is every line in one perpendicular to the other? Are all perpendiculars to one plane at the intersection in the other plane ? Where is the pro- jection of lines in one plane on the other plane? What is the defi- nition of a line perpendicular to a plane? How many lines of the plane does it use ? Can the line be proved perpendicular to the plane with fewer lines ? State two methods that use parallels to prove a line perpendicular to a plane. State two methods that use perpen- dicular planes to prove a line perpendicular to a plane. Define dihedral angle, and its plane angle. In what two ways is the plane of a measuring angle determined ? Explain in full in what sense and why the plane angle of a dihedral angle measures it. Can the projection of a straight line upon a plane ever be a curve? Can the projection of a curve ever be a straight line ? Where must a pro- 274 LINES AND PLANES jected line lie, in order that its projection upon a plane shall be a straight line () if it is straight? (6) if it is not straight? If a line when pro- jected upon each of two planes makes straight lines on both, what must be true of it? (Two cases.) Give the determining lines that must be drawn in order to construct a plane (1) through a point (a) parallel to a given plane, (ft) parallel to two non-coplanar lines, (c) perpendicular to a line, (d) perpendicular to a plane ; (2) through aline (a) parallel to a non-coplanar line, (ft) perpendicular to a plane. State two methods that have been used to prove two lines perpendicular. What methods have been found to prove angles equal ? unequal? sects equal? un- equal ? sects proportional ? Upon what plane geometry locus theorems can the following be made to depend : locus of points equidistant from two points? three points? two lines? two planes? State all the propositions that prove that there can be a line or a plane that fulfills certain conditions ; all propositions that prove that there can be but one line or one plane that fulfills certain conditions. GENERAL EXERCISES 82. If two perpendiculars to a plane from two points on the same side of it are equal, the line joining those points is parallel to the plane. If three or more such perpendiculars are equal, the points determine a plane parallel to the given plane. 83. If two planes are drawn perpendicular respectively to two non-coplanar lines, their intersection is perpendicular to any plane parallel to the given lines. 84. Find the locus of points in a given plane, equidistant from two given planes NOTE. Two points equidistant from a plane, on the same perpen- dicular to that plane, are called symmetric with regard to that plane. 85. Prove that two points symmetric to a plane are equidistant from any point in the plane. 86. If A and B are on the same side of plane M, find the shortest path between A and B that includes one point of M. (Use the point symmetric to either A or B.} This is simply finding the shortest way to go from A to M and back to B. It is the path that a ray of light travels when it meets a mirror and is reflected; it is the path followed by a billiard ball striking a cushion and bounding back, or GENERAL EXERCISES 275 by a tennis ball bounding from the ground, unless, in either of these cases, the ball is affected by some motion, such as whirling, other than the mere rebound. 87. If two balls are in positions A and B on a billiard table, show how the point on a cushion could be determined so that if A strikes the cushion at that point, it will rebound to B. Show how a point could be determined so that A would strike two cushions and re- bound to B', three cushions; all four cushions. Note that in the last cases symmetry is being used in regard to two or more planes. 88. Find the shortest path in two intersecting planes between a point in one of the planes, and a point in the other plane. 89. Find the shortest path in the surface of a box from one corner to the opposite corner. 90. If two lines in one of two intersecting planes make equal angles with the intersection, they make equal angles with the other plane, and conversely. 91. Perpendiculars from a point to a set of parallel lines re co- planar. 92. The arms of an angle are equally inclined to any plane through its bisector. 93. The arms of an angle are equally inclined to any plane through a coplanar line perpendicular to its bisector. 94- If a line is perpendicular to one of two intersecting planes, the projection upon the other plane is perpendicular to the intersection. 95. If any number of planes perpendicular to the same plane have a common point, they intersect in a common line. 96. If a plane and a line are parallel, a plane perpendicular to the line is also perpendicular to the plane. Is the converse also true? 97. If three or more lines are drawn from points on one of two non-coplanar lines perpendicular to the other line, the sects cut off on the two non-coplanar lines by these perpendiculars are propor- tional. 98. If a common perpendicular is drawn to two lines that are either parallel or non-coplanar, the plane perpendicular to that common per- pendicular at its midpoint bisects every transversal of the two lines. 276 LINES AND PLANES 99. If two congruent rectangles are placed so that they have an edge in common but are not coplanar, their common edge is perpen- dicular to two of the planes determined by other edges, and is paral- lel to a third plane. State the general case. 100. When will lines perpendicular to two intersecting planes meet? Prove ib. 101. Planes perpendicular to intersecting lines meet. What can be told about their intersection ? 102. Given three lines, no two of which are coplanar : (a) pass a plane through one of them so as not to be parallel to either of the others ; (6) draw a line through all three of them. Can a line be drawn through one of them parallel to the other two? a plane? 103. Given a line lying within and parallel to the edge of a dihe- dral angle, how could a plane be passed through this line, intersect- ing the faces of the angle, so that the parallels so formed would be equidistant from the given line? 104. Given a line outside a dihedral angle parallel to its edge, how could a plane be passed through this line meeting the faces of the dihedral angle so that the distance apart of the parallel intersections shall equal the distance of one of them from the given line. 105. If a circle is inscribed in a triangle, the lines from any point in the perpendicular erected to the plane of the circle at the center to the points of contact are perpendicular to the sides of the triangle. 106. The lines from any point in the perpendicular to a plane at the incenter of a given triangle, perpendicular to the sides of the triangle, meet them at the points where they are tangent to the in- scribed circle. 107. If in each of two intersecting planes two lines are drawn parallel to the intersection, and such that all four lines are the same distance from the intersection, those lines determine two pairs of parallel planes. 108. If four lines determine more than four planes, the lines are either all concurrent or all parallel. 109. If two pairs of parallel planes intersect, the other two planes determined by their intersections meet in a line parallel to the given planes and to their intersections, and equidistant from each opposite pair of intersections. GENERAL EXERCISES 277 110. If one pair of parallel planes is perpendicular to a second pair of parallel planes, the distance between one pair of opposite in- tersections equals the distance between the other pair of opposite intersections. 111. If three planes meet in parallel lines, and two of them make equal angles with the third, then the bisector of the dihedral angle between those two planes is perpendicular to the third plane. 112. If two pairs of parallel planes intersect, and the consecutive pairs of their intersections are the same distance apart, the diagonal planes of the figure formed are perpendicular to each other. 113. If two parallel planes are crossed by two non-parallel planes that make equal angles with one of them, (a) those planes make equal angles with the other of the parallel planes ; (6) the opposite intersections are equidistant; (c) the non-parallel planes meet in a line parallel to the given intersections, and equidistant from those in either of the parallel planes ; (<^J the given intersections in one of the non-parallel planes are the same distance apart as the intersections in the other of those planes. (e) State and prove the converse of (W). 114- If two triangles (as ABC and XYZ} in different planes are so placed that each corresponding pair of sides intersect (as, AB and XY, BC and YZ, CA and ZF), then the lines through their corre- sponding vertices (as AX, BY and CZ) are either parallel or concur- rent. BOOK VII. POLYHEDRONS, CYLINDERS, AND CONES SECTION I. THE PRISM AND THE CYLINDER 97. Polyhedrons. A geometric solid bounded by planes is 'A polyhedron. The polygons that bound it are its faces, their intersections are its edges, and the intersections of its edges are its vertices; the faces taken together make up its surface. The area of its surface is the area of the polyhedron, and the amount of space that it occupies is the volume of the polyhedron. See also 135, which explains how volume is measured. A POLYHEDRON A TETRAHEDRON 98. Number of Faces. It. is evident that, in order to inclose space, a polyhedron must have at least three faces about each vertex, and at least four faces in all. A solid bounded by four faces is called a tetrahedron; by five faces, a pentahedron ; by six faces, a hexahedron ; by eight faces, an octahedron; by ten faces, a decahedron; by twelve faces, a dodecahedron; and by twenty faces, an icosahedron. The corresponding prefixes are used for the polyhedrons of any number of faces, but those here men- tioned include the most common. 278 THE PRISM AND THE CYLINDER 279 99. Polyhedral Angles. When three or. more planes meet in a point, they include & polyhedral angle, or a solid angle. In the figure, the planes OVW, OWX, OXY, OYZ, OZV meet in O, and form a polyhedral angle O, or more defi- nitely O-VWXYZ. The common point O is its vertex, the portions of planes that bound it are its faces, and the intersection lines of its faces are its edges. The faces and edges extend indefi- nitely from the vertex,, but, for convenience in dealing with a polyhedral angle, a plane, as ABODE, is sometimes considered as having cut its edges. 100. Dihedral and Face Angles. A polyhedral angle evidently has as many dihedral angles as edges, and as many face angles, that is, angles about the vertex in its faces, as Z AOB, /.BOG, etc., as faces. The dihedral angles and face angles are sometimes called its parts, just as the sides and angles of a triangle are referred to as its parts. A polyhedron has a polyhedral angle at each vertex, a dihedral angle at each edge, and face angles about each vertex, the number of face angles depending on the kind of polyhedral angle at that point. 101. Sections. When a plane is passed through a solid, the figure in the plane bounded by its intersection with 280 POLYHEDRONS, CYLINDERS, AND CONES the surface of the solid is called a plane section, or simply a section, of the solid. It is evident that any section of a solid must be bounded by a closed line, or lines. Why ? Similarly, a plane passed through all the edges of a pylyhedral angle, but not through the vertex, forms a polygon of as many sides as the polyhedral angle has faces. In the diagram of 99, ABODE is a section of the polyhedral angle O. 102. Convex Figures. Either a polyhedral angle, or a polyhedron, is said to be convex if no part of the plane of any one of its faces lies within the space inclosed by its faces. As only convex figures will be dealt with in this book, the word convex will be understood whenever polyhedral angles or polyhedrons are used. 103. The General Polyhedron. Elementary geometry is concerned mostly with certain classes of polyhedrons, such as prisms, parallelepipeds, and pyramids, all of which will be defined when they are to be used. The general poly- hedron is discussed to some extent in 323. 104. The Prismatic Surface and the Cylindrical Surface. If a moving straight line always remains parallel to its original position, and always intersects a given straight line, it evidently describes, or generates, a plane. Why ? In the figures on p. 281, AB represents the original position of the moving line (or generatrix), while XY rep- resents the guiding line (or directrix). A straight line that always remains parallel to its origi- nal position, and always intersects a line that is in one plane, but is not coplanar with the moving line, (a) if the guiding line is broken, generates a prismatic surface; which is evidently composed of a number of por- THE PRISM AND THE CYLINDER 281 PLANE PRISMATIC SURFACE CYLINDRICAL SURFACE tions of planes intersecting in parallel lines, which are the edges of the prismatic surface. (5) if the guiding line is curved, generates a cylindrical surface, of which no part is plane, for if it were, that part of the guiding line would be straight. The moving line in each of its various positions is called an element of the cylindrical surface. For some purposes, it is convenient to let " element " also apply to prismatic surfaces. A non-coplanar guiding line may give a combination of prismatic and cylindrical surfaces, and need not be con- sidered in an elementary course. 105. Closed Surfaces. If the guiding line is a closed line, the prismatic or cylindrical surface is also closed, and the space inclosed is called prismatic or cylindrical space. X U 106. Sections. If a plane that is not parallel to the moving line meets a closed prismatic or cylindrical surface, 282 POLYHEDRONS, CYLINDERS, AND CONES it cuts all elements and so cuts the surface in a closed line. The figure bounded by this closed line is called a section of the prismatic or cylindrical space. If it is perpendicu- lar to an edge or element, it is a right section ; otherwise it is an oblique section. When two sections are made by parallel planes, they are called parallel sections. In the figure, ABCD and EFGH are parallel sections of the pris- matic space ; XT is a section of the cylindrical space. * 107. The edges of a prismatic surface, or the ele- ments of a cylindrical surface, between parallel sections are equal. * 108. Parallel sections of a prismatic space are con- gruent polygons. 109. Theorem I. Parallel sections of a cylindrical space are congruent. Let planes M and N make parallel sections of a cylin- drical space. Take points A and B on the perimeter of section 3f, and let P represent any third point of that perimeter. Draw the elements through A, B, and P to A f , #', and P' on the perimeter of section jy, and draw THE PRISM AND THE CYLINDER 283 AB, AP, BP, A'B', A'P', B'P'I Then AABP^AA f B r P f (why ?), and if section M is superposed on section N with AB coinciding with A'B', P falls on p 1 . But P may be any point on the perimeter of section M, so each point of section M falls on a corresponding point of section N, and conversely, and the two sections coincide and are congruent. 110. Prisms and Cylinders. The part of a prismatic space between two parallel sections is called a prism. The part of a cylindrical space between two parallel sections is called a cylinder. The parallel sections are the bases, while the prismatic or cylindrical surface is the lateral surface of the figure. The area of the lateral sur- face is the lateral area of the figure. In the case of the prism, the planes forming the lateral surface are lateral faces ; their intersections are lateral edges. A prism or a cylinder is right or oblique according as its bases are right or oblique sections. 111. Altitude. The altitude of a prism or a cylinder is the perpendicular between its bases. 112. Classification of Prisms according to Number of Faces. A prism is triangular, quadrangular, pentagonal, etc., according as its bases have three, four, five, etc., sides. 113. Regular Prisms. A right prism whose bases are regular polygons is a regular prism. 114. Circular Cylinders. A cylinder whose bases are circles is a circular cylinder; if its bases are also right sections, it is a right circular cylinder, or since it might be generated by revolving a rectangle around one side as an axis, it is sometimes called a cylinder of revolution. The cylinders considered in elementary solid geometry are almost always circular cylinders. See Appendix, 321. 284 POLYHEDRONS, CYLINDERS, AND CONES 115. Axis and Radius ofa Circular Cylinder. The line joining the centers of the bases of a circular cylinder is called its axis. The radius of the base of a circular cylinder is called the radius of the cylinder. 115. Find the locus of points at a fixed distance from a given line. 116. The Cylinder the Limiting Case of a Prism. A prism is said to be inscribed in, or circumscribed about a cylinder if its bases are respectively inscribed in, or cir- cumscribed about, the bases of the cylinder. The follow- ing statement is assumed without proof. Its truth is manifest, but it is difficult to prove. Theorem II. If a prism, the base of which is a regu- lar polygon, is inscribed in, or circumscribed about, a given circular cylinder, the lateral area of the prism approaches the lateral area of the cylinder as its limit, and the volume of the prism approaches the volume of the cylinder as its limit, as the number of faces of the prism is increased indefinitely. It follows that the circular cylinder is the limiting case of the prism having a regular base, and that theorems true for the one will usually be true for the other. This is why the two solids are here treated to- gether. 117. Theorem III. A section of a cylinder made by a plane containing an element is a parallelogram. 116. A section of a prism made by a plane parallel to an edge is a parallelogram. 118. Tangent to a Cylinder. A plane is tangent to a cylinder if it meets its surface in one element, and in no point outside that element. THE PRISM AND THE CYLINDER 285 * 119. If a plane is tangent to a circular cylinder, its intersection with the planes of its bases are tangent to the bases. For the intersection has but one point on the circum- ference. * 120. The plane determined by a tangent to a base of a circular cylinder, and the element from the point of contact, is tangent to the cylinder. For if it met the cylinder in a point outside the ele- ment what would follow ? 121. Theorem IV. The lateral area of a prism is the product of an edge and the perimeter of a right section. What form would this take for a right prism ? 122. COR. The lateral area of a right circular cylinder is the product of an element and the circum- ference of the base. Inscribe a regular prism of lateral area _L', edge e, and perimeter of the base p, in the right circular cylinder of lateral area L, element e, and circumference of the base c. Then L' = ep. If the number of faces of the prism is increased without limit, L = L f , and p = c. Why ? But if a variable approaches a limit, its product by a constant will approach the product of its limit by that constant, therefore ep = ec. Also, since L 1 and ep are two variables that are constantly equal, and are approaching the limits L and ec, those limits are equal, that is, L = ec. If the cylin- der is of radius r and altitude A, its lateral area = 2 Trrh. 117. Prove, without using the formula for the lateral area, that the lateral areas of two prisms cut from the same prismatic space and having an equal lateral edge are equivalent. 286 POLYHEDRONS, CYLINDERS, AND CONES 118. Prove, without using the formula for the lateral area, that the lateral areas of two cylinders cut from the same cylindrical space and having an equal element are equivalent. 119. Find the lateral area of a right prism of altitude 5 in., and having an equilateral triangle of side 4 in. as its base. 120. Find the total area (lateral area and bases) of a regular hex- agonal prism of edge (base and lateral) 6 in. 121. Find the lateral area of an oblique prism of edge 10 ft., hav- ing as its right section a square of side 2.5 ft. 122. What is the lateral area of a right circular cylinder of ele- ment 12 in. and base of radius 3 in.? 123. The pillars of a colonial house are regular hexagonal prisms of base edge 1 ft., and height 30 ft. How much surface must be painted on four of them ? 124- How much is the surface of a log 14 in. in diameter and 12 ft. long diminished by cutting the largest possible square timber out of it? 125. The circular right section of a cylinder has an area of 54 sq. ft., its altitude is 25 ft., and its elements make a 60 angle with the base. Find the lateral area. 123. Theorem V. Two right prisms, or two right cylin- ders, are congruent if they have congruent bases and equal altitudes. Superpose. 124. Equivalence. Two solids are said to be equivalent when their surfaces inclose the same amount of space. 126. Two cylinders cut from the same space and having an equal element are equivalent. Show that the solid from the lower base of one to the lower base of the other can slide along the cylindrical space so as to coincide with the solid between the upper bases, etc. 127. An oblique cylinder is equivalent to a right cylinder whose base is its right section, and whose altitude is its element. 125. The Parallelepiped. A prism whose bases are parallelograms is called a parallelepiped. It is manifest from this definition that all the faces of a parallelepiped THE PRISM AND THE CYLINDER 287 are parallelograms, and that it is therefore bounded by three pairs of parallel planes, and has three sets of four equal parallel edges. Explain in full. * 126 . Any pair of opposite faces of a parallelepiped may be considered to be its bases. For the parallelepiped fulfills the conditions of the definition of a prism whichever faces are used as bases. Therefore, the opposite faces of a parallelepiped are congruent. 127. Right and Rectangular Parallelepipeds. A parallel- epiped whose lateral edges are perpendicular to its bases is, by the definition of a right prism, a right parallelepiped ; otherwise it is oblique. If, in addition to being right, it has rectangular bases, it is called a rectangular parallel- epiped, for all its faces are rectangles. Many shorter names for rectangular parallelepipeds have been suggested but no one is in general use. The term cuboid seems as convenient as any. 128. Cube. A rectangular parallelepiped having all its edges equal, and therefore all its faces square, is called a cube. 129. Diagonal. The sect joining any vertex of a par- allelepiped to the one not in the same face is called a diagonal of the parallelepiped. 130. Dimensions. In a rectangular parallelepiped the lengths of the three edges drawn from one vertex are the dimensions of the figure. 131. Ratio between Solids. By "the ratio of one solid to another " is meant " the ratio of the amounts of space 288 POLYHEDRONS, CYLINDERS, AND CONES occupied by them." This can of course, be worked out only in terms of congruent parts of the solids. NOTE. At the discretion of the teacher,' 132, 133, and 134, may be omitted and 136 may be assumed as the definition of the volume of a rec- tangular parallelepiped. The author does not recommend this course, but it has the sanction of some authorities. 132. Theorem VI. Two rectangular parallelepipeds having two dimensions in common are to each other as their third dimensions, For the commensurable case, divide the third dimen- sions by a common divisor, pass planes through the points of division dividing the given parallelepipeds into con- gruent parts, and prove the proportion. The incommensurable case can be proved by limits, but it is not of Gen required in an elementary course. 133. Theorem VII. Two rectangular parallelepipeds having one dimension in common are to each other as the products of their other dimensions. Compare each with a third parallelepiped so con- structed as to make this possible. 134. Theorem VIII. Two rectangular parallelepipeds are to each other as the products of their three dimen- sions. Use the method of Theorem VII. 128. Prove Theorem VIII without using Theorem VII, by con- structing two auxiliary parallelepipeds with which to compare the given ones. 135. Volume: Cubic Unit. The unit of measure for solid figures is the cubic unit, or unit of volume, which is a cube whose edge is a linear unit, and whose faces are therefore units of area. Since volume is always expressed in cubic units, the volume of any solid is its ratio to a cubic unit, or, in other words, the number of cubic units it contains. THE PKISM AND THE CYLINDER 289 136. Theorem IX. The volume of a rectangular par- allelepiped is the product of its three dimensions. Let B be a parallelepiped of dimensions a, 5, c, and let U be a unit of volume, of edges equal to the linear unit u. Then compare the two parallelepipeds : E a b c = - X - X -, U u u u But is the volume of the parallelepiped, and -, -, and - U u u u are the three dimensions in length units, therefore, The volume of a rectangular parallelepiped (in cubic units) equals the product of its three dimensions (in length units). Evidently this volume might also be expressed as the product of its base and its altitude. Why ? 129. How many cubic yards of earth must be removed in digging a rectangular ditch 6 ft. deep, 3 ft. wide, and one quarter of a mile long ? 130. A log 2 ft. in diameter and 24 ft. long has been sawed in two lengthwise, making two semicircular timbers. If the largest possible square timber is cut out of each of these parts, how many board feet (1 ft. square and 1 in. thick) will they contain ? 131. In ex. 130, if the largest possible square log had been cut from the original log, how would the number of board feet in it compare with the number in the two timbers ? 137. Theorem X. If two solids contained between the same two parallel planes are such that their sections by any third plane parallel to these two planes are equiva- lent, the two solids have the same volume. This is " Cavalieri's Theorem," and such solids as those described are called Oavalieri bodies. The theorem de- pends on limits for its proof, but the proof is too difficult at this point. See Appendix, 319. To prove two solids equivalent in volume by this theo- 290 POLYHEDRONS, CYLINDERS, AND CONES rem, it is necessary (1) to show that the solids can be placed between the same parallel planes ; (2) to pass any third parallel plane through the solids, and to prove that the sections obtained are equivalent. 138. Theorem XI. Prisms cut from the same pris- matic space and having equal lateral edges are equiva- lent in volume. For the part of the prismatic space between their upper .bases can coincide with the part between their lower bases ; and after taking each of these parts from the whole figure formed by the prisms, equivalent prisms remain. Draw the prisms so that their bases do not intersect, as this would unnecessarily complicate the figure. They may be entirely outside each other^ or may have some common space. Note the similarity between this proof and one method of proving that parallelograms having equal bases and lying between parallels are equivalent. 139. Theorem XII. A parallelepiped is equivalent to a rectangular parallelepiped of equivalent base and equal altitude. [V 1 N t. / 5 1 \ M \ i; / : ' JV 1 * \ c\ -H ;/j^"- / ? A B t JT I ^y For, given the oblique parallelepiped B, with the base ABCD, and the altitude h, pass two planes M and N, whose distance apart, XF, equals AB, through the prismatic space of which E is a part, so as to be perpendicular to AB. THE PRISM AND THE CYLINDER 291 Then these planes cut from the prismatic space a new parallelepiped S equivalent to .R, since XY = AB, and having its rectangular base XTZW equivalent to ABCD (why?), and the same altitude h. In S, the planes M and N are perpendicular to XYZW, but the other lateral faces may or may not be perpendicular to the base. The same operation applied to S by extending FZ and its parallels to form the edges of a prismatic space, and passing planes perpendicular to YZ and the distance FZ apart, would form a third parallelepiped equivalent to 8 and therefore to #, and having a base equivalent to the bases of those figures, and the same altitude h. But this parallelepiped would, by construction, be rectan- gular, for its base would be a rectangle, and the lateral planes would all have been constructed perpendicular to the base. Therefore it is possible to transform any paral- lelepiped into an equivalent rectangular parallelepiped with an equivalent base and the same altitude. But one of the transformations is shown in this figure, because the attempt to repeat the operation in the same figure makes an unnecessarily complicated diagram. Alternative method : Use Cavalieri's Theorem. 140. COR. The volume of any parallelepiped is the product of its base and its altitude. 141. Theorem XIII. A plane through two opposite edges of a parallelepiped divides it into equivalent tri- angular prisms. If the parallelepiped is right, the prisms are congruent. If not, it is equivalent to a right parallelepiped cut from the same space, and the parts into which the plane cuts it are equivalent to the halves of that right parallelepiped. 292 POLYHEDRONS, CYLINDERS, AND CONES 142. COR. 1. The volume of a triangular prism is the product of its base and its altitude* 143. COR. 2. The volume of any prism is the prod- uct of its base and its altitude. 144. COR. 3. The volume of a circular cylinder is the product of its base and its altitude. For it is the limit approached by the inscribed prism of regular base when the number of faces is increased indefinitely. (See 116.) The volume of a cylinder of radius r and altitude h is 7rr 2 h. 182. Which would be lighter and how much, circular pillars 23 in. in diameter and 12 ft. high, or octagonal pillars 24 in. in diameter and 12 ft. high, if they were made of cement weighing 150 pounds to the cubic foot? 133. What per cent of the lumber is cut off in cutting a square timber out of a log of radius r? in cutting out a hexagonal timber ? 134. The diagonals of a parallelepiped are concurrent in the mid- point of each. 135. Any section of a parallelepiped through two pairs of parallel planes is a parallelogram. 136. If the diagonals of the faces of a parallelepiped are drawn, the lines joining the intersection points of the diagonals in opposite faces are concurrent in the midpoint of each. 137. The diagonal of a cube equals the square root of three times its edge. 138. The sum of the squares on the diagonals of a parallelepiped equals the sum of the squares on its edges. 139. The square on a diagonal of a rectangular parallelepiped equals the sum of the squares on its three edges. 140. Find the volume of a circular cylinder of radius 5 cm., whose axis is 1.5 m. long, and makes an angle of 45 with its bases. 141. An irrigation ditch is 4 ft. wide, 3 ft. deep for its whole width, with a semicylindrical bottom below this 3 ft. If it is filled to within 6 in. of the top, how much water does it contain to the mile ? SECTION II. THE PYRAMID AND THE CONE 145. The Pyramidal Surface and the Conical Surface. If a moving straight line always contains a fixed point, and always intersects a given straight line, it generates a portion of a plane. Why ? PLANE SURFACE PYRAMIDAL, SURFACE CONICAL SURFACE In these figures, AB represents the original position of the moving line (or generatrix), P being the fixed point. XY represents the guiding line (or directrix). A moving straight line that always contains a fixed point, and always intersects a line that lies entirely in one plane, but is not coplanar with the moving line : (a) if the guiding line is broken, generates a pyramidal surface, which is evidently composed of portions of a num- ber of planes having one common point, with each succes- sive pair intersecting in lines that are the edges of the surface. (>) if the guiding line is curved, generates a conical sur- face, of which no part is plane, for if it were, that part of the guiding line would be straight. (Why?) The mov- 293 294 POLYHEDRONS, CYLINDERS, AND CONES ing line in each of its various positions is called an element of the conical surface. Non-coplanar guiding lines may give combinations of pyramidal and conical surfaces, and need not be con- sidered in an elementary course. 146. Vertex. The fixed point through which the mov- ing line passes is called the vertex. 147. Closed Surfaces. If the guiding line is a closed line, the pyramidal, or coni- cal, surface also is closed, and the space inclosed is called pyramidal, or conical, space respectively. 148. Sections. If a plane cuts all the edges of a closed pyramidal surface, or all the elements of a closed conical surface, but not at the vertex, it cuts the surface in a closed line, and the figure bounded by this closed line is called a section of the pyramidal or the conical space. When two sections are made by parallel planes, they are called parallel sections. Sj and s z are parallel sections of the closed pyramidal space P, and of the closed conical space O. In each case F is the vertex. In the pyramidal space the section must be a polygon. Why ? *149. The edges from the vertex of a pyramidal sur- face, or the elements from the vertex of a conical surface, are cut proportionally by parallel sections. 150. Theorem XIV. Parallel sections of a pyramidal space are similar polygons, and their areas are propor- tional to the squares of their distances from the vertex. THE PYRAMID AND THE CONE 295 142. Which are the corresponding points on the perimeters of parallel sections? If a line is drawn from the vertex of a conical space through two parallel sections, show that the sects from the points thus obtained to any two corresponding points on the perimeters of the sections will be proportional to the distances of those sections from the vertex. 151. Pyramids and Cones. That part of a pyramidal space between the vertex and a section is called a pyramid. That part of a conical space between the vertex and a section is called a cone. The section is called the base, and the pyramidal or conical surface is called the lateral surface of the figure, while the area of the lateral surface is called the lateral area. In the case of the pyramid, the portions of planes forming the lateral surface are called lateral faces, and their intersections are called lateral edges. 152. Truncated Figures. That part of a prismatic or cylindrical space between two non-parallel sections that do not intersect on the surface is called a truncated prism or cylinder. That part of a pyramidal or conical space between two sections on the same side of the vertex that do not inter- sect on the surface is called a truncated pyramid or cone. 153. Frustums. A truncated pyramid or cone between parallel sections is called a frustum of the pyramid or cone. 154. Lateral Faces and Edges. The definition of lateral surface, lateral faces, and lateral edges given in 151 holds also for frustums and truncated figures. The parallel sec- tions in a frustum are its bases, but only one of the sections is in general considered the base of the truncated figure. 155. Altitude. The altitude of a pyramid or a cone is the perpendicular from the vertex to the base. In a frus- 296 POLYHEDRONS, CYLINDERS, AND CONES turn, it is the perpendicular between the bases. A trun- cated figure cannot be said to have an altitude ; a triangular truncated prism, however, is sometimes said to have three altitudes, namely, the perpendiculars from the three ver- tices of one section to the other section. 156. Classification of Pyramids according to Number of Faces. A pyramid, like a prism, is triangular, quad- rangular, pentagonal, etc., according to the number of sides of its base. 157. Regular Pyramids. A pyramid whose base is a regular polygon, and whose altitude meets the base at its center, is called a regular pyramid. The altitude of such a pyramid is called its axis. The lateral faces of a regular pyramid are congruent. (Why?). The altitude of one of the lateral faces is called the slant height of the pyramid. 158. Circular Cones. A cone whose base is a circle is a circular cone. The line joining the vertex of a circular cone to the center of its base is called the axis of the cone, and if the axis is perpendicular to the base, the cone is a right circular cone. The radius of the base is called the radius of the cone. The cones considered in elementary solid geometry are usually circular cones. See Appendix, 321. The right circular cone, since it might be generated by revolving a right triangle about one leg as an axis, is sometimes called a cone of revolution. An element of a right circular cone is sometimes called its slant height. 143. Find the locus of a point such that its distances from a given plane and a line perpendicular to that plane shall have a given fixed ratio. THE PYRAMID AND THE CONE 297 159. The Cone the Limiting Case of the Pyramid. A pyr- amid is said to be inscribed in or circumscribed about a cone if the pyramid and cone have the same vertex and the base of the pyramid is respectively inscribed in or cir- cumscribed about the base of the cone. The following theorem is assumed without proof : Theorem XV. If a pyramid, the base of which is a regular polygon, is inscribed in or circumscribed about a given circular cone, the lateral area of the pyramid approaches the lateral area of the cone as its limit, and the volume of the pyramid approaches the volume of the cone as its limit, as the number of sides of the base of the pyramid is increased indefinitely. The cone is then a limiting case of the pyramid ; there- fore the two figures will be treated together. 160. Theorem XVI. A section of a cone made by a plane containing an element is a triangle. 144. In Theorem XVI, would it be sufficient to say "by a plane through the vertex " ? 145. A section of a pyramid made by a plane through the vertex is a triangle. 161. Theorem XVII. A. section of a circular cone made by a plane parallel to the base is a circle, the center of which is the point where the axis meets it. Show that the sects from this point to the perimeter of the sections are in proportion to the radii of the base. 162. Tangant to a Cone. A plane is tangent to a cone if it meets its surface in one element, and in no other point. * 163. If a plane is tangent to a circular cone, its inter- section with the plane of the base is tangent to the base. 298 POLYHEDRONS, CYLINDERS, AND CONES * 164. The plane determined ~by a tangent to the base of a circular cone and the element from the point of contact, is tangent to the cone. 165. Theorem XVIII. The lateral area of a regular pyramid is equal to one half the product of the slant height and the perimeter of the base. 166. COR. The lateral area of a right circular cone is equal to one half the product of the slant height (or element} and the circumference of the base. If the cone is of radius r and slant height s, the lateral area = irrs. 167. Theorem XIX. The lateral area of a frustum of a regular pyramid is equal to one half the product of the slant height and the sum of the perimeters of the bases. What kind of plane figures are its faces ? 168. COR. 1. The lateral area of a frustum of a right circular cone is equal to one half the product of the slant height and the sum of the circumferences of the bases. If the radii of the bases are r 1 and r v and the slant height is s, the lateral area is 7r(r 1 + r 2 ). 169. Midsections. When a solid has parallel bases, the section parallel to the bases and half way between them is called the midsection. 170. COR. 2. The lateral area of a frustum, of a right circular cone is equal to the product of the slant height and the circumference of its midsection. If the radius of the midsection is r m , lateral area = 2 7rr m s. 171. COR. 3. The lateral area of a frustum of aright circular cone is equal to the product of the altitude and the 1 circumference of a circle whose radius is the per pen- THE PYRAMID AND THE CONE 299 dicular erected at the midpoint of an element, and termi- nated by the axis. For if the perpendicular is a and the altitude A, by similar triangles - = , and ah can be sub- "> r m stituted for sr m in the formula of 170, giving 2 irah. 146. Find the surface of a hexagonal tower of base edge 10 ft. and altitude 20 ft. surmounted by a pyramid of height 15 ft. 147. What area does a line generate if it is revolved around a coplanar axis, its projection upon the axis being 10 in. and the length of its perpendicular bisector when extended to the axis being 6 in.? 172. Prisms Inscribed in, and Circumscribed about, a Pyr- amid. If the altitude of a triangular pyramid is divided into equal parts by planes passed parallel to the base, 1. The prisms within the pyramid having the sections formed by the parallel planes as bases, and each having one of the divisions of a certain edge of the pyramid as one of its lateral edges, are called a set of inscribed prisms. In Fig. 1, U, flf, and T are inscribed prisms, the common edge being VC. FIG. 1. FIG. 2. 2. The prisms having the base of the pyramid and the parallel sections as bases, and each having one of the divi- sions of a certain edge of the pyramid as one of its lateral edges, are called a set of circumscribed prisms. In Fig. 2, X, r", Z, and W are circumscribed prisms, the common edge being VC, 300 POLYHEDRONS, CYLINDERS, AND CONES When a set of circumscribed prisms and a set of in- scribed prisms are formed in a pyramid by the use of the same lateral edge and the same parallel planes, they are spoken of as corresponding sets of inscribed and circum- scribed prisms. * 173. The volume of a triangular pyramid is greater than that of any set of inscribed prisms, and less than that of any set of circumscribed prisms. *174. In any triangular pyramid, the difference in volume between a set of circumscribed prisms and the corresponding set of inscribed prisms is the circum- scribed prism on the base of the pyramid. As, in the figures, R = F, S = Z, T = TF, therefore (x + r + z + w) - OR + 8 + r) = x. * 175. If the number of equal parts into which the alti- tude of a triangular pyramid is divided by planes par- allel to the base is increased indefinitely, the volumes of the sets of inscribed and circumscribed prisms formed with the sections as bases will approach the volume of the pyramid as a limit. For, since their difference is a prism on a constant base (ABC) whose altitude can be made indefinitely small, the difference between the two sets, and therefore between either and the pyramid, can be made to approach zero. 176. Theorem XX. Two triangular pyramids having equivalent bases and equal altitudes are equivalent in volume. Inscribe sets of prisms, using the same number of divisions of the altitudes. What is known of the volumes THE PYRAMID AND THE CONE 301 of correspondingly placed prisms in the two pyramids ? What is true of the volumes of the sets of prisms ? In- crease the number of divisions of the altitude. Then what is true of the sets of prisms ? Therefore what is true of the pyramids ? Why ? Alternative Method. The two pyramids are Cavalieri bodies. 177. Theorem XXI. A triangular prism can be di- vided into three equivalent triangular pyramids. Pass two planes through vertices of the prism so as to divide it into three pyramids. See whether they are equi- valent by examining their bases and altitudes. It is pos- sible to prove any one equivalent to any other. When no easier method seems possible, choose the bases of the two to be examined so that they are in the same plane. 178. COR. 1. The volume of a triangular pyramid equals one third the product of its base and its altitude. 179. COR. 2. The volume of any pyramid equals one third the product of its base and its altitude. 180. COR. 3. The volume of a circular cone equals one third the product of its base and its altitude. V = J Trr^h. 148. Find the volume of a right circular cone of radius 8 in. arid altitude 15 in., and of the regular square pyramid circumscribed about it. 149. If square pyramids whose lateral faces are equilateral triangles are placed on all the faces of a cube, find the volume of the resulting figure. 150. If a rectangular parallelepiped of base 6 in. by 8 in., and altitude 14 in. is hollowed out along the diagonal planes from its bases to the intersection of its diagonals, find the volume of the remaining figure. 302 POLYHEDRONS, CYLINDERS, AND CONES 151. If the corners are cut off a cube by passing planes through the one-third points on the three edges from each vertex, what part of its volume is left ? 152. Find the volume of the solid formed by passing planes through the midpoints of the three edges of each trihedral angle of a cube. 181. Prismatoids. A polyhedron all of whose vertices lie in two parallel planes is called a prismatoid. Its bases are obviously polygons in parallel planes, and its lateral faces are quadrilaterals or triangles. 182 The Prismatoid Formula and its Extension to Curved Surfaces. The volume of any prismatoid is expressed by the formula : where h is the altitude, b and b 2 are the bases, and m is the midsection. Extension of the Formula. The formula applies also to solids having curved surfaces, if (1) their bases are circles in parallel planes ; (2) their generating elements are straight lines or arcs of circles. This extends the formula to cylinders and cones, and to certain parts of spheres that have not yet been studied ; it also applies to other figures with which elementary geometry is not concerned. ' The proof of the Prismatoid Formula is not given, but for the case of polyhedrons it will be found in the Appendix, 320. It will not be used to any extent to prove other formulas, but will be shown to agree with the formulas found for the various figures by other methods. It serves, then, as a summary for almost all the volume formulas of THE PYRAMID AND THE CONE 303 solid geometry, and might well be used for all the volumes to which it applies were it not for the fact that it is often harder to apply than the particular formula for the figure in question. For example, in the cone, the midsection is the base divided by 4 and one base (considering the cone a pris- matoid) is zero, so the prismatoid formula would be which agrees with the regular formula. 183. Theorem XXII. The volume of the frustum of a pyramid equals one third the product of its altitude and the sum of its two bases and the mean proportional be- tween them. h being the altitude, and b 1 and 6 2 the lower and upper bases, the one farther from the vertex being considered, for convenience, the lower base. FIRST PROOF. Continue the lateral edges through the vertices of the frustum to the vertex of the pyramid, and consider the frustum as the difference of two pyramids. The altitude of the smaller pyramid can be obtained from the proportion -L = - , , where x is the unknown 2 x* altitude. Solve this for x by taking the square root and applying division, finding x = *-= Express the V5j - V6 2 volume of the frustum as the difference of two pyramids in terms of A, #, 5 2 , b v and substitute this value of x. 304 POLYHEDRONS, CYLINDERS, AND CONES SECOND PROOF. Divide a triangular frustum into three triangular pyramids, one of which has the lower base of the frustum and the same altitude, a second of which has the upper base of the frustum and the same altitude, and prove that the third is the mean proportional between these two. In order to do this take the ratio of the third pyramid to each of the others, and reduce these ratios to simpler form. Use the same method as in Theorem XXI, but the bases of the pyramids, instead of being equal, will have equal altitudes only, and so, in turn, will be pro- portional to their bases. This proof must be extended to any frustum. THIRD PROOF. Use the Prismatoid Formula, express- ing the midsection in terms of the bases, to which it is simi- lar. This is done by taking the ratio of each of the bases to the midsection and adding the square roots of the equa- tions obtained. It is recommended that all of these proofs be worked out, for they are all valuable applications of the various methods used. 184. COR. The volume of a frustum of a circular cone is equal to one third the product of its altitude and the sum of its two bases and the mean proportional be- tween them. where h is the altitude, and ^ and r 2 are the radii of the bases. 153. A monument is in the form of a pillar of granite 1 ft. in di- ameter and 12 ft. high, surmounted by the frustum of a cone 1 ft. high with an upper base of radius 10 in., and a cone of height 6 in. How much does the stone weigh at 175 Ib. to the cubic foot? SUMMARY OF PROPOSITIONS 805 185. SUMMARY OF PROPOSITIONS I. THK STRAIGHT LINE : (1) Sects equal: The edges of a prism are equal (107). The elements of a cylinder are equal (107). (2) Sects proportional : The edges of a pyramidal space cut by parallel planes are proportional (149). II. TANGENT PLANES AND SECTIONS : (1) Tangent planes : The plane determined by an element and a tangent to the base of a cylinder (or a cone) is tangent to the cylinder (or the cone), and conversely (119, 120, 163, 164). (2) Sections: (a) of a prismatic space : Parallel sections are congruent polygons (108). Any pair of opposite faces of a parallelepiped may be considered its bases (126) ; its opposite faces are congruent (126). (b) of a cylindrical space : Parallel sections are congruent (109). A section through an element of a cylinder is a parallelogram (117). (c) of a pyramidal space : Parallel sections are similar polygons (150) ; their areas are proportional to the squares of their dis- tances from the vertex (150). (e?) of a cone : A section parallel to the base of a circular cone is a circle (161). A section through an element is a triangle (160). III. LIMITING CASES: If the number of faces of a prism (or a pyramid) of regular base, inscribed in or circumscribed about a circular cylinder (or a cone), is increased indefinitely, the lateral area and the volume 306 POLYHEDRONS, CYLINDERS, AND CONES of the prism (or the pyramid) approaches as its limit the lateral area and the volume of the cylinder (or the cone) (116, 159). If the number of equal parts into which the altitude of a trian- gular pyramid is divided by planes parallel to the base is in- creased indefinitely, the volumes of the sets of inscribed and circumscribed prisms formed with the sections as bases ap- proach the volume of the pyramid as a limit (175). IV. LATERAL AREA : (1) of a prism = edge x perimeter of a right section (121); (2) of a right circular cylinder = altitude x perimeter of the base = 27rrA, where h is the altitude, r is the radius (122); (3) of a regular pyramid = ^ slant height x perimeter of the base (165) ; (4) of a right circular cone = | slant height x perimeter of the base = TITS, where r is the radius, s is the slant height (166); (5) of a frustum of a regular pyramid = | slant height x the sum of the perimeters of the bases (167); (6) of a frustum of a right circular cone = | slant height x the sum of the perimeters of the bases = ITS (ri + r 2 ), where r\ and r z are the radii, 2?rsr w , where r m is the radius of the midsection, 2 irdh, where a = the perpendicular bi- sector of an element extended to the axis (168, 170, 171). V. VOLUMES : (1) Congruent : Right prisms or cylinders having congruent bases and equal altitudes are congruent (123). (2) Equivalent : Two Cavalieri bodies are equivalent (137). Prisms cut from the same prismatic space and having equal lateral edges are equivalent (138). An oblique parallelepiped is equivalent to a rectangular parallelepiped having an equivalent base and an equal altitude (139). The parts of a parallelepiped made by passing a plane through two opposite edges are equivalent (141). SUMMARY OF PROPOSITIONS 307 Triangular pyramids of equivalent bases and equal altitudes are equivalent (176). A triangular prism is equivalent to three equivalent pyra- mids (177). (3) Proportional: Rectangular parallelepipeds having two dimensions equal are to each other as the third dimension (132). Rectangular parallelepipeds having one dimension equal are to each other as the products of the other two dimensions (133). Rectangular parallelepipeds having no dimensions equal are to each other as the products of all three dimensions (134). (4) The Pyramid and its inscribed and circumscribed prisms : A triangular pyramid is less than the circumscribed, and greater than the inscribed prisms (173). The difference between the sets of circumscribed and in- scribed prisms is the circumscribed prism on the base of the pyramid (174). (5) Formulas for volume : (a) of a rectangular parallelepiped = the product of its di- mensions = base x altitude (136); (6) of any parallelepiped = base x altitude (140); (c) of a prism = base x altitude (142, 143); ( Two lines tangent to a sphere at a point determine the plane tangent to the sphere at that point. 201. Circumscribed Polyhedrons. A polyhedron is said to be circumscribed about a sphere, and the sphere to be inscribed in the polyhedron, if the sphere is tangent to all the faces of the polyhedron. * 202. One, and but one, sphere can be inscribed in any given tetrahedron. For there is one point equi- distant from its faces. 203. Theorem I. Every plane section of a sphere is a circle whose center is the foot of the perpendicular from the center of the sphere to its plane. Plane sections of a sphere are called circles of the sphere. "Circle " is usually used for "circle of a sphere." 215. If a secant plane is gradually moved farther from the center of a sphere, describe the change in its intersection with the surface. 318 POLYHEDRAL ANGLES AND THE SPHERE 216. A line tangent to a sphere is tangent to every circle through the point of contact, in whose plane it lies. 204. Axis and Poles of a Circle. The diameter perpen- dicular to a circle is called its axis ; the points where the axis cuts the surface of the sphere are called the poles of the circle. 205. COR. 1. A line perpendicular to a circle at its center, or one joining the center of the circle to the center of the sphere (unless they are the same point}, is the axis of the circle. 206. COR. 2. Sections through the center of a splwre are all equal, and are the largest circles of the spJiere. Of others, two that are equidistant from the center are equal, and conversely ; and, of two not equidistant from the center, the nearer is the greater, and conversely. 207. Great and Small Circles. A section through the center of a sphere is called a great circle ; any other sec- tion is called a small circle. In the diagram, G is a great circle, and 8 is a small circle. The axis of S is PP f , its poles being P and P'. What relation to one another have r (the radius of the sphere), r f (the radius of the small circle), and d (its distance from the center)? SECANTS AND TANGENTS 319 The circles of latitude and the meridians on the earth's surface are familiar examples of circumferences of circles of a sphere. The equator and the meridians are circum- ferences of great circles. *208. The center of a great circle is the center of the sphere. * 209. Any two great circles of a sphere intersect in a diameter. * 210. A great circle bisects the sphere and its surface. *211. Two points on the surface (not extremities of a diameter} and the center of the sphere determine a great circle ; any three points on the surface determine a circle. 212. COR. 3. Parallel circles of a sphere have the same axis and the same poles. The circles of latitude on the earth's surface have the axis of the earth as their axis, and its poles as their poles. 213. COR. 4. A great circle through the poles of an- other circle is perpendicular to it ; a great circle perpen- dicular to another circle contains its axis and its poles. This is sometimes stated : For a great circle to be per- pendicular to another circle, it is necessary, and it is sufficient, that it contain its poles. 217. A great circle that contains one of the poles of another circle must contain the other pole also. 214. COR. 5. The locus of points in a sphere equidis- tant from all points on the circumference of a circle is the axis of the circle ; the poles of a circle are equidistant from all points on its circumference. 320 POLYHEDRAL ANGLES AND THE SPHERE 215. Polar Chords and Polar Distances. The chords from the pole of a circle to points on its circumference are called polar chords; the great circle arcs from its pole to points on its circumference are called polar dis- tances. Unless otherwise stated the chords and distances from the nearer pole are meant. * 216. The polar distances of the same, or of equal, cir- cles on a sphere are equal. 217. A Quadrant. One quarter of the circumference of a great circle is called a quadrant of the sphere. The degree measure of a quadrant is 90. * 218. The polar distance of a great circle is a quad- rant, and conversely. Use the central angles subtended by the polar distances. * 219. If two points on a spherical surface are a quad- rant's distance apart on the great circle arc through them, each is the pole of a great circle through the other. Using either as a pole, draw the axis, then 'draw the great circle of which it is the pole. * 220. If a point on the surface of a sphere is at a quadrant's distance from each of two other points of the surface, it is the pole of a great circle through them. Proceed as in 218 or 219. What cases are there? 221. Angles between Arcs. The angle between two arcs is the angle between the tangent lines at the point of intersection. 218. Two coplanar circles are perpendicular to each other when a radius of one is tangent to the other. Must a radius of each arc be tangent to the other circle? 222. Spherical Angles. The most important class of SECANTS AND TANGENTS 321 angles between arcs is that having great circle arcs as the arms of the angle. Such an angle is called a spherical angle. As two great circles intersect in a diameter, the tangents are both perpendicular to this diameter, and therefore are the arms of a measuring angle of the dihe- dral angle between the great circles. * 223. The angle between two great circle arcs equals the measuring angle of the dihedral angle between their planes. * 224. A great circle arc through a pole of another great circle is perpendicular to its circumference. 225. Measurement of Spherical Angles. The most con- venient way to measure a spherical angle is by the meas- uring angle of the corresponding dihedral angle, drawn A at the center of the sphere. This angle is evidently in the great circle perpendicular to the edge of the dihedral angle, and therefore perpendicular to the planes in which the arms of the spherical angle lie. The spherical angle XTY equals Z ATB or Z XOF, measuring angles of the dihedral angle between the planes TXT f and TYT' . But ZXOF might be measured by arc XY.\ therefore it follows that 322 POLYHEDRAL ANGLES AND THE SPHERE * 226. ji spherical angle is measured by the subtended great circle arc having its vertex as a pole. 219. Explain how a figure can be drawn so that it will be bounded by three great circle arcs, each perpendicular to the others. THE MATERIAL SPHERE * 227. To find the radius of a given circumference on a given material sphere. Select three points on the circumference, and consider a triangle, with these three points as vertices, as inscribed in the circle. Construct, in some plane, the triangle having these three sides (using an ordinary compass to carry the lengths, for it will measure a chord of a circle on a sphere just as well as a sect on a plane). Circum- scribe a circle about this triangle, and it will equal the given circle. Why ? * 228. To describe a circumference on a sphere with a given pole, and a given polar chord. * 229. The polar chord of a circle of a sphere is the mean between the diameter cf the sphere and its own projection upon the axis of the circle; the radius of a circle is the mean between the two sects it makes on the axis of that circle. Pass a great circle through the pole, and work in that plane. 230. Theorem II. To find the diameter of a given material sphere. With an} 7 pole, and any polar chord that is of con- venient length, describe a circumference on the sphere ; find the radius of this circle, and from these two lengths, construct the right triangle considered in 229. THE MATERIAL SPHERE 323 220. On a sphere of known diameter, to construct a circumference such that a triangle inscribed in it shall be congruent to a given triangle. 221. If the diameter of a material sphere is known, to construct on it a circumference of given radius. 222. To construct a circumference on a given material sphere so that it will be a given distance from the center. 223. To construct parallel circumferences on a material sphere. 224-. Given any circumference on a sphere, to find any number of points on the circumference of the great circle perpendicular to the given circle at any given point. 231. Relative Position of Two Spheres. Two spheres either lie entirely outside each other ; lie outside each other except for one point in common ; have more than one point in common, without either being entirely con- tained in the other; or one is entirely contained in the other. Only the second and third of these possibilities are of special interest in solid geometry. 232. Center Line. The line through the centers of two spheres is called their line of centers. The sect between the centers is called their center sect. 233. Tangent Spheres. Two spheres whose surfaces have one point in common are called tangent spheres, the point of tangency being called the point of contact. They are said to be externally or internally tangent, according as they are outside each other, or one is contained in the other. Unless otherwise stated, tangent will be used to mean externally tangent. *234. Two spheres whose surfaces meet in the center line are tangent. *235. Two spheres tangent to the same plane at the tame point are tangent. 324 POLYHEDRAL ANGLES AND THE SPHERE 236. Theorem III. If the surfaces vf two spheres meet at a point not on the center line, they meet in the circumference of a circle that is perpendicular to the center line. If a plane is passed through the point and the two cen- ters, the two great circles in which this plane intersects the spheres intersect in two points and have a common chord. Rotate these circles to generate the spheres, and examine the figure generated by their common points. 225. If one sphere is inside another, and their surfaces meet on the center line, the spheres are tangent. 226. If the center sect of two spheres is less than the sum, but greater than the difference of the radii, the surfaces of the spheres intersect in the circumference of a circle. 227. If the center sect of two spheres is less than the difference of the radii, how must the spheres lie? 228. If two equal spheres meet, the circle in whose circumference their surfaces intersect is equidistant from their centers. 229. Find the center of a sphere whose surface passes through the circumference of a given circle, and contains a given point not co- planar with the circle. 230. If from a point within a sphere, three chords are drawn, each perpendicular to the other two, the sum of the squares of the sects of the chords equals half the sum of the squares of the diameters of the circles determined by the chords. SECTION II. SPHERICAL POLYGONS AND POLYHEDRAL ANGLES 237. Spherical Polygons. A portion of a spherical sur- face bounded by three or more arcs of great circles is called a spherical polygon. The bounding arcs are the sides of the polygon, the spherical angles between the sides are its angles, and the vertices of the angles are its vertices. 238. Convex Polygons. A spherical polygon is convex if no side when extended can cut the surface of the poly- gon. This is sometimes stated, a spherical polygon is convex if when any side is extended to form a circum- ference, the whole polygon lies on the surface of one of the hemispheres bounded by this circle. Unless other- wise stated, only convex polygons will be considered, and the word "convex" should be understood whenever necessary. 239. Spherical Triangles. A spherical polygon of three sides is called a spherical triangle. The meanings of isos- celes, equilateral, scalene, right-angled, and any other such terms, are the same as when applied to plane triangles. 240. Relation between Polyhedral Angles and Spherical Polygons. (See 99-102.) Any spherical polygon is subtended by a polyhedral angle whose vertex is at the center of the sphere, for its sides are arcs of great circles whose planes intersect at the center, and so bound a polyhedral angle. Conversely, 325 326 POLYHEDRAL ANGLES AND THE SPHERE any polyhedral angle whose vertex is at the center of a sphere subtends a spherical polygon on the surface, for its faces cut the surface in great circle arcs, which are the sides, and its edges cut the surface in points, which are the vertices, of such a polygon. 0-ABCD, and o-A f B'c f D' are poly- hedral angles ^subtending the spheri- cal polygons ABCD and A'B'C'D' . 241. Central Polyhedral Angles. A polyhedral angle whose vertex is at the center of a sphere is called a cen- tral polyhedral angle, and the subtended spherical polygon is spoken of as its polygon. * 242. Each face angle of a central polyhedral angle is measured by the subtended side of its spherical polygon. * 243. Each dihedral angle of a central polyhedral angle has the same measure as the corresponding spher- ical angle of its spherical polygon. 244. Corresponding Propositions on Polyhedral Angles and Spherical Polygons. From the relations existing be- tween the parts of a central polyhedral angle and the parts of its spherical polygon, it is evident that for every proposition concerning the face angles and the dihedral angles of a polyhedral angle, there must be a correspond- ing proposition concerning the sides and the angles of a spherical polygon, and conversely. In the propositions dealing with these two kinds of figures, this relation is shown by stating the propositions in pairs, one about polyhedral angles, the other about spherical polygons. The one that is more convenient to SPHERICAL POLYGONS 327 prove independently is stated first, the other follows from the relations shown in 242 and 243. 245. Theorem IV. (V) The sum of any two face angles of a trihedral angle is greater than the third face angle. (ft) The sum of any two sides of a spherical triangle is greater than the third side. FIKST PROOF: Let the trihedral angle be V-ABC. Pass a plane through CV perpendicular to plane AVB, meeting it in c'v. Then c'v contains the projections of BV and AV upon the plane CVC 1 '. Why ? Therefore Z AVC 1 < Z. AVC, Z BVC' < Z BVC. Why ? And Z AVC' + Z BVC' < Z AVC + Z BVC. But Z AVC' + Z BVC' =, or >, Z ^4F, according as VC f falls within or outside the angle AVB, so Z. AVB < Z. AVC + Z. BVC. SECOND PROOF: Let the trihedral angle be V-KLM, and let Z KVL be its largest face angle. On this angle, cut off Z RVL = Z LMV. Cut off VX on VR, and VC = VX on VM. Pass a plane through X and C cutting VK at A and FI* at JB. Since Z #FC = Z XF.B, it is only necessary to prove that Z. CVA > Z AVX (by comparing their tri- angles) to obtain the required conclusion. Explain in full why (ft) follows from (#). 231. Any side of a spherical triangle is greater than the difference of the other two sides. To what proposition on polyhedral angles does this correspond ? 328 POLYHEDRAL ANGLES AND THE SPHERE 246. COR. If two circles on a sphere meet at a poiwt on the great circle arc through their poles, they cannot meet again. For, if they did, on drawing the polar distances to that point, the sum of two sides of the triangle formed would equal the third side. Could two circles meet at a second point on the circumference of the same great circle ? 247. Theorem V. The shortest line from one point to another on the surface of a sphere is the minor arc of the great circle through them. Let PQ be the minor arc of a great circle through the given points P and Q ; then the shortest line from P to Q on the surface must contain every point of PQ (there- fore must be PQ), for if there is even one of the points of PQ that it does not contain, it can be made still shorter. Suppose the shortest line from P to Q does not contain T. With P and Q as poles, and PT and QT as polar distances, describe circles, which will therefore meet at T. Then the circles meet only at T, and any line from P to Q not through T must intersect the circles at different points E and L. But if circles P and Q were rotated so that JT and L were at T, the line PKLQ would be shortened by the length KL, its other parts remaining the same. Therefore any line not containing T is not the shortest line from p to Q. SPHERICAL POLYGONS 329 248. Distance on a Spherical Surface. The distance be- tween two points on a spherical surface means the length of the minor arc .of the great circle through them, because the great circle arc is the shortest line between two points on the surface of a sphere and therefore corresponds to the straight line on a plane surface. Unless otherwise stated, only great circle arcs will be used. 249. Theorem VI. (a) The sum of the face angles of any polyhedral angle is less than a perigon. (b) The sum of the sides of any spherical polygon is less than the circumference of a great circle. VA Let ABCDEF be a plane section of the polyhedral angle with vertex F. Then atA,^.FAV + Z VAE > Z. FAB of the the polygon ABCDEF (why ?), and similarly at each ver- tex of the polygon. Therefore the sum of the angles in the faces of the polyhedral angle at A, B, C, > (n 2) st. angles. Why ? But the sum of all the angles in the face triangles = ? Therefore the sum of the angles at V = ? Or, join a point O within ABCDEF to its vertices. The faces of the polyhedral angle contain the same number of triangles as ABCDEF, and so the sum of the angles is the same. But the angles in the faces at A, J5, C, D, E, F are greater than the angles in the polygon. What follows ? 330 POLYHEDRAL ANGLES AND THE SPHERE Explain fully why (b) follows from (a). If it is pre- ferred, (5) can be proved independently, and (a) can be deduced from it, the proof for any case following the method shown here for a quadrilateral. To show that the sum of the sides of any quadri- lateral ABCD is less than a circumference, extend AB and AD to their other intersection A . Extend DC to meet AA' at x. Now use BC) Two trihedral angles are either congruent or symmetric if they have the three face, angles of one respectively equal to the three face angles of the other. 272. Theorem XII. Two symmetric spherical trian- gles are equivalent. Draw a circle through the vertices of each triangle and prove the circles equal. Draw the polar distances to the vertices of the triangles, and prove them equal. Show that SPHERICAL POLYGONS 341 the three triangles thus formed in one of the given tri- angles are respectively congruent to the three triangles formed in the other triangle, in other words, that sym- metric triangles can be divided into congruent parts, and so are equivalent. 249. The median to the base of an isosceles spherical triangle divides the triangle into equivalent triangles. 250. If the opposite sides of a spherical quadrilateral are equal, either diagonal divides it into equivalent triangles ; the diagonals bi- sect each other. 273. Polar Triangles. If for each side of a spherical triangle that one of its poles is taken that is on the same hemispherical surface as the opposite vertex (the circle of which that side is an arc being considered to form the hemispheres), and if these three points are joined by minor arcs of great circles, the resulting triangle is called the polar triangle of the original triangle. In the figure, if A* ', B f , c', are the poles of BC, CA, and AB respectively, since A and A 1 are on the same hemispherical surface with reference to BC, B and B r are on the same hemispherical surface with reference to CA, and C and c f are on the same hemispherical surface with reference to AB, then the great circle arcs A'B', B f C r , and C'A' form the polar triangle of triangle ABC. 342 POLYHEDRAL ANGLES AND THE SPHERE 274. Theorem XIII. If the first of two spherical tri- angles is the polar triangle of the second, the second is also the polar triangle of the first. How can a point be proved the pole of an arc ? No construction lines need be drawn. Two triangles, such that each is the polar of the other, are called polar triangles. 275. Theorem XIV. In two polar triangles, each angle of one is measured by the supplement of the side oppo- site to it in the other. It is necessary to prove that Z c is measured by 180 A'B '. Since C is the pole of A f B f , by what arc can Z c be measured? Show that this arc is 180 A'B' by using the fact that A' and B f are poles of the great circles through BC and CA respectively. They therefore have what polar distances to those circumferences? 251. The polar triangle of a right triangle has one side a quadrant. 252. If the angles of a triangle are 135, 97, and 88, find the op- posite sides of the polar triangle. 253. Show that if one side of a spherical polygon is more than 180, the polygon is not convex. 254. If the sides of two polar triangles meet, in how many points can they intersect ? 255. Two triangles, such that one side of each is a quadrant (quad- rantal triangles}, are congruent or symmetric if they have two angles of one respectively equal to two angles of the other. SPHERICAL POLYGONS 343 276. Theorem XV. (#) Two triangles on the same sphere are either congruent or symmetric if they have the three angles of one respectively equal to the three angles of the other. (ft) Two trihedral angles are either congruent or symmetric if they have the three dihedral angles of one respectively equal to the three dihedral angles of the other. For the sides of their polar triangles are equal (why ?), so the polar triangles are congruent or symmetric. Show that the sides of the given triangles must therefore be respectively equal. 256. If two angles of a spherical triangle are equal, the opposite sides are equal. (Use the polar triangles.) State this for a trihedral angle. 277. Theorem XVI. The sum of the angles of a spheri- cal triangle is greater than one, and less than three, straight angles. Let the angles be A, B, (7, the opposite sides be a, ft, c, and the angles and sides of the polar triangle be A*, B 1 ', C', and a', b r , c' : then 4 + B + C = 540-(a'+6' + 850( 103 o) would represent the triangle having angles of 97, 85, and 103. It is convenient also to denote a polygon of more than three sides by P with a subscript to show its angles. 264- On a sphere of radius 10 ft., find the area of (a) a triangle of angles 84 30', 111 28', 35 17' ; (ft) a pentagon each angle of which is 120. 265. If an equiangular triangle is one tenth of the spherical sur- face, find its angles. 266. A spherical triangle whose angles are in the ratios 1:2:3 is equivalent to a zone of altitude -. Find its angles. 3 267. A spherical polygon on a sphere of radius 6 in. has each angle 160, and its area is 44 sq. in. How many sides has it? 268. Find the area of T^ggo, 38, m) in terms of r. 269. Find the area of P ( i68, 178, 107, m so-, 177 14-, 154 is-) on a sphere of radius 1 ft. SECTION IV. VOLUME OF A SPHERE AND ITS PARTS 300. One method of finding the volume of a sphere is to consider it the limit approached by a circumscribed polyhedron as the number of its faces is increased indefi- nitely. This corresponds to the method used to find the area of a circle. As in dealing with the circle, there must be assumed a statement that the figure approaches a limit as the number of its faces is increased. If each vertex of a circumscribed polyhedron is joined to the center of the sphere, and planes are passed tangent to the sphere at the points where these lines meet the surface (thus forming, with the intercepted parts of the faces of the original polyhedron, a new polyhedron of a larger number of faces), the area and the volume of the polyhedron ap- proach the area and the volume of the sphere as limits, as the number of faces is increased indefinitely. 301. Theorem XX. The volume of a sphere is one third the product of its radius and its area. FIRST METHOD. The volume of a circumscribed poly- hedron is one third its area times the radius (why?). Increase the number of faces indefinitely. SECOND METHOD. Show that a sphere of radius r, and a cylinder of radius r and altitude 2 r that has been hol- lowed out from each base to the center in the form of right circular cones of radius r and altitude r, are Cavalieri 350 VOLUME OF A SPHERE AND ITS PARTS 351 bodies, since the section of each at a distance d from the center is irr 2 Trd 2 . Then find the volume of the hol- lowed cylinder. 302. Solids Extending from the Center of the Sphere to its Surface. The first method of proof is applicable to all solids that extend from the center of the sphere to some part of its surface, if they are bounded laterally by planes or conical surfaces, for all such solids can be considered as limits of the sum of an indefinitely large number of pyramids with vertices at the center of the sphere, and bases tangent to the sphere. 303. Spherical Cones and Spherical Sectors. That part of a sphere included in a conical space whose vertex is at the center of the sphere is called a spherical cone (as 1^). It could be generated by revolving a plane sector about one of its bounding radii as an axis, so it is considered as one kind of spherical sector. That part of a sphere generated by revolving a plane sector about a diameter is called a spherical sector (as J? 2 )> ^ ^ revolves about a diameter other than one of its own boundaries, it is evidently the 352 POLYHEDRAL ANGLES AND SPHERES difference of spherical cones. The base of any kind of spherical sector is evidently a zone. The spherical cone 0-S l is cut out of sphere by the conical space 0OD. * 304. The volume of a spherical* cone or a spherical sector is one third its base times the radius of the sphere. F=f 7rr 2 A (.where h is the altitude of the zone). 270. Express the formula for the volume of a spherical cone in terms of the distance of the plane cutting off the zone from the center of the sphere, instead of in terms of h. 271. Express the formula for the volume of a spherical cone in terms of the radius of the circle instead of in terms of h. 272. Find the volume of a spherical sector in a sphere of radius 12 in. if the radii of the two circles of its base are 6 in. and 8 in. 305. Spherical Pyramids. That part of a sphere in- cluded in a pyramidal space whose vertex is at the center of the sphere is called a spher- ical pyramid. The base of a spherical pyramid is evidently a spherical polygon. The spherical pyramid, like any other pyramid, can be tri- angular, quadrangular, etc. O-ABC is a triangular spheri- cal pyramid. * 306. The volume of a spherical pyramid is one third its base times the radius of the sphere. 3 720 where E is the spherical excess of the base. 273. State the volume of a spherical pyramid in form correspond- ing to that used for the area of a spherical triangle in 297. VOLUME OF A SPHERE AND ITS PARTS 353 307. The Spherical Wedge. That part of a sphere between two great semicircles is called a spherical wedge or an ungula. Its curved surface is evidently a lune. * 308. The volume of a spherical wedge is one third its base times the radius of the sphere. 3 360 where A is the angle between the circles. 274- State the volume of a spherical wedge in form corresponding to that used for a lune in 295. 309. Spherical Segments. The part of a sphere that lies between parallel planes is called a spherical segment. The planes can both be secant planes, one can be a tan- gent plane, or, in the limiting case of the whole sphere, both can be tangent planes. A spherical segment will have one base or two bases according as it is included between a secant plane and a tangent plane, or between two secant planes. The spherical surface of a spherical segment is a zone. 310. Volume of a Spherical Segment. The second method of proof used for finding the volume of a sphere is appli- cable also to spherical segments, for the spherical segment and the corresponding part of the hollowed out cylinder are Cavalieri bodies, and the volume of the hollowed cylinder can be found by the formulas for the cylinder and the cone. The volume of a spherical segment of one -in base is expressed by the formula r=^ (3r ^), and 3 that of two bases is expressed by the formula 354 POLYHEDRAL ANGLES AND SPHERES where h is the altitude, and /^ and r 2 are the radii of the bases. These formulas, especially the second, are not easy to deduce, and are not necessary for calculation, as the Prismatoid Formula applies to spherical segments, and is more convenient since it can be used in both of these cases as well as for the other solids already noted. 275. A sphere is inscribed in a cylinder. Find the ratios of their areas and of their volumes. 276. A cylindrical pail of radius 10 in. is full of water. If a sphere of radius 12 in. is set in the top of the pail, how much water is spilled, counting 231 cu. in. to the gallon ? 277. A sphere of radius 10 in. is hollowed out so. as to leave a hollow bounded by the lateral surface of the frustum of a right cir- cular cone of radii 6 in. and 8 in. with a diameter as axis. Find the volume of the remaining solid. 278. A sphere of 10 in. radius is hollowed out so as to leave a hollow bounded by a conical surface whose vertex is at one end of the diameter that is the axis of the cone. If the radius of the base of the cone is 6 in., find the volume of the remaining figure. 279. In building some decorations, certain trirectangular corners were filled in with blocks terminated by spherical surfaces of radius 3 in. and having the corners as centers. What was the spherical area and the volume of each block ? 280. If the number of square units in the area of a sphere is equal to three times the number of cubic units in its volume, find the radius. 281. A plane passes through a sphere so that it cuts off one third the surface. What part of the volume does it cut off? 282. The rays of light from a point illumine one fourth of the sur- face of a sphere of radius 4 ft. How far is the point from the sur- face? 283. The volume of the moon is approximately fa that of the earth. What is the diameter of the moon if that of the earth is 7916 miles ? VOLUME OF A SPHERE AND ITS PARTS 355 284. What is the radius of a sphere such that the number of cubic units in its volume equals the number of square units in the area of a great circle? such that it equals the number of linear units in the circumference of a great circle ? 285. What is the area of a sphere whose area contains the same number of square units that there are linear units in the circumfer- ence of a great circle ? 286. How many marbles one half inch in diameter can be made by melting a sheet of glass 4 ft. by 3 ft. and f in. thick ? 287. Show that the volume of a sphere inscribed in a cube of edge E is approximately .7854 E s . 288. Show that the volume of a cube inscribed in a sphere of radius r is approximately 1.6396 r 3 . 289. Which is the better bargain, apples at 20^ a dozen, or apples one and a half times as great in diameter at 60 1 a dozen ? 290. The diameter of the sun is 111^ times that of the earth. How does its volume compare with that of the earth ? its area ? 291. If the diameter of the moon is 2000 miles, and that of the earth 8000 miles, how far from the surface of the earth would the moon have to be to shine on one fourth of the earth's surface at a time? 292. Using the same diameters as in the last exercise, find on what part of the earth's surface the moon shines if the distance between the centers of the earth and the moon is 238,000 miles. 293. What is the volume of a sphere if a circle of radius 4 in. has a polar distance of 45? of 30? of 60? of 90? 294- If a spherical triangle of angles 90, 85, and 125 has an area of % 5 TT sq. in., what is the volume of the spherical pyramid on that base? 295. If the volume of a spherical wedge of 72 is - 3 /7r, find the area of a lune of 72 on the same sphere. 311. The Degree of Area and Volume Formulas. In plane geometry, it has been found that the formulas for the areas of all plane figures are of second degree in terms of lengths of sects, and that the areas of similar figures are proportional to the squares' of corresponding sheets. 356 POLYHEDRAL ANGLES AND SPHERES Similarly, the formulas for the areas of all solids studied are of second degree, either involving the second degree of one length, as, the area of a sphere equals 4 ?rr 2 , or the product of two lengths, as, the lateral area of a cylinder equals 2 Trrh. (Note that TT is a number, and so does not affect the degree.) The formulas for the volumes of all the solids studied are of third degree, either involving the cube of one length, as, the volume of a sphere equals ^Trr 3 ; the square of one length times another length, as, the vol- ume of a right circular cone equals -J- 7rr*h ; or the pro- duct of three lengths, as, the volume of a rectangular parallelepiped equals the product of its three dimensions. 312. Similar Solids. Polyhedrons are said to be similar when they are bounded by the same number of correspond- ingly similar faces, and their corresponding polyhedral angles are equal. It is evident that any one pair of cor- responding sects would be proportional to any other pair of corresponding sects. Right circular cones or cylinders are called similar if their axes are proportional to the radii of their bases. Any two spheres are similar, in that any pair of corresponding sects are proportional to any other pair of corresponding sects. The proportionality of corresponding sects shown in all of these definitions is the idea that is of the most importance in elementary geometry. The definitions are sometimes summed up by saying that any two solids are similar if they can be placed so that any two corresponding points will cut a sheaf of lines in the same ratio as any other pair of corresponding points. Show that solids fulfilling the conditions of this definition also satisfy the conditions of the other defini- tions. SUMMARY OF PROPOSITIONS 357 * 313. The areas of similar solids are proportional to the squares of any pair of corresponding sects. * 314. The volumes of similar solids are proportional to the cubes of any pair of corresponding sects. 296. If two similar solids have the area ratio 4, what is their vol- ume ratio ? 297. What relation have the volume ratio, the area ratio, and the line ratio, of two similar solids? 298. Name six kinds of solids such that any two of the same kind are similar. 299. If the volumes of two similar solids are 3141.6 cu. ft. and 25,132.8 cu. ft., and the area of the first is 1256.64 sq. ft., what is the area of the second ? 300. Given a material sphere, construct the radius of a sphere having twice as great an area ; any given number of times as great an area. 301. On different spheres, zones of the same altitude are propor- tional to the radii of the spheres. In order that zones on different spheres may fulfill the conditions of the general definition of similarity, what would have to be true of their altitudes ? 302. Given a regular tetrahedron, construct another having half as great an area. What is true of their volumes? 315. SUMMARY OF PROPOSITIONS I. DETERMINATION: of a sphere, by four points (194) ; of a great circle, by the center and two points on the sur- face (211); of a small circle, by three points on the surface (211). II. RELATIVE POSITIONS ; TANGENTS AND SECANTS : (1) A point and a sphere: The point is within, on the surface, or outside of a sphere, according as its distance from the center is less than, equal to, or greater than a radius (193). 358 POLYHEDRAL ANGLES AND SPHERES (2) A line and a sphere: The line is a secant, a tangent, or does not meet the sphere, according as its distance from the center is less than, equal to, or greater than a radius (196). (3) A plane and a sphere : ' The plane cuts the sphere in a circle (203), is tangent to it (198), or does riot meet it (197), according as its distance from the center is less than, equal to, or greater than, a radius, and conversely. The perpendicular to a tangent plane at its point of con- tact passes through the center of the sphere (200). (4) Two spheres are tangent if the center sect equals the sum of their radii (or if they meet on the line of centers) (234) ; are tangent if they are tangent to the same plane at the same point (235) ; intersect in a circle if they meet at a point not on the line of centers (236). III. CIRCLES OF A SPHERE: (1) The great circle : Its center is the center of the sphere (208). It is the largest circle of a sphere (206). Two great circles on the same sphere intersect in a di- ameter (209). The great circle bisects the sphere and its surface (210). The great circle is determined by the center and two points on the surface (211). The great circle is perpendicular to a circle if it contains one of its poles, and conversely (213). Its arc is perpendicular to a circumference of a great circle if it contains one of its poles (224). Its polar distance is a quadrant, and conversely (218). The shortest line on the surface between two points is the minor great circle arc (247). SUMMARY OF PROPOSITIONS 359 (2) Axes and poles : The line perpendicular to a circle at its center, or joining its center to the center of the sphere, is its axis (205). Parallel circles have the same axis and poles (212). A pole is equidistant from all points on the circumference (214). Polar distances of the same or of equal circles are equal (216). Either of two points a quadrant's distance apart is the pole of a great circle through the other (219). A point at a quadrant's distance from each of two other points is the pole of a great circle through them (220). A polar chord is the mean between the diameter and its own projection upon the axis (229). The radius of a circle is the mean between the sects of its axis (229). (3) Miscellaneous: Circles equidistant from the center of a sphere are equal, and conversely (206). Circles not equidistant from the center are unequal, the one farther from the center being smaller, and con- versely (206). If two circumferences meet at a point on the great circle circumference through their poles, they cannot meet again (246). IV. PROPERTIES OF THE SPHERE : Radii or diameters of the same or of equal spheres are equal (190, 191). Spheres of equal radii are congruent (191). One, and but one, sphere can be circumscribed about a tetrahedron (195). One, and but one, sphere can be inscribed in a tetra- hedron (202). Its surface is the locus of points at a fixed distance from a given point (its center) (192). 360 POLYHEDRAL ANGLES AND SPHERES V. MEASUREMENT OP A SPHERICAL ANGLE: by the measuring angle of the dihedral angle between the planes in which its arms lie (223) ; by the subtended part of the great circle circumference of which it is a pole (226) ; when it is an angle of a triangle, by the supplement of the opposite side of the polar triangle (275). VI. POLYHEDRAL ANGLES AND SPHERICAL POLYGONS : (1) Relations between polyhedral angles and spherical polygons : The face angles of a central polyhedral angle are measured by the subtended sides of its spherical polygon (242). The dihedral angles of a central polyhedral angle are measured by the same measuring angle as the corre- sponding angles of its spherical polygon (243). (2) Properties of a spherical polygon (polyhedral angle). The sum of the sides (face angles) of a spherical polygon (polyhedral angle) is less than a circumference (peri- gonj (249). In a spherical triangle (trihedral angle) the sum of the spherical angles (dihedral angles) is more than one, and less than three, straight angles (277). In a spherical triangle the sum of two sides (face angles) is greater than the third (245). In an isosceles spherical triangle (trihedral angle) the base angles (opposite dihedral angles) are equal (268). In an isosceles spherical triangle the bisector of the vertex angle, the median to the base, the altitude to the base, and the perpendicular bisector of the base, are all the same line (269). (3) Congruence and symmetry : Vertical spherical polygons (polyhedral angles) are sym- metric (261, 262). Spherical triangles (trihedral angles) are congruent or symmetric if the following parts of one are equal to the corresponding parts of the other : SUMMARY OF PROPOSITIONS 361 (a) two sides and the included angle (two face angles and the included dihedral angle) (266) ; (&) two angles and the included side (two dihedral angles, and the included face angle) (267) ; (c) three sides (three face angles) (271) ; (d) three angles (three dihedral angles) (276) ; Isosceles symmetric spherical triangles (trihedral angles) are congruent (270). Symmetric spherical triangles are equivalent (272). (4) Polar triangles : If one triangle is the polar of a second, the second is also the polar of the first (274). An angle of one of the two polar triangles and the op- posite side of the other have supplemental measures (275). VII. Locus OF POINTS : The locus of points at a fixed distance from a given point is the spherical surface with the point as center and the distance as radius (192). The locus of points in a sphere equidistant from all points on the circumference of a circle is its axis (214). VIII. THE MATERIAL SPHERE : To describe a circumference with a given pole and a given polar chord (228). To find the radius of a given circumference (227). To find the diameter of a given sphere (230). IX. LIMITING CASES : The zone is the limit approached by the area generated by a broken line of equal sects inscribed in the generating arc of the zone (282). The area and the volume of a sphere are the limits ap- proached by the area and the volume of a circumscribed polyhedron (300). 362 POLYHEDRAL ANGLES AND SPHERES X. REGULAR POLYHEDRONS AND SIMILAR SOLIDS : There can be but five regular polyhedrons (251). The areas of similar solids are proportional to the squares of corresponding sects (313). The volumes of similar solids are proportional to the cubes of corresponding sects (314). XL LUNES : Limes are equal if their angles are equal, and conversely (290, 291). Adding or subtracting lunes adds or subtracts their angles (292). Multiplying a lime multiplies its angle by the same factor (293). Lunes are proportional to their angles (294). A lune is the same part of a spherical surface that its angle is of a perigon, or 360 (295). XII. AREA ON A SPHERE: (1) of the sphere = 4 Trr 2 (284) ; (2) of a zone = 2 -nrh (283) ; (3) of a lune = r ^ (4irr 2 ), where A is its angle in degrees (295) ; (4) of a spherical triangle = that of a lune whose angle is one half its spherical excess, or -^ (4 Trr 2 ) (296, 297) ; (5) of a spherical polygon = -^ (4 Trr 2 ) (298). XIII. VOLUME OP A SPHERE AND ITS PARTS: (1) of the sphere = | area x radius = f Trr 8 (301) ; (2) of the spherical cone (304), spherical sector (304), spherical pyramid (306), spherical wedge (308), = radius x area of its spherical base = ^ br ; (3) of a spherical segment. Use the Prismatoid Formula (310). 316. ORAL AND REVIEW QUESTIONS State the theorems in this book about spherical triangles that correspond to theorems about plane triangles. What important difference is there between the angles of a plane triangle and those ORAL AND REVIEW QUESTIONS 363 of a spherical triangle? Why is this so? How many points deter- mine a sphere ? Why? How many points 013. the surface determine a great circle? Why? a small circle? Why are circles through the center called great circles ? Why are circles equidistant from the center equal? Define axis, pole, polar distance. What is the polar distance of a great circle ? In what form is the converse of this fact used ? How is the diameter of a material sphere found ? In a sphere of radius 10 in., how far from the center is a circle of 6 in. radius? What is its polar chord? Which methods of proving triangles con- gruent or symmetric use superposition ? What are the other methods? Can one triangle be polar to a second, and the second be polar to a third? Explain fully which poles are taken in forming a polar tri- angle. What numerical relation holds between parts of two polar triangles? If the three sides of one of two polar triangles are 97, 83, 69, find as many parts of the other triangle as possible. How is the formula for the area of a sphere obtained? What part of the surface has its formula derived in the same way ? How is the area of a lune found ? of a spherical triangle ? of a spherical polygon ? What group of solids has its volumes expressed by the formula -^, and for what does b stand in each case ? What other parts of a 3 sphere are there, and how are their volumes found? What is the intersection of a sphere and a plane? Why? What is the intersec- tion of two spheres? Why? When is a plane tangent to a sphere? When are two spheres tangent to each other ? Of what is a spherical surface the locus? What is the locus of points at a fixed distance from a point, and at the same time equi- distant from that point and another point? at a fixed distance from each of two given points? What is the only distinction between con- gruence and symmetry of spherical triangles? If a great circle is perpendicular to a small circle, what must it contain ? Is this suffi- cient to make it a perpendicular? What is known about the sum of the face angles of a polyhedral angle ? What spherical theorem fol- lows from this? What is known about the face angles of a trihedral angle ? What spherical theorem follows ? What are the limits of the sum of the sides of a spherical polygon ? of the sum of the angles of a spherical triangle? How many equilateral triangles can be used to form a solid angle? Why? What are the regular polyhedrons and why are they the only ones ? State the area and volume formulas for the sphere and all its parts. How many of the volume formulas can 364 POLYHEDRAL ANGLES AND SPHERES be derived from the Prismatoid Formula? Which could be proved by the use of Cavalierfs Theorem, and how? Upon what power of the lengths of sects do area formulas depend ? volume formulas ? If the solids are similar, what rule holds for the area and the volume ? If the line ratio is 3, what are the area and volume ratios? If the volume ratio is 3, what are the line and area ratios? If the area ratio is 3, what are the line and volume ratios ? State all the ways in which spherical angles can be measured. What relation is there between polyhedral angles and spherical polygons? Explain why. How is this relation used ? What is meant by " distance " on a spherical sur- face, and why ? Explain how a spherical triangle can have three right angles, and tell what its polar triangle is. How are symmetric tri- angles proved equivalent? GENERAL EXERCISES 803. If the strength of material is in proportion to its cross section, what is the effect of doubling the diameter of a wire ? If a wire of | in. radius will support an iron ball of radius 2 in., how large a wire will support a ball of radius 4 in.? What effect has doubling the diameter of a tree on the amount of wood in it ? 304- How many tangents can be drawn from an external point to a sphere ? How could one .be constructed ? How could a plane be drawn from an external point tangent to a sphere ? 305. Find the locus of the centers of all spheres that are tangent to a given plane at a given point. 306. Find the locus of the centers of all spheres of given radius that are tangent to a given plane. 307. Find the locus of the centers of all spheres that are tangent to the faces of a trihedral angle. 308. A cylindrical hole of radius 6 in. was bored through a sphere of radius 10 in., the axis of the cylinder being a diameter of the sphere. What was the weight of the remaining part of the sphere, if its entire weight was 35 Ib. ? What ratio had the total area of the remaining figure to the area of the sphere ? 309. To inscribe a cube in a given sphere. 310. To inscribe a regular tetrahedron in a given sphere. 311. To inscribe a regular octahedron in a given sphere. GENERAL EXERCISES 365 812. How large an iron shell 1 in. thick would a sphere of radius 3 in. make if melted and recast ? 313. A cup is in the form of a segment of a sphere of internal radius 3 in., with a flat base of radius 2 in., and a cylindrical top of radius 2J in., and height 2 in. How much water will it hold ? (231 cu. in. = 1 gal.) 314. Why is a plumb line, in general, perpendicular to the surface of the earth ? 315. In mixing paint, it is important to know of what sizes to grind the spherical particles of coloring matter in order to cover the surface evenly. If spheres of radius r are tangent to each other and to a plane, find what size other spheres must be so that they can lie between each three of those spheres and be tangent to them and to the plane. 316. If spheres of radius r are piled up in pyramidal form so that each one rests on three others that are tangent to one another, show that the centers of the outside spheres lie in the faces of a regular tetrahedron. Find the height from the highest point to the ground if cannon balls of radius 6 in. are piled three deep. * 317. If the edge of a cube is doubled, what is the effect on its area? on its volume ? If the area is doubled, what is the effect on its edge ? on its volume? If its volume is doubled, what is the effect on its edge? on its area? 318. Prove that the shortest sect from a given point to the surface of a given sphere is along a line to the center. (Two cases.) 319. Circles are inscribed in, and circumscribed about, an equi- lateral triangle. Find the ratios of the areas and the volumes of the solids generated by revolving the triangle and the circles about an altitude of the triangle as an axis. 320. If a right circular cylinder of altitude equal to its diameter and a right circular cone whose slant height equals its diameter are circumscribed about a sphere, show that the total area of the cylinder is the mean proportional between the areas of the cone and the sphere, and that the volume of the cylinder is the mean proportional between the volumes of the cone and the sphere. 321. How many square inches of gold leaf are required to gild a sphere 6 ft. in diameter, no allowance being made for waste? 366 POLYHEDRAL ANGLES AND SPHERES 322. A manufacturer of marbles uses a sheet of white glass 3 ft. square and 1 in. thick, a sheet of green glass in the form of a cylin- drical surface of element 2 ft., thickness f in., and length along a line perpendicular to the elements 28 in., and a piece of yellow glass which when placed in a tank 3 ft. deep that is a circular frustum of radii 2 ft. and 3 ft., raises the water from 1 ft. deep to 2 ft. 3 in. deep. How many marbles f in. in diameter can be made by mixing these glasses, and what proportion of each kind of glass will be in each? 323. If all possible lines are drawn from a point to a sphere, the product of two sects from .the vertex to the surface on any one secant equals the product of the sects from the vertex to the surface on any other secant, and any tangent is the mean proportional between the sects of any secant. 324' To construct a sphere of given radius tangent to a given plane at a given point. 325. Find the locus of a point such that the ratio of its distances from two given points is constant. 326. In filling a measure with fruit or vegetables of approximately spherical shape, would a larger quantity be obtained with those of lesser radius, or with those of greater radius ? Why ? 327. If cannon balls of 1 ft. diameter are piled four deep so that each one rests on four others that are tangent to each other, find the height of the pile. 328. A cylindrical fire extinguisher is 8 in. in diameter and 2 ft. long with a spherical segment 3 in. high on its end as a great circle. How much liquid is required to charge it? How long will it take to empty it by a hose \ in. in diameter if the liquid flows through the hose at the rate of 25 ft. per second ? GENERAL SUMMARY OF THE FORMULAS OF SOLID GEOMETRY 317. MEANING OP THE LETTERS USED IN THE FORMULAS h = altitude, the perpendicular from the vertex to the base, or between the two bases. s = the slant height in a regular pyramid or its frustum, the element in a cylinder, and in a right circular cone or its frustum, the edge in a prism, ft, />, r : when a figure has but one base, b is its area, p is its perimeter, and, if it is circular, r is its radius. &! and 6 2 , p 1 and p 2 , r^ and r 2 , m : when a figure has parallel bases ^ and 2> 2 are their areas, p l and p 2 are their perimeters, and, if they are circular, r t and r z are their radii ; m is the area of the midsection. p rt and r rt are the perimeter and radius of a right section. I. AREA. (1) Lateral Area. (a) Frustum of a pyramid = g (-Pi + Pv (derived from trap- ezoid). Applies to the frustum of a regular pyramid, the entire pyramid (/> 2 = 0), the frustum of a right circular cone, the entire cone (jo 2 = 0), the right prism and the right circular cylinder (p l = jt? 2 ) For figures with circular bases, it becomes Trs(r l + r a .) (&) Pyramid = ~ (derived from the triangle). Applies to the regular pyramid and the right circular cone. For the cone it becomes irrs. (c) Prism = sp rt (derived from the parallelogram). Applies to the prism (including parallelepiped), and cylinder. For the cylinder it becomes 2 Trr rt s. 307 3G8 GENERAL SUMMARY OF FORMULAS (2) Area on a Sphere. (a) Sphere = 4 Trr 2 (rotating a semicircle about its diameter). (6) Zone = 2 Trrh (rotating an arc about its diameter). (c) Lune = - (4 Trr 2 ) where A is the angle of the lune 360 U (derived from its ratio to the surface of the sphere). (d) Spherical polygon = - - (4 Tr 2 r) where E is the spher- ical excess of the polygon (derived from the fact that a spherical triangle is equivalent to half the lune of its spherical excess). II. VOLUME. (1) Prismatoid = (^ + 6 2 + 4 ro) Applies to any solid whose bases are in parallel planes, if it is bounded laterally by planes, or by curved surfaces gen- erated by revolving a straight line sect or an arc of a circle about an axis. Applies to the prismatoid, prism, cylinder, pyramid (b 2 = 0), cone (6 2 = 0), frustum of a pyramid, frustum of a cone, sphere (ft x = 6 2 = 0), spherical segment, spherical wedge (&i = &, = 0). Does not apply to spherical cone, spherical sector, spherical pyramid. (2) Frustum of a Pyramid or a Cone = ^(b l + b 2 + V6^) 3 (derived from the difference of two pyramids). Applies to the frustum of a pyramid, the entire pyramid (b z = 0), the frustum of a circular cone, the entire cone (& 2 = 0), a prism and a circular cylinder (b l J 2 ). For circular figures it becomes -^ (r^ + r 2 2 + ry 2 )' o (3) Pyramid or Cone = o (a triangular prism is composed of three equivalent tri- angular pyramids). For the cone it becomes ~ . 3 (4) Prism or Cylinder = bh (derived from the rectangular parallelepiped). For the cylinder it becomes GENERAL SUMMARY OF FORMULAS 369 (5) Sphere and its Parts. All parts from the center to the surface, bounded laterally by planes or conical surfaces, = where b is the spherical 3 surface and r is the radius of the sphere (derived by taking the limit approached by pyramids from the center to the faces of a circumscribed poly- hedron). Applies to the sphere, spherical cone, spherical sector, spher- ical pyramid, spherical wedge. (a) sphere = f Trr 3 ; (b) spherical cone or sector = f -rrr^h ; 77/1 \ (c) spherical pyramid = - - f - irr s J , where E is the spher- ical excess of the base ; ( If a plane is passed through one of the diagonals of a parallelo- gram, the perpendiculars to the plane from the extremities of the other diagonal are equal. 395. A point moves so that, if any secant be drawn from it to a fixed sphere, the product of the whole secant and its external segment is constant. What is its locus? 396. If h is the altitude of a segment of one base in a sphere of radius r, the volume of the segment is equal to Trk 2 (r J. 397. How far from the surface of a sphere of radius 2 ft. must a light be placed to illuminate one third of its entire surface ? 398. Find the volume of the portion of a sphere, lying outside of an inscribed right circular cylinder, whose altitude is h and radius r. 399. A regular hexagonal pyramid whose base is inscribed in a circle of radius 3 in. has an altitude of 4 in. Find the dimensions of a similar pyramid such that its lateral area shall be equal to the lateral area of the first increased by the surface of a sphere of radius 4 in. Answer need not be simplified. 400. A solid right circular cone of altitude 3 in., radius of base 2 in., rests base downwards in an inverted hollow cone, similar to the solid cone, the axes of both cones being vertical and the joint between the two water-tight. If the vertex of the solid cone just reaches the level of the rim of the hollow cone, find how many cubic inches of water can be contained in the vessel so formed. Answer need not be simplified. 376 COLLEGE EXAMINATION QUESTIONS 401. State and prove the theorems true of the figure formed by cutting through a pyramid by a plane parallel to the base. 402. What is the ratio of the volume of a cube to the volume of a second cube whose edge is a diagonal of the first cube? 403. Find the locus of the center of a sphere which is tangent to three given planes. 404. How many rubber balls 2 in. in diameter and r \ in. thick can be made from a flat circular sheet of rubber 1 in. thick and 2 ft. in diameter ? 405. Find the portion of the earth's surface bounded by a geodetic triangle whose sides are 60, 90, and 100 respectively. 406. Name and describe the regular polyhedrons. 407. What portion of the surface of the earth is included between the parallels of 30 north and south latitude? 408. State (without proof) the theorem for the area of any spheri- cal triangle, and illustrate by computing the area of an equilateral spherical triangle, each of whose angles is 70, on a sphere of radius 5 feet. 409. What kind of figure is any section of a cone made by a plane passing through the vertex and cutting the base ? by a plane parallel to the base ? Prove your answers. 410. If the altitude of a cylinder of revolution equals the diameter d of a given sphere, determine the radius of the base of the cylinder so that (a) the total surfaces shall be equal, (b) the volumes shall be equal. 411. Find the volume of a spherical shell bounded by two concen- tric spheres whose surfaces are 20 TT and 15 TT. 412. Enumerate the various cases of equal spherical triangles, and prove any one of them. 413. In a right triangle whose sides are a and b, a line c is drawn bisecting a and parallel to b. Find the ratio of the surfaces generated by c and the hypotenuse of the original triangle when the figure is revolved about b. 414. Two truncated prisms are equal if three faces including a trihedral angle of one are respectively equal to three faces similarly including a trihedral angle of the other. COLLEGE EXAMINATION QUESTIONS 377 415. If an equilateral triangle and its inscribed circle are revolved about an altitude of the triangle, then the ratio of the volumes of the cone and the sphere thus generated are as 9 to 4. 416. What is the shortest line which can be drawn on the surface of a sphere between two points ? 417. To determine a point in a given straight line which shall be equidistant from two given points in space. 418. If a plane is passed through a diagonal of a parallelogram, the perpendiculars to it from the extremities of the other diagonal are equal. 419. A cylinder of revolution is inscribed in a sphere of radius 6 in. The altitude of the cylinder is twice the radius of the base. Find its total surface. 420. To determine a point in a given plane which shall be equi- distant from three given points in space. 421. A projectile has the shape of a cylinder of revolution sur- mounted by a conical cap. Its total length is 24 in., and the cylindri- cal part is three times as long as the conical end. The greatest diameter is 10 in. Find the volume and the total surface of the projectile. 422. A plane is passed through a sphere, bisecting at right angles one of the radii. Compare the areas of the two portions into which the spherical surface is divided. 423. To construct a sphere passing through two given points in space, and having its center in a given line. 424- Given a sphere 10 in. in diameter. A regular square pyramid is inscribed and its altitude is 8 in. Find the volume and total sur- face of the pyramid. 425. If the diameter of a sphere is doubled, in what ratios are the surface and the volume increased? 426. Given two intersecting planes in space, to determine the locus of the center of a sphere of given radius which shall be tangent to each of the given planes. 427. One and but one sphere can be inscribed in any tetrahedron. 378 COLLEGE EXAMINATION QUESTIONS 428. In a trapezoid ABCD, AB and CD are parallel and AC is perpendicular to CD. The dimensions are, AR = 4 in., CD = 12 in., A C = 6 in. The figure is revolved about A C. Find the total surface and the volume of the conical frustum generated. 1$9. The diameter of a sphere of radius 5 in. is increased by 1 in. How much are the surface and the volume increased? 430. A point and any two non-intersecting straight lines are given in space. Show how to construct a straight line which shall pass through the given point and intersect each of the given lines. 431. An equilateral triangle and its inscribed circle are revolved about an altitude of the triangle, generating a cone of revolution and its inscribed sphere. Find the area of the circle along which the cone and sphere are tangent. 432. Find the area of a spherical triangle whose angles are 90, 120, 45, on a sphere whose radius is 6 in. 433. Prove that if the three faces including a trihedral angle of a prism are respectively equal to the three faces including a trihedral angle of another prism, and are similarly placed, the two prisms are equal. 434' In a sphere whose radius is 13 is inscribed a cylinder whose altitude is 24. Find the ratio of the area of the surface of the cyl- inder to that of the sphere. 435. On a sphere whose radius is 12, find the altitude of a zone whose area is equal to the area of the spherical triangle who.se angles are 120, 110, and 80. 436. Prove that the four diagonals of a parallelepiped meet in a point, which is the midpoint of each. 437. Prove that every section of a parallelepiped made by a plane intersecting all its lateral edges is a parallelogram. 438. Construct a spheric surface with a radius r which shall pass through two given points and be tangent to a given plane. Prove your construction. 439. Let PQ be a line drawn from a given point P, meeting a given plane in Q. Find the locus of the point midway between P and Q when Q moves arbitrarily in the given plane. Give proof. 440- On a sphere of radius 10 in. find the area of a zone with altitude 3 in, COLLEGE EXAMINATION QUESTIONS 379 441. A right cone and a cylinder of revolution on the same base with radii 3 in. have the same altitude 4 in. Find the ratio of their volumes and of their lateral areas. 44%> Find the capacity of a circular pail 15 in. high, the radius of the lower base being 4 in. and the radius of the upper base 6 in. 443. The spire of a church is a right pyramid on a regular hexagonal base, each side of the base is 10 ft., and the height is 50 ft. There is a hollow part which is also a right pyramid on a regular hexagonal base, the height of the hollow part being 45 ft., and each side of the base 9 ft. Find the number of cubic feet of stone in the spire. 444- The volumes of two similar polyhedrons are 64 cu. ft. and 216 cu. ft. respectively. If the area of the surface of the first is 112 sq. ft., what is the area of the surface of the second? 445- Find the volume of a cone of revolution the area of whose total surface is 200 TT sq. ft., and whose altitude is 16 ft. 446- Find the area of a spherical triangle whose angles are 100, 120, 140, the diameter of the sphere being 16 in. 44^' Iu any trihedral angle the common intersection of the three planes bisecting the three dihedral angles is a straight line. 448- To what extent is a straight line determined by the condition that it is perpendicular to (a) two intersecting straight lines, (6) two non-intersecting straight lines? \449- If a section parallel to the base of a pyramid is equal to one ninth of the base in area, the altitude of the pyramid being 6 ft., how far from the vertex is the plane of the section ? * 450. A solid is in the form of a right circular cone, standing on the flat base of a hemisphere of equal base, radius 10 inches; the volume of the cone is equal to that of the hemisphere. Determine the* surface and the volume of the smallest right circular cylinder that will contain this solid, the axis of the cylinder being the same as that of the solid. How much waste space is there in the cylinder? 451. Explain why a circular table with three legs, symmetrically placed, will be perfectly steady if placed on a plane floor, even though one of the legs is slightly longer than the other two, while a table with four legs, of which one is a little longer than the others, will not be steady. 380 COLLEGE EXAMINATION QUESTIONS '452. Two angles of a spherical triangle are 80 and 120. Find within what limits the third angle must lie; and prove that the greatest possible area the triangle can have is four times the least possible area, the sphere on which it is drawn being given. 453. Define a regular polyhedron, and prove that there cannot be more than five regular polyhedrons. Assuming that there are five, state for each (a) the nature of the faces, (ft) the number meeting at a vertex, and (c) the number of faces. How many vertices has each of the regular polyhedrons ? how many edges ? " 454- Prove that any right circular cone can be produced by the revolution of a triangle about one side as an axis. What kind of triangle must be used? 455. An irregular portion (less than half) of a material sphere is given. Explain how the radius can be found, compass and ruler being allowed. 456. Give expressions for the surface and volume of a sphere of radius r and prove one of them. 457. A solid sphere of metal of radius 18 in. is recast into a hollow sphere. If the cavity is spherical, of the same radius as the original sphere, find the thickness of the shell. 458. The volumes of a hemisphere, right circular cone, and right circular cylinder, are equal. Their plane bases are also equal, each being a circle of radius 10 in. Find the height of each. 459. A sphere of radius 5 ft. and a right circular cone on a base also of radius 5 ft. stand on a plane. If the height of the cone is equal to the diameter of the sphere, find the position of the plane that cuts the two solids in equal circular sections. Find the area of these sections. ~ 460. Find the area ^of a spherical triangle whose angles are 100, 120, 75, if the radius'of the sphere is 7 in. 461. The vertices of one regular tetrahedron are the centers of the faces of another regular tetrahedron. Find the ratio of the volumes. 462. Find the angle of an equiangular spherical hexagon equiva- lent to the sum of three equiangular spherical hexagons whose angles are equal to 130. COLLEGE EXAMINATION QUESTIONS 381 ^463. The radii of two spheres are 13 ft. and 15 ft. respectively, and their line of centers equals 14 ft. Find the volume of the solid common to both spheres (spherical lens). 464. Find the altitude of a frustum of a circular cone, if its volume equals 190 cm., and the radii of the bases are respectively 2 cm. and 3 cm. 465. Assuming the earth to be a sphere with a diameter equal to 8000 mi., find the area of a zone bounded by the parallels of 30 north and 30 south latitude. 466. The lateral surface of a right circular cone is equal to three times the area of the base. If the radius of the base equals r, find the altitude and the volume of the cone. 467. Two opposite angles A and C of a spherical quadrilateral A BCD are equal, and AB and CB are produced through B to meet the opposite sides in E and F respectively. If angle E equals angle F, prove that AB = BC. Is the corresponding proposition for plane geometry correct? Give reasons. 468. State a proposition that may be used to construct a plane parallel to a given line. State a proposition that may be used to construct a plane perpendicular to a given plane. Through a given point to construct a plane parallel to a given line and perpendicular to a given plane. 469. From the vertices of a triangle ABC perpendiculars AA f , BB', and CC 1 are dropped upon a straight line XY in the same plane. A A' = 2, BB' = 3, CC' = 1, A'B' = 2, B'C' = 1, and C'A' = S. Find the volume generated by the revolution of triangle ABC about X Y as an axis. 470. Find the volume of a pyramid whose base contains 30 sq. cm. if one lateral edge is 5 cm., and the angle formed by the edge and the plane of the base is 45. 471. Prove that two triangles on the same sphere that are mutu- ally equiangular are mutually equilateral. Are such triangles neces- sarily equivalent? Are they necessarily equal? 472. If a spherical quadrilateral is inscribed in a small circle, prove that the sum of two opposite angles is equal to the sum of the other two angles. 382 COLLEGE EXAMINATION QUESTIONS / 473. On the base of a right circular cone a hemisphere is con- structed so that it lies outside the cone, and the surface of the hemi- sphere equals the surface of the cone. If r is the radius of the hemisphere, find (a) the slant height of the cone, (b) the inclination of the slant height to the base, (c) the volume of the entire solid. 474- Find the total surface and the volume of a regular tetrahe- dron whose edge equals 8 cm. 475. Let AB and CD be two lines not in the same plane, and let E be any point on AB and F any point on CD. What is the locus of the middle point of the straight line EFt Prove the truth of your answer. 476. The altitude of a cone of revolution is 12 cm. and the radius of its base is 5 cm. Compute the radius of the sector of paper which, when rolled up, will just cover the convex surface of the cone; and compute the size of the central angle of this sector in degrees, minutes, and seconds. 477. Regarding the earth as a sphere, show that J of its volume is included between the planes of the small circles of 30 north lati- tude and 30 south latitude. 478. If the area of a zone of one base is n times the area of the circle which forms its base, prove that the altitude of the zone is n ~ times the diameter of the sphere. n 479. A variable sphere is tangent to a fixed plane at a fixed point of the plane, and a plane is drawn tangent to the sphere and parallel to a second fixed plane. What is the locus of the point of contact? J 480. Prove that the edge of a regular octahedron is approximately 2.45 times the radius of the inscribed sphere. 481. Prove that if the four sides of a spheric quadrilateral are equal, its diagonals bisect each other. 482. Define : (a) a perpendicular to a plane ; (b) the distance from a point to a plane ; (c) the angle that a line makes with a plane ; (d) a right circular cylinder ; (e) a spherical polygon. 483. Prove that if a plane and a line not in that plane are per- pendicular to the same plane, they are parallel to each other. COLLEGE EXAMINATION QUESTIONS 383 484. If from a point P of a perpendicular PA to a plane MN a perpendicular PD is drawn to a line EC of the plane, any line DA that joins the foot D of the second perpendicular to any point A of the first perpendicular will itself be perpendicular to the line BC of the plane MN. 485. If two lines in space are non-intersecting, are their projections on a plane necessarily non-intersecting ? Illustrate with a figure. 486. Is it true that if two lines are perpendicular to each other, any plane passed through one of the lines is perpendicular to the other? Give the reason for your answer. 487. Is it possible for two planes to be perpendicular to each other if both are perpendicular to a third plane ? Illustrate your answer by a figure. 488. Prove that if two planes are parallel to a third they are parallel to each other. Is this theorem true if the word "parallel" is replaced by " perpendicular " ? 489. Prove that if a straight line is perpendicular to a plane, every plane passed through that line is perpendicular to the plane. Is it true that, if a plane is perpendicular to a second plane, every line of the first plane is perpendicular to the second plane ? Illustrate by a figure. Is it true that, if a plane is parallel to a second plane, every line of the first plane is parallel to the second plane ? Give the reasons. 490. Prove that the lateral area of a prism is equal to the product of the perimeter of a right section of the prism and a lateral edge. Is this product the same as the product of the perimeter of the base and an altitude ? Give the reason for your answer. 491. Show how to cut a tetrahedron by a plane so that the section shall be a parallelogram. In what three ways may the cutting plane be passed ? When is it possible to get a rectangular section ? \i 492. The vertices of a tetrahedron are four vertices of a cube, no two of which lie on the same edge of the cube. Prove that the volume of the tetrahedron is one third the volume of the cube. 493. Prove that if a pyramid is cut by a plane parallel to the base, the section is a polygon similar to the base. Where would such a section be placed if its area were one half that of the base? 384 COLLEGE EXAMINATION QUESTIONS 494. If two non-coplanar lines are cut by three parallel planes, prove that the corresponding segments are proportional. If a tri- angular pyramid is cut by two parallel planes whose distances from the vertex are 2 ft. and 5 ft., what can you say about the area of the sections thus formed? 495. Assuming that the volume of a rectangular parallelepiped is equal to the product of its base by its altitude, state in logical order, without proof, the theorems that lead to the volume of any cylinder. 49$. From a cylinder of revolution whose altitude is 12 in. and whose diameter is 10 in. is cut out a cone. The base of the cone coincides with one base of the cylinder, and its vertex is at the center of the other base of the cylinder. Compute the total surface and the volume of the solid that remains. 497. The area of a certain small circle on a sphere is equal to the difference of the areas of the zones into which the circle divides the spherical surface. Find the distance from the center of the sphere to the plane of the small circle, if the radius of the circle is r. If a cone is tangent to the sphere along the given circle, show that the distance from the center of the sphere to the vertex of the cone is ( V5 x 2)r. 498. From a hemisphere whose radius is 3 in. a solid is cut by means of a cone of revolution whose base coincides with the base of the hemisphere, and whose altitude equals the radius of the sphere. Find the volume and the surface of the solid that is left, correct to two decimal places. 499. A hemispherical dome 140 ft. in diameter is divided into two parts by a horizontal plane that lies seven eighths of the way from its base to the summit. What will be the expense of gilding the lower part at a cost of 25 cents per square foot V 500. Show that if a man ascended in a balloon to a height equal to the earth's radius, he would see one quarter of the earth's surface. 501. Find the volume of a spherical segment of one base whose altitude is one half the radius of the sphere. 502. Derive the formula for the volume of a solid included be- tween a spherical surface of radius r, and a conical surface tangent to the sphere, the vertex of the cone being distant 2 r from the center of the sphere, COLLEGE EXAMINATION QUESTIONS 385 503. A certain dome is in the form of a spherical zone of one base. The base is a circle 24 ft. in diameter, and the highest point is 9 ft. above the plane of the base. Find the number of square feet in the surface of the dome, correct to two decimal places.- 504- What proportion of the earth's surface lies between the equator and latitude 45? Draw the figure. 505. Find the area of a spherical quadrilateral whose angles are 135, 90, 80, and 70, on a sphere of radius 10 ft. 506. The dihedral angles of a trihedral angle are 100, 65, 87. The trihedral angle is closed by a portion of a spherical surface whose radius is 4 in., and whose center is at the vertex of the trihedral angle. Compute to two decimal places the area qf this portion of the spherical surface. 507. Prove that if two arcs of great circles intersect on the surface of a hemisphere, the sum of the opposite spherical triangles which they form is equivalent to a lime whose angle is the angle between the arcs in question. In what connection is this proposition im- portant? 508. Assuming the area of a triangle as known, give, without de- tailed proof, the logical steps that lead to the formula for the lateral area of a right circular cone. Does this formula apply to an oblique cone? Give your reasons. 509. If a rectangle revolves about its shorter side , and then about its longer side A, which of the cylinders thus generated will have the greater volume? 510. Assuming the volume of a cube as known, give, without de- tailed proofs, the logical steps which lead to the formula for the volume of a right circular cylinder. Does this formula apply also to an oblique circular cylinder? Give your reasons. 511. The chimney of a factory has the shape of a frustum of a regular pyramid. It is 180 ft. high, and its upper and lower bases are squares whose sides are 10 ft. and 16 ft. respectively. The flue is throughout a square whose side .is 7 ft. How many cubic feet of material does the chimney contain ? J 512. The corner of a cube is cut off by a plane passed through the outer extremities of the three edges meeting at the given corner. What part of the volume of the cube is thus removed? 386 COLLEGE EXAMINATION QUESTIONS 513. How far from the vertex of a pyramid must a section parallel to the base be passed in order that the product of this section by the altitude of the pyramid shall equal the volume of the pyramid ? 514> The volumes of two given spheres are in the ratio 1 : 2. Find the ratio of the total surfaces of their inscribed cubes. 615. Show how to construct a regular tetrahedron. 516. A right triangle whose legs are 3 and 4 revolves about the longer side. . Show that the total surface thus generated is less than the surface of a sphere whose diameter equals the hypotenuse of the given triangle. 517. State and prove the theorems true of the figure formed by cutting through a pyramid by a plane parallel to the base. 518. In a sphere whose diameter is 14 in. the altitude of a zone of one base is 2 in. Find the altitude of a cylinder of revolution whose base is the base of the zone, and whose lateral area equals the area of the zone. 519. Given two points A and B in two intersecting planes M and N. To find Z in the line of intersection of M and N such that A Z + XB shall be a minimum. J 520. What are the ratios of the radii and the volumes of two spheres, one having 81 times the surface of the other? What are the ratios of the areas and radii of two spheres, one having 81 times the volume of the other. 521. Two planes perpendicular to the same plane P, and containing two lines AB and A'B' parallel to each other, are parallel. Show also that this is not true if the lines AB and A'B 1 are perpendicular to plane P. 522. Similar cylinders are to each other as the cubes of their alti- tudes, or as the cubes of the diameters of their bases. 523. Show that the projections of parallel lines on the same plane are parallel, but that the converse is not true. 624- The area of the surface generated by the base AB of the isosceles triangle OAB, which revolves about a fixed axis A' Flying in its plane and passing through its vertex without cutting the triangle, is equal to the circumference which has for its radius the altitude OT of the triangle, multiplied by the projection of the base A B on the axis AT. COLLEGE EXAMINATION QUESTIONS 387 525. If, from any point on a sphere as a pole, with a polar distance equal to one third of a quadrant, \ve describe a circle on the sphere, the radius of this circle will be half the radius of the sphere. 526. If M and N are the feet of the normals to two planes from a point P, show that the line of intersection of the two planes is normal to the plane MNP. 527. The base of a prism of altitude 7 in. is a regular hexagon whose side is 4 in. The edges of the prism make angles of 00 with the altitude. Find the lateral area and the volume of the prism. 528. Through a point A are drawn three mutually perpendicular lines, AB, AC, and AD, B, C, and D being any points on the three lines. The points C and D are joined, and from A a perpendicular is dropped to this line CD, meeting it in E. Prove that the line joining E to any point in AB is perpendicular to CD. 529. The three lines joining the middle points of the three pairs of opposite edges of a triangular pyramid meet in a point. 530. The area of the entire surface of a frustum of a right circular cone is 120 TT, and the diameters of the parallel bases are 8 and 14. Find the lateral area and the volume of the frustum. 531. The surfaces of a sphere, the circumscribed right cylinder, and the circumscribed right cone whose axial section is an equilateral triangle are in the ratio 4:6:9. The same is true of the volumes. 532. In grading the site for a summer house, a mound is to be made in the shape of a truncated right circular cone. The lower base has a diameter of 12 meters, the upper base (4 meters higher), of 8 meters. How many cubic meters of earth will be necessary ? 533. If two lines intersect at right angles, under what conditions will their projections upon a given plane be perpendicular ? 534. Find the volume of the frustum of an oblique pyramid from the following data : the lower base is a square of side 4 in. ; the upper base a square of side 2 in. ; one of the inclined edges, which is 8 in. long, has as its projection upon the lower base one of the diagonals of that base. 535. Find the length of the projection upon a plane of a line 8 in. long (a) parallel to the plane, (b) making an angle of 60 with the plane, (c) perpendicular to the plane. COLLEGE EXAMINATION QUESTIONS 536. Find the number of cubic feet of concrete in a dam 250 ft. long, 31 ft. high, 33 ft. wide at the bottom, and 5 ft. wide at the top. 537. Find the volume of a regular tetrahedron whose slant height is V3. 538. Find the length of wire ^ in. in diameter that can be made from a cubic foot of copper. 539. The volume of a regular square pyramid is 43f cubic ft. ; its altitude is twice one side of the base. Find (a) the lateral area of the pyramid, (6) the area of a section made by a plane parallel to the base and 1 ft. from the base. 540. In a triangle, the sides including an angle of 120 are respec- tively a and 2 a. Find the volume of the solid generated by revolving the triangle about the shorter side as an axis. \ 5Jj.l. A solid glass ball 6 in. in diameter is expanded by a glass blower till the glass is 1 in. thick. Find the outer diameter of the hollow globe. 542. The altitude of a cone, the diameter of its base, and the edge of a given cube are equal. Find the ratio of the volume of the cone to the volume of the cube. 543. The sides of a parallelogram which are 12 in. and 8 in. re- spectively form an angle of 60. Find the volume and the convex surface of the solid generated by the revolution of the parallelogram about one of its longer sides as an axis. 544. Three plane angles are 20. 80, 105. Will these angles form a trihedral angle ? Why ? 545. A cylindrical water pipe has a diameter of a inches and a ve- locity of flow of b ft. per second. How many cubic feet of water will be discharged in one minute at a given point, no allowance being made for friction ? Also, what would be the effect upon the quantity if the diameter of the pipe were doubled ? 546. A circular sector whose angle is 60, and whose radius is 6, revolves about a diameter perpendicular to one of the limiting radii. Find the volume of the spherical sector generated. 547. The volumes of two similar cones of revolution are to each other as 343 is to 512. What is the ratio of the lateral surfaces? ^ 648. Define symmetry as to a center, and show that the symmetrical figure of a dihedral angle is an equal dihedral angle. COLLEGE EXAMINATION QUESTIONS 389 549. The total surface of a regular tetrahedron is 60 sq. m. Find its volume. 550. Find the locus of those points of a plane at which a given straight line, not lying in that plane, subtends a right angle. Show when the locus becomes a point and disappears. 551. The volume of a sphere inscribed in a regular tetrahedron is 2 cu. in. What is the volume of a circumscribed sphere? 552. Show how to cut a given polyhedral angle of four faces so that the section shall be a parallelogram. 553. What part of the surface of a sphere is illumined by a lamp placed at a distance of a diameter from the surface of the sphere ? J 554. Two tetrahedrons which have a trihedral angle of one equal to a trihedral angle of the other are to each other as the products of the three edges of the equal trihedral angles. 555. A right circular cone the altitude of which is three times the radius of its base, and a sphere, the radius of which is equal to the radius of the base of the cone, are immersed in a rectangular cistern of water the base of which is 9 ft. by 11 ft. If they are removed, the water level is lowered by 2 ft. Find their dimensions. 556. A cylindrical tank 10 ft. long and 5 ft. in diameter lies with its axis horizontal. If it contains gasoline to a depth of 15 in., how many gallons are in the tank ? (231 cu. in. to the gallon.) 557. Find the point the great circle distances of which from the sides of a spherical triangle are equal. State your construction. 558. Four of the six planes determined by the diagonals of a parallelepiped divide the parallelepiped into six quadrangular pyra- mids. Prove that these six pyramids are equivalent. 559. Describe a spherical surface with a given radius that shall pass through two given points and be tangent to a given plane. 560. The axis of a right cylinder passes through the centers of all sections parallel to the base. 561. The base of a pyramid is a right triangle whose base is 12 in. and whose hypotenuse is 20 in. The altitude of the pyramid is 15 in. Find the volume of the frustum of this pyramid cut off by a plane 3 in. above the base. COLLEGE EXAMINATION QUESTIONS 562. The base of a right prism is a rhombus, one side of which is 10 in., and the shorter diagonal is 12 in. The altitude is 15 in. Find the volume. 563. A regular hexagonal pyramid is cut into two parts of equal volume by a plane parallel to the base. What is the distance from the vertex to this plane, in terms of the altitude? 564- Find the volume of a right pyramid whose slant height is 13 ft., and whose base is an equilateral triangle inscribed in a circle whose radius is 10 ft. 565. The volume of the frustum of a square pyramid is 74 cu. in. ; the edges of the bases are 3 in. and 4 in. respectively. Find the altitude. 566. The altitude of a regular pyramid is 2 a, and the base is a triangle inscribed in a circle of radius a. Find the lateral area of the pyramid. 567. Find the area of a zone on a sphere of radius r that is illumined by a lamp placed at a distance a from the surface. 568. A cone of wood has its vertex angle equal to 60, and the radius of its base equal to 2 in. A cylindrical hole of radius 1 in. is bored through the entire cone, the axes of the two coinciding. How much of the cone goes into chips ? 569. The distance between two parallel planes is 16 in. A line 24 in. long has an extremity in each of these planes. Find the length of the segments into which this line is divided by a plane parallel to the given planes and 4 in. from one of them. 570. What is the greatest number of faces that a convex polyhedral angle can have if each face angle is 60? Why? 571. If the area of a section is one third that of the base, what is the ratio of the segments into which the altitude is divided ? 572. Define polar triangle of a spherical triangle. If a is the side of a spherical triangle, and A' the opposite angle of the polar triangle, prove that A' + a = 180. 573. From a point 6 ft. from the surface of a sphere, one quarter of its surface is visible. Find the radius of the sphere. COLLEGE EXAMINATION QUESTIONS 391 574- A triangle whose sides are respectively 15 in., 13 in., and 4 in., revolves about the shortest side as an axis. Find the volume of the solid generated by the revolving triangle. 575. The volume of a regular hexagonal prism is 81 \/3; the altitude of the prism is equal to the longest diagonal of the base. Find the total area of the prism. 576. Find the volume of a hemisphere whose entire surface equals S. 577. Find the locus of a point on a sphere that is equidistant from two given points on the surface. 578. The volume of a sphere is 4500 TT cu. in. Find its surface. 579. Find the surface and the volume of the solid generated by a door 3 ft. wide and 8 ft. high swinging in an arc of 144. 580. The hypotenuse of a right triangle is 5 in., one of its legs is 3 in. Find the volume of the solid generated by revolving the triangle on its hypotenuse as an axis. 581. Assuming that the radius of the earth is 4000 mi. and that the crust is 30 mi. thick, find the volume of the crust of the earth. 582. Prove that if a line is parallel to one plane and perpendicular to another, the two planes are perpendicular to each other. 583. Find the weight of 52,800 linear feet of copper wire T \ of an inch in diameter. (1 cu. ft. of copper weighs 556 Ib.) 584- Find the number of cubic feet of earth in a railway embank- ment 2500 ft. long, 10 ft. high, 12 ft. wide at the top, and 42 ft. wide at the bottom. 585. Find the cost, at $2.50 a square foot, of gilding a hemispheric dome whose diameter is 50 ft. ^,J586. A sphere of lead 10 in. in diameter is melted and cast into a cone 10 in. high. Find the diameter of the base of the cone. 587. Find the capacity in cubic inches of a berry box in the form of a frustum of a pyramid 5 in. square at the top, and 4^ in. square at the bottom, and 2f in. deep. 588. Two tanks are in form similar solids ; one holds 128 gal., the other 250 gal. If the first is 20 in. deep, find the depth of the second. 589. The total surface of a cube is 450 sq. in. Find the volume. 392 COLLEGE EXAMINATION QUESTIONS 590. Define as loci (1) the intersection of two planes; (2) the bisecting plane of a dihedral angle ; (3) the plane which is the per- pendicular bisector of a given sect ; (4) the plane perpendicular to a given line at a given point. 591. Define (1) the angle between a line and a plane; (2) a paral- lelepiped ; (3) symmetric polyhedral angles ; (4) the volume of a geo- metric solid. 592. Find the number of cubic yards of dirt to be excavated in digging a canal, 50 ft. wide at the top, 30 ft. wide at the bottom, 14 ft. deep on the average, between two locks 2.6 mi. apart. 593. Compute the volume of a regular tetrahedron whose slant height is \/3. 594- The sides of a parallelogram, which are 12 in. and 8 in. re- spectively, form an angle of 60. Find the volume and the convex surface of the solid generated by the revolution of the parallelogram about one of its longest sides as an axis. 595. Prove that the smallest section of a sphere made by a plane passing through a given point within the sphere is that made by a plane perpendicular to the radius through the given point. 596. The lateral area of a right cylinder is 48 TT ; the volume is 96 TT. Find the radius and the height of the cylinder. 597. An isosceles trapezoid revolves about its longer base as an axis ; the bases are respectively 14 in. and 8 in., the legs each 5 in. Find the surface of the solid generated. 598. The radius of a sphere is 20 in. Find the area of a section made by a plane 5.6 in. from the center of the sphere. 599. Find all possible locations of a point that is equidistant from two given points in space and at a given distance from a third point. 600. A cone 5 ft. high is cut by a plane parallel to the base and 2 ft. from the base; the volume of the frustum formed is 294 cu. ft. Find (a) the volume of the cone, (6) the volume of the part cut off by the plane. 601. A cylindrical tank 20 ft. long and 9 ft. in diameter lies with its axis horizontal. If it contains gasoline to a depth of 6 ft., how many gallons are in the tank ? (231 cu. in. to the gallon.) APPENDIX 318. Statements whose Conditions and Conclusions may have Two or More Parts. If the condition or the con- clusion of a statement is composed of two or more parts, the consideration of its related statements can be made very complicated by taking those parts in all their dif- ferent combinations as condition and conclusion. The full discussion of such possibilities has no place in Geom- etry, but belongs in a course on Logic. The following illustration will, however, show how the different combi- nations can be used to find new propositions. It will be considered only in reference to the converses. In an isosceles triangle, a line through the vertex parallel to the base bisects the exterior angles at the vertex. CONDITIONS : The triangle is isosceles, The line passes through the vertex, The line is parallel to the base. CONCLUSION : The line bisects the exterior angles at the vertex. (Since the bisector of one angle also bisects the other, this is considered as one part.) COMPLETE CONVERSE : The interchange of the entire condition with the entire conclusion would give If a line bisects the exterior angles at the vertex of a tri- angle, (1) the triangle is isosceles; (2) the line passes through the vertex ; (3) the line is parallel to the base. 393 394 APPENDIX Of these three conclusions, (2) obviously is true ; (1) and (3) are not true from the condition alone, for the bisector of the angle cannot affect the shape of the triangle, neither is a bisector always parallel to the base of a triangle, no matter how the triangle is drawn. If, however, either (1) or (2) is taken as part of the condition, and the other is taken as the conclusion, a new proposition results, as, In an isosceles triangle, the bisector of the exterior angle at the vertex is parallel to the base. Or, If the bisector of an exterior angle of a triangle is parallel to the base, the triangle is isosceles. In applying logic to elementary geometry, it is seldom necessary to consider the general description of the figure as a part of the condition. For example, in If two sides of a triangle are equal, the opposite angles are equal. The fact that a triangle is being considered is regarded as a general description, and the statement is therefore one having a single condition, and its converse is If two angles of a triangle are equal, the opposite sides are equal. 319. Cavalieri's Theorem. If two solids contained between the same two parallel planes are such that their sections by any third plane parallel to those two planes are equivalent, tlxe two solids have the same volume. APPENDIX 395 i Divide the common altitude of the two solids into n equal parts, and through the points of division pass planes parallel to the base planes. Let M and N be two con- secutive parallel planes, and let M cut the solids in the equivalent sections 8 l and 8 2 . At some point in the perimeter of 8-^ erect a line between M and N perpendicular to Jf, and let it generate a cylindrical or prismatic surface, or a combination of them, by moving always parallel to its original position, with 8 1 as a guiding line. Do the same with S 2 , thus generating two surfaces which include right solids between the planes M and jy. These solids have equivalent bases and the same altitude, and are therefore equivalent in volume. If the same operation is performed with each consecutive pair of parallel planes, two sets of right solids will be formed, such that each corresponding pair are equivalent in volume, and their sums are also equivalent. Their sums differ somewhat from the two given solids, but if the number of divisions of the altitudes is increased without limit, the sums of the right figures will approach the given solids as limits. But, since the sums are equal variables, their limits are also equal, and the given solids are equivalent in volume. Among the various sets of Cavalieri bodies are the follow- ing: prisms, cylinders, and combinations of prisms and cyl- inders, of equivalent bases and equal altitudes ; pyramids, cones, and combinations of pyramids and cones, of equiva- lent bases and equal altitudes ; a sphere of radius r, a right cylinder of radius r and altitude 2 r, hollowed out in conical form from each base to the center, and a tetrahedron with two of its edges in two planes tangent to the sphere at the ends of a diameter, and with its midsection parallel to those edges and equivalent to a great circle of the sphere. 396 APPENDIX 320. The Prismatoid. The statement of the formula for the volume of a prismatoid, and the figures of elemen- tary geometry to which it applies, were given in 182. It is now required to prove the formula for the prismatoid itself. Given the prismatoid of bases ^ and 5 2 , midsection m, and altitude h ; to prove F = -(^ -f > 2 + 4ra). Draw a diagonal in each quadrilateral face (as KL), thus dividing the entire lateral surface into triangles. Take P, any point in m, and by pass- ing planes through P and each lateral edge of the figure, including the diag- onals drawn in the faces, form pyra- mids from P as a vertex to each face, or triangle formed in a face, as a base. Then the prismatoid is divided into pyramids of which it is the sum ; that is, P-b 1 -f P-> 2 + P-RST 4- pyramids from P to each other triangular lateral face = the prismatoid. On considering any one of the pyramids from P to a lat- eral face, as P-RST, where PXY is the part of m cut off by the faces of the pyramid : P-RST = 4 (P-XFT), because RST 4 XYT. (Why ?) But P--X"Fr, with T as the vertex, = - PXY, SO P-RST = ~ PXY. 6 6 Since each pyramid from P to a lateral triangle equals - times its part of m, their sum equals - m. D o K S APPENDIX 397 But pyramid P-b l = - b v and pyramid P-6 2 = - 5 2 , therefore F = - (^ + 2 -f 4 w). Besides this formula for the prismatoid, there is another formula that uses but one base, and instead of the mid- section, uses the section parallel to the bases, and two thirds the distance from the base used in the formula. If the section at the two-thirds point of the altitude is called s, this formula is - (b 1 + 3 s). 1 321. Circular Cylindrical and Conical Surfaces. A cylin- drical surface whose right section is a circle is a circular cylindrical surface. The surface of an oblique circular cylinder is not circular ; and conversely, an oblique cylin- der cut from the space inclosed by a circular cylindrical surface is not a circular cylinder. The two ideas, that of the surface being circular, and that of the solid being circular, are therefore distinct. The formula for the area of a right circular cylinder can be extended to any cylinder by making the formula the element times the perimeter of a right section. It will not, however, use the circle in this case. Similarly, a conical surface is circular if a section per- pendicular to its axis of symmetry is a circle. The surface of an oblique circular cone is not circular, and an oblique cone cut from the space bounded by a circular surface is not a circular cone. The area formula does not hold for the general case of the cone. The volume formulas for the circular cylinder and the circular cone can be extended to all cylinders and cones. 1 From Metrical Geometry, by Dr. George Bruce Halsted. 398 APPENDIX The formula for the cylinder is bh, and for the cone is , o where b is the area of the base and h is the altitude. If a cylinder or a cone that is cut from a space whose surface is cir- cular, has not circular bases, its bases are ellipses, and the area can be obtained by the formula 7r#5, where a and b are the two half diameters (or axes) of the ellipse. 322. Comparison between Plane Geometry and Spherical Geometry. For many of the plane geometry propositions concerning rectilinear figures corresponding propositions have been proved, in spherical geometry concerning great circle arcs. Corresponding propositions do not exist for all such plane geometry propositions because two straight lines must either be parallel or intersect in one point, while two great circle circumferences must meet in two points. This difference eliminates those theorems which depend on parallels, including the propositions about similar fig- ures, and those which depend upon the fact that two non- parallel lines can intersect in but one point. That it does not necessarily eliminate all those which use the single intersection is shown by the fact that two spherical tri- angles are congruent if they have two angles and the in- cluded side of one equal to the corresponding parts of the other, although in plane geometry the proof uses the fact that two lines meet in but one point. In the spherical tri- angles the proof still holds because the two great arcs can meet in but one point on the hemispherical surface, and in superposing the triangles, but one of the intersection points can therefore be used. One of the groups of theorems most affected by the two APPENDIX 399 intersections is the one concerned with the sum of the angles of a triangle or other polygon. The foundation proof of this group is the proposition. An exterior angle of a triangle is greater than either in- terior angle not adjacent to it. That this does not hold in spherical geometry can be shown as follows : In plane geometry, the line AR is drawn from A through M, the midpoint of BC, to R, so that MR = AM ; and BR is drawn. Then A MBR ^ and Z MBR ^. C. But since AR has met both arms ofZ CBX, it cannot meet either again, and R and BR are within that angle, and so Z MBR is part of, and is smaller than, Z CBX, which proves the exterior angle greater than Z C. But, in spherical geometry, while all the rest of the proof holds, it is not true that AM extended cannot meet BX again, for it must meet it at a distance from A equal to a semicircumference. Therefore 1. If AM < a quadrant, R is within Z CBX, as in posi- jr c \jif tion R v and Z C < Z CBX. A~~~~ " \* 2. If AM = a quadrant, R ^ \*^^ >R > is on AX, as in position J? 2 , B and Z C = Z CBX. 3. If AM > a quadrant, R is outside Z CBX, as in posi- tion J? 3 , and Z C > Z CBX. To show that the exterior angle of a spherical triangle is always less than the sum of the non-adjacent interior angles, and therefore the sum of the angles of a spherical triangle is greater than a straight angle : 400 APPENDIX If the exterior angle in the last figure is equal to, or less than, the interior angle, no farther proof is needed ; if it is greater than the one interior angle, that is, if AM is less than a quadrant, let the three great circumfer- ences AC, AM, and AB meet again at A 1 ', and continue BE to Kon AC A', Z CAM = Z MRS (cor. pts. of ^ A) and = Z KA' R (A of a lune). Z MRB = Z KRA' (vertical A); . * . Z KRA f = Z KA* R, and KA' = KR. .' . KA' < KB, and Z KB A' st. Z. This is also shown by the use of polar triangles in 277. 323. The General Polyhedron. Theorem. In a polyhe- dron of f faces, e edges, and v vertices, If the numbers of vertices and edges of any polyhedron are counted, beginning with any face, and then taking a second face, a third face, and so forth, always choosing a face that has one or more edges in common with faces already counted, and so placed that the common edges, if more than one is common, form one broken line, the count- ing, after the first face, is simply counting the sects and vertices, exclusive of the ends, of broken lines. This can be readily seen by experiment with a box, the walls of a room, or any physical solid that is a polyhedron, In a broken line, if the ends are not counted, there is one more sect than vertex. Therefore there are as many more edges than vertices in the potyhedron as there are APPENDIX 401 broken lines to count. But there is one broken line for every face except the first face, where the line is closed and there are the same number of edges and vertices, and the last face, where there are no edges or vertices to count as they have all been counted in preceding faces* Therefore, e = v+(/-2), or e+2=f+v. 324. Radian Measure. There is another system of angular measurement that is sometimes more conven- ient than measurement by degrees. In this measure, the unit is a radian, or angle whose subtended arc, on a circle described with its vertex as center, is equal in length to the radius of the circle. Show that 2 TT radians are equal to a perigon, and TT radians to a straight angle. Since the circumference of a circle has a length equal to 2 TT times a radius (or 6.2832 r), the arc of a radian is the circumference divided by 6.2832, and the number of degrees in a radian is , or 57.2958. In practical 6.2832 use, this number is seldom necessary, for the fact that a straight angle equals TT radians, and also equals 180, gives a more convenient comparison between the two systems. The use of the radian in solid geometry will be shown by two examples. 1. Using radian measure, find the area of a spherical triangle of angles 90, 110, and 120, on a sphere of radius 10 in. In radian measure, the angles, compared with a straight angle which equals TT radians, are J TT, -J--J TT, f TT, and their sum is -g 6 -7r. But the sum of the angles of a plane triangle is TT radians, so the spherical excess of this triangle is - ?r radians. But the area, A, is to the area 402 APPENDIX of the sphere, 400 TT sq. in., as the spherical excess is to 720, or 4 TT radians. Therefore x /inA N 700 (400 TT) = - TT sq. in. 2. If, on a sphere of radius 10 in., the sides of a spheri- cal triangle are 12 in., 10 in., and 15 in., find the area of the polar triangle (to two decimal places). The sides of this triangle are 1.2, 1, and 1.5 radii re- spectively ; therefore the angles of the polar triangle are TT 1.2, TT 1, TT 1.5, radians, and its spherical excess is the sum of these angles minus TT radians, or (2?r 3.7) radians. The area is therefore 2 ?r -3. 7^^ ^ = 20Q ^ _ 37Q or 258.32 sq. in. 4 7T In the first example, it is probably easier to use degree measure, while in the second, there is little doubt that radian measure presents less difficulty. In most cases where the lengths of arcs are expressed in length units, radian measure will be found simpler. 602. Find the area of a lime of angle radians, on a sphere of radius 5. 603. Find the area of Ps n 5,,. 7^ ZJT\ on a sphere of radius 1. Vs' e ' s ' 3 ) 604- On a sphere of radius 12 in., a spherical triangle of sides 6 in., 8 in., and 10 in. subtends a trihedral angle at the center. Find the face angles in radians, and in degrees. 605. In the triangle in the last exercise find the angles of its polar triangle, using radian measure. 606. How many radians must the angle of a lune equal if its area is half that of the sphere V one third that of the sphere ? APPENDIX 403 325. Symmetry. 1. Two points are symmetric with tegard to a center when they are on a line through the center, and are equidistant from it. A figure is symmetric with regard to a center when for each point of the figure there is a corresponding point symmetric to it. 2. Two points are symmetric with regard to a line (or axis) when they are on a perpendicular to that line and are equidistant from it. A figure is symmetric with regard to a line (or axis) when for each point of the figure there is a corresponding point symmetric to it. 3. Two points are symmetric with regard to a plane when they are on a perpendicular to the plane, and are equi- distant from the plane. A figure is symmetric with regard to a plane when for each point of the figure there is a corresponding point symmetric to it. 4. Two figures are symmetric with regard to a plane, line, or center, if for each point of one there is a symmetric point of the other. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. REC'D LD MAY 2 9 1959 REC'D LD NOV 3 DEAD LD 21-100m-9,'48(B399sl6)476 FEB 18W83 LD VB \. 304096 UNIVERSITY OF CALIFORNIA LIBRARY