M !< i DI i :. ,-;<; ,i . , IN MEMORIAM FLORIAN CAJOR1 SOLID GEOMETRY BY HERBERT E. HAWKES, PH.D. PROFESSOR OF MATHEMATICS IN COLUMBIA UNIVERSITY WILLIAM A. LUBY, M.A. HEAD OF THE DEPARTMENT OF MATHEMATICS IN THE JUNIOR COLLEGE OF KANSAS CITY AND FRANK C. TOUTON, PH.D. LECTURER IN EDUCATION, DEPARTMENT OF EDUCATION UNIVERSITY OF CALIFORNIA GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1922, BY HERBERT E. HAWKES, WILLIAM A. LUBY AND FRANK C. TOUTON ALL RIGHTS RESERVED 322.4 AJORI GINN AND COMPANY PRO- PRIETORS BOSTON U.S.A. PREFACE During recent years the study of solid geometry has occu- pied a somewhat less commanding position in the mathematical curriculum than formerly. Important and essential as its sub- ject matter is admitted to be, it has been little more than an appendage to plane geometry, both in the methods of its presentation and in its scientific results. The authors of this text feel that the subject is much more vital than such a tendency would indicate. Not only are the bare truths gained from a study of solid geometry essential to the student of science, but through its medium a multi- tude of mathematical ideas can be presented and elucidated in a natural and convincing manner. In fact, no subject of elementary mathematics can be compared to solid geometry as a climax and capstone of mathematical study for the stu- dent who pursues the subject no farther. It not only utilizes and applies much that he has learned in other courses, but serves as a point of vantage from which may be gained many glimpses of scientific fields which he is not to enter. In this text the authors have presented the subject in such form that a minimal course as prescribed by the colleges and the various examining boards may be covered. At the same time it affords at every turn a richness of suggestion and development for those who have the time and the inclination to do more than that minimum. One of the important opportunities afforded by the study of solid geometry is that of using and developing the scientific iii 91833 1 iv SOLID GEOMETRY imagination. This text, through its hundreds of queries, aims to encourage the student to regard the subject not merely as a logical sequence of theorems but as a subject inviting reflection and the play of speculation. These queries should be used in class as a basis for discussion and will be found to render the more formal work not only more interesting but more intelligible. If time does not permit any attention to the queries, they may be omitted from the class assignments without disturbing the continuity of the subject. Their use, however, is strongly urged by the authors. Exercises illustrating or dependent upon the various theo- rems are scattered throughout the text and afford as much drill of this kind as many teachers can profitably use. The collections at the end of each book may be regarded as sup- plementary. Great care has been exercised to provide a col- lection of originals that is fresh, interesting, not too difficult, but illustrating all parts of the subject. The assumption of Cavalieri's Theorem as a basis for the theorems on measurement is the result of many years of class- room experience. The simplicity and power of this procedure should commend it both to teachers and to students. The geometry of the sphere and its relation to plane geom- etry is also elaborated with care and in such a manner as to give the student an insight into the meaning of geometrical science. CONTENTS PAGE BOOK VI. LINES AND PLANKS 303 Parallel Lines and Planes 311 Perpendicular Lines and Planes . 324 Angles between Planes 339 Projections , 349 Skew Lines 354 BOOK VII. POLYHEDRONS, CONES, AND CYLINDERS . . . 358 Volumes 365 Cylinders 375 Pyramids 387 Cones 405 Polyhedrons ....,.< 419 BOOK VIII. THE SPHERE 443 General Properties of the Sphere 443 Measurement of the Sphere 452 Geometry on the Sphere 464 INDEX 493 REFERENCES FROM PLANE GEOMETRY POSTULATES AND AXIOMS 19. Postulate I. There is only one straight line through two points. 20. Postulate II. Any geometric figure may be moved from one place to another without changing its size or shape. 31. Axiom I. If equals are added to equals, the results are equal. 32. Axiom II. (1) Two numbers or magnitudes each equal to a third are equal to each other. (2) Two figures congruent to a third are congruent to each other. 36. Postulate III. All straight angles are equal. 39. Axiom III. If equals are divided by the same number, the results are equal. 41. Postulate IV. At a given point of a line, one and only one perpendicular can be drawn to the line. 45. Postulate V. The postulate of parallels. Through a given point outside a line, one line parallel to it exists, and only one. 51. Axiom IV. If equals are subtracted from equals, the results are equal. 65. Axiom V. A number may be substituted for its equal in any operation on numbers. 124. Axiom VI. If equals are multiplied by equals, the results are equal. 136. Axiom VII. The whole is greater than any of its parts. 137. Axiom VIII. If the first of three magnitudes is greater than the second and the second is greater than the th'ird, the first is greater than the third. viii SOLID GEOMETRY 139. Axiom IX. If the same number, positive or negative, is added to or subtracted from each member of an inequality, the results are unequal in the same order. 140. Axiom X. If both members of an inequality are multi- plied or divided by the same positive number, the results are unequal in the same order. 141. Axiom XI. If the corresponding members of two or more inequalities which are in the same order are added, the sums are unequal in the same order. 142. Axiom XII. If unequals are subtracted from equals, the results are unequal in the reverse order. 146. Postulate VI. Any side of a triangle is less than the sum of the other two sides. DEFINITIONS 15. Angle. A plane angle (symbol Z.) is the figure formed by two rays which meet. 16. Triangle. A triangle (symbol A) is a portion of a plane bounded by three straight lines. 24. Congruence. Two geometric magnitudes are congruent if their boundaries can be made to coincide. 30. Isosceles triangle. An isosceles triangle is a triangle which has two equal sides. 37. Perpendicular. If one straight line cuts another so as to make any two adjacent angles equal, each line is perpendicular (symbol _L) to the other. 43. Parallel lines. Parallel lines are lines that lie in the same plane and do not meet however far they are produced. 49. Hypotenuse. The hypotenuse of a right triangle is the side opposite the right angle. 52. Vertical angles. Two angles are vertical angles if the sides of one are the prolongations of the sides of the other. REFERENCES FROM PLANE GEOMETRY ix 54. Transversal. A transversal is a line that crosses (cuts or intersects) two or more lines. 61. Supplementary angles. One angle is the supplement of another if their sum equals two right angles (or 180). 79. Regular polygon. A regular polygon is a polygon all of whose angles are equal and all of whose sides are equal. 80. Diagonal. A diagonal of a polygon is a line joining any two nonconsecutive vertices. 83. Parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel. 84. Rectangle. A rectangle is a parallelogram whose angles are right angles. 107. Trapezoid. A trapezoid is a quadrilateral two and only two of whose sides are parallel. 119. Concurrent lines. Three or more lines which have one point in common are said to be concurrent. 154. Circle. A circle is a closed plane curve every point of which is equally distant from a point in the plane of the curve. 187. Tangent. A tangent to a circle is a straight line which, however far it may be produced, has only one point in common with the circle. 233. Altitude of a triangle. An altitude of a triangle is a per- pendicular from any vertex to the side opposite, produced if necessary. 242. Locus. A locus is a figure containing all the points, and only those points, which fulfill a given requirement. 267. Similar polygons. Two polygons are similar (symbol ^) if the angles of one are equal respectively to the angles of the other and the sides are proportional each to each. 310. Area. The area of a plane figure is the number which expresses the ratio between its surface and the surface of the unit square. x SOLID GEOMETRY 354. Center of polygon. The center of a regular polygon is the common center of its inscribed and circumscribed circles. 363. Definition of IT. The number TT (pronounced pi), used in calculations on the circle, is the number obtained by dividing the circumference of a circle by its diameter ; that is, TT = From the above, C = TrD or C = 2 irR. PROPOSITIONS 25. If two sides and the included angle of one triangle are equal respectively to two sides and the included angle of another, the two triangles are congruent. 27. Corresponding parts of congruent figures are equal. 29. If two sides of a triangle are equal, the angles "opposite them are equal. 33. If the three sides of one triangle are equal respectively to the' three sides of another, the triangles are congruent. 42. There is only one perpendicular from a point to a line. 44. Two lines perpendicular to the same line are parallel. 46. If a line intersects one of two parallel lines, it intersects the other also. 47. If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also. 50. Two right triangles are congruent if the hypotenuse and an. adjacent angle of one are equal respectively to the hypotenuse and an adjacent angle of the other. 53. If two straight lines intersect, the vertical angles are equal. 57. If two parallel lines are cut by a transversal, the alternate- interior angles are equal. 66. The sum of the angles of any triangle is two right angles. 76. If two angles have their sides perpendicular each to each, they are equal or supplementary. REFERENCES FROM PLANE GEOMETRY xi 82. If a side and the two adjacent angles of one triangle are equal respectively to a side and the two adjacent angles of another, the triangles are congruent. 85. The opposite sides of a parallelogram are equal. 88. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. 90. The diagonals of a parallelogram bisect each other. 97. Two right triangles are congruent if the hypotenuse and another side of the first are equal respectively to the hypotenuse and another side of the second. 98. Two right triangles are congruent if a side about the right angle and an acute angle of the first are equal respectively to a side about the right angle and a corresponding angle of the second. 116. If a point is on the mid-perpendicular of a line, it is equi- distant from the ends of the line. 117. If a point is equidistant from the ends of a line, it is on the mid-perpendicular of the line. 118. Two points each equally distant from the extremities of a line determine the mid-perpendicular of the line. 121. If a point is on the bisector of an angle, it is equally distant from the sides of the angle. 122. If a point is equally distant from the sides of an angle, it is on the bisector of the angle. 125. If n is the number of sides of a convex polygon and s is the sum of its interior angles, then s = (2 n 4) right angles. 128. If two lines are parallel to a third line, the two lines are parallel to each other. 130. If a line bisects one side of a triangle and is parallel to another side, it bisects the third side. xii SOLID GEOMETRY 131. The line which joins the mid-points of two sides of a tri- angle is parallel to the third side and equal to one half of it. 133. The line joining the mid-points of the nonparallel sides of a trapezoid is parallel to the bases and equal to half their sum. 143. The medians of a triangle are concurrent in the basal point of trisection of each. 145. If two sides of a triangle are unequal, the angles opposite them are unequal and the greater angle lies opposite the greater side. 149. The perpendicular from a point outside a straight line is the shortest line from the point to the line. 151. If two triangles have two sides of one equal respectively to two sides of the other and the third side of the first greater than the third side of the second, the included angle of the first is greater than the included angle of the second. 171. In the same circle or in equal circles if two central angles are equal, their intercepted arcs are equal. 174. In the same circle or in equal circles, if two arcs are equal, they subtend equal central angles. 178. In the same circle or in equal circles, if two arcs are equal, their chords are equal. 179. In the same circle or in equal circles, if two chords are equal, their subtended arcs are equal. 182. If a line passes through the center of a circle and is perpendicular to a chord, it bisects the chord and the arcs subtended by it. 183. In the same circle or in equal circles, if two chords are equal, they are equally distant from the center. 191. If a line is perpendicular to a radius at its outer extremity, it is tangent to the circle. REFERENCES FROM PLANE GEOMETRY xiii 192. If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency. 193. If two tangents are drawn to a circle from an outside point, (1) the tangents are equal ; (2) the line joining the outside point to the center bisects the angle between the tangents and the angle between the radii drawn to the points of contact. 201. If two circles intersect, the line of centers bisects their common chord at right angles. 202. If two circles are tangent externally or internally, the centers and the point of tangency are in a straight line. 211. A central angle is measured by its intercepted arc. 216. An inscribed angle is measured by one half the number of degrees in its intercepted arc. 217. An angle inscribed in a semicircle is a right angle. 224. If three points are not in the same straight line, one circle and only one can pass through them. 260. A line parallel to one side of a triangle and cutting the other two divides them into four corresponding segments which are proportional. 263. If a line parallel to one side of a triangle cuts the other two sides, the two sides are in proportion to their corresponding segments. 265. If a line divides two sides of a triangle into proportional corresponding segments, it is parallel to the third side. 269. Corresponding sides of similar polygons are in proportion. 270. If two triangles are mutually equiangular, they are similar. 271. A line cutting two sides of a triangle and parallel to the third side forms a second triangle similar to the first. 277. In two similar triangles any two homologous sides are proportional to (1) two corresponding altitudes ; (2) two corre- sponding medians ; (3) the bisectors of two corresponding angles. xiv SOLID GEOMETRY 282. If a perpendicular is drawn from the vertex of the right angle to the hypotenuse of a right triangle, (1) the two triangles formed are similar to each other and to the given triangle ; (2) the perpendicular is a mean proportional between the segments of the hypotenuse ; and (3) the square of either side about the right angle equals the product of the whole hypotenuse and the seg- ment adjacent to that side. 284. In any right triangle the square of the hypotenuse equals the sum of the squares of the other two sides. 292. If from a point without a circle a secant terminating in the circle and a tangent be drawn, the square of the tangent equals the whole secant times its external segment. 312. The area of a rectangle is the product of its base and altitude. 319. The area of a parallelogram is the product of its base and altitude. 320. The area of a triangle is one half the product of its base and altitude. 322. The area of a trapezoid is one half the product of its altitude and the sum of its bases. 325. If two triangles have an angle of one equal to an angle of the other, their areas are to each other as the product of the sides including the angle of the first is to the product of the sides including the angle of the second. 326. The areas of two similar triangles are to each other as the squares of any two homologous sides. 327. The areas of two similar convex polygons are to each other as the squares of any two homologous sides. 359. The area of a regular polygon is one half the product of its perimeter and its apothem. 369. The area of the sector of a circle equals one half the product of its radius and its arc. REFERENCES FROM PLANE GEOMETRY xv 371. The area of a circle is TrR 2 . 372. The areas of two circles are to each other as the squares of their radii. 373. The area of a sector is to the area of the circle as the angle of the sector is to four right angles. CONSTRUCTIONS 232. Construct a triangle, given the three sides. 234. At a given point in a line construct a perpendicular to the line. 235. From a given point outside a line construct a perpen- dicular to the line. SOLID GEOMETRY BOOK VI LINES AND PLANES 377. Introduction. Solid geometry is concerned with the properties and relations of figures which occupy space. This does not imply that the cubes, cylinders, and other bodies con- sidered in this text are made of wood or other material sub- stance, any more than that the squares and triangles of plane geometry are made of chalk or of the carbon from our pencils. In both cases the diagrams or solids which we construct are merely rough aids to our imagination, helping us to visualize the properties of the real figures, which are in every case objects of thought without material existence. Since, however, many of the objects around us very closely approximate the form of the purely geometric solids, the subject of solid geom- etry finds abundant application in the affairs of everyday life. 378. Assumptions. The assumptions of plane geometry fall into two classes, the axioms and the postulates. Axioms I-XII do not refer either to the plane or to space, but to relations between numbers, and will be assumed without discussion in what follows. The following assumptions of geometric con- tent have been made in plane geometry. Postulate I. There is only one straight line through two points (19)- Postulate II. Any geometric figure may be moved from one place to another without changing its size or shape (20). 303 304 , V SOLID GEOMETRY Postulate .V> ,' T'lirouyJi a given point outside a line one line parallel to it exists, and only one (45). These postulates are assumed to bold in space. Postulate I may also be restated as follows : Two points deter- mine a straight line. One point does not determine a line, because more than one line passes through a fixed point. Nor do three points determine a line, because no line can be found which passes through any three points taken at random. The significance of the word determine in this statement is that there is one and only one line which contains the two points. In several theorems of plane geometry, Postulate II is assumed to hold in space. For example, in proving that two triangles are congruent if three sides of one are equal respectively to three sides of the other ( 33), the triangle may have to be lifted out of ..the plane and turned over in space, in order to make it take the .desired position. The word move in Postulate II does not imply that one can lift the figures of solid geometry, as one would lift stone blocks, and carry them from place to place. One cannot be expected to move in this sense anything that is merely an object of thought. To move a figure in geometry is to transfer our attention from a figure in one position to another figure exactly like it somewhere else. Hence congruent figures in space as well as in a plane may be considered as the same figure in different positions. 379. Definitions from plane geometry. The definitions given in plane geometry will be taken over to solid geometry with- out change. One must observe, however, that in solid geom- etry the emphasis in certain definitions is entirely redistributed. For example, parallel lines are defined ( 43) as lines that lie in the same plane and do not meet however far they are pro- duced. In the study of plane geometry it is never necessary to emphasize the clause " that lie in the same plane," because in every case the whole figure lies in one plane. But in solid geom- etry, where our figures lie anywhere in space, in proving two BOOK VI 305 lines parallel it is frequently more difficult to prove that they lie in the same plane than it is to show that they cannot meet however far they are produced. QUERY 1. Are two horizontal lines necessarily parallel? Illustrate. QUERY 2. Are two vertical lines necessarily parallel? Illustrate. QUERY 3. Is every pair of lines in space either intersecting or paral- lel ? Illustrate. 380. Undefined terms. It is assumed that the terms figure, line, curve, surface, and solid are familiar. The word line will be used to denote a straight line extend- ing indefinitely in both directions. To avoid ambiguity, the portion of a line between two of its points will often be called a line-segment. 381. The plane. A. plane is a surface such that if any two points in it are taken, the straight line containing them .lies wholly in the surface. Since a line extends indefinitely in both directions, a plane is unlimited in extent. On account of the size of our page it is impossible to show the whole plane by a drawing. Hence it is customary to draw a parallelogram to represent a plane or that part of a plane with which we are particularly concerned. If it is desired to emphasize the fact that the plane extends farther in a certain direction, one side of the parallelogram .may be replaced by a wavy line. QUERY 4. What postulate is assumed in the definition of 381. QUERY 5. Show from the definition that two walls of a room do not form a single plane. QUERY 6. Is the surface of a perfectly calm sea a plane? QUERY 7. Do two points exist on the curved surface of a straight flag- pole such that the line passing through them lies entirely in the surface of the pole ? If so, why is this surface not a plane ? QUKRY 8. Does a plane have any edges? 306 SOLID GEOMETRY 382. Postulate VII. Two intersecting lines lie in one and in only one plane. This postulate plays about the same role in solid geometry that Postulate I does in plane geometry. QUERY 1. Can more than one plane pass through a given* line ? Illustrate. Does one line determine a plane ? QUERY 2. Can you hold two pencils in such positions as to show that a plane cannot contain any two lines taken at random in space? QUERY 3. Does a set of three concurrent lines necessarily determine a plane? Illustrate. QUERY 4. Is a triangle necessarily a plane figure? Why? Theorem 1 383. Three points not on the same line lie in one and in only oneplane. Given three points A, , and C which do not lie on the same line. To prove that A, B, and C lie in one and in only one plane. Proof. Draw the lines AB and AC. Denote by M the plane containing them both. 382 A , B, and C lie in the plane M, since the lines containing them lie in that plane. A, B ) and C cannot lie in any other plane than M, since the lines containing them cannot lie in another plane. 382 Hence A, B, and C lie in one and in only one plane. * When a figure is referred to as ff given," it is understood that the figure is supposed to be fixed both in size and in position. Thus, a given circle is one of definite size which is assumed to be fixed in position during the dis- cussion. Of course, a line or a plane can be fixed only in position, since by definition they are indefinite in extent. BOOK VI 307 384. Corollary 1. A line and a point not on the line lie in one and in only one plane. Given the line a and the point P not on a. To prove that P and a lie in one and in only one plane. Proof. Let K and L be any two points on a. Denote by M the only plane containing P, K, and L. 383 The line a must lie in M. 381 Hence both a and P lie in M and in no other plane. 385. Corollary 2. Two parallel lines lie in one and in only one plane. Given the parallel lines a and b. To prove that a and b 'lie in one and in only one plane. Proof. a and b lie in one plane. 43 If they could lie in two planes at once, then one of them and any point of the other would lie in two planes at once, which is impossible. 384 Hence a and b lie in one and in only one plane. 386. Restatement. The foregoing results may be stated as follows: 1. Two intersecting lines determine a plane. 2. Three points not in the same line determine a plane. 3. A line and a point outside the line determine a plane. 4. Tivo parallel lines determine a plane. 308 SOLID GEOMETRY 387. Perspective. In the diagrams of solid geometry the figure is usually supposed to be either to the right or the left of the eye of the observer or directly in front of and slightly below it. When a figure is drawn as it appears to the eye, it is said to be drawn in perspective. QUERY 1. What is the position of the three preceding figures with respect to the eye of the observer ? QUERY 2. How would a horizontal plane look if the eye were just level with it? QUERY 3. How does the shape of the top of a level table seem to change if you stand directly in front of it, a few feet away, first with the eye level with the top, and then rise to your full height? What difference do you observe if you repeat the process but do not stand directly in front of the table ? Draw figures showing the different forms that the table top presents from your various positions. 388. Coplanar. Lines or points which lie in the same plane are said to be coplanar. QUERY 4. Are two parallel lines necessarily coplanar? QUERY 5. If three points are collinear, can more than one plane be found which will contain all of them? Illustrate. Do three collinear points determine a plane? QUERY 6. Do any four points taken at random in space determine a plane ? Illustrate. QUERY 7. Why does a three-legged stool stand firmly on a level floor, while a four-legged one is likely to be unsteady? QUERY 8. What does a moving line generate if it always passes through a given point and always intersects a given line ? QUERY 9. What does a moving line generate if it always intersects two intersecting lines? BOOK VI 309 EXERCISES 1. Any transversal of two parallel lines lies in the plane of those lines. 2. If a line cuts three concurrent lines at points other than their intersection, the four lines are coplanar. 3. If a plane contains one of two parallel lines and one point of the other, it must contain both of the parallels. HINTS. Use 385, 384. 389. Postulate VIII. If two planes have one point in common, they must have at least two points in common. 390. Intersection. The intersection of two lines, curves, or surfaces comprises those points, and only those, which they have in common. Intersections of geometric figures fall into several classes, which are defined as follows : If two intersecting lines, curves, or sur- faces pass through each other, they are said to cut each other. If one of two intersecting lines, curves, or surfaces is terminated at their intersection, it is said to meet the other. If both of two figures terminate at their in- tersection, they are said to meet each other. If a line either cuts or meets a plane, the intersection consists of only one point. If a line lies entirely in a plane, one frequently says that the plane passes through or contains the line. QUERY 1. Do the two sides of an angle cut each other? QUERY 2. In the figure above is it correct to say that the line meets the plane or that the plane meets the line ? QUERY 3. Do the two planes above meet or cut each other? QUERY 4. Hold two sheets of paper so that they meet each other. 310 SOLID GEOMETRY Theorem 2 391. If two planes intersect, their intersection is a straight line. Given two planes MN and PQ which intersect. To prove that their intersection is a straight line. Proof. MN and PQ must have at least two points, as A and B, in common. 389 Hence both MN and PQ contain the line AB determined by these points. 381 But Jl/JVand PQ cannot have any point outside AB in common, else the planes would coincide. 384 Therefore the intersection of MN and PQ is a straight line. QUERY 1. Can you imagine two planes which do not have ap- points in common ? Illustrate. QUERY 2. How many planes can be passed, meeting a given plane in a given line ? Illustrate. QUERY 3. If three planes in space are taken at random, what is their intersection ? Give an example. QUERY 4. What is the least number of planes that can inclose a space ? Give an example. QUERY 5. What relations other than the one given in the answer to Query 3 may three planes bear to each other ? Give examples. QUERY 6. If two surfaces intersect in a straight line, is it necessary that both of them be planes ? Give examples. QUERY 7. Is the converse of Theorem 2 true ? BOOK VI 311 PAEALLEL LINES AND PLANES 392. Parallel planes. Two planes which do not meet how- ever far they are produced are said to be parallel.* Theorem 3 393. If a p lane intersects two parallel planes, the inter- sections are parallel lines. Given the plane M parallel to the plane N, and both cut by the plane Q y in lines a and b respectively. To prove that a is II to b. Proof. a and b are straight lines. 391 In order to prove a II to b, we must show, first, that they lie in the same plane ; second, that they cannot meet. Now a and b lie in Q. Given Also, a and b cannot meet, since, if they did meet, the planes M and N would have a point in common, which is impossible. 392 Therefore a is II to b. 43 QUERY 1. A point, a line, and a plane are given. What is the inter- section of the plane with a moving line which contains the given point and cuts the given line ? Mention any special cases that may occur. * In this definition it is implied that if planes meet they are not parallel. A similar remark applies to most of the definitions in this and other texts. 312 SOLID GEOMETRY QUERY 2. If two parallel planes are given, is it certain that there is a plane which intersects both of them ? How many such planes are there ? Are these planes necessarily parallel? QUEKY 3. Where is the figure of Theorem 3 situated with respect to the eye ? QUERY 4. Draw a figure for Theorem 3 which appears to be below and to the right of the eye. 394. Corollary. Parallel line-segments intercepted between parallel planes are equal. Given a and &, parallel line-segments intercepted between the planes M and N. To prove that a is equal to b. HINTS. Pass the plane determined by a and b intersecting M and N in x and y respectively. Prove that a parallelogram is formed. QUERY 5. Is a parallelogram necessarily a plane figure? QUERY 6, Is every closed four-sided figure necessarily a plane figure ? Illustrate. QUERY 7. If two parallel planes cut off equal lengths on two lines, are the lines necessarily parallel ? Illustrate. QUERY 8. If two parallel planes cut off equal lengths on two lines, do the lines necessarily intersect if sufficiently produced ? Illustrate. EXERCISE 4. Given two parallel planes which cut off equal segments on two intersecting lines. Pass the plane of the inter- secting lines and show that two isosceles triangles are formed. BOOK VI. 313 395. Parallel lines and planes. A line and a plane that do not meet however far they are produced are said to be parallel. QUERY 1. If a line is parallel to two planes, are the planes necessarily parallel to each other ? Illustrate. QUERY 2. If two planes are parallel, is a given line in one plane parallel to every line in the other ? Is it parallel to some line in the other? QUERY 3. How many lines are there through a given point parallel to a given plane ? Illustrate. QUERY 4. In what kind of surface do you think all the lines of Query 3 would lie? QUERY 5. Hold a pointer so that it is parallel to a side and an end wall of the room. How many straight lines are there through a given point parallel to each of two intersecting planes ? Theorem 4 396. If a plane contains only one of two parallel lines, it is parallel to the other line. Given the line a parallel to the line &, and the plane M con- taining b hut not containing a. To prove that M is II to a. Proof, a meets M, if at all, in some point X which is not on b. Why ? Through X draw c in M II to b. 43 Then we have through X two lines, a and c, both II to b, which is impossible. 45 Hence a cannot meet M and is II to it. 395 314 SOLID GEOMETRY 397. Constructions. For the present a construction in solid geometry means the building of a figure by application of the following actual or imagined operations: 1. The passing of planes ( 382, 383, 384, 385). 2. The determination of lines by the intersection of planes ( 391). 3. The use of ruler and compasses in planes. The third operation refers to the constructions of plane geome- try performed in the planes afforded by the first process. Instead of having only one method of determining a line, as was the case in plane geometry, we now have two : a pair of points and a pair of nonparallel planes. Construction 1 398. Through a point outside a plane construct a line to the plane. Given the point P outside the plane M. Required to construct a line through P II to M. Construction. Through the point P pass a plane N intersecting M in some line, as a. In the plane N draw a line b through P II to a. 45 Then b is II to M. Proof. b is II to a. Const. Therefore I is II to M. 396 BOOK VI 315 Theorem 5 399. If a line is parallel to a plane, the intersection of the j)lanv with a plane passed through the line is parallel to the line. A B Given the line AB parallel to the plane M\ and the plane AL containing AB and intersecting M in KL. To prove that AB is II to KL. Proof is left to the student. QUERY 1. To how many lines in a given plane may a line be parallel ? QUERY 2. Under what conditions is a straight stick parallel to its own shadow on the ground? QUERY 3. Under what conditions may a line be parallel to each of three planes? Illustrate. QUERY 4. How many planes are there through a given point parallel to a given line ? Illustrate. . EXERCISES 5. If a line and a plane are parallel, a line containing a point of the plane and parallel to the given line lies wholly in the plane. HINTS. Let AB be parallel to M and let PR meet M in P and be parallel to AB. Pass a plane determined by AB and PR, meeting plane M in PQ. Show that PQ and PR are both II to AB. 6. Through a given point construct a line parallel to a given plane and meeting a given line. Is this construction always possible ? 316 SOLID GEOMETRY 7. Two intersecting planes are each parallel to a given line. What is the relation between the intersection of these planes and the line ? Prove your statement. HINT. Pass a plane determined by the given line and any point of the intersection of the planes. Theorem 6 400. If two intersecting lines are parallel to a plane> their plane is parallel to the plane. Given the lines a and &, both parallel to the plane M\ and Q, their plane. To prove that Q is II to M. Proof. If Q should intersect M in a line x, then a and b would each be II to x. 399 We should then have through the point P two lines, each II to the same line, which is impossible. 45 Therefore Q does not meet M, and is II to it. Why? 401. Corollary. If two intersecting lines are respectively parallel to, but not coplanar with, two other intersecting lines, the plane of the first pair is parallel to the plane of the second pair. HINTS. Let N be determined by c and d. Then a and b are each II to ^V by 396. BOOK VI 317 Theorem 7 402. If a plane intersects one of two parallel lines, it intersects the other also. 10 R Given the line a parallel to the line &, and the plane M in- tersecting b Sit the point 0. To prove that M also intersects a. Proof. Pass the plane N determined by a and I, intersecting M in RP. 385 Then RP must intersect a. 46 Hence M, the plane in which RP lies, must intersect a. NOTE. It is always desirable to. observe whether the reason for a given step is taken from plane geometry, and, if so, to note the plane containing the figure to which the reference is made. In the foregoing proof the figure to which 46 applies lies in the plane of a and b. 4C3. Corollary 1. If a line intersects one of two parallel planes, it intersects the other also. HINTS. Pass a plane determined by a and any point P of M. Apply 46. 404. Corollary 2. If a plane intersects one of two parallel planes, it intersects the other also. HINT. . In the cutting plane draw a line cutting the line of intersection of the two planes. 318 SOLID GEOMETRY 405. Corollary III. If tiuo planes are par- allel to the same plane they are parallel to each other. /K / HINT. Let M be II to N and to P. If N should yjT~ ~~7 intersect P, show that it would also intersect M. Theorem 8 406. If two lines are parallel to the same line, they are parallel to each other. Given the lines b and c each parallel to a. To prove b and c parallel to each other. Proof. Pass the plane M containing I and one point of c. 384 Now M either cuts c or contains it. If M cuts c, it will cut a and b. 402 But M cannot cut b for it contains b. Const. Hence M cannot cut c. Therefore M contains c, since the only other possibility leads to a contradiction. Therefore b and c lie in the same plane. Since they are each II to a, Given b cannot meet c. 45 Hence b is II to c. . 43 NOTE. It should be observed that 45 is assumed to be true in space as well as in a plane. BOOK VI . 319 Theorem 9 407. If two angles not in the same plane have their sides parallel and extending in the same direction from their vertices, they are equal. -B Given the angles LAK and RGS in the planes M and N respectively, with sides AL and AK parallel respectively to sides GR and GS. To prove that Proof. Construct A B = GF, and A C = Gil, and draw A G, CH, and 1>F. Now EG and CG are parallelograms. 88 Hence BF and CH are each II to A G. Why ? Consequently BF is II to CH. 406 Also BF = CIL Why ? Hence the figure BCIIF is a parallelogram, 88 and CB = HF. Why? In ABA C and FGH, BA = FG and 4 C = GH, Const. and we have proved CB = HF. Therefore the A are congruent, Why? and Z.BAC = Z.FGH. Why? QUERY 1. If the angles of Theorem 9 have their sides parallel each to each, but extend in opposite directions from their vertices, what is the relation between the angles ? 320 SOLID GEOMETRY Construction 2 408. Through a point not in a plane, construct a plane parallel to that plane. 57 7 Given the point P and the plane M. Required to construct a plane containing P and parallel to M. Construction. Construct PR and PK II to M. 398 Pass plane N determined by PR and PK. Hence N is II to M. Proof. Since PR and PK are each li to M, N is II to M. 400 Theorem 10 409. Through a point not lying in a plane, one and only one plane can be passed parallel to that plane. 7 Given the point P and the plane M. To prove that one and only one plane through P is II to M. BOOK VI 321 Proof. Denote by N the plane constructed II to M and con- taining P. 408 Any plane through P other than N, such as R, would cut N. 391 Therefore R would cut M, 404 which is contrary to the hypothesis. Hence N is the only plane through P II to M. QUERY 1. How many planes are there parallel to two given parallel planes ? Illustrate. QUERY 2. How many planes are there through a given point parallel to two given parallel planes ? Illustrate. QUERY 3. Two lines are given. If a moving line is always parallel to one of the given lines and always intersects the other, what surface is generated and what "is its position? QUERY 4. In the statement "Through a given point outside a one parallel to it can be drawn, and only one," fill the blank spaces with the words line and plane in each of the four possible ways. Which of the resulting statements are true ? In what section is each proved or assumed ? QUERY 5. If the angles of Theorem 9 have their sides parallel each to each, and if one pair extend in the same direction from the vertices while the other pair extend in opposite directions, what is the relation between the angles? EXERCISES 8. If two lines are not in the same plane, one plane and only one can be passed containing one of these lines and parallel to the other. 9. Construct a line through a given point parallel to two given intersecting planes. HINT. Pass the plane determined by the point and the intersection of the planes. 10. If one of two parallel lines is parallel to a plane, the other is also. 11. Through a given point one and only one plane can be passed parallel to two given nonparallel lines in space. 322 SOLID GEOMETRY Theorem 11 410. If two straight lines are cut by three parallel planes, the corresponding segments are proportional. Given three parallel planes M, N, R, cutting two lines AD and CD in the points -A, F, B and C, H, D respectively. To prove that |f=||- Proof. Draw AD; pass the planes of DC and DA, and of AB and AD. Denote the intersections with the given planes by AC, GH, FG, BD. 393 In AADC. In AABD, Therefore AC is II to Gil, and FG is II to BD. CH _AG II D ~ GD ' AF _AG FB ~ GD' AF _ CH ~FB~ HD' Why? Why ? Why? 411. Corollary. If two parallel planes cut a series of con- current lines, the corresponding segments are proportional. HINT. Pass the plane through the point common to the lines and parallel to one of the given planes ( 409). BOOK VI 323 QUERY 1. In what case are F, G, and // of Theorem 11 in a straight line ? Under what conditions are A C and BD parallel ? QUERY 2. Describe the positions of the planes that are drawn through a fixed point so as to contain a set of parallel coplanar lines ; of parallel lines, not all coplanar; of concurrent coplanar lines; of concurrent lines, not all coplanar. REVIEW EXERCISES 12. A quadrilateral is a plane figure if two of its sides are parallel. 13. If any number of parallel lines meet a given line, they are all coplanar. 14. If each of three lines meets the other two, the three are either coplanar or concurrent. 15. If two lines are not coplanar, show that it is impossible to draw two parallel lines each cutting both the given lines. 16. Given two lines which do not meet and are not parallel. Through a given point construct a third line meeting both the given lines. Discuss the special cases. 17. If the foot of a ten-foot pole is placed on the bottom of a body of water 8 feet deep, and the top of the pole is at the sur- face, will the middle of the pole always lie in the same plane ? Prove your statement. 18. If a line in one of two intersecting planes is parallel to a line in the other, both lines are parallel to the intersection of the planes. 19. Given four lines in space, only two of which are parallel. Construct a line cutting all four lines. Discuss the special cases. 20. The top of a ten-foot pole is placed in the corner of a room at the ceiling. The foot of the pole is found to be on the floor 6 feet from the corner. How high is the room ? 21. If three parallel planes cut off equal segments on one transversal, they cut off equal segments on every transversal. 324 SOLID GEOMETRY 22. In one of two parallel planes three lines are drawn which are parallel, each to each, to lines in the other plane. Are the triangles formed in the two planes necessarily similar? Prove your statement. Discuss special cases. 23. To which of the preceding theorems is the following state- ment equivalent ? " If a line is parallel to a line in a plane, it is either parallel to or contained by the plane." 24. Construct a line parallel to a given plane and meeting each of two given lines. Discuss special cases. 25. Construct a line which cuts three given lines. Is more than one such line possible ? Discuss special cases. 26. Through a given point pass two planes, one parallel to each of two given intersecting planes. What can you say of the inter- section of the two planes so drawn ? Prove your statement. 27. Construct a plane which shall pass through a given line and cut two given planes in parallel lines. PERPENDICULAR LINES AND PLANES 412. Foot of a line. The point where a line intersects a plane is called the foot of the line. QUERY 1. Can a line have two feet in a given plane and at the same time cut the plane ? QUERY 2. Are there any lines which have no foot in a given plane ? Illustrate. QUERY 3. At a given point in a line, how many perpendiculars to the line are there? QUERY 4. How are the planes arranged which are determined by a given line and the various perpendiculars at a point on the line? QUERY 5. If two lines are perpendicular to the same line, are they necessarily parallel? Illustrate. QUERY 6. Keeping in mind that one must always provide a plane in space in which to perform a construction of plane geometry, how would you construct two lines perpendicular to a given line at the same point? BOOK VI 325 Theorem 12 413. If a line is perpendicular to two lines at their point of intersection, it is perpendicular to any line in their plane through that point. Given OX and OF, two lines in the plane Af, each perpen- dicular to OP at 0, and let OZ be any other line through in M. To prove that OP is _L to OZ. Proof. Draw any line in M not through 0, cutting OX, Y, and OZ at A, B, and C respectively. Produce PO to A", making OK = OP. Draw lines PA, PB, PC, KA, KB, and KG. In the plane determined by PK and OA, PA = KA. In the plane determined by PK and OB, PB = KB. Therefore &PAB and KAB are congruent. Hence Z PB C = Z KB C. Also CB = CB. Therefore APBC = AKBC. Consequently, KC =PC. 116 Why? Why? Why? Why? 27 Hence OZ contains two points, and C, equidistant from P and K, and is therefore _L to PK at 0. 118 Therefore OP is _L to OZ. 414. Perpendicular to a plane. A line is perpendicular to a plane if it is perpendicular to every line in the plane drawn through its foot. 326 SOLID GEOMETRY If a line intersects a plane and is not perpendicular to it, it is said to be oblique to the plane. From this definition it appears that if a line makes an oblique angle with any line of a plane, it cannot be perpendicular to the plane. But one cannot show directly from the definition that a line is perpendicular to a plane without testing the angle which is found with every line through its foot, a process which could never be completed, since it would require an infinite number of operations. The great importance and power of Theorem 12 consists in two facts : first, it shows that a line can be perpendicular to all of the lines in a plane through its foot ; second, it replaces the infinite number of operations mentioned above by only two, mak- ing it possible to show that a line is perpendicular to all lines in the plane drawn through its foot if it is found to be perpendicular to just two of them. 415. Corollary. If a line is perpendicular to two lines at their point of intersection, it is perpendicular to their plane. This follows immediately from 413, 414. QUERY 1. If one line is perpendicular to another, is a plane con- taining the first line sure to be perpendicular to the second ? QUERY 2. How could you determine by use of a carpenter's square whether a square post is perpendicular to a level floor? How many operations are necessary? QUERY 3. In the figure for Theorem 12, which point is nearer the eye, A or 0? B or (9? C or 0? QUERY 4. Where is the triangle OAB with respect to the eye? QUERY 5. Stand a pencil on end on a sheet of paper which lies on a level table. Draw several lines on the paper through the end of the pencil. Hold up a book so that one corner is between the eye and the point where the pencil meets the paper, and one edge of the book is in line with the pencil. In this way test which of the right angles formed should appear as right angles in a drawing. QUERY 6. May a right angle ever be correctly represented by an obtuse angle? an acute angle? a straight angle? an angle of zero degrees ? Explain. QUERY 7. Can two pointers at right angles be held in such a position that the angle which they form appears to the class to be obtuse ? BOOK VI 327 Construction 3 416. Construct a plane containing a given point and perpendicular to a given line. Given the point P and the line a. Required to construct a plane containing P and _L to a. Case I. When P is on the line a. Construction. At P construct two lines, PR and PS, each 234 382 _L to a. Pass the plane M determined by PR and PS. M is _L to a at P. Proof. a is _L to PR and to PS. Therefore a is J_ to M. Case II. When P is not on the line a. Const. 415 Construction. From P drop a _L to a and denote the intersection by S. 235 At S draw ST, another _L to a. 234 Pass the plane M determined by SP and STY 382 M is _L to a. Proof is left to the student. 328 SOLID GEOMETRY Theorem 13 417. One and only one plane can be passed containing a given point and perpendicular to a given line. K B Given the point P and the line AB. To prove that one and only one plane can be passed con- taining P and _1_ to AB. Case I. When P is on AB. Proof. Denote by Afa plane _L to AB at P. 416 Suppose there were another plane, as KL, also _L to AB at P. Let PT be any line in M through P. Pass the plane A T determined by P7 7 and AB, cutting KL in the line PS. Then PT and PS are both _L to AB at P. 414 But PT and PS are both in the plane A T. Therefore PT and PS must coincide. But P T is drawn as any line in M through P. Hence M coincides with KL. Therefore only one plane can be _L to AB at P. 41 382 QUERY. If a line a meets a plane M,to which it is not perpendicular, at a point /*, does a line exist in M through P to which a is perpen- dicular? Does more than one such line exist? BOOK VI 829 Case II. When P is not on AB. Proof. Denote by M a plane containing P and _L to AB. Sup- pose there were another plane, N, containing P and also _L to AB. Now M and N could not both intersect AB at the same point. Case I Let T and S be the intersec- tions of M and N respectively with AB. Then PT and PS are both in the plane determined by A B and P, and are both _L to AB. 414 Therefore PT and PS must coincide, and M coincides with N. 42 Case I Hence only one plane can be _L to AB through P. 418. Corollary. All of the lines perpendicular to a line at a point lie in a plane perpendicular to that line at that point. HINTS. "Let BK and BR be any two Js to A B. Pass plane MN through BK and BR. Let EL be any other JL to AB at B. Pass plane through BL and BK and show that the two planes coincide. QUERY 1. In order to prove that the plane perpendicular to a line at a given point is the locus of lines perpendicular to the given line at that point, what two facts must be established? Are sections 414 and 418 sufficient for this purpose ? QUERY 2. If a right angle be rotated about one of its sides, what does the other side generate ? QUERY 3. What kind of surface does a spoke of a wheel, which is not dished, generate in its rotation ? QUERY 4. Can a plane always be passed parallel to one of two lines in space and perpendicular to another ? EXERCISE 28. If two planes are perpendicular to the same line, the planes are parallel. 830 SOLID GEOMETRY Theorem 14 419. If a line is perpendicular to one of two parallel planes, it is perpendicular to the other also. Given the parallel planes M and N and the line AB perpen- dicular to the plane M at the point 0. To prove that AB is A. to N. Proof. A B intersects N in some point S. 403 Through the point S draw two lines in the plane JV, as SD and SF. Pass the planes SH and SG, determined by SD and AB, and by SFand AB respectively. Let the intersections of these planes with M be called OH and OG respectively. OG is II to SF, and OH is II to SD. 393 But AB is _L to OH and to OG. Why? Hence AB is _L to SD and to SF. 47 Therefore AB is _L to N. Why? QUERY 1. Will a pointer held perpendicular to the floor of a room also be perpendicular to the ceiling? QUERY 2. To how many planes is a given line perpendicular? Illustrate. BOOK VI 331 Construction 4 420. Construct a line perpendicular to a given plane and containing a given point. Given the point P and the plane M. Required to construct a line containing P and _L to M. Case I. When P lies in M. Construction. Draw any line PA in M through P, and pass the plane KL J_ to PA at P. In KL draw PC _L to the intersection PL at P PC is _L to M. Proof. PC is J_ to PL. PC is _L to A P. Therefore PC is _L to M. 417 Const. 414 415 Case II. When P does not lie in M. Construction. Construct the plane N containing P and II to M. At P construct PR _L to N. Then PR is _L to M. Proof. PR is _L to N. Therefore PR is _L to M. 408 Case I Const. 419 QUERY. Why could not Case II be proved as follows: "Draw any line in M, and drop a perpendicular from P to this line." 332 SOLID GEOMETRY Theorem 15 421. One and only one line can be drawn perpendicular to a given plane and containing a given point. C X Given the plane M and the point P. To prove that one and only one line can be drawn JL to M and containing P. Case I. When P lies in M. HINTS. Denote by CP a line containing P to M ( 420). If possible, let PX also be JL to M at P. Pass the plane of CP and XP, and show that CP and XP coincide. p Case II. When P does not lie in M. The proof, which is left to the student, may follow the hints for Case I, but referred to the adjacent figure. / 422. Corollary. The perpendicular is the shortest line that can be drawn from a point to a plane. Given (see figure for Case II above) the plane M y and PC the perpendicular to M from P. To prove that PC is the shortest line from P to the plane M. Let PX be any other line from P to M. In A PCX, Z C is a right angle. 414 Hence PC is shorter than P.Y. 149 QUERY. Is the shortest distance from a point to a given line in a plane necessarily the shortest distance from the point to the plane ? BOOK VI 333 Theorem 16 423. The locus of points in space which are equidistant from two given points is the plane which bisects perpen- dicularly the line-segment joining the points. Given two points, A and 5, and the plane M, which is perpen- dicular to the line AB at its middle point R. To prove that M is the locus of points in space equidistant from A and B; or (1) that every point in Mis equidistant from A and B, and (2) that every point in space which is equidistant from A and B lies in M. Proof. (1) Let P be any point in M. Draw PA, PB, and PR. A PAR is congruent to APBR. Why ? Therefore PA = PB. Why ? (2) Let K be any point such that KA = KB. Draw KR. AKAR is congruent to AKBR. W T hy ? Therefore Z KRA = Z KRB, Why ? and KR is _L to AB. 37 Hence KR lies in the plane _L to AB at R, 418 and consequently K lies in that plane. Therefore M is the locus of points in space equidistant from A and B. 334 SOLID GEOMETRY QUERY 1. What is the locus of points which lie in a given plane and which are equidistant from two points not in that plane ? QUERY 2. What is the locus of points equidistant from three given points which are not in the same straight line ? QUERY 3. Determine a point in a given plane which is equidistant from three points in space. Discuss the various special cases. 424. Logical relation between propositions. If the hypothesis and the conclusion of a proposition are interchanged, the resulting proposition is called the converse of the original one. The relation between a proposition and its converse may be expressed in terms of symbols as follows : (1) Direct proposition : If A is B, then C is D. (2) Converse proposition : If C is 7), then A is B. If the negative of both the hypothesis and the conclusion is taken, the resulting proposition is called the opposite of the original. Using the same notation as above, (3) Opposite proposition : If A is not B, then C is not D. If a direct proposition and its converse are true, then the oppo- site proposition is true. Let us assume the truth of (1) and of (2) and prove that the truth of (3) follows. Now if A is not B, it follows that C is not D. For if C were D, then A would be B, by (2). But this contradicts the hypothesis of (3). Hence (3) is true. In proving a locus theorem it is sufficient to prove a propo- sition and its converse (cf. 423). In this text locus theorems are established by this method. From the preceding discussion it follows that another theorem, namely, the opposite, follows immediately from the proof of a theorem and its converse. For example, from Theorem 16 it follows that if a point is not in the plane which bisects the line-segment joining A and B, it is not equidistant from A and B. QUERY 4. State the opposite of Theorem 5. Is it true? QUERY 5. State the opposite of'(o) 413; (b) 407; (c) 400. QUERY 6. Is the opposite of a true proposition necessarily true? Illustrate. EXERCISE 29. Prove that if a proposition and its opposite are true, then the converse is true. BOOK VI 335 Theorem 17 425. (1) If equal oblique line-segments are drawn from a point to a plane, they meet the plane at equal distances from the foot of the perpendicular to the plane from that point. (2) If oblique line-segments from a point to a plane meet the plane at equal distances from the foot of the perpendicular, the line-segments are equal. A /AV /M B / (1) Given PT perpendicular to the plane Af, and PA equal toPB. To prove that TA = TB. HINT. Compare the &PTA and PTB. (2) Given PT perpendicular to AT, and TA equal to TB. To prove that PA = PB. Proof is left to the student. QUERY 1. What is the locus of points in space equidistant from all the points of a given circle ? QUERY 2. What is the least number of rigid iron braces that will hold a pole in vertical position? What is the least number of guy ropes? QUERY 3. The ceiling of a room is 10 feet high. How would you determine by means of a 12-foot pole and a pair of compasses a point in the floor directly under a given point in the ceiling ? QUERY 4. What is the logical relation of the proposition in 425 (1) to that in (2) ? QUERY 5. State the opposite of 425 (1). Is it true? Why? 336 SOLID GEOMETRY QUERY 6. State the opposite of 425 (2). Is it true? Why? QUERY 7. What is the locus of the points in a plane a fixed distance from a given point not in the plane ? EXERCISES 30. If two oblique lines drawn from a point in a perpendicular to a plane cut off unequal distances from the foot of the perpen- dicular, the more remote is the greater. 31. Given a circle and a line perpendicular to its plane at its center. Prove that a line drawn to the circle from any point of the perpendicular is perpendicular to the tangent to the circle through its foot. HINT. Draw any other line from the given point to the tangent and apply Exercise 30. Theorem 18 426. If one of two parallel lines is perpendicular to a plane, the other line is also perpendicular to the plane. Given the line AB parallel to the line KF and perpendicular to the plane M. To prove that KF is _L to M. Proof. KF intersects M. 402 Draw FB, and any other line in M through F, as FG. Draw BH in M II to FG. AB is _L to BF. Why? Hence KF is J_ to BF. 47 BOOK VI 337 It remains to prove KF _i_ to FG. Now AB is II to KF and EH is II to FG. Why ? Hence Z.ABH=^.KFG. 407 But ABH is a right angle. Why? Therefore KF is _L to FG. Why? Hence /iCF is _L to 717. Why ? QUERY 1. In the preceding proof why is it not sufficient to prove AB and KF both perpendicular to BF, and then to refer to 44 ? 427. Corollary 1. If two lines are perpendicular to the same plane, they are parallel. HINTS. Given a and b JL to M. Draw c through the foot of b and II to #. Then c is _L to M ( 426). Therefore b and c coin- cide ( 421). 428. Distance to a plane. The dis- tance from a point to a plane is the length of the perpendicular from that point to the plane. By 422 the distance from a point to a plane is the shortest distance from that point to the plane. The distance between two parallel planes is the distance from a point of one to the other. 429. Corollary 2. The distances to one of two parallel planes from any two points of the other are equal. QUERY 2. What is the locus of points a given distance from a given plane ? QUERY 3. What is the locus of points equidistant from two given parallel planes? QUERY 4. How would you locate the points in a given plane which are 6 inches from another given plane ? In what kind of figure would the points lie ? 338 SOLID GEOMETRY REVIEW EXERCISES 32. Two points on opposite sides of a plane and equally dis- tant from it determine a line-segment which is bisected by the foot of the line. 33. Two points on the same side of a plane and equally dis- tant from it determine a line that is parallel to the plane. 34. A plumb-line 6 feet long is suspended from the ceiling of a room 8 feet high. The lowest point of the line is 30 inches from a given point on the floor. What is the distance from this given point to the point in the floor, directly under the plumb-line ? 35. A point is 16 inches from a given plane. What is the perimeter of the circle which contains the points of the plane which are 20 inches from the point. 36. A line parallel to a plane is everywhere equidistant from the plane. 37. Construct a line perpendicular to a given pair of parallel lines, which shall also meet a given third line. Is there any case in which the construction is impossible ? 38. Every line perpendicular to a line which is given perpen- dicular to a given plane is parallel to the plane. 39. Let denote the center of an equilateral triangle ABC whose side is a. At a line-segment OP is drawn perpendicular to the plane of the triangle, so that the angle APB is a right angle. How long is OP ? 40. A point is 10 inches from a given plane. Lines 20 inches long are drawn from the point to the plane. What is the area of the circle on which the feet of these lines lie ? 41. What is the locus of points in space equidistant from two parallel lines ? Prove your statement correct. 42 . What condition must be satisfied by two lines in order that it may be possible to construct a plane containing one of them and perpendicular to the other ? Assuming that the condition is satisfied, perform the construction. BOOK VI 339 ANGLES BETWEEN PLANES 430. Dihedral angle. A dihedral angle is the figure formed by two planes which meet each other ( 390). QUERY 1. Can you hold your book so as to form a dihedral angle? QUERY 2. How many dihedral angles are formed by the walls, ceiling, and floor of a square room ? 431. Parts of a dihedral. The portions of the planes which form a dihedral angle are called the faces of the angle. The intersection of the faces of a dihedral angle is called its edge. A dihedral angle may he designated by its edge when there is no ambiguity. Thus, we may designate the adjacent figure as the dihedral AB. It may also be designated as K-AB-R. QUERY 3. Can two dihedral angles have one face in common ? Illustrate with your book. QUERY 4. Can more than one dihedral angle have the same edge ? 432. Plane angle. The plane angle of a dihedral angle is formed by two lines, one in each face, each perpendicular to the edge at the same point. In the above figure CDE and FGII each represent a plane angle of the dihedral AB. EXERCISE 43. Any two plane angles of the same dihedral angle are equal to each other. QUERY 5. Is the plane of ED and DC _L to ABt QUERY 6. Can angles other than plane angles be formed by two lines, one in each face of the dihedral angle and intersecting its edge at the same point? Illustrate. QUERY 7. Can you find very small and also very large (nearly 180) angles in illustrating Q.uery 6 ? QUERY 8. What relations do the planes of the plane angles of a dihedral angle bear to the edge of the dihedral? 340 SOLID GEOMETRY Theorem 19 433. If two dihedral angles are congruent, their plane angles are equal. Given the congruent dihedrals AB and CD and their plane angles GFE and KLH respectively. To prove that Z GFE = Z KLH. Proof. Since the dihedrals are congruent, we may consider them as different positions of the same figure. 378 Hence, if they are brought into coincidence so that the points F and L coincide, FE will coincide with LH and FG with LK. 41 Therefore A EFG and HLK coincide and are equal. Theorem 20 434. Two dihedral angles are congruent if their plane angles are equal. Given (see figure for Theorem 19) the dihedrals AB and CD, having equal plane angles GFE and KLH. To prove that the dihedral angles AB and CD are congruent. Proof. Bring the equal plane A into coincidence. 20 Then the plane of GFE will coincide with that of KLH. Why ? BOOK VI 341 The edge AB is _L to the plane of GFE at F, and the edge CD is _L to the plane of KLH at L. 415 Then the edge AB coincides with the edge CD, 421 and the faces A G and CK, and AE and CH, coincide. 382 Therefore the dihedrals AB and CD are congruent. Why ? QUERY 1. Following the analogy of the definition of vertical angles in plane geometry, can you give a definition of vertical dihedral angles ? QUERY 2. Why are vertical dihedrals equal ? QUERY 3. If two parallel planes are cut by a third plane, how would you pass a plane which would determine the plane angles of all eight dihedrals of the figure ? QUERY 4. Can you state several theorems giving relations between the dihedral angles formed when two parallel planes are cut by a third plane? 435. Perpendicular planes. Two planes are perpendicular to each other if they form a dihedral angle whose plane angle is a right angle. 436. Right dihedral. A dihedral angle whose plane angle is a right angle is called a rigid dihedral angle. NOTE. If two equal dihedral angles are placed adjacent to each other, doubling of the dihedral angle also doubles the plane angle. In general two plane angles are in the same proportion as their corresponding dihedrals. EXERCISES 44. Following the analogy of the corresponding definitions of plane geometry, define the following kinds of dihedral angles : acute, obtuse, adjacent, supplementary, complementary. 45'. Construct a dihedral angle having a plane angle of (a) 90, (b) 45, (c) 60. 46. Construct a right dihedral with a given line as edge and a given plane containing that line as face. 47. If two planes cut and are perpendicular to each other, the four dihedrals formed are all right dihedrals. 342 SOLID GEOMETRY Theorem 21 437. If a line is perpendicular to a plane, every plane containing the line is perpendicular to the plane. Given the line AB perpendicular to the plane Af, and PQ any plane containing AB. To prove that PQ is to M. Proof. In M draw EC J_ to the edge BQ at B. Then AB is _L to BQ and to BC. Why? But ARC is the plane Z of the dihedral A-BQ-C. Why? Therefore PQ is J_ to M. 435 438. Corollary. If a line is perpendicular to a plane, every plane parallel to the line is perpen- dicular to the plane. HINT. Apply 399, 420, and 437. QUERY 1. In Theorem 21 would it have been correct to draw BC in M perpendicular to AB and to Q? QUERY 2. If a line is parallel to one ^ plane and at the same time perpendicular to another, what relation must the planes bear to each other? Illustrate. QUERY 3. In the figure above what is the plane angle of the dihedral formed by PQ and the plane of AB and B C? QUERY 4. How many planes are there which contain a given per- pendicular to a plane ? BOOK VI EXERCISES 343 48. If a plane is perpendicular to the intersection of two planes, it is perpendicular to each of the planes. 49. If three lines are perpendicular to each other at a common point, what is the relation to one another of the three planes determined by the three pairs of lines ? Prove your statement. Theorem 22 439. If two planes are perpendicular to each other, a line drawn in one perpendicular to their intersection is perpendicular to the other. Given the plane PQ perpendicular to the plane M, and the line AB in PQ perpendicular to the intersection BQ. To prove that AB is J_ to M. Proof. Draw BC in M _L to BQ at B. Then ABC is the plane Z of dihedral A-BQ-C. Why? Hence ABC is a right Z. 436 Therefore ABis_LtoM. Why? QUERY 1. If two planes are perpendicular to each other, any line perpendicular to one of them is how related to the other? QUERY 2. What condition must be fulfilled in order that there may be a plane perpendicular at the same time to a given plane and to a given line? 344 SOLID GEOMETRY EXERCISES 50. If a line and a plane are both perpendicular to the same plane, they are parallel. (Assume that the line does not lie in the first plane. See Theorem 21.) 51. Construct a plane which contains a given point, is parallel to a given line, and is perpendicular to a given plane. Theorem 23 440. If two planes are perpendicular to each other, a line drawn from any point in one, perpendicular to the other, lies in the first. Given the plane PQ perpendicular to the plane Af, and the point A in PQ, from which a line AB is drawn perpendicular to M. To prove that AB lies in PQ. Proof. From A draw AB' in PQ J_ to the intersection RT. Then AB' is to M. 439 Then AB and AB' are the same line. 421 Hence AB lies in PQ, since it coincides with ,4 B', which was constructed in PQ. EXERCISE 52. Construct a figure and devise a proof of Theo- rem 23 for the case where the point A is in the intersection, RT, of the plane PQ and M. BOOK VI 345 Theorem 24 441. If two intersecting planes are perpendicular to a third plane, their line of intersection is perpendicular to that plane. p m R Given the planes PQ and /?S, each perpendicular to M, and AB, their intersection. To prove that AB is J_ to M. Proof. From A, a point of AB, drop a _L AK to M. Then AK lies in PQ and also in RS. 440 Hence it is their intersection and coincides with AB. 391 Consequently AB is _L to M, since it coincides with the line A K, which is drawn _L to M. QUERY. A plane revolves about a fixed line which lies in it (as the lid of a chest revolves about the line of its hinges). Find a plane to which the revolving plane is always perpendicular. How many such planes are there ? 442. Bisector of a dihedral angle. A plane is said to bisect a dihedral angle if it contains the edge of the dihedral angle, and if the dihedral angles which it forms with the faces are equal. EXERCISES 53. Construct a plane bisecting- a given dihedral angle. 54. If one plane meets another, forming two adjacent dihedral angles, the planes which bisect these angles are perpendicular to each other. 346 SOLID GEOMETRY Theorem 25 443. The bisector of a dihedral angle is the locus of points equidistant from the faces of the dihedral. Given the dihedral M-QO-N, and the plane PQ bisecting this dihedral. To prove (1) that every point in PQ is equidistant from M and N ; (2) that every point equidistant from M and N lies in PQ. Proof. (1) Let A be any point in PQ. Draw AB and AD _L to M and to N respectively. Pass the plane determined by AB and AD, cutting 717, PQ, and N in BC, AC, and CD respectively. The plane ABCD is _L to M and to N. 437 Hence OC is _L to plane ABCD. 441 Therefore OC is _L to CB, CA, and CD. Why ? Therefore A CB and A CD are the plane angles of their respective .dihedrals. 432 Hence ^ACB=Z.ACD. 433 Therefore A ABC is congruent to A DC A. 50 Hence AB=AD. Why? (2) Let A be any point such that the J,4Z) and AB are equal. Pass the plane PQ determined by A and OQ. BOOK VI 347 Also pass the plane determined by AB and AD, cutting A/, PQ, and N in BC, A C, and CD respectively. AABC is congruent to AACD. Why ? Therefore Z .4 CD = Z .1 C. Why ? But these angles may be proved to be the plane angles of their respective dihedrals by the same method as that used earlier in this demonstration. Hence dihedrals A-CQ-B and A-CQ-D are equal, 434 and PQ bisects M-QO-N. 442 Hence A lies in the bisecting plane. Theorem 26 444. If a line is not perpendicular to a plane, one and only one plane can be passed containing the line and perpendicular to the plane. Given the line AB, not perpendicular to the plane M. To prove that one and only one plane can be passed containing AB and _L to M. Proof. From any point C of AB draw CD _L to M, and pass the plane A Q determined by AB and CD. A Q is to M. 437 Any other plane through AB _L to M would contain CD, 440 and hence would coincide with A Q. 382 Therefore one and only one plane can be passed, containing AB and J_ to M. 348 SOLID GEOMETRY QUERY 1. If two planes cut each other, forming four dihedrals, what is the locus of points equidistant from the faces of these dihedrals ? QUKRY 2. How can you find a point in a given line equidistant from the faces of a given dihedral ? Discuss any special cases. QUERY 3. What is the locus of points in a given plane equidistant from the faces of a dihedral angle ? Discuss any special cases. QUERY 4. What is the locus of points equidistant from the faces of a dihedral and also equidistant from two given points ? QUERY 5. What is the locus of points equidistant from the faces of a dihedral angle and also equidistant from three given points? QUERY 6. How would you find the locus of points 4 inches from each of the faces of a given dihedral ? QUERY 7. What is the locus of points equidistant from two parallel planes and also equidistant from the faces of a given dihedral? EXERCISES 55. If three or more planes intersect in a common line, the lines perpendicular to them from any given external point are coplanar. 56. If from any point within a dihedral angle lines are drawn perpendicular to the faces, the angle between these lines is the supplement of the plane angle of the dihedral. 57. All the perpendiculars to a plane erected from points of a line in that plane lie in a plane perpendicular to the given plane. 58. Draw a figure and devise a proof of Theorem 26 for the case where AB lies in M. 59. A line is perpendicular to the bisector of a dihedral at the point C and intersects the faces of the dihedral at A and B. Show that AB is bisected at C. 60. If the bisecting planes of two adjacent dihedrals are per- pendicular to each other, the exterior faces of the adjacent angles form one plane. 61. Through a point O of the edge of a dihedral angle a line OA is drawn in one of the faces. Construct in the other face a line OB such that the angle A OB will be a right angle. BOOK VI 349 PROJECTIONS 445. Projection of a point. The projection of a point on a plane is the foot of the perpendicular from the point to the plane. The perpendicular which contains the point and its pro- jection is called the projecting line of the point. QUERY 1. Does a given point have more than one projection on a given plane ? QUERY 2. Under what conditions may two points have the same projection on a given plane? QUERY 3. If two points are projected on the same plane, what is the relation between their projecting lines ? QUERY 4. If a point is projected on two planes, how must the planes be situated in order that the projecting lines may be identical ? 446. Projection of a line. The projection of a line (or of a curve) on a given plane is the locus of the projections of its points on that plane. The plane containing a given line, and perpendicular to a given plane ( 444), is called the projecting plane of the line on the given plane. P is the projecting plane of the line a on the plane M. QUERY 5. Can a complete line have a line-segment for its projection? Illustrate. QUERY 6. What is the projection on a plane of a line perpendicu- lar to it? QUERY 7. In what part of the heavens is the sun if the shadow of a line is its projection on the ground? QUERY 8. Between what limits may the length of the projection of a given line-segment vary ? Illustrate. QUERY 9. Under what conditions is the projection of a circle an equal circle ? a line-segment ? QUERY 10. Under what conditions does the projection of a plane figure form a straight line ? 350 SOLID GEOMETRY Theorem 27 447. If a line is not perpendicular to a given plane, its projection on that plane is the intersection of its projecting plane with the given plane. Given the plane M, the line AB not perpendicular to M, and the projecting plane SP, which intersects M in OP. To prove (1) that every point of AB lias its projection in OP; (2) that every point of OP is the projection of some point of AB. Proof. (1) Let C be any point of AB } and let F be the foot of the J_ from C to M. F is the projection of C on M. CF lies in SP. Therefore F is in OP, the intersection of M and SP.- (2) Let G be any point of OP. Draw GH _L to M. Then GH lies in SP. Hence GH must either cut A B or be II to it. If they were II, AB would be _L to M, which contradicts the hypothesis. Therefore GH cuts AB in some point K. Hence G is the projection of A'. QUERY 1. Which of the two parts of the proof given in 447 would it be possible to carry out in an attempt to show that the projection of a circle whose plane is perpendicular to the given plane is a complete line? 445 440 390 440 426 445 BOOK VI 351 QUERY 2. Is the projection of a corkscrew on a plane ever straight? QUERY 3. By holding a right angle in various positions, what angles may be obtained as its projection on a given plane ? QUERY 4. In what case does a rectangle project into a rectangle ? QUERY 5. Is the following statement correct? "A plane is deter- mined by a line and its projection on a given plane." Illustrate. EXERCISES 62. Construct the projection of a given line-segment upon a given plane. 63. If a line-segment is parallel to a plane, it is parallel and equal to its projection on the plane. 64. The projection of a square upon a plane which is parallel to that of the square is a congruent square. Theorem 28 448. The acute angle which a line makes with its pro- jection on a plane is the least angle which it makes with any line drawn in the plane through its foot. B Given the line AB, and AC its projection on the plane M. Let AK be any other line in M through the point A. To prove that Z.BAC } draw the projecting line LF, and draw LR, making AR = AF. In&LAF &nd LAR, AL =AL and AF = AR. Why? But LFy planes parallel to and at the same dis- tance from their respective bases are always equal, the solids have the same volume. It should be emphasized that the planes mentioned in the theorem may be any distance from the bases of the two figures, but must be the same distance from the base of each. In terms of the piles of cards, this means that cards the same distance up in each pile must have the same size in order that the piles may have equal volumes. NOTE. The Italian mathematician Bonaventura Cavalieri was born in 1598 and died in 1647 at Bologna. The principle known under his name was first used by him for the case of plane figures. That is to say, he stated the theorem which is given in the preceding section in such a way as to include not only the volume of certain solids but also the area of certain plane figures. The form in which he stated his theorem is as follows : Plane and solid figures are equal in content when sections drawn at the same height from the base produce equal lines or areas. A rigorous demonstration of the validity of this assumption can be given by use of the methods of the calculus. 370 SOLID GEOMETEY QUERY 1. If two prisms have equal bases, are the sections of the two figures made by planes parallel to the bases necessarily equal ? QUERY 2. How manygparallelograms are there with a given base and altitude? AVhat can be said about the volumes of the parallele- pipeds having the same altitude, which are constructed on these paral- lelograms as bases? QUERY 3. If a pyramid and a prism have congruent bases and equal altitudes, would sections the same distances from their bases be con- gruent ? Do you think that the two figures have the same volume ? Theorem 2 482. Two prisms having equal bases and equal altitudes are equal in volume. Given any two prisms P and /?, having equal altitudes a, and bases B and C, which are equal in area. To prove that P and R are equal in volume. Proof. Pass planes II to the bases of the prisms P and R, at any distance, k, from the corresponding bases. Call the areas of the sections thus formed S and T respectively. Then S = B and T = C. 459 But B=C, Given Hence S = T. 32 Therefore P = R. 481 BOOK VII 371 483. Corollary. A plane which passes through the opposite parallel edges of a parallelepiped divides it into two equal triangular prisms. EXERCISE 23. Two right prisms are equal if three faces which meet at a vertex of one are respectively equal to three faces similarly placed, which meet at a vertex of the other. HINT. The faces which are given equal must include among them a base of each prism. Theorem 3 484. The volume of any prism is equal to the product of its base and altitude. 7 1 i * i____ 4 ___ B * Given any prism P, with base B and altitude a. To prove that volume P = B a. Proof. Let K be a rectangular solid having its base equal to B and its altitude equal to a. Volume K = B- a. 476 But the base and altitude of P equal those of K. Const. Hence volume P = volume A'. 482 Therefore volume P = B - a. Why ? 485. Corollary. The volume of a parallelepiped equals the product of the area of any face and the altitude on that face. 372 SOLID GEOMETKY QUERY 1. Is the volume of any parallelepiped equal to the product of its three altitudes ? Give an example. QUERY 2. In what case is the volume of a parallelepiped equal to the product of three of its edges ? QUERY 3. In what kind of a triangular prism would the volume be equal to one half the product of its three edges? EXERCISES 24. Construct a rectangular solid whose base is equal in area to that of a given triangular prism, and which has an altitude equal to that of the prism. Give a reason for each step. 25. Prove that the volume of a triangular prism is equal to one- half the product of the area of a given lateral face and the distance to that face from the opposite lateral edge. Theorem 4 486. The lateral area of a prism equals the product of a lateral edge and the perimeter of a right section. X . B Given any prism A-X, of which HLMOP is a right section, and one of whose lateral edges is AR. To prove that lateral area of A~X = perimeter of H~0 > AR. Proof. Plane H-0 is _L to .47?, BS, CU, etc. 466 AR is _L to HL, BS to LM, CU to MO, etc. Why ? BOOK VII 373 Hence the lateral faces of the prism are parallelograms, each having one of the line-segments HL, LM, MO, etc. as its altitude, and each having a base equal to AR. 465 Hence area AY=AR . PH, area,BR=BS.HL, Why? areaCS = CU ML. Adding, area (/I Y+BR+CS-\ )=AR(PH+HL+LM-{ ), or lateral area of A -X = perimeter of H-0 x AR. 458 487. Corollary. The lateral area of a right prism is equal to the product of its altitude and the perimeter of its base. QUERY. If the shape of a prism is varied by moving the upper base about in its plane while the lower base remains fixed in position, does the volume of the prism vary ? Does the lateral area vary ? EXERCISES In the following exercises, h represents the altitude, n the num- ber of sides of the base, b one side of the base, B the area of the base, V the volume, S the lateral area, and T the total area, of a prism. 26. Given V= 50, B= 25. Find h. 27. Given h = 8j, B=6. Find V. In Exercises 28-35 the prism is regular. 28. Given h = 12, b = 3, n = 3. Find S. 29. Given h = 10, b = 4, n = 4. Find V. 30. Given h 6f , S= 80, n = 4. Find V. 31. Given V= 56, 5 = 14. Find h. 32. Given V= 300, n = 3, h = 20. Find b. 33. Given n = 4, h = 6, 7= 1536. Find I. 34. Given n = 3,b = 6,k = 8. Find T. 35. Given n = 6, b = 8, V= 128. Find T. 374 SOLID GEOMETKY 36. Find the lateral area of a right prism whose base is a rec- tangle 6x8 inches and whose altitude is 1 foot. Find also the total area. 37. Find the lateral area of a prism whose base is an equilateral triangle 4 inches on a side, and whose altitude is 10 inches. Find also the total area. 38. The total area of a cube is 216 square inches. Find the edge and the volume. 39. Sand lies against a vertical wall, reaching a point 3 feet high. If the sand just lies at rest with its surface at an angle of 30 with the horizontal, how many cubic feet of sand are there in a pile 20 feet long ? Assume that the ends of the pile are perpendicular to its length. 40. What is the weight of a block of ice 24 x 24 x 18 inches if ice weighs 92 per cent as much as water ? (Cubic foot of water weighs 62.5 pounds.) 41. A railway cut 25 feet deep is to be made with one side vertical and the other inclined at an angle of 45 to the vertical. The bottom is to be 30 feet wide. How many cubic feet of mate- rial must be removed per running foot of track ? 42. Express T in terms of V for a cube. 43. If d is the diagonal of a cube, express V in terms of d. 44. Express V in terms of b and h for a triangular prism whose base is equilateral. 45. Find correct to two decimal places the edge of a cube equal in volume to a prism whose base is a regular hexagon 4 inches on a side and whose altitude is equal to an edge of the cube. 46. The altitude and base of a parallelepiped are 6 and 8 inches respectively, and a lateral edge whose length is 10 inches makes an angle of 60 with the base. Find the volume. 47. A regular hexagonal prism whose lateral edge makes an angle of 60 with the plane of its base has an altitude of 12 feet, while one edge of the base is 4 feet. What is its lateral area ? BOOK VII 375 CYLINDERS 488. Cylindrical surface. If a line moves, always remain- ing parallel to its first position and always cutting a fixed plane curve not situated in the plane of the line, the surface generated is called a cylindrical surface. The moving line is called the gener- ator ; in one of its positions it is called an element ; the fixed curve is called the directrix of the surface. From the definitions just given it follows that the elements of a cylinder are parallel ( 406). In this text the directrix will be assumed to be a closed curve, although in more advanced mathematical work the directrix is often an open curve, or even a curve consisting of several branches. A convex curve is one which a straight line can cut in no more than two points. A convex cylindrical surface has a convex curve as directrix. Only convex cylindrical surfaces are studied in this text. 489. Corollary. Each point of a cylindrical surface lies on one and only one element of the surface. HINT. Suppose a certain point were contained by two elements. 490. Cylinder. The solid bounded by a cylindrical surface and two parallel planes cutting the elements is called a cylinder. The terms altitude, base, right section, lat- eral area, volume, and right cylinder are defined similarly to the corresponding terms applied to the prism ( 458). 491. Circular cylinder. If a cylinder has a circular right section, it is called a circular cylinder. EXERCISE 48. Define the terms mentioned in 490. 376 SOLID GEOMETRY Theorem 5 492. The sections of a cylindrical surface by two paral- lel planes , one of which cuts an element, are congruent. Given any cylindrical surface AB, of which one element AT is cut by the plane M, which is parallel to the plane N. To prove that the sections formed are congruent. Proof. The plane M cuts all the elements of A B. 402 Take any three points P, R y S at random in the intersection of the surface by M, and let PA', R Y, SZ be the elements of the surface containing these points. 489 Let the elements PX, RY, and SZ cut .ZV in the points /, K, and L respectively. 403 Let the intersections of M and N with the planes determined by the parallel lines PJ, RK, SL be PS, JL, etc. PS is II to JL. 393 Hence PL is a parallelogram, and PS = JL. 83, 85 Similarly, SR = LK and RP = KJ. Therefore APRS is congruent to AJKL. Why? Since P, R, and S were taken at random, it follows that if the upper section is applied to the lower one so that PS coincides with JL, any other point, as R, of the upper section will coincide with the point of the lower section which is on the same element. BOOK VII 37T Hence any point of the upper section coincides with some point .of the lower. Similarly it can be shown that any point of the lower section coincides with some point of the upper. Therefore the upper and lower sections are congruent. 493. Corollary. The bases of a cylinder are congruent. Theorem 6 494. The section of a cylinder made ~by a plane which contains an element of the cylinder and a point of the cylindrical surface not in this element is a parallelogram. Given any cylinder C and a plane M which contains an ele- ment AB and a point O of the cylindrical surface not on AB. To prove that the section formed is a parallelogram. Proof. Denote by FH the element through 0. Now FH is II to AB. If FH cut M, AB would also cut M. Therefore FH lies in M and also in the surface 'of C. Let M intersect the bases of C in H B and FA. HB is II to FA. Hence ABHFis a parallelogram each side of which lies in both the plane and the surface of the cylinder. Therefore the section of C by M is a parallelogram. 488 402 Why? 378 SOLID GEOMETRY QUERY 1. Would the section of the cylinder made by the plane determined by an element and any point in one of the bases be a parallelogram ? QUERY 2. What kind of cylinder may have a square as a section? a rectangle ? a rhombus ? EXERCISES 49. Every section of a right cylinder made by a cutting plane perpendicular to the base is a rectangle. 50. The intersection of a right cylinder and a plane which passes through an element is a rectangle. 495. Inscribed prism. If the bases of a prism are inscribed in the bases of a cylinder, and the lateral edges of the prism are elements of the cylinder, the prism is said to be inscribed in the cylinder. QUERY 1. Can a regular prism be inscribed in any given right circular cylinder? QUERY 2. Can a regular prism be inscribed in any given circular cylinder ? QUERY 3. Can a rectangular solid be inscribed in any given right circular cylinder? EXERCISES 51. Construct a triangular prism inscribed in a given right circular cylinder, giving a reason for each step. 52. Construct a regular hexagonal prism inscribed in a given right circular cylinder, giving a reason for each step. 496. Volume of cylinder. If the base of a prism which is inscribed in a cylinder has a very large number of sides, and each side is very short, then the area and the perimeter of the base of the prism are approximately equal to the area and the perimeter respectively of the base of the cylinder. By taking the sides sufficiently short, as close approximations as may be desired can be obtained. BOOK VII 379 The inscribed prism whose base is almost the same in area as that of the circumscribing cylinder will have a large num- ber of very narrow parallelograms -as its lateral faces, and its lateral area and volume will differ very little from the lateral area and volume respectively of the cylinder. But the volume of the prism, however many lateral faces it may have, is the product of its base and its altitude, and the lateral area of the prism is the product of the perimeter of a right section and a lateral edge. Since the volume and the base of the cylinder become very nearly equal to the volume and the base respectively of the inscribed prism, when the number of its faces is sufficiently increased, and since the altitudes of the two figures are iden- tical, we are led to the following statements, which we here assume without proof, but which can be demonstrated by the theory of limits. Theorem 7 497. The volume of a cylinder equals the product of the area of its base and its altitude. 498. Corollary. The volume of a right circular cylinder is equal to 7rr 2 h, where r denotes the radius of the base and h the altitude of the cylinder. Theorem 8 499. The lateral area of a cylinder is equal to the prod- uct of the perimeter of a right section and an element. 500. Corollary 1. The lateral area of a right circular cylinder is equal to 2 Trrh, where r denotes the radius of the base and h the altitude of the cylinder. 380 SOLID GEOMETRY 501. Corollary 2. The total area of a right circular cylinder is equal to on ?,o 20 ^7 , -\ $=2 Trrh + 2 TTT = 2 ?rr (7i -f r). NOTE. Many theorems concerning the prism are equally true when applied to the cylinder. Theorems 7 and 8 illustrate this important fact, which may be stated in general terms as follows : 'Any theorem regarding the prism which does not depend on the number of lateral faces is true for the cylinder. QUERY 1. The lateral surface of a cylinder is cut along an element and rolled out flat. What kind of figure is obtained ? QUERY 2. What is the locus of points a given distance from a given line? QUERY 3. What is the locus of points a given distance from two given parallel lines? QUERY 4. What is the locus of points a given distance from a given cylinder ? QUERY 5. What is the locus of points a given distance from a given line and equidistant from two parallel planes to which the given line is perpendicular? What if the planes are not perpendicular to the given line? EXERCISES In the following exercises, all of which relate to right circular cylinders, // represents the altitude, r the radius of the base, S the lateral area, T the total area, and V the volume : 53. Given r = 2, h = 2. Find T. 54. Given r = 4, h = 9. Find T. 55. Given r = 5, h = 21. Find S. 56. Given r = 4, F=42. Find S. 57. Given T= 235, r = 3. Find V. 58. Given S = 121, r = 2. Find h. 59. Given r = 2, V= 64. Find h. 60. Given V= T, r=7. Find S. 61. Given S = V, h=r. Find r. BOOK VII 381 Theorem 9 502. If a rectangle is revolved about one side as an axis, the figure formed is a right circular cylinder. Given any rectangle ABKL which revolves about AB as an axis. To prove that the figure generated is a right circular cylinder. Proof. AL and BK each generate a plane _L to^lZ?. Therefore the planes generated are parallel. The points K and L describe circles in these planes. KL generates a cylindrical surface. 418 Why ? Why ? 488 Hence the figure 0-L is a right circular cylinder. Why ? 503. Cylinder of revolution. A right circular cylinder is often called a cylinder of revolution when it is desired to emphasize the foregoing method of generation. 504. Axis of cylinder. The line joining the centers of the bases of a right circular cylinder is called the axis of the cylinder. EXERCISES 62. The axis of a cylinder of revolution is equal and parallel to the elements of the cylinder. 63. The axis of a cylinder of revolution passes through the center of every plane section of the cylinder parallel to its base. 505. Similar cylinders. If two cylinders of revolution are generated by the rotation of similar rectangles about corre- sponding sides, the cylinders are said to be similar. 382 SOLID GEOMETRY Theorem 10 506. If two cylinders of revolution are similar, (a) their volumes are proportional to the cubes of their altitudes or of their radii ; (b) their lateral surfaces, or their total surfaces, are pro- portional to the squares of their altitudes or of their radii. Given any two similar cylinders of revolution with radii r and r x , and altitudes h and h lt respectively. Let V and V v S and S v T and 7 7 1 represent the volumes, lateral surfaces, and total surfaces, respectively. 3 8 To prove - i r ) () === Proof. Trr-h But Hence, from (1) and (2), V r 2 (1) 498 (2) 505,268 (*) T = r! 2 7rrh r h hi h 2 = 7-f = -4 = 71 ( 3 ) 500 and T_ 27rr(r + h) _ r_ ^_ 7T1\ (l\ + 7 tl ) 'I + />! 501 BOOK VII 383 But from (2), r + h r*l -f- AJ (5) 257,262 Hence, from (4), (5), and (2), T_ _r_ r + h T' ~ r, r , -\- h, 507. Theorem of Pappus. If a circle is revolved about a line lying in its plane but not intersecting it, a figure called an anchor ring or torus is formed. If one imagines the torus cut along one of the generating circles and straightened out so as to form a cylinder, one would expect that the altitude of the cylinder would be the line described by the center of the generating circle in its revolution. That this is the case can be proved by means of the calculus; in fact, a much more general theorem can be demon- strated, which is called The Theorem of Pappus. If a closed curve (or polygon) rotates about an axis lying in its plane, the volume of the ring described equals that of the cylinder (or prism) whose base is the generating figure and whose altitude is the length of the curve described by the center of gravity of the generating figure. The area of the surface of the ring generated equals the lateral area of the cylinder (or prism) described. 384 SOLID GEOMETRY NOTE. This theorem was first discovered by Pappus of Alexandria (third century of the Christian era). His work was forgotten for more than twelve hundred years, until the interest in the subject was revived at the end of the sixteenth century by the works of Kepler and Guldin. Kepler (1571-1630) investigated a number of solids generated by the rotation of a plane figure and succeeded in finding rules for computing their volumes in certain particular cases. All of his rules are special cases of the Theorem of Pappus, although Kepler never announced the theorem in its general form. Among the solids treated by Kepler were the torus and the solid termed by him w the apple," which is formed by revolving a segment greater than half a circle around its chord as an axis, and " the lemon," which is formed by revolving a segment of a circle with an arc less than 180 around its chord as an axis. The Jesuit, Paul Guldin (1577-1643), who was a professor of mathematics in Rome and in Graz, was the first after Pappus to give a statement of the theorem in its general form in a work published in 1640. He failed, however, to give a satisfactory proof of the general case. The Theorem of Pappus was later generalized by Leibniz and Euler. Leibniz (1646-1716) noticed that the theorem holds true when the moving plane figure moves along any path to which it always remains perpendicular. EXERCISES 64. The center of a circle of radius 4 inches is 10 inches from the axis about which it rotates to form a torus. Find (1) the volume, (2) the surface of the torus. Solution. Applying the Theorem of Pappus, we have V = 2 TrlO TT= 320 -rr 2 cubic inches. S = 2 TrlO 2 7r4 = 160 vr square inches. 65. How many cubic feet of water will a 90 elbow of an 8-inch water main contain if the axis about which the generating circle of the elbow rotates is 12 inches from the center of the circle ? 66. How much will an iron elbow of Exercise 65 weigh if the iron is one-half inch thick ? (A cubic foot of iron weighs about 480 pounds.) BOOK VII 385 67. A rectangle 2x4 inches rotates about a line parallel to the nearer one of the shorter sides and 6 inches from it. Find (1) the volume, (2) the area of the solid generated. 68. Compare the results of the foregoing exercise with those obtained if the same rectangle is rotated about an axis 6 inches from the longer side. 69. The cross section of the rim of an iron flywheel 3 feet in diameter is a rectangle 12 x 2 inches. How much does it weigh ? 508. Tangent plane. A plane which contains an element of a cylindrical surface and meets the surface nowhere else is said to be tangent to the cylinder. 509. Circumscribed prism. A prism is said to be circum- scribed about a cylinder when its bases are polygons circum- scribed about the bases of the cylinder and its lateral faces are tangent to the cylinder. EXERCISES 70. The ratio of the volume of a cylinder to that of its inscribed or circumscribed prisms is equal to the ratio of their correspond- ing bases. 71. A cylinder is generated by the revolution of a rectangle 2x6 inches about the shorter side. Find the volume and the total area. 72. A cylinder is generated by the revolution of a rectangle 2x6 inches about the longer side. Find the volume and the total area. 386 SOLID GEOMETRY 73. Two cylinders are generated by the revolution of a rec- tangle a x b inches, first about the side a, then about the side b. Find the ratio (1) of the volumes, (2) of the total areas, (3) of the lateral areas, of the cylinders formed. 74. The edge of a cube is 8 inches. Find the volume of the circumscribed cylinder. 75. The lateral area of a cube is 54 square inches. Find the lateral area of the inscribed cylinder. 76. A tank is 2 x 3 x 4 feet. It is desired to make a cylindrical tin tank, without a lid,- of the same volume and having an altitude of 3 feet. Find the number of square feet of tin in the new tank. 77. A cylindrical gasoline tank 48 inches long and of diameter 16 inches is on its side in a horizontal position. The greatest depth of the gasoline is 1 foot. How many gallons are required to fill the tank ? (231 cubic inches = 1 gallon.) 78. Find the volume of a cylinder of revolution inscribed in a regular hexagonal prism whose altitude is 8 inches and whose lateral area is 288. 79. Find the diameter of a cylinder of revolution if the volume is equal numerically to (1) the total area, (2) the lateral area. 80. The radius of a cylinder of revolution is 7 inches, and its altitude is 15 inches. Find the distance from the axis of the cylin- der to the plane parallel to the axis which makes a section of the cylinder equal in area to the base. (Use TT = 22/7.) 81. What is the ratio of the diameter to the altitude of a cylinder of revolution if the area of the greatest section made by a plane through an element is equal to that of the base ? 82. A cylinder of revolution is of radius 8 inches. Two tan- gent planes are drawn to the cylinder through a point 8 inches from the surface of the cylinder. What is the angle between the planes ? 83. What is the diameter of a cylindrical quart measure which is 4 inches high ? BOOK VII 387 PYRAMIDS 510. Pyramid. A pyramid is the solid bounded by a polygon and triangles having the sides of the polygon as their bases and having a common vertex. The polygon is called the base of the pyramid ; the triangles are called the lateral faces ; and the common vertex of the lateral faces is called the vertex. The intersections of the lateral faces are called the lateral edges of the pyramid. Pyramids are triangular, quadrangular, hexagonal, etc., accord- ing as their bases have three, four, six, etc. sides. 511. Lateral area. The sum of the areas of the lateral faces is the lateral area of the pyramid. 512. Altitude. The altitude of a pyramid is the perpendicular distance from the vertex to the plane of the base. QUERY 1. Can the altitude of a pyramid be equal to one of the lateral edges? QUERY 2. To how many lateral edges can the altitude of a pyramid be equal? 388 SOLID GEOMETRY QUERY 3. For what kind of pyramid may any one of the faces be taken as the base ? QUERY 4. From the adjacent figure name the various parts of the pyra- mid which have been defined. QUERY 5. How many pyramids are there with a given base and a given altitude? What is the locus of their vertices ? QUERY 6. As the vertex of a pyra- mid becomes very remote from the base, what relation to each other do the lateral edges approach ? 513. Pyramidal surface. The lateral faces of a pyramid may be generated by the motion of a line always passing through a fixed point, the vertex, and continually meeting the perimeter of a fixed polygon. If we consider the figure generated by the entire line, we have not merely the lateral surface of the pyramid, but a surface extending indefinitely in both directions from the vertex. This sur- face is called & pyramidal (py ram 'I dal) surface. The terms generator, element, and directrix are applied to the pyramidal surface in the same sense as that defined in 488. 514. Another definition of a pyramid. A second definition of a pyramid may be given which is identical in meaning with that of 510 but which is more useful in dealing with certain problems. A pyramid is a solid bounded by a pyramidal surface and a plane which cuts all of the elements on the same side of the vertex. BOOK VII 389 Theorem 11 515. If a pyramid is cut by a plane parallel to the base, (a) the lateral edges and the altitude are divided pro- portionally^ (b) the section is a polygon similar to the base. Given any pyramid 0-CG, cut by a plane M parallel to the base making the section P-L. Let the altitude OS intersect M at the point R. OH_OK_OL_ _OR } ~OA~~OB~~OC~ ~~OS' (&) PL is a polygon similar to G~C. OH OK OL OR Proof, (a) _ = _ = _ = ... = _. 411 (ft) P# is II to (7/1; HK is II to AB; KL is II to BC\ etc. Why? Therefore Z PHK = Z GA B ; Z HKL = Z A B C, etc. Why ? Also AOAB is similar to AOHK-, A OB C is similar to AOKL; etc. 271 He - e S-3' '-.!*.' 269 Therefore ^ = 5 etc. Why? Hence polygon P-L is similar to polygon G-C. 267 390 SOLID GEOMETRY Theorem 12 516. If two pyramids having equal bases and altitudes are cut by planes parallel to their bases and. equidistant from their vertices, the sections formed are equal. Given any two pyramids, with equal bases -Band B', and with equal altitudes h. Let them be cut by planes parallel to the bases and distant a from the vertices. Let the sections be called S and S'. To prove that 8 = 8'. Proof. EF VF But Therefore VF VK~ EF a OU h~ OR TU TU OU Wh ? EF' TU GK PR ' " -QK 2 -pj? But S is similar to B, and AS" is similar to B'. Hence Consequently EF GK' S' Why ? 515 (ft) 326 Why? But B = B '. Given Therefore S = S'. Why ? QUERY. In Theorem 12, under what circumstances would the sec- tions made by the planes be congruent? BOOK VII 391 517. Corollary. If a pyramid is cut by two parallel planes, the areas of the sections are proportional to the squares of their distances from the vertex. QUERY 1. If a plane parallel to the base of a pyramid is halfway be- tween the base and the vertex, what is the ratio of the section to the base ? QUERY 2. How large a shadow will a book 6x8 inches, held 4 feet from a light, cast on a wall parallel to the book and 8 feet away from the light ? QUERY 3. How much larger would the shadow in Query 2 be if the wall of the room were (a) 12 feet, (b) 16 feet from the light? QUERY 4. Explain how the law that the intensity of a light varies inversely with the square of its distance is consistent with 517. QUERY 5. If a pyramidal surface is cut by two parallel planes on opposite sides of the vertex (above and below in figure of 513), are the sections similar ? QUERY 6. If a pyramidal surface is cut by two parallel planes on opposite sides of the vertex and at equal distances from it, are the sec- tions equal ? Are they congruent ? If the sections are triangles, are they congruent? EXERCISES 84. A square pyramid has all of its lateral faces equilateral triangles whose sides are each 12 feet. Find the altitude of the pyramid. 85. A pyramid of altitude 6 feet is cut by a plane which makes a section half the area of the base. How far from the vertex of the pyramid is the plane ? 86. If a pyramid is cut by two parallel planes, the correspond- ing sides of the sections are in the same ratio as the distances of the planes from the vertex. 87. If two pyramids having bases of equal perimeter and equal altitudes are cut by planes equidistant from their vertices, the sections formed have equal perimeters. 392 SOLID GEOMETRY Theorem 13 518. Two pyramids having equal bases and equal alti- tudes are equal in volume. Given any two pyramids P and P', with equal altitudes h and equal bases B and B' in the same plane. To prove that P = P 1 . Proof. At any distance d from the bases pass planes bases of the pyramids, making sections S and S'. to the Then But Therefore d is any distance less than h. P = P'. 516 481 QUKRY 1. Is Theorem 13 valid for two pyramids whose bases have different numbers of sides ? QUERY 2. Is the converse of Theorem 13 true? EXERCISES 88. If a pyramidal surface is cut by two parallel planes, each cutting all the' edges on opposite sides of the vertex and at equal distances from it, the pyramids formed are equal in volume. 89. What relation do the two plane sections of Exercise 88 bear to each other ? Prove your statement. BOOK VII Theorem 14 393 519. The volume of a triangular pyramid equals one third of the product of its base and altitude. R Given any triangular pyramid 0-ABC, with altitude h. To prove that the volume 0~ABC = 1 h X ABC. Proof. At B and C construct BR and CS II to OA. Pass planes determined by pairs of parallels A O, CS ; A 0, BR', BR, CS. Pass plane ORS through II to ABC. 408 The complete figure formed is a triangular prism having ABC as its base and h as its altitude. Why ? Pass plane ORC, dividing the pyramid O-RBCS into the two triangular pyramids 0-RCS and 0-RBC. AABC = AORS. Therefore 0-ABC = C-RSO. Also ARCB=ARCS. Therefore O-RBC = 0-RCS. But 0-RSC C-RSO. Therefore 0-ABC = 0-RCS = 0-RBC. Hence O-ABC = ^ prism ABCORS. But prism ABCORS = hx ABC. Therefore volume O-ABC = i A X ^C. Why? 518 Why? Why? Identical Why? Why? 394 SOLID GEOMETRY NOTE. The relation between the three pyramids into which a prism may be divided, which is demonstrated in the foregoing theorem, is found in Euclid's Geometry and was known by a geometer of even earlier date. It is one of the comparatively few theorems in our modern treat- ment of solid geometry which has been contained in nearly all the text- books from the earliest times to the present. The Greeks were more interested in the logic of geometry than in its application, and conse- quently paid more attention to plane geometry than they did to solid. Theorem 15 520. The volume of any pyramid equals one third of the jiroduct of its base and altitude. Given any pyramid P, with base B and altitude h. To prove that volume P= \ B h. Proof. From any vertex V, draw all the diagonals of the base B, dividing it into triangles B lt B 2 , B^ etc. Pass planes determined by these lines of division and the vertex, dividing the pyramid into several triangular pyramids P 1? P a , P g , etc. h is the common altitude of P 1? P a , P 8 , etc. 512 Now P = i B . h. 519 Adding, )> or P=$B.?i. BOOK VII 395 EXERCISES 90. Two pyramids have bases of equal areas, but one is 14 feet high and the other is 10 feet high. Compare their volumes. 91. Two pyramids have equal altitudes. The base of one is a square 6 feet on a side ; that of the other is an equilateral triangle 9 feet on a side. Compare their volumes. 92. Two pyramids have the same altitude. Their bases are respectively a square and a regular hexagon which can be in- scribed in the same circle. Compare the volumes. 521. Frustum of pyramid. If a plane is passed parallel to the base of a pyramid and cutting the edges, the solid in- cluded between the base and the cutting plane is called a frustum of a pyramid. The section formed by the cutting plane is called the upper base, the base of the original pyramid is called the lower base, and the distance between the bases is called the altitude, of the frustum. In case the cutting plane is not parallel to the base of the pyramid, but cuts all of the lateral edges, the solid obtained is called a truncated pyramid. QUERY 1. Why are the bases of a frustum of a pyramid similar polygons ? QUERY 2. How many lateral faces of a truncated pyramid may be trapezoids ? QUERY 3. If a pyramid with a base a few inches on a side has its vertex 10 miles above the base, what would a frustum of the pyramid near the base resemble? 396 SOLID GEOMETRY Theorem 16 522. The volume of a frustum of a pyramid equals one third of the product of the altitude and sum of the upper base, the lower base, and the mean proportional between the two bases. Given F, any frustum of a pyramid, with the upper base U, the lower base L, and the altitude h. To prove that volume F= 1 h ( L + U + V 7" x L ). Proof. Construct the pyramid P, of which F is a frustum, and let its altitude be h + x. Denote by S the small pyramid which has U for a base and x for an altitude. It is required to express F in terms of L, U, and h. Now F = P 6' = ^ L (Ji -\- x) ^ U ' x = ^ (JiL -+- Lx Ux) = J [h L + x(L- C7)]. (1) 520 We must now eliminate x, that is, express x in terms of L, U, and h, and substitute the value obtained in (1). x 2 U WTW = Extracting the square root, 517 Vu vz Solving this equation for x, x = BOOK VII 397 Substituting in (1), we obtain F= = i- \1iL + h (since L- U = ( VI - Vu)( VZ+ VZ/)) NOTE. The foregoing rule for the volume of the frustum of a pyra- mid was first given by Heron of Alexandria in substantially the same form as in this text. A Hindu mathematician named Brahmagupta (about 650) gave a rule for the volume of the frustum of a pyramid with square bases of sides S 1 and S 2 as follows : QUERY 1. A pyramid may be considered as a frustum of a pyramid with its upper base equal to zero. Verify the formula for the volume of a frustum of a pyramid in this case. QUERY 2. Verify the formula for the frustum of a pyramid for the case where the upper base equals the lower base. What is a more familiar name for this solid ? 523. Tetrahedron. A polyhedron hav- ing four faces is called a tetrahedron. The terms tetrahedron and triangular pyramid are interchangeable. Two skew edges of a tetrahedron are called opposite edges. EXERCISES 93. If the base of a regular pyramid 10 feet high contains 16 square feet, what is its volume ? 94. What is the volume of a square pyramid 12 feet high if the base is 6 feet on a side ? 95. The base of a pyramid is an equilateral triangle 6 inches on a side, and its altitude is 8 inches. Find the volume. 398 SOLID GEOMETRY 96. What is the volume of the frustum of a pyramid whose bases are equilateral triangles, 3 feet and 6 feet on a side respec- tively, and whose altitude is 5 feet ? 97. The altitude of a pyramid with a square base is 8 inches. The volume is 128 cubic inches. Find a side of the base. 98. The base of a pyramid is an isosceles triangle whose sides are 10, 10, and 6 inches. The altitude of the pyramid is 15 inches. Find the volume. 99. The sides of the base of a tetrahedron are 10, 17, and 21 inches, and its altitude is 6 inches. Find its volume. 100. The base of a pyramid is twice that of a section parallel to it and 3 feet above it. Find the altitude of the pyramid. 101. A pyramid is 18 inches in height. A section 1 foot from the base and parallel to it has an area of 56 square inches. Find the volume of the pyramid. 102. The base of a pyramid contains 64 square inches, and its altitude is 8 inches. How far from the vertex must a plane par- allel to the base be passed in order to get a section one quarter of the area of the base? 103. A frustum of a pyramid whose square lower base is 40 feet on a side consists of earth which rests at an angle of 45. How high is the frustum if the top contains 100 square feet? How many cubic feet are there in the frustum ? 104. A miller wishes to make a hopper in the form of an in- verted frustum of a square pyramid, with the bases 6 and 3 feet en a side, respectively. How deep shall he make it in order that it shall hold 15 bushels of grain ? (A bushel = 1.24 cubic feet.) 105. Cleopatra's Needle, the Egyptian obelisk in New York City, consists of a frustum of a pyramid whose lower base is a square 1\ feet on a side, whose upper base is 4 feet on a side, and whose altitude is 61 feet, surmounted by a pyramid whose base is the upper base of the frustum and whose altitude is 1\ feet. Find the weight of the Needle in tons. (One cubic foot of the stone weighs about 170 pounds.) BOOK VII 399 524. Regular pyramid. A regular pyramid is one whose base is a regular polygon and whose vertex lies in the perpendicular erected at the center of the base. (The center of a regular polygon is the center of its circumscribing circle.) EXERCISE 106. Prove that the perpendicular from the vertex to the base of a regular pyramid passes through the center of the circle which circumscribes the base. 525. Regular tetrahedron. A regular triangular pyramid has an equilateral triangle as base ( 524). Its altitude may have any length. There may be, therefore, any number of regular triangular pyramids on the same base. The regular triangular pyramid which has all its lateral faces equilateral triangles is called a regular tetrahedron. QUERY. Are the lateral faces of a regular tetra- hedron congruent to the base ? Prove each of the following propositions concerning a regular pyramid: 526. The lateral edges of a regular pyramid are equal. 527. The lateral faces of a regular pyramid are congruent isosceles triangles. 528. The altitudes of the lateral faces of a regular pyramid are equal. 529. If a regular pyramid is cut by a plane parallel .to its base, the pyramid cut off has equal lateral edges. 530. Slant height. The slant height of a regular pyramid is the altitude of any of its lateral faces. 400 SOLID GEOMETRY Theorem 17 531. The lateral area of a regular pyramid equals the product of one half the perimeter of the base and the slant height. Given any regular pyramid P, with the base ABCDFG and the slant height /. To prove lateral area of P = 1 1 . perimeter AB CDFG. Proof. Area of face OAB=%l*AB. 320 Area of face OBC = %l . EC. 528 Adding and factoring, Area (OAB + OBC -\ )= %l(AB + BC -\ ), or lateral area P = % I - perimeter ABCDFG. QUERY 1. Does a pyramid which is not regular have any slant height ? How could its lateral area be found? QUERY 2. If two regular pyramids, one having a square base and the other having a hexagonal base, have equal altitudes, and if their bases are inscribed in the same circle, which one has the greater slant height ? QUERY 3. If two regular pyramids have equal altitudes, and bases of equal area but of different numbers of sides, are their lateral areas equal?. Discuss completely. QUERY 4. If two pyramids have equal lateral areas, are their volumes necessarily equal? QUERY 5. Can a pyramid with very small lateral area have a great volume? Can one with a small volume have a great lateral area? BOOK VII : :,,,,,, 401 Theorem 18 532. The laleral faces of a frustum of a 'regular 'pi/rk^ mid are congruent trapezoids. Given a frustum F of a regular pyramid, and let ABDC and SKHR be any two lateral faces. To prove that ABDC and SKHR are congruent trapezoids. Proof. RH is II to SK and CD is II to AR. Why ? Hence ^(7 and SKHR are trapezoids. 107 Let be the vertex of the pyramid. To prove ABDC and KACH congruent, fold over OAK on OA as axis, so that it coincides with OAB. 20 Then AK coincides with AB, 79, 527 and therefore CH coincides with CD. 45 Therefore ABDC and KACH coincide throughout. Similarly, any lateral face can be proved congruent to an adja- cent one, and hence to any lateral face of the frustum. 32 (2) 533. Slant height of frustum. The slant height of a frustum of a regular pyramid is the altitude of any of its faces. QUERY 1. Would a truncated pyramid have any slant height? QUERY 2. Would the frustum of a pyramid which is not regular have any slant height? 402 SOLID GEOMETRY ^.... . /, Theorem 19 '" ti&: : T]ie lateral area of the frustum of a regular pyra- mid is equal to one half the product of the sum of the perimeters of the bases and the slant height. Proof is left to the student. EXERCISES 107. The lateral area of a pyramid is greater than the area of its base. 108. If the regular tetrahedron of the adja- cent figure has an edge 6 inches long, compute the lines AL, KL, AR, and KR, where L is the mid-point of CB and KR is the altitude of the tetrahedron. HINT. See Exercise 106, page 399. 109. If the frustum of a regular pyramid in the adjacent figure has slant height 10 and sides of the bases 24 and 12 respectively, compute RA, KA, AL, where KR is the slant height and KA the altitude of the frustum. In Exercises 110-116, all of which relate to pyramids, li denotes the altitude, B the area of the base, b one side of the base, n the number of sides of the base, I the slant height, S the lateral area, and V the volume. 110. Given F= 46, h =113/4. Find B. 111. Given B = 12, A = 15. Find V. In Exercises 112-116 the pyramid is regular. 112. Given b = 6, n = 4, h = 4. Find S. 113. Given V= 25, h = 3, n = 4. Find S. 114. Given S = 100, b = 4, n = 5. Find I. 115. Given I = 4, n = 3, b = 6, Find A, BOOK VII 403 116. Given n= 3, = 16, 1=5. Find T, 117. The area of the base of a regular quadrangular' pyr.'inn-l is 36 square inches. A lateral edge is 10 inches. Find (1) the lateral area, (2) the volume. 118. One side of the base of a regular tetrahedron is 8 feet. Find the lateral area. 119. The frustum of a regular quadrangular pyramid has bases with edges 12 and 18 respectively. Its altitude is 4. Find the lateral area. 120. The frustum of a regular triangular pyramid has bases with sides 6 and 12 inches respectively, and an altitude of 4 inches. Find the lateral area. 121. A regular quadrangular pyramid has an altitude of 12 feet, and a base 24 feet on a side. Find the angle between a face and the base. 122. A regular quadrangular pyramid has a base 6 inches on a side, and a lateral face makes an angle of 60 with the base. Find (1) the lateral area, (2) the volume. 123. It is desired to make a regular triangular pyramid equal in volume and altitude to a given regular hexagonal pyramid whose base is 6 feet on a side. How long is a side of the base of the new pyramid ? 124. The bases of a frustum of a regular pyramid are equi- lateral triangles whose sides are 4 and 6 inches respectively. The slant height of the frustum is 12 inches. Find (1) the lateral area and (2) the total area of the frustum. 125. Prove that the altitude of a regular tetrahedron is 1 V6 times the edge. HINT. The altitude meets the base at the intersection of its medians. 126. Prove that for a regular tetrahedron whose edge is a the a*V2 volume is equal to 4.Q4 . :::: SOLID GEOMETRY 127.. t .Lf t the. base of/a pyramid is a parallelogram, and the section byii-I^arie not?' parallel to the base has one side parallel to one side of the base, prove that the section is a trapezoid. 128. If each pair of opposite edges of a given tetrahedron are equal, and the surface of the tetrahedron is cut along three concurrent edges and opened put flat, prove that the figure formed is a triangle. 129. How many right angles in the sum of the angles of all the faces of a pyramid whose base has n sides ? 130. In a tetrahedron 0-ABC the points P, R, S, T, which bisect the edges OA, OB, BC, AC, are coplanar. 131. In a tetrahedron, if a plane is passed parallel to two oppo- site edges, the section is a parallelogram. 132. In the tetrahedron 0-ABC let D and F be the points where the medians of the faces OBC and ABC respectively meet. Prove (1) that the lines AD and OF meet, (2) that the triangle R OA is similar to RFD. 133. Prove that the three lines drawn from the vertices of a tetrahedron to the intersections of the medians of the oppo- site faces meet in a point which divides each of the lines in the ratio 3 : 1. These lines are called the medians of the tetrahedron. NOTE. The point located in Exercise 133 is called the center of gravity of the tetrahedron. It is the point about which the solid would exactly balance if it could be supported at that point. It is interesting to note that the center of gravity of a triangle divides the medians in the ratio 2 :1, while that of the tetrahedron divides the medians in the ratio 3:1. BOOK VII 405 CONES 535. Conical surface. A conical surface is generated by a moving line which always passes through a given fixed point and continually intersects a given fixed curve. The fixed point is called the vertex, the fixed curve the directrix, and the moving line the generator, of the conical surface. The surface consists of two parts, or nappes, connected only at the vertex. 536. Cone. The solid bounded by a conical surface and a plane whose section with the surface is a closed curve is called a cone. In this text only convex cones, that is, cones whose directrices are convex curves, are treated. The terms base, altitude, and frustum are defined similarly to the corresponding terms applied to the pyramid. 537. Element. The generator of a conical surface in any of its positions is called an element of the conical surface. There is one and only one element containing the vertex and a given point of the directrix. 538. Element of a cone. The segment of the element of a conical surface included between the vertex of the cone and the base is called an element of the cone. QUERY 1. What kind of surface would be generated if the vertex and the directrix of a conical surface were in the same plane ? QUERY 2. What is the locus of lines through a given point which make a given angle with a given plane ? QUERY 3. What is the locus of lines making a given angle with a given plane at a given point? What if the given angle is 90? QUERY 4. Define base, altitude, and frustum as applied to the cone. 406 SOLID GEOMETRY Theorem 20 539. If one of two parallel planes intersects a conical surface in a circle, the other does also, and the vertex of the conical surface is in a straight line with the centers of the two circles. Given a conical surface with vertex cut by the two paral- lel planes M and N. Let the intersection with N be a circle with center X. Let OX intersect N at the point P. To prove that the section with M is a circle whose center is P. Proof. Let OF and OH be the elements through any two points R and S respectively on the section. Pass planes determined by XO and FO, and by XO and //(>, intersecting N in XF and XH, and M in PR and PS, respectively. XF is II to PR ; XH is II to PS. PS^_OP_. PR _ OP XH~ OX' XF~ OX' PS PR Why? 271, 269 Therefore But Therefore XH = XF. PS = PR. But R and S were any points on the section. Therefore the section is circular, with P as its center. Why? Why? Why? Why? BOOK VII 40T 540. Corollary. If either base of a frustum of a cone is cir- cular, both bases are circular. NOTE. It appears from Theorem 20 that if a plane intersects a conical surface in a circle, all the planes which are parallel to it also cut the surface in circles. All other planes cut the surface in curves which are called conic sections. The names of these conic sections are the ellipse, the hyperbola, and the parabola. Of these the ellipse is closed, while the others are open curves. If the cutting plane is not quite parallel to the circular base of the cone, the section is a closed curve a little longer in one direction than in another. If the plane is passed near the vertex, and barely cutting all the elements, a very narrow closed curve is obtained. Any one of the closed sections of a conical surface whose directrix is a circle is called an ellipse, except, of course, those that are circular. If the cutting plane is parallel to one of the elements, it cuts all the elements except this one, and hence is an open curve, called a parabola. If the plane cuts both nappes of the conical surface, making a curve with two separate branches, it is called a hyperbola. The Greeks studied the properties of all the conic sections by methods similar to those of this text, and derived many important properties. These results were regarded as mathematical curiosities without any application to nature until Kepler (1571-1630) discovered that the planets move around the sun in ellipses, and that the properties 408 SOLID GEOMETRY of the curves discovered so long before were useful in explaining their motions. Since the time of Kepler thousands of applications of the properties of the conic sections have been discovered in all branches of science, and a thorough knowledge of them is an important part of the equipment of the modern scientist. This is acquired nowadays most effectively through the study of analytic geometry, a branch of mathemetics whose early development was due to the French philosopher Descartes (1596-1650). The fundamental ideas of analytics are now usually taught in connec- tion with algebra and are included in the work on graphs. But the complete elaboration of the subject includes extensive studies of all kinds of curves and surfaces, of which the simplest and the most useful are the conic sections. 541. Circular Cone. A circular cone is one which has a cir- cular section such that the line joining the vertex of the cone to the center of the circle is perpendicular to the plane of the circle. A circular cone does not necessarily have a circular base. In the adjacent figure the plane M makes a circular section of the cone of which T is the center, and OT is perpendicular to M. 542. Right circular cone. A right circular cone is a circular cone with a circular base. 543. Axis. The line joining the vertex of a right circular cone with the center of its base is called the axis of the cone. 544. Corollary. The elements of a right circular cone are equal. HINT. Apply 425. 545. Slant height. The length of an element of a right circular cone is called its slant height. BOOK VII Theorem 21 409 546. The section of a cone made by a plane which passes through the vertex and cuts the base is a triangle. Given any cone O-KR, and a plane which contains the vertex and cuts the base in AB. To prove that the section OAB is a triangle. Proof. AB is a straight line. Why? The straight line joining and A lies in the given plane 381 and also in the conical surface. 535 Therefore the intersection OA is this straight line. 390 Similarly, OB is a straight line. Therefore the section A OB is a triangle. 16 547. Inscribed pyramid. A pyramid whose base is inscribed in the base of a given cone, and whose vertex is the vertex of the cone, is said to be inscribed in the cone. QUERY 1. If the vertex of a cone is remote from the base, a frustum of the cone near the base resembles what solid ? QUERY 2. To what theorem regarding the cylinder does Theorem 21 correspond? QUERY 3. Why will the lateral edges of an inscribed pyramid be elements of the cone ? 410 SOLID GEOMETRY QUERY 4. Will the slant height of an inscribed pyramid ever be equal to an element of the cone ? QUERY 5. Which will be greater, the slant height of a regular tri- angular pyramid or that of a quadrangular pyramid, if both are inscribed in the same cone ? QUERY 6. If a pyramid is inscribed in a right circular cone, what kind of triangles form its lateral faces? QUERY 7. Will the altitudes of all the triangles mentioned in Query 6 necessarily be equal? EXERCISES 134. Construct a regular hexagonal pyramid inscribed in a right circular cone, giving a reason for each step. 135. Given a right circular cone with altitude 8 and radius 6, find the following for the inscribed regular quadrangular pyramid : (1) area of base, (2) slant height, (3) volume, (4) lateral area. 136. Find the parts required in Exercise 135 for a regular hexagonal pyramid inscribed in the same cone. 548. Volume of cone. If the base of a pyramid which is inscribed in a cone has a very large number of sides, so that each side is very short, then the area of the base of the pyra- mid is approximately equal to the area of the base of the cone. By taking the sides sufficiently short, thereby increasing their number, as close an approximation as desired may be obtained. The volume of the inscribed pyramid whose base is almost the same in area as that of the circumscribed cone will differ very little from that of the cone. But the volume of the pyramid is one third the product of its base and altitude, however many lateral faces it may have. Since the volume, and the area of the base, of the cone can be made to differ as little as we may desire from the same features of the pyra- mid, and since their altitudes are identical, we shall make the statements of theorems 22, 23, and 24 without proof. They can be rigorously demonstrated by the theory of limits. BOOK VII 411 Theorem 22 549. The volume of a cone is equal to one third the product of the area of its base and its altitude. 550. Corollary. The volume, V, of a cone with a circular base is V=7rr% where r is the radius of the base and h is the altitude of the cone. Theorem 23 551. The volume, V, of a frustum of a cone is where h is the altitude and L and U the areas of the lower and upper bases respectively of the frustum. 552. Corollary. The volume, V, of a frustum of a circular cone is where h is the altitude and r and r l are the radii of the lower and upper bases respectively of the frmtum. HINT. Apply 551. Theorem 24 553. The volumes of two prisms, cylinders, pyramids, or cones (1) are to each other as the products of their bases and altitudes ; (2) having equalbases are to each other as their altitudes; (3) having equal altitudes are to each other as their bases. HINT. Denote the altitude by h and the base by B for each solid, and use the appropriate formula for the volume. QUERY 1. What is the ratio of the volume of a cylinder to that of a cone with the same base and altitude ? 412 SOLID GEOMETRY QUERY 2. If two cones have equal bases and equal slant heights, are their volumes necessarily equal ? QUERY 3. What is the locus of the vertices of cones which have the same base and equal altitudes? QUERY 4. Does every cone have a slant height? Explain. EXERCISES 137. The altitude of a right circular cone is 16 feet, and its slant height is 29 feet. Find its volume. 138. What is the radius of the base of a circular cone whose volume is 100 cubic inches and whose altitude is 5 inches ? 139. The radius of a right circular cone is 6 inches. Find the area of the section formed by a plane containing the axis of the cone if the element is 12 inches long. 140. A right circular cone with altitude 10 inches and radius 6 inches is cut by a plane through the vertex so as to make a section having an area just half as great as that of the greatest triangular section. How far from the center of the base is the intersection of this plane with the base ? 141. The frustum of a right circular cone has an altitude of 12 inches and radii of 4 and 6 inches respectively. Find its volume. 142. The slant height of a frustum of a right circular cone is 20 feet. The radii are 2 and 8 feet respectively. Find the volume. 554. Lateral area of a cone. The lateral area of a cone is the area of the conical surface of the cone. We cannot conveniently apply the unit of area to such a curved surface as a conical surface so as to estimate the number of times it is contained in the surface. Yet the fact that some number exists which tells how many times the unit area is contained in the conical surface is so clear that no one but a student of advanced mathematics would ever think of requiring a proof of the statement. Such discussions would be out of place here. BOOK VII 413 Since the lateral area of a regular pyramid equals one half the product of its slant height and the perimeter of its base, whether there are few or many lateral faces, and since, when the lateral faces become very numerous and at the same time very narrow, the lateral area, the perimeter of the base, and the slant height, of a regular pyramid inscribed in a right circular cone are approximately equal to the same features of the cone, we may make the following statements, which will be assumed without proof. Theorem 25 555. The lateral area of a right circular cone is equal to one half the product of the perimeter of the base and the slant height. Expressed in symbols, 2 -rrrl where S is the lateral area, r is the radius, and I is the slant height, of the cone. NOTE. The student can further convince himself of the truth of this formula by roll- ing about its vertex and on a plane surface a right circular none which rests on one of its elements. If the cone is rolled contin- uously in the same direction until the ele- ment comes in contact with the surface for the second time, it will appear that the part of the plane with which the cone has come in contact is a sector of a 414 SOLID GEOMETRY circle the arc of which equals 2 irr, the perimeter of the base of the cone, and the radius of which equals /, the slant height of the cone. Now the area of this sector equals one half of the product of the arc which forms its base, 2 TTT, and the radius, /, of the circle ( 369). Here again we have 2 TTT I S = - = irrl. 556. Corollary. The lateral area of a right circular cone is equal to the product of the slant height and the perimeter of the circle halfway between the base and the vertex. 557. Lateral area of frustum of cone. Since the lateral area of the frustum of a regular pyramid equals one half the product of the sum of the perimeters of the bases and the slant height, whether there are few or many equal isosceles trapezoids as lateral faces, and since, when these trapezoids become very numerous and at the same time very narrow, the lateral area, the perimeter of the base, and the slant height of the frustum of a regular pyramid inscribed in the frustum of a right circular cone are approximately equal to the same features of the frustum of the* cone, we may assume the following : Theorem 26 558. The lateral area of a frustum of a right circular cone is equal to one half the product of the sum of the perimeters of the bases times the slant height. Expressed in symbols, where S is the lateral area, I the slant height, r and r l the radii of the bases of the frustum. NOTE. Rolling a frustum of a cone in the manner described in the pre- ceding note shows that the lateral area of the frustum equals an incomplete sector of a circle. BOOK VII 415 Let x denote the portion of the slant height between the upper base of the frustum and the vertex of the completed cone. Then r = ~ Area rolled out = TTT (/ + #) irr^x = irrl + irrx From (1), rx = r^l + r^x. Substituting in (2) we obtain, area of frustum = irrl + irr^l = Trl (r + r^. Hence S = 7rl(r + r t ). (2) 559. Corollary. The lateral area of a frustum of a right circular cone is equal to the product of the slant height by the perimeter of a circle halfway between the bases. QUERY. How do you interpret the formulae for the volume and the lateral area of a frustum of a cone when r = 0; when r = r t ? EXERCISES 143. A right circular cone is 14 inches high and has a radius of 5 inches. Find its lateral area. 144. A frustum of a right circular cone is 10 inches high, and its radii are 4 and 6 respectively. Find the total area. 416 SOLID GEOMETRY 145. A tent of radius 16 feet and 14 feet high is in the form of a cylinder surmounted by a cone. The height of the conical part is 8 feet. Find the number of square yards in the surface of the tent. 146. Find the lateral area and the total area of a right circular cone whose volume is 24 TT cubic feet and whose radius is 3 feet. 147. Prove that the areas of circular sections of a cone made by parallel planes are proportional to the squares of the distances of their planes from the vertex. 148. The radii of circular sections of a cone made by parallel planes are proportional to their distances from the vertex. Theorem 27 560. A right circular cone is generated by the revolution of a right triangle about one of its sides as an axis. Given any right triangle XOA revolving about the side OX as an axis and generating the figure OAK. To prove that OAK is a right circular cone. Proof. XA generates a plane _L to XO at X. A generates a circle with center at A'. OA generates a conical surface. Therefore XOA generates a right circular cone. Why? Why? 535 542 561. Cone of revolution. In accordance with Theorem 27, a right circular cone is frequently called a cone of revolution. BOOK VII 417 562. Similar cones. Two cones of revolution which are generated by similar right trian- gles with corresponding sides as axes are similar cones of revolution. QUERY 1. Would two cones of revo- lution ever be similar if they were formed by the revolution of similar triangles about sides that were not corresponding ? QUERY 2. Would a cone of revolution be formed by the revolution of a triangle about its altitude as an axis? Theorem 28 563. If two cones of revolution are similar, (1) their volumes are proportional to the cubes of their altitudes, of their radii, or of their slant heights ; (2) the lateral areas or the total areas are proportional to the squares of their altitudes, of their radii, or of their slant heights. Let F, S, T, h, r, and I stand for volume, lateral area, total area, altitude, radius, and slant height, respectively. Then * ___________ * *' The proof, which is left to the stu- dent, is similar to that of Theorem 10. 564. Tangent plane to a cone. A tan- gent plane to a cone is one which meets the cone only along an element. A tangent plane to a cone is deter- mined by a tangent to the base of the cone and the element drawn to the point of contact. 418 SOLID GEOMETRY 565. Circumscribed pyramid. A pyramid is said to be cir- cumscribed about a cone when its base is circumscribed about that of the cone and its lateral faces are tangent to the cone. QUERY 1. What point do all the tangent planes to a cone have in common? QUERY 2. How many planes tangent to a given circular cone can be passed through a given exterior point? QUERY 3. Can a right triangle be found which will generate any given right circular cone? QUERY 4. What does a right triangle generate when revolved about its hypotenuse? QUERY 5. Describe the solid generated by the revolution of an obtuse triangle about one side of the obtuse angle. QUERY 6. Describe the solid generated by an isosceles trapezoid when revolved about (1) the line joining the mid-points of the parallel sides, (2) the longer parallel side, (3) the shorter parallel side. QUERY 7. What does a square generate when revolved about a diagonal ? QUERY 8. What does a rectangle generate when revolved about an axis parallel to one of the sides of the rectangle and in its plane but lying entirely outside the rectangle? QUERY 9. Similar polyhedrons have not yet been denned in this text. But from analogy with the ratio of the areas of similar polygons, and with the results of Theorems 10 and 28 in mind, what do you think the ratio of the volumes of similar polyhedrons probably is ? EXERCISES In Exercises 14-157, .which refer to cones of revolution, h represents the altitude, r the radius of the base, I the slant height, S the lateral area, T the total area, and V the volume. 149. Given r = 5J, h = 8f Find V. 150. Given r = 6, I = 5. Find S. 151. Given r = 5, I = 8. Find V. BOOK VII 419 152. Given I = 6, S = 132. Find T. 153. Given T = 55, r = 3. Find S. 154. Given V= 110, r = 4. Find S. 155. Given T = 2 S, r = 5. Find V. 156. Given F = S, I = 4. Find V. 157. Given J = 2 r, /*, = 2 V3. Find r. 158. The sides of a right triangle are 6, 8, and 10 inches. Find (1) the volume, (2) the lateral area, and (3) the total area of the cone generated by revolving the triangle about its shortest side. 159. The triangle of Exercise 158 is revolved about the side 8. Find (1) the volume, (2) the lateral area, and (3) the total area, and compare these values with the results of Exercise 158. 160. Find (1) the volume and (2) the area of the solid formed by the rotation of an equilateral triangle about one of its altitudes. 161. The parallel sides of an isosceles trapezoid are 12 and 18 inches respectively. The altitude is 4 inches. Find (1) the volume, (2) the area of the solid obtained by rotating it about the longer of the parallel sides. 162. What is the area of the largest section which can be made by a plane through the vertex of a right circular cone whose alti- tude is 10 inches and whose radius is 6 inches. 163. Prove that a cone circumscribed about a regular pyramid is a cone of revolution. POLYHEDRONS 566. Polyhedral angle. The figure formed at the vertex of a pyramid by one nappe of a pyramidal surface (see 513) is called a polyhedral angle. The principal distinction between the polyhedral angle and the pyramidal sur- face lies in the fact that in the former we fix the attention on the portion of the figure near the vertex. 420 SOLID GEOMETRY The terms face, edge, vertex^ and dihedral angle are applied to the parts of the polyhedral angle in the same sense as they are applied to the parts of a pyram- idal surface. 567. The angle between two edges of a polyhedral angle is called a face angle. 568. Trihedral angle. A polyhedral angle which has only three faces is called a trihedral angle. Theorem 29 569. If two trihedral angles have their face angles equal and arranged in the same order, ike corresponding dihedral angles are equal. Given any trihedral angles and O f , in which the face angle XOY equals X'0'Y>, YOZ equals Y'O'Z', ZOX equals Z'O'X'. To prove the dihedrals OX=0'X', OY=0'Y', OZ=0'Z'. Proof. Lay off OA, OB, OC, equal respectively to O'A ', O'B', O'C'. Pass planes through ABC and A'B'C'. BOOK VII 421 Lay off A K equal to A 'K', and pass KLM and K'L'M' J_ respec- tively to OA and O'A'. AK is _L to KL and KM, while A 'K' istoK'L' and K'M'. 414 Hence the angles LKM and L'K'M 1 are plane A of dihedrals OA and O'A 1 respectively. 432 We shall prove the plane angles equal by proving A LKM congruent to A L'K'M'. A A OB is congruent to AA'O'B'. 25 Hence Z KA L = ^K'A'L '. Why ? Therefore right A. 4 KL is congruent to right A AWL'. Why? Similarly, A A KM is congruent to AA'K'M'. Hence KL = K'L' and KM = K 'M'. 27 It remains to show that LM= L'M\ Now AB = A'B'. Why? Similarly, BC = B'C' and CV1 = CU'. Hence A ABC is congruent to AA'B'C' Why? and ZC4=ZC"4''. Why? Furthermore, A L = A 'L' and A M = A 'M'. Why ? Therefore AMAL is congruent to AMM'Z' 27 and LM=L'M'. Why? Hence ALKM is congruent to AL'K'M' Why? and ^ LKM = AL'K'M!. Why? Therefore dihedral ,40= dihedral ^ '0'. Why ? After similar constructions on the other edges of the trihedral it can be proved by the method just employed that dihedrals OB and OC = dihedrals O'B' and 0'C" respectively. Hence the three dihedrals of one trihedral angle are equal respectively to the three corresponding dihedrals of the other. 422 SOLID GEOMETRY EXERCISES 164. If the face angles of two trihedrals are equal each to each and are arranged in the same order, the trihedrals are congruent. 165. If two trihedral angles have a face angle of one equal to a face angle of the other, and the corresponding adjacent dihedrals equal, prove by superposition that the trihedrals are congruent. 166. If the edges of two trihedral angles are parallel each to each and extend in the same directions from the vertices, the dihedrals are congruent. QUERY 1. Can each of three planes meet the other two without forming a trihedral angle ? QUERY 2. How small may the sum of the face angles of a trihedral angle be? How large? QUERY 3. How does a trihedral angle look if the sum of its dihedral angles is nearly equal to six right angles ? How if the sum of its dihe- dral angles is nearly as small as two right angles ? QUERY 4. Referring to the propositions proved in Plane Geometry, it appears that Theorem 29 and Exercise 165 above correspond to 33 and 82, respectively, of the Plane Geometry if we relate the face angles and dihedrals of the trihedral to the sides and angles of the triangle respectively. Read the list of theorems proved in Plane Geometry regarding the triangle, and state those to which you think a corresponding theorem for trihedral angles seems likely to be true. Consider, for example, 25, 29, and 66. QUERY 5. If isosceles trihedral angles are defined as those, two of whose face angles are equal, study the correspondence between the theorems on isosceles triangles and isosceles trihedrals. NOTE. In plane geometry, if two triangles have three angles of one equal respectively to three angles of the other, the triangles are not congruent but merely similar. The corresponding problem relating to trihedral angles would be that of finding the relation between two trihedrals whose dihedrals are respectively equal. Asa matter of fact such trihedrals are congruent, not similar, as one would expect from plane geometry. This fact cannot be proved simply until a little later, in connection with our study of the sphere. BOOK VII 423 Theorem 30 570. The sum of two face angles of a trihedral angle is greater than the third. O Given any trihedral angle 0-XYZ. To prove that Z.ZOY+ ^XOZ Proof. There is no necessity for proof if either XOZ or ZOY is greater than or equal to XOY. In the face XOY draw OK, making the /.XOK = /.XOZ. On OZ lay off OC = OK. Through K and C pass a plane which does not contain 0, as ABC. In A 0,4 # and OAC, AO = AO, OK= OC, ^AOK = Z.AOC. Why? Hence A A OK is congruent to AAOC. Why? Therefore AK = AC. in AABC, AC + BOAK + KB. 146 Subtracting the equals AC and AK, we have BC > KB. 139 In the A 0KB and OBC, OK= OC, OB = OB, CB > KB. Therefore Z.BOC > Z.KOE. 151 Hence, adding to this inequality the equal angles A OC and A OK, we have ^LAOC + Z.BOC >Z.AOK + ^KOB. 139 571. Corollary. Any face angle of a polyhedral angle is less than the sum of the remaining face angles, 424 SOLID GEOMETRY Theorem 31 572. The sum of the face angles of any convex poly- hedral angle is less than four right angles. X \Z Given any polyhedral angle P with face angles XPY, YPZ, etc. To prove that XPY + YPZ+ . . . < 4 right angles. Proof. Pass a plane forming the section A ROD. Let the sum of the face angles of P be denoted by F. Let the sum of the remaining angles in the lateral faces, as A HP, PRO, etc., be denoted by L. Let the sum of the angles of the polygon ARCD be denoted by 5. Then if the polyhedral angle has n faces, F+L = 2ni-t.A. (1) 66 Also S = (2 n - 4) rt. A. (2) 125 Now ARP + PRC.>ARC. 570 Since a similar inequality holds at each vertex of the polygon A-D, we have L>g ^ Hence L > (2 n - 4) rt. A. (3) Subtracting (3) from (1), F < 4 right angles. 142 573. Regular polyhedron. A convex polyhedron is said to be regular if its faces are all congruent regular polygons and its polyhedral angles are all congruent. BOOK VII Theorem 32 425 574. No more than five regular polyhedrons are possible. Proof. Each of the polyhedral angles of a regular convex polyhedron is included by three or more faces which are regular polygons. Consider in turn the possibilities when the faces are (1) equilateral triangles, (2) squares, (3) pentagons, (4) hexagons. (1) Each angle of an equilateral triangle contains 60. Hence there may be convex polyhedral angles with three, four, or five such faces, but not with six or more. 572 Hence there can be no more than three regular polyhedrons with triangular faces. (2) Each angle of a square contains 90. There may be convex polyhedral angles with three square faces, but not with four or more. Hence there can be no more than one regular polyhedron with square faces. (3) Each angle of a regular pentagon con- tains 108. Hence there may be convex poly- hedral angles with three pentagons for faces but not with four or more. Why ? Hence there can be no more than one regular polyhedron with pentagonal faces. (4) Each angle of a hexagon contains 120. Hence there can be no regular polyhedrons with hexagonal sides. In the case of polygons of more than six sides, no polyhedral angle can be formed. Why ? Therefore no more than five regular polyhedrons are possible. 426 SOLID GEOMETRY 575. The five regular bodies. It does not follow from the foregoing theorem that there are necessarily five regular poly- hedrons, but merely that this is the maximum possible num- ber. In order to show the existence of polyhedrons of. the various kinds it is necessary to prove the possibility of con- structing each type. As a matter of fact, each of the five cases referred to in Theorem 32 does actually correspond to a regular polyhedron, as shown in the following diagrams: TETRAHEDKON HEXAHEDRON (Cube) OCTAHEDRON DODECAHEDRON ICOSAHEDRON NOTE. The names of the regular polyhedrons are derived from the Greek words for four, six, eight, twelve, and twenty respectively, which reference to the diagrams will show to be the numbers of the faces of the various polyhedrons. Construction 1 576. Construct a regular tetrahedron. Construction. Construct an equilateral A ABC, and find its center, P. At P construct a line _L to the plane of ABC. 420 BOOK VII 427 With a vertex of ABC as center, and a side of ABC as radius, determine 0, in this J_, so that AO = AB. Draw OA, OB, OC. Then OA B C is a regular tetrahedron. Proof. OA = OB = OC. But AB = AO. Hence A OB is an equilateral A. Similarly, the other faces are equilateral A. Therefore all the trihedral A are congruent. Hence the tetrahedron is regular. Construction 2 577. Construct a regular hexahedron, or cube. M L 425 Const. 569 573 .XT Construction. Construct a square, A BCD, and at each of the four vertices erect lines _L to the plane of the square. 420 Lay off along these Js AP = BK = CL = DM, each = AB. Pass planes PKB, KLC, LMD, and MPA. 385 Through P pass a plane II to AC. 408 The resulting figure is a cube, Proof is left to the student, 428 SOLID GEOMETRY Construction 3 578. Construct a regular octahedron. The Construction and Proof are left to the student. HINT. Construct two square regular pyramids having the same base, with vertices on opposite sides of the base and with lateral edges equal to the sides of the base. NOTE. The constructions of the dodecahedron and icosahedron are much more complicated and will not be given here. V_A_/ kx> Models of all the regular solids can easily be made by cutting card- boards in the forms of the above diagrams, then c,utting half through the cardboard along the dotted lines, folding along the half-cuts, and closing the model by pasting strips of paper along the open edges. BOOK VII 429 EXERCISES 167. Prove that any two pairs of opposite vertices of a regular octahedron are the vertices of a square. 168. The side of a cube is 6 inches. Find the side of a cube of twice the volume. NOTE. To Hippocrates of Chios (about 430 B.C.) is due the proof that the solution of the problem of duplicating the cube can be reduced to the finding of two mean proportionals between two given lines, of which one is the side of the given cube and the other is twice that side. If x and y are two mean proportionals between a and 2 a, wehave a:x = x:y = y:2a. Then x 2 = ay and y 2 = 2 ax. Squaring the first, a; 4 = a' 2 ?/ 2 . Substituting value of ?/ 2 from the second, x* = a 2 2 ax = 2 a 3 x. Hence x 3 = 2 a 3 . That is, the volume of the cube of edge x will be double that of a cube with edge a. The geometric procedure for the duplication of the cube has been carried out in a variety of ways, usually by finding the intersection of various curves such as parabolas or hyperbolas. Plato (400 B.C.) is said to have solved the problem by means of a mechanical device, but to have rejected the method as not being geometric. 169. One edge of a regular octahedron is 8 inches. Find the volume. 170. A cube and a regular tetrahedron have the same edge. What is the ratio of their volumes ? HINT. See Exercise 126. 171. The base of a regular hexagonal pyramid is 66 square inches, and its altitude is 10 inches. A plane is passed parallel to the base 5 inches from it. Find the ratio of the volume of the original pyramid to that of the one cut off by this plane. 430 SOLID GEOMETRY Theorem 33 579. If two tetrahedrons have a trihedral angle of one congruent to a trihedral angle of the other, their volumes are in the same ratio as the products of the edges of these trihedral angles. Given 0-ABC and P-KLM, with trihedrals and P congruent. , , volume 0-ABC OA OB OC To prove that = volume P-KLM PK PL PM Proof. Apply the trihedral to P so that the tetrahedron O-ABC takes the position P-RST. Let TX and MY be the altitudes of the two tetrahedrons. volume P-RST PRS TX PRS TX (1) 553 Now volume P-KLM PKL -MY PKL MY TX is II to MY, 427 and the plane determined by TX and MY contains PM and hence the APTX and PMY. Therefore APTX is similar to A PMY. 271 TX PT _, Therefore In the APRS and PKL, we have PRS _ PR . PS _ PR PKL ~ PK PL "~ 'PK PS PL (2) Why? (3) 325 BOOK VII 431 or Hence, substituting (3) and (2) in (1), volume P-RST _ PR PS PT volume P-KLM PK PL PM* volume 0-ABC _ OA OB . PC volume P-KLM ~~ 1'K - PL - PM 580. Symmetric trihedrals. If two trihedral angles have their parts equal each to each, but arranged in the opposite order, they are called symmetric. For example, the trihedral angles in the two nappes of a triangular pyram- idal surface are sym- metric. For if one thinks of oneself as stationed at the vertex and looking first in the direction of the faces of one angle and then in that of the faces of the other, the equal dihedral angles OA, OB, 0(7 in the diagram follow each other in clockwise order on the lower nappe and in counterclockwise order on the upper nappe. QUERY 1. Can two triangles whose parts are equal but arranged in opposite order be superposed so as to coincide throughout ? QUERY 2. Can two symmetric trihedral angles be superposed so as to coincide throughout? QUERY 3. Are the dihedral angles of a regular polyhedron all equal ? EXERCISE 172. If two trihedral angles have their edges paral- lel, and all the edges of one trihedral extending in directions from the vertex opposite to the corresponding edges of the other tri- hedral, the trihedrals are symmetric. 432 SOLID GEOMETRY 581. Similar polyhedrons. Two polyhedrons are similar if their faces are similar, each to each, and similarly placed, and if their corresponding polyhedral angles are congruent. It should be observed that similar solids have the same shape but different sizes. Theorem 34 582. The ratio of any two corresponding edges of two similar polyhedrons is equal to the ratio of any other corresponding pair. Proof is left to the student. QUERY 1. Are all regular polyhedrons with the same number of faces similar? QUERY 2. Are all rectangular solids similar? EXERCISES 173. The areas of two corresponding faces of two similar poly- hedrons are in the same ratio as the squares of any two corre- sponding edges. 174. Two tetrahedrons are similar if three faces of one are similar and similarly placed to three faces of the other. 175. Two tetrahedrons are similar if a dihedral in one is equal to a dihedral in the other, and if the faces including the dihedral are similar and similarly placed. 176. If a plane is passed parallel to the base of a pyramid, the pyramid cut off is similar to the original one. BOOK VII 433 Theorem 35 583. The volumes of two similar tetrahedrons are in the same ratio as the cubes of two corresponding edges. AkV>C B Given two similar tetrahedrons 0-ABC and 0'-A' r C f , with volumes Fand V respectively. V O4 3 , 4 To prove that V Q'A' S Proof. Trihedral is congruent to trihedral O'. 581 V PA-OB- PC S ._ Q T'^O'A'.O'B'.O'C' OA OB PC ~ O'A 1 ' O'B 1 ' O'C"' The A OAB and O'A'B', OA C and O'A 'C', etc. are similar. 581 OA OB PC Substituting (2) in (1), we obtain 584. Assumption. Any two similar polyhedrons can be divided into the same number of tetrahedrons, similar each to each and similarly placed. 434 SOLID GEOMETRY The foregoing statement, a proof of which is rather involved, appears evident if we consider two corresponding points (one in the interior of each polyhedron) and with these points as ver-tices construct pyramids having as their bases the various faces of the polyhedrons. The pyramids with similar faces for bases are sim- ilar, and hence the tetrahedrons into which they may be divided are similar and similarly placed. Theorem 36 585. The volumes of two similar polyhedrons are in the same ratio as the cubes of any two corresponding edges. Given two similar polyhedrons P and P r , with volumes V and V 1 and corresponding edges E and E' respectively. V E* To prove that - -= = . Proof. Divide P and P' into tetrahedrons which are similar each to each and similarly placed. 584 Let the volumes of the similar tetrahedrons be denoted by T ly T[; T i, T i\ T z, TZ, etc., and let corresponding edges of th'ese tetra- hedrons be denoted by E lt E[-, E^ E^ E B , E' 3J etc. Then = ; = |4' etc. 583 2 & t = 582 Therefore = = . . . = ~ Why? Hence T 2+ Ti V 1 or BOOK VII 485 586. Similar figures. We can now summarize the whole doctrine of similar figures by the following statements: Similar plane figures or similar surfaces are in the same ratio as the squares of any two corresponding lines. Similar solids are in the same ratio as the cubes of any two corresponding lines. In similar figures of any kind, pairs of corresponding lines are in the same ratio. EXERCISES 177. If one edge of a polyhedron is 6 inches, what is the length of the corresponding edge of a polyhedron of twice the volume ? 178. If the base of one pyramid has nine times the area of the base of a similar pyramid, what is the ratio of their volumes ? 179. If a cube has an edge 8 inches long, what is the diagonal of a cube of four times the volume ? 180. A regular tetrahedron has a volume of 27 cubic inches. What is the volume of a similar tetrahedron whose edges are each half as long ? 181. If the strength of two steel wires varies directly as their cross section, what is the ratio of two weights that can just be supported by wires, one of which is three times as great in diameter as the other ? 587. Prismatoids. A prismatoid is a polyhedron all of whose vertices lie in two parallel planes. The lateral faces of prismatoids are either triangles or quadrilaterals. Pyramids, prisms, and frustums of pyramids are special cases of prismatoids. The terms base, altitude, etc. are defined for the prismatoid similarly to the corresponding terms as applied to the prism. 436 SOLID GEOMETRY Theorem 37 588. The volume of a prismatoid equals the product of one sixth the altitude by the sum of the upper base, the lower base, and four times the mid-section. A C JC Given the prismatoid ABCDEFG in which the upper base, the lower base, the mid-section, and the altitude are denoted by bj B, M, and h respectively. h To prove that V= - (6 + ^ + 4 Jf). Join any point in the mid-section, as } with the various vertices of the mid-section and of both bases by the lines OB, OR, OF, etc. The planes passed through these lines divide the figure into pyramids with a common vertex at 0, like 0-ABC, 0-DEFG, which have as their bases b and .B respectively, and 0-BCF, whose base is one of the triangular lateral faces of the prismatoid. The volumes of these pyramids are found separately. (a) Volume. 0-ABC = J - \ - 1 = ~- O 2i D (/>) Volume 0-DEFG = - | . B = ^ - o 2i D (c) Volume OBCF is divided into two parts by the mid-section. Since R S joins the mid-points of the sides of BCF, AFRS=AFBC, and volume 0-FRS = J 0-FBC. 553 BOOK VII 437 } ORS = p (ORS). But 0-FRS = F-ORS = -(-h o Therefore volume 0-FBC = 4 (0-FRS) = ^ ORsJ^h . ORS. D O In a similar manner the volume of the other portions of the figure not considered under (a) and (b) can be shown to equal - h times the area of the mid-section included in it. Hence by adding together the portions of the prismatoid not considered in (a) and (b) we obtain as their volume h M. Adding (a), (b), and this last result, NOTE. It can be proved by the integral calculus that the foregoing formula is valid for a much more extensive class of solids than prism a- toids. It is usually called the Prismoidal Formula. Among these solids are included cones, cylinders, and spheres. In fact, the class of solids whose volume can be found by means of (1) is so extensive that the for- mula finds wide practical application. The formula was first stated by Thomas Simpson (1710-1701), who also gave a list of solids for which it gives accurate results. It may, for example, be assumed that a pile of earth 'or sand or the excavation for a foundation or a railway cut is so nearly in the form of such a solid that its volume may be computed by the use of (1) without appreci- able error. For example, suppose an excavation 100 feet square at the top and 10 feet deep is to be made in sand so that the sides of the excavation form an angle, of 45 with the horizontal. The bottom of the excavation is then a square 80 feet on a side, and the area of the section halfway down is a square 90 feet on a side. Hence V= ^ (80 2 + 100 2 + 4 x 90 2 ) D = IPX 48,800 = 81; , 33c ,,i, icfeet . 438 SOLID GEOMETRY EXERCISES 182. Show that the expression for the volume of a pyramid is a special case of (1). In the case of the pyramid, b = o, B B, M= i B. Hence (1) becomes F= | (0 + B + B) = ^j-- 183. Show that the expression for the volume of the prism is given by the Prismoidal Formula. 184. Show that the volume of a cone, the volume of the frustum of a cone, and the volume of the frustum of a pyramid are each given by the Prismoidal Formula. 185. An irregular pile of earth is 15 feet high and covers 500 square feet. Its mid-section and level top are estimated to con- tain 400 and 200 square feet respectively. Find the cost of re- moving it at 50 cents per load, if the truck measures 3x4x9 feet. 186. A circular pile of sand which stands at rest at an angle of 45 is 6 feet high. How many cubic feet does it contain ? 187. An excavation must be carried 20' deep in earth which at rest stands 60 to the horizontal. The bottom must be a square 40' x 40'. (1) How large are the top and the mid-section of the excavation ? (2) How many cubic feet of earth must be removed ? REVIEW EXERCISES 188. A cubic foot of water weighs about 62 pounds. What must be the side of a square refrigerator pan 6 inches 'high in order to hold the water from 50 pounds of ice ? How many square inches of sheet tin are required to make the pan, allowing half an inch along each seam for folding ? 189. A trough is formed by nailing together, edge to edge, two boards 12 feet in length, so that the right section is a right angle. If 15 gallons of water is poured into the trough and it is held level so that a right section of the water is an isosceles right triangle, how deep is the water ? (231 cubic inches = 1 gallon.) BOOK VII 439 190. Two water tanks in the form of rectangular solids whose tops are on the same level are connected by a pipe through their bottoms. The base of one is 6 inches higher than that of the other. Their dimensions are 4 x 5 x 2 feet and 4x7x3 feet respectively. How deep is the water in the larger tank when the water they contain equals half their combined capacity ? 191. If a right triangle is revolved first about the longer leg and then about the shorter, determine in which case (1) the vol- ume, (2) the lateral area, (3) the total area, is the greater. HINT. Let the shorter side be a ; the longer side, a + Ji. Then F! = J Tra 2 (a + h) = $ TT (a 3 + a 2 /*). V 2 = i TT (a + A) 2 a = TT (a 3 + 2 a"h + air}. 192. Find (1) the volume and (2) the total area of the solid formed by the rotation of an equilateral triangle about one side. 193. The tensile strength of wire is proportional to the area of its cross section. If a certain wire is just strong enough to sup- port a cube of iron 8 inches on a side, what must be the diameter of the wire which will just support a cube of iron 16 inches on a side ? 194. An irregular-shaped body is placed in a cylindrical vessel of water whose radius is r inches. The water rises a inches. What is the volume of the body ? 195. The volumes of two cubes are in the ratio of 4 : 5. Find the ratio of (1) their surfaces, (2) their edges. 196. If the total area of one solid is twice that of a similar solid, what is the ratio of (1) their edges, (2) their volumes ? 197. The dimensions of a box are 2x3x4 feet. Find the dimensions of a box of the same shape in order that it may hold three times as much. 198. The bases of two similar pyramids are in the ratio 2 : 3. What is the ratio of (1) their edges, (2) their volume ? 199. How long a wire T ^ of an inch in diameter can be drawn from a block of copper 2x4x6 inches ? 440 SOLID GEOMETRY 200. Find the total area of a rectangular solid whose base is 5x9 feet and whose volume is 900 cubic feet. . 201. Eind the edge of a cube whose total surface is numerically equal to its volume. 202. A sector having an angle of 90 is cut from a circle 8 inches in radius, and the edges of the cut are joined so as to form the lateral area of a cone. Eind the volume of this cone. 203. A cone of revolution has a slant height of 10 inches, and its radius is 4 inches. How large an angle must be cut from a circle to form its lateral surface ? 204. If the slant height of a cone of revolution is twice the radius, find the angle that must be cut from a circle in order to form its surface. 205. What per cent of a square stick is wasted if the largest possible cylindrical stick is turned out of it on a lathe ? ' 206. What per cent of the volume of a circular cylindrical log of uniform diameter is the volume of the greatest square timber which can be cut from it ? 207. The slant height of a cone of revolution makes an angle of 45 with the base. The altitude of the cone is 25 inches. Find the volume. 208. A pipe f -inch inside measure conducts water from a spring to a house 300 feet distant. It is desired to empty the pipe after the water has been turned off at the spring. Will a 10-quart pail hold the water ? 209. A tank in the form of a rectangular solid 2x3x4 feet can be filled through a pipe f-inch in diameter in 30 minutes. How many feet of water flow through the pipe per second ? 210. It is desired to cut off a piece of lead pipe 2 inches in outside diameter and i-inch thick, so that it will melt up into a cube of edge 6 inches. How long a piece will it take ? 211. The volume of a rectangular solid is 1296 cubic inches, and its dimensions are in the ratio 1:2:3. Find the dimensions. BOOK VII 441 212. If a cone and a cylinder of revolution have the same base and equal altitudes, what is the ratio of their lateral areas ? 213. If a child 2^ feet in height weighs 30 pounds, what would be the weight of a man 6 feet tall of the same proportions ? 214. Which is the more heavily built, a man 5^ feet tall who weighs 160 pounds, or one 6 feet tall who weighs 200 pounds ? 215. A wooden cone of revolution has a diameter of 6 inches and an altitude of 10 inches. An auger hole 1 inch in diameter is bored through it, the axis of the cone and the auger being co- incident. (1) What is the volume of the small cone which is cut out from the top of the larger cone ? (2) What is the volume of the cylindrical hole ? 216. The radius of the lower base of a frustum of a cone of revolution is twice that of the upper base. The slant height is inclined at an angle of 45 to the base. If the altitude of the frus- tum is 6 feet, find (1) the lateral area, (2) the volume, of the frustum. 217. A circular cone is 4 feet high. The shortest and the longest elements are 6 and 10 feet respectively. Find the volume. 218. How many board feet of lumber is a stick 4" x 4" at one end, 2" x 12" at the other end, and 16' long ? (One board foot is 1' x 1' X 1"; that is, it contains 144 cubic inches.) 219. If a lead pipe \ inch thick has an inner diameter of 1^ inches, find the number of cubic inches of lead in a piece of pipe 12 feet long. 220. How many cubic yards of material is needed for the foundation of a barn 40 x 80 feet if the foundation is 2 feet wide and 12 feet high ? 221. A pail 12" high is in the form of a frustum of a cone. If the diameters of the bases are 10" and 12" respectively, find its capacity in quarts. 222. An 80-foot flagpole has upper and lower diameters of 4 and 16 inches respectively. Find the cost of painting it at 5 cents per square foot. 442 SOLID GEOMETRY 223. A reservoir is in the form of an inverted square pyramid with bases 100 and 90 feet on a side respectively. How long will it require an inlet pipe to fill it if it pours in 400 gallons per minute ? 224. Compute the cost of the lumber necessary to resurface a footbridge 16' wide and 150' long with 2" plank if lumber is $40 per 1000 board feet. 225. A druggist sells a certain kind of powder in a rectangular box 4 x 2 x 1 inches for 25 cents, and in a cylindrical can 2i inches high and 2i inches in diameter for 20 cents. Which is it more economical to buy ? 226. A concrete dam is 4' wide at the top and 12' wide at the lowest point. It is 72' long ; it is 8' high at one end and 12' high at the other ; at the lowest point, 24' from the latter end, it is 21' high. What would it cost to build it at |4.60 per cubic yard ? 227. A railway embankment across a valley has the following dimensions : width at top, 20 feet ; width at base, 45 feet ; height, 11 feet ; length at top, 1020 yards ; length at base, 960 yards. Find its volume. BOOK VIII THE SPHERE GENERAL PROPERTIES OF THE SPHERE 589. Sphere. A sphere is a closed surface every point of which is equidistant from a fixed point called the center. 590. Radius. A line connecting any point of the sphere with its center is called a radius of the sphere. Since the sphere is closed, every point in space is either inside, outside, or on, the sphere, according as its distance from the center is less than, greater than, or equal to, the radius. Terms such as diameter, chord, and secant are used in the same sense for the sphere as they are for the circle. By volume of the sphere we mean the volume of the solid inclosed by the sphere. 591. Another definition of the sphere. A sphere may also be defined as the sur- face generated by the complete rotation of a semicircle about a diameter. QUERY 1. If a line is known to have one point inside a sphere, in how many points must C it cut the sphere? QUERY 2. The distances from the center of a sphere to lines which (1) cut, (2) do not cut, the sphere have what relation to the length of the radius? 443 444 SOLID GEOMETRY QUERY 3. If two spheres are congruent, is it certain that their cen- ters coincide when the spheres coincide? QUERY 4. If two spheres are congruent, what can you say of their radii ? QUERY 5. If the radii of two spheres are equal, are the spheres congruent ? QUERY 6. What is the locus of points five inches from a given point ? QUERY 7. What is the locus of points two inches from a sphere which is six inches in diameter? EXERCISES 1. Prove that any diameter of a sphere is twice as long as the radius. 2. Prove that a diameter is longer than any other chord of a sphere. Theorem 1 592. If a plane is perpendicular to a radius of a sphere at its outer extremity, it has no other point in common with the sphere. Given the sphere S with center 0, and the plane M which is perpendicular to the radius OP at P. To prove that M meets S only at P. Proof. Let K be any point in M other than P. Then OK > OP. 422 Therefore K is outside the sphere. 590 Therefore no point of M except P is on S. BOOK VIII 445 593. Tangency. If a sphere and a plane, a sphere and a line, or two spheres have one and only one point in common, they are said to be tangent to each other. The common point is called the point of contact in each case. 594. Corollary. If a plane is tangent to a sphere, it is per- pendicular to the radius drawn to the point of contact. HINT. Show that the distance from the center of the sphere to the point of contact is shorter than any other line from the center to the plane. Then apply 422. QUERY 1. How many planes are tangent to a sphere at a given point ? QUERY 2. What is the locus of the centers of spheres of fixed radius which are tangent to a given plane ? QUERY 3. What is the locus of the centers of spheres which are tan- gent to both faces of a given dihedral angle ? QUERY 4. What is the locus of the centers of spheres of radius two inches which are tangent to both faces of a dihedral angle ? QUERY 5. What is the locus of the centers of spheres which are tangent to a plane at a given point? QUERY 6. How many lines are there in each tangent plane to a sphere which have one and only one point in common with the sphere ? QUERY 7. On what kind of surface do all of the tangent lines to a sphere from a fixed exterior point lie ? If the point approaches very close to the sphere, what does the surface approach in form ? As the point recedes farther and farther from the sphere, what does the surface approach in form? EXERCISES 3. Prove that a line which is tangent to a sphere is perpen- dicular to the radius drawn to the point of contact. 4. At a point on a sphere construct a plane tangent to the sphere, giving a reason for each step. 5. If two spheres are tangent to a plane at the same point, prove that their line of centers passes through the point of contact. 446 SOLID GEOMETRY Theorem 2 595. If the distance from the center of a sphere to a plane is less than the radius of the sphere, the plane cuts the sphere in a circle whose center is the foot of the perpen- dicular drawn from the center of the sphere to the plane. Given any sphere S, with center O and radius r, and a plane M whose distance OC from O is less than r. To prove that the section in which M cuts S is a circle with center C. Proof. Connect any two points of the section, as L and A', with C. Draw OL and OK. A OCL is congruent to A OCR. Why ? Therefore CK = CL. Why ? Hence the intersection of S and M is a circle of which C is the center. 154 596. Great circle. A circle on a sphere whose plane passes through the center of the sphere is called a great circle. 597. Corollary I. All great circles of a sphere are equal and have for their common center the center of the sphere. 598. Corollary II. Any two points on a sphere except the extremities of a diameter determine a great circle of the sphere. BOOK VIII 447 599. Corollary III. Any three points on a sphere determine a circle of the sphere. 600. Small circle. A circle on a sphere whose plane does not pass through the center of the sphere is called a small circle. QUERY 1. What can you say of the distance, from the center of a sphere, of a plane (1) which does not cut the sphere, (2) which cuts it, (3) which cuts it in a great circle ? QUERY 2. If a plane through a given tangent line to a sphere swings around this line as an axis, what kind of sections with the sphere are obtained ? QUERY 3. What relation do the circles obtained in Query 2 bear to the given tangent line ? QUERY 4. Is a given tangent line to a sphere tangent to all the great circles and also to all the small circles of the sphere through the point of contact ? QUERY 5. How many equal circles can a set of parallel planes cut from a sphere? QUERY 6. On M T hat kind of surface are all the radii of a sphere which meet a given small circle ? QUERY 7. What is the intersection of a circular conical surface and a sphere whose center is the vertex of the conical surface ? EXERCISES 5. A great circle divides the surface of a sphere into two congruent parts. HINT. Prove by superposition. 6. Any two great circles on the same sphere bisect each other. 7. Prove that the distance from the center of a sphere to any one of its tangent lines is the radius of the sphere. HINT. Pass a plane and apply 192. 8. Prove that the distances from a fixed point outside a sphere to the points of contact of the lines tangent to the sphere drawn through that point are all equal. 448 SOLID GEOMETRY 9. Prove the relation 1 st = s 2 -f- d 2 , where ?*, s, and e denote, respectively, the radius of a sphere, the radius of a small circle, and the distance of the plane of the circle from the center of the sphere. From this relation deduce the following properties of the sphere : (1) The radius of a small circle is less than that of the sphere. (2) Circles on a sphere whose planes are equidistant from the center of the sphere are equal. (3) The converse of (2). (4) The plane of the larger of two unequal circles of a sphere is nearer the center of the sphere than the plane of the smaller. (5) The converse of (4). 601. Hemisphere. One of the equal parts into which a sphere is divided by a great circle is called a hemisphere. Theorem 3 602. If two spheres cut each other, their intersection is a circle whose, plane, is perpendicular to their line of centers. Given two spheres S and S' which cut each other and whose centers are and T respectively. To prove that their intersection is a circle whose plane is per- pendicular to OT. BOOK VIII 449 Proof. Through any point P of the intersection pass a plane M _L to the line of centers OT at K. 417 Then M cuts S in a circle whose center is K and whose radius is KP. 595 M also cuts S 1 in a circle whose center is K and whose radius is KP. These two circles coincide, since they lie in the same plane and have the same center and radius. Therefore the intersection of S and S' is a circle. Since M is _L to T by construction, the plane of this circle is _L to the line of centers of S and S 1 . EXERCISES 10. The radius of a sphere is 14 inches. What is the radius of a circle whose plane is 4 inches from the center of the sphere ? 11. Two spheres whose centers are 24 feet apart have radii 9 and 18 feet respectively. Find the area of the circle of intersection. 12. The diameter of a sphere is 16 inches. 'Find the distance from the center of this sphere to the plane of a circle whose area is half that of a great circle. 603. Inscribed sphere. If a sphere is tangent to each of the faces of a polyhedron, it is said to be inscribed in the polyhedron. 604. Circumscribed sphere. If a sphere passes through each of the vertices of a polyhedron, it is said to be circumscribed about the polyhedron. 450 SOLID GEOMETRY Construction 1 605. Construct a sphere circumscribed about a given tetrahedron. Given any tetrahedron PRST. To construct a sphere which contains the points P, R, S, and T. Construction. At O, the center of the circle circumscribed about the face RST, construct a line OK to RST. 420 At A, the middle point of PR, construct a plane M _L to PR. 416 Then the point of intersection C of OK and M is the center of a sphere which passes through the points P, R, S, and T. Proof. It is first necessary to prove that the line OK and the plane M intersect. If OK were II to M, M would be _L to the face RST. 438 Therefore RP would lie in the plane RST 440 and the figure PRST would not be a tetrahedron. Hence OK and M must meet in some point C. Since C lies in OK, it is equidistant from R, S, and T. 425 Since C lies in M, it is equidistant from P and R. 423 Therefore C is equidistant from the four points P, R, S, and T, and is the center of a sphere on which they lie. BOOK VIII 451 QUERY 1. How many points are required to determine a sphere on which they all lie ? QUERY 2. If the center is known, how many additional points are necessary to determine a sphere ? QUERY 3. How many spheres contain a given circle? QUERY 4. What is the locus of the centers of the spheres which contain a given circle ? QUERY 5. How many spheres can be passed through three given points ? QUERY 6. What is the locus of the centers of the spheres which pass through two given points ? QUERY 7. What is the locus of the centers of the spheres with given radius which pass through two points ? QUERY 8. What is the locus of points at a given distance from a given point and equidistant from two given points ? QUERY 9. Two points, A and B, are a distance c apart. What is the locus of points a distance m from A and n from Bl Discuss for all cases. QUERY 10. Under what circumstances will two circles in different planes determine a sphere ? QUERY 11. What is the locus of points a given distance from a given point and at the same time equidistant from all the points of a given circle ? Discuss special cases. QUERY 12. What is the locus of the centers of the spheres which are tangent to the faces of a given dihedral angle ? EXERCISES 13. Construct a cube circumscribed about a given sphere. 14. Construct a sphere circumscribed about a given cube. 15. Prove that the lines which are perpendicular to the faces of a tetrahedron at the centers of their circumscribed circles meet in a point. 16. Construct a sphere of given radius passing through three given points. 17. Construct a sphere inscribed in a given tetrahedron. 18. The edge of a regular tetrahedron is 8 inches. What is the radius of the circumscribing sphere ? 452 SOLID GEOMETRY MEASUREMENT OF THE SPHERE Theorem 4 606. The area generated by the base of an isosceles tri- angle rotated about an axis which lies in its plane and contains the vertex, but which does not cut the triangle, equals the product of the projection of the base of the tri- angle on the axis and the circumference of a circle whose radius is the altitude of the triangle. Case I. The base of the triangle does not meet and is not parallel to the axis. Given any isosceles triangle, OAB, whose base AB does not meet and is not parallel to the line RS y which lies in the plane of OAB and contains the vertex of OAB. Let MO be the altitude of the triangle, and let XY be the projection of AB on RS. And let MZ be the projecting line of M, the midpoint of AB. To prove that area generated by AB e'quals XY 2 TrMO. Proof. AX is _L to RS, and BY is _L to RS. 445 Therefore XABY is a trapezoid Why ? and generates a frustum of a cone when rotated about R S as an axis. Therefore area generated by AB = AB - 2 TrMZ. (1) 559 Draw AC II to RS. BOOK VIII 453 In AA/0Z and BAG, MZ is _L to RS. MO is _L to AB. OZ is _L to BC. Hence the A of AMOZ = respectively the A of ABA C and &MOZ and BAC are similar. Hence AB/MO = AC/MZ. That is, A B - MZ = A C - MO = XY or AB 2 TrMZ = XY>2 TrMO. Hence area generated by AB = XY 2 TrMO. MO, 445 Why? Why? 76 Why? Why? Why? 124 32 Case II. The base of the triangle terminates in the axis. HINTS. Compare the triangles MOZ and ABY, as in Case I, apply 556, and prove that area generated by AB = A Y R X M Y S Case III. The base of the triangle is parallel to the axis. HINTS. Apply 502, 500 and prove that area generated by AB = XY - 2 TrMO. 454 SOLID GEOMETRY Theorem 5 607. If an arc of a semicircle is divided into a number of equal parts, and chords are drawn joining the points of division in order % and the figure is rotated about the diam- eter of the semicircle as an axis, the area generated by the chords equals the product of the sum of their projections on the axis and the circumference of a circle whose radius is their common distance from the center. Given the arc AF. Let the points B, C, and D divide it into equal parts ; let VW, WX, XY, and YZ be the projections of the equal chords AB, BC, CD, and DF, respectively, on the diameter RS ; and let MO be the common distance of the chords from the center of the circle. To prove that the area generated by the broken line ABCDF is equal to VZ - 2 TT M 0. Proof. ABO, ECO, CDO, and DFO are equal isosceles A. Why ? The projections of their bases on RS are VW, WX, etc. Given Area generated by AB equals VW - 2 irMO. 606 Area generated by BC equals WX 2 TrMO. Area generated by CD equals XY -2 TrMO. Area generated by DF equals YZ - 2 TrMO. Adding, area generated by broken line ABCDF is equal to (VW+ WX+XY+ YZ} 2 irMO = VZ 2 TrMO. BOOK VIII 455 608. Zone. The portion of a sphere included between two parallel planes which intersect the sphere is called a zone. The distance between the planes is called the altitude of the zone. The two circular sections of the sphere which are made by the parallel planes are called the bases of the zone. 609. Zone of one base. If one of the parallel planes is tan- gent to the sphere, the zone is said to have one base. If both planes are tangent to the sphere, the zone is the sphere itself, and the altitude of the zone is the diameter of the sphere. 61.0. Generation of a zone. Since a sphere is generated by the rotation of a circle about a diameter, a zone is generated by the rotation of an arc of a circle about a diameter of the circle. If the arc meets the axis of rotation, the zone has only one base. If the arc is a semicircle rotating about its diameter the entire sphere is generated. 611. Area of zone. If in 607 the number of chords in- scribed in the arc is increased without limit while each becomes very short, the figure generated approaches a zone the expression for the area of which may be inferred from the result of Theorem 5. It is observed that as the number of equal chords and arcs increases, the line MO of Theorem 5 increases in length until for very short chords it becomes almost equal to the radius of the sphere. By the theory of limits it is possible to prove the following statement; but since a rigorous proof is too difficult to include here, and since the truth of the proposition is evident after 607, we shall assume it without further demonstration. 456 SOLID GEOMETRY Theorem 6 612. The area of a zone equals the product of its alti- tude and the perimeter of a great circle of the sphere. 613. Corollary I. If r denotes the radius of a sphere and h the altitude of a zone whose area is Z, then 614. Corollary II. If r denotes the radius of a sphere and S its area, then HI-NT. In 613, let h = 2 r. NOTE. The simplicity of this expression for the area of the sphere is one of the most remarkable results to be found in the whole field of elementary mathematics. That the area of the curved surface of the sphere should turn out to be just 4 (not 4 plus some bothersome irra- tional number) times the area of the largest section of the sphere sug- gests a harmonious relationship which would seem even more astonishing if the symmetry of the subject had not led us to take such results as a matter of course. QUERY 1. If different diameters are used as axes, will a given arc always generate zones with the same altitude ? QUERY 2. In what position do you think that the axis should lie in order that a given arc may generate (1) the zone of least area, (2) the zone of greatest area ? EXERCISES In the following exercises r and S denote the radius and sur- face, respectively, of a sphere, and Z denotes the area and h the altitude of a zone. 19. Given r = 4. Find S. 20. Given r = 1/2. Find S. 21. Given S == STT. Find r. 22. Given r = 6, h = 2. Find Z. 23. Given r = 9.4, k = 5J. Find Z. BOOK VIII 457 24. Given S = 72 TT, h = 2. Find Z. 25. Given Z = irr. Find h. 26. On the same sphere or equal spheres the areas of two zones are in the same ratio as their altitudes. 27. The areas of two spheres are in the same ratio as the squares of their radii. 28. The area of a zone of one base equals the area of the circle whose radius is the chord of the generating arc of the zone. HINT. To prove that Z = irPA 2 , pass a section through PA making a great circle, and consider the triangle PP'A. 29. A zone of one base is 36 TT square inches in area, and the chord of its generating arc is 4 inches from the center of the sphere. Find the surface of the sphere. 30. Find the area of the greatest zone which can be generated by an arc whose chord is 9 inches long, on a sphere of radius 16 inches. 31. Prove that half the surface of the earth is included between the parallels 30 K and 30 S. 32. What portion of a sphere is visible from an exterior point at a distance from it equal to the radius ? 33. The radius of a sphere is 5. Find the radius of a sphere having an area three times that of the given sphere. 34. The radius of a given sphere is 8. Find the radius of a sphere having an area one half that of the given sphere. 35. The radius of a given sphere is 10 inches. What is the altitude of a zone whose area is one fourth that of the sphere ? 36. The radius of a given sphere is r. What is the radius of a zone of one base whose area is one third that of the sphere ? 37. The radius of a given sphere is r. A zone whose area is one fourth that of the sphere has one base twice the radius of the other. How far from the center of the sphere is the larger base ? 458 SOLID GEOMETRY Theorem 7 615. The volume of a sphere equals the radius of the sphere. 4-rrr* where r denotes FIG. 1 FIG. 2 Given any sphere of radius r and volume V. To prove that V= Proof. Consider the figure consisting of a cylinder of revolu- tion the radius of whose base is r and whose altitude is also r, having had removed from it the cone of revolution whose base is the upper base of the cylinder and whose vertex is at the center of the lower base. Let the hemisphere of radius r have its base in the same plane with the base of the other figure. Pass a plane through both figures parallel to the common plane of their bases at any dis- tance PR = OK from the base. This plane will cut the sphere in a circle whose radius is RT, and the other figure in a ring whose outer radius is KS and whose inner radius is KC. In Fig. 1, RT' 2 = PT 2 - PR' In Fig. 2, OA/AB = OK/KC. But OA = AB. Hence, OK = KC = PR. Also PT=KS = r. (1) 284 263 Given Given BOOK VIII 459 Substituting KS for PT and KC for PR in (1), or 7TR T 2 = TrKS 2 - 7TKC*. The left-hand member of this equation is the area of the circle cut from the sphere, while the right-hand member is the area of the ring in Fig. 2. Hence the area of a plane section of Fig. 1 at any distance from the base equals the area of a plane section of Fig. 2 at the same distance from its base. Therefore the volume of the hemisphere equals the volume of the cylinder with the cone removed. 481 But the volume of the cylinder minus that of the cone = Trr 2 r - 1/3 irr 2 . r = 2/3 Trr 3 . Therefore the volume of the hemisphere = 2/3 Trr 3 , and the volume of the entire sphere equals 4/3 Trr 3 . 498, 550 EXERCISES 38. The radius of a sphere is 4 inches. Find (1) the surface, (2) the volume. 39. The surface of a sphere is 616 square inches. Find the volume. 40. A zone whose altitude is 6 inches has an area of 3696 square inches. Find (1) the area, (2) the volume of the sphere. 41. A zone of one base is 36 TT square inches in area, and the chord of its generating arc is 4 inches from the center of the sphere. Find the volume of the sphere. 42. Prove that the volumes of any two spheres are propor- tional to the cubes of their radii. 43. If one sphere has twice the surface of another, find the ratio of their volumes. 44. If one sphere has twice the volume of another, find the ratio of their surfaces. 460 SOLID GEOMETRY 616. Spherical segment. The solid bounded by a zone and the planes of its bases is called a spherical segment of two bases. The altitude of a spherical segment is the altitude of its zone. If the zone has only one base, the segment is said to have one base. 617. Spherical sector. The solid generated by the rotation of a sector of a circle about an axis which passes through the center of the circle, but which does not cut the sector, is called a spherical sector. The bounding surfaces of a spherical sector are a zone, which is called the base of the sector, and either one or two conical surfaces, according as the zone has one or two bases. 618. Spherical cone. If the base of a spherical sector is a zone of one base, the sector consists of a cone and a spherical segment of one base. This figure is some- times called a spherical cone. The spherical sector in the adjacent figure may be looked upon either as a spherical seg- ment with two cones removed or as the entire solid sphere with two spherical cones removed. BOOK VIII 461 619. Volume of spherical cone. Consider the pyramids whose common vertex is the center of the sphere and whose bases have their vertices on the base of the spherical cone. These pyramids can be taken numerous enough, and their bases can each be taken small enough, so that each altitude is nearly equal to the radius of the sphere, and the sum of their bases is almost exactly equal to the base of the spherical cone. The sum of the pyramids themselves forms a solid approximating the spherical cone as closely as may be desired. Now the volume of each pyramid equals one third the product of its base and altitude, and it can be proved that as the number of pyramids becomes very large and each of their bases becomes very small, the volume of the sum of the pyramids approaches one third the product of the base of the spherical cone and the radius. But since the sum of the pyramids approaches the spherical cone, we may infer the truth of the following theorem, a rigorous proof of which is beyond the scope of this book. Theorem 8 620. The volume of a spherical cone equals one third the product of the area of the zone ivhich forms its base, and the radius of the sphere. 621. Corollary I. The volume of any spherical sector equals one third the product of the area of the zone ivhich forms its base, and the radius of the sphere. 622. Corollary II. If h denotes the altitude of the zone which forms the base of a spherical sector on a sphere of radius r, the volume, F, of the sector is V= o 462 SOLID GEOMETRY 623. Corollary III. The volume of a spherical segment of one base is where h is the altitude of the segment and r is the radius of the sphere. HINTS. The segment is equal to a spherical cone less the ordinary cone whose base is the base of the segment and whose vertex is at the center of the sphere. Let a denote the radius of the base of the segment. Thei 2,rr 2 V 7m 2 (r-/Q 3 3 But a 2 = h(2r-h). 282 Hence V= J[2 r*k - h (2 r - h) (r - /*)] = (3 r - A). o 3 QUERY 1. Explain how the solid bounded by a hemisphere and a plane can be considered as a special case either of a spherical segment or of a spherical sector. QUERY 2. What is the locus of the centers of spheres of given radius which are tangent to two given intersecting planes? QUERY 3. What relation must exist between the lengths of the radii and the lengths of the line of centers of two spheres in order that one may lie entirely inside the other ? QUERY 4. What are some examples of great circles on the earth ? QUERY 5. What are some examples of small circles on the earth ? QUERY 6. If a sphere is viewed from a finite distance, can the observer see an entire great circle ? QUERY 7. What kind of figure is the shadow of a sphere cast on a horizontal plane by the sun when it is directly overhead? EXERCISES 45. Find the volume of the segment on a sphere which is cut off by a plane 2 inches from the center of the sphere whose radius is 6 inches. 46. What is the volume of a segment of one base of a sphere whose radius is r if its altitude is one half that of r ? BOOK VIII 463 47. The water in a hemispherical bowl 18 inches across the top is 6 inches deep. What per cent of the capacity of the bowl is filled ? 48. A 4-inch auger hole is bored through a 10-inch sphere, the axis of the hole coinciding with a diameter of the sphere. Find the volume remaining. REVIEW EXERCISES 49. Prove that two circles which have two points in common, but which do not lie in the same plane, determine a sphere. 50. Prove that a circle and a point not in its plane determine a sphere. 51. Prove that the surface of a sphere and the lateral area of the circumscribed cylinder of revolution are equal. 52. Find the ratio of the volume of a sphere to that of the cir- cumscribed cylinder of revolution. 53. A piece of lead pipe is 50 feet long. Its outer radius is 2 inches, and it is ^-inch thick. Into how many spherical bullets ^-inch in diameter can it be melted ? 54. A cylinder of revolution is capped on each end by a hemi- sphere. Show that the total surface of the figure equals the prod- uct of its entire length and the circumference of the base of the cylinder. 55. Find the ratio of the volumes of a sphere and a cube if their surfaces are equal. 56. Through a point 6 inches from a sphere of radius 4 inches all the tangent lines to the sphere are drawn. Find the lateral area of the conical surface included between the point and the sphere. 57. The volume and the surface of a sphere are expressed by the same number. Find the radius. 58. Find the volume of a cone whose vertex angle is 60 and which is inscribed in a sphere whose radius is 10 inches. 464 SOLID GEOMETRY 59. The outside diameter of a spherical iron shell 2 inches thick is 14 inches. Find its weight if a cubic inch of iron weighs 4.2 ounces. 60. A wooden sphere whose radius is 15 inches rests in a cir- cular hole in a board the radius of which is 5 inches. How far below the upper surface of the board does the sphere extend ? 61. Find the volume of a cube inscribed in a sphere of radius r. 62. Find the volume of a regular octahedron inscribed in a sphere of radius r. 63. Prove that two lines tangent to a sphere at the same point determine the tangent plane to the sphere at that point. 64. Referring to the Prismoidal Formula in 588, prove that the volume of a sphere is given by that Formula. HINT. The areas of the two extreme sections are each zero, while the mid-section is a great circle. 65. Prove that the volume of a spherical segment of one base can be found by the Prismoidal Formula. 66. A cylindrical glass of radius 1.5 inches and altitude 6 inches is filled with water to a depth of 2 inches. If three spheres each 1 inch in diameter are dropped into the glass, by how much is the level of the water raised ? GEOMETRY ON THE SPHERE 624. Practical importance of spherical geometry. We shall now study briefly the properties of figures drawn on a sphere. The fact that in the sciences of Geodesy, Astronomy, Naviga- tion, and to a certain extent Civil Engineer- ing, the theorems of Spherical Geometry find important application gives practical significance to this part of our subject. 625. Spherical polygon. The portion of a sphere bounded by arcs of great circles is called a spherical polygon. BOOK VIII 465 626. Spherical angle. The figure formed on a sphere at the point where two arcs of great circles meet p \ each other is called a spherical angle. ^^g^T 627. Measure of a spherical angle. The numerical measure of a spherical angle is equal to the numerical measure of the l angle between the tangents to the great circles at their point of intersection. In this text it is not necessary to define or to discuss the angle between two small circles or other curves which may be drawn on the sphere. Terms which are used in spherical geometry in the same sense as in plane geometry will not be defined again. QUERY. Does it make any difference which point of intersection of two great circles is taken in denning the measure of the angle between them? EXERCISES 67. Prove that the angle between two great circles is equal to the angle between their planes. 68. Construct the arc of a great circle making an angle of (a) 30, (b) 90, (c) 45 with a given great circle at a given point. 628. Relation between spherical polygons and polyhedral angles. If radii of the sphere are drawn from the vertices of a spherical polygon, and the planes determined by successive pairs of these radii are passed, a polyhedral angle is formed which is intimately related to the polygon. Prove each of the following proposi- tions relating to the spherical polygon: 629. The face angles of a polyhedral angle are measured by the: arcs which form the sides of the corresponding spherical polygon. 466 SOLID GEOMETRY 630. The dihedral angles of a polyhedral angle are equal to the angles of the corresponding spherical polygon. HINT. Apply 627 and 432. 631 . The sum of the sides of a convex spherical polygon is less than a great circle. HINT. Apply 629, 572. 632. Any side of a spherical triangle is less than the sum of the other two sides. HINT. Apply 570. 633. Congruence. Two spherical triangles are congruent if their sides and their angles are equal each to each and arranged in the same order. As in the case of all other geometric fig- ures, if two spherical triangles are identical in every respect, they may be looked upon as merely different positions of the same figure and may, by 20, be superposed. 634. Symmetric spherical triangles. Two spherical triangles are symmetric if their parts are equal each to each but arranged in opposite orders when both triangles are viewed from the center of the sphere. From an inspection of the figure it C'j appears that the triangles determined by vertical trihedral angles ( 580) whose vertex is at the center of the sphere are symmetric, since their angles and their sides are equal each to each ( 629, 630) and the corre- sponding parts are arranged in opposite orders when viewed from the center of the sphere, BOOK VIII 467 Theorem 9 635, Two spherical triangles on the same sphere or on equal spheres, which have the three sides of one equal respec- tively to the three sides of the other, are either congruent or symmetric, according as the equal sides are arranged in the same or in opposite orders. Case I. When the equal sides are arranged in the same order. Given the two spherical triangles ABC and A'B'C', in which the corresponding sides are equal and are arranged in the same order. To prove that A ABC is congruent to AA'B'C'. Proof. Construct the trihedral A corresponding to ABC and A'B'C'. Then the face A of 0-ADC are equal to the face A of 0-A'B'C 1 and are arranged in the same order. 629 Hence the dihedral A OA = OA', OB = OB', OC = OC', 569 anaZ.ABC=^A'B'C',^ACB=Z.A'C'B',Z.BAC=Z.B'A'C 1 . 630 Therefore AABC is congruent to AA'B'C'. 633 Case II. When the equal sides are arranged in opposite order. Given the two spherical triangles ABC and A'B'C', whose corre- sponding sides are equal and arranged in opposite order. To prove that AABC is symmetric to AA'B'C'. Proof. The demonstration is identical, except that the tri- hedral angles, and therefore the triangles, are symmetric. 634 468 SOLID GEOMETRY Theorem 10 636. The angles opposite the equal sides of an isosceles spherical triangle are equal. A Given the spherical triangle ABC having the side AB equal to^C. To prove that Z_fi = Z.C. Proof. Construct the mid-point of the arc BC and denote it by M. Let AM IQQ the arc of the great circle determined by A and M. 598 Then the spherical A A MB and AMC 1 are symmetric. 634 , Therefore Z7J = ZC. 635 QUERY 1. Are isosceles spherical triangles whose parts are equal each to each necessarily congruent ? QUERY 2. Are isosceles spherical triangles whose parts are equal each to each necessarily symmetric ? QUERY 3. If you try to superpose two symmetric spherical triangles by turning one of them over so that the equal parts are arranged in the Same order, why are you unable to do it? Is the same difficulty met in the case of plane triangles ? QUERY 4. If two triangles are symmetric to the same triangle, are they necessarily congruent? QUERY 5. If from any point of AM in the figure above two arcs of great circles are drawn to B and C respectively, are these arcs neces- sarily equal ? QUERY 6. Can A in the figure above be so situated that the angles B and C are both right angles ? BOOK VIII 469 637. Poles. The poles of a circle on a sphere are the points where a tine perpendicular to the plane of the circle at its center intersects the sphere. QUERY 1. How many poles does a circle on a sphere have? QUERY 2. How can one obtain a set of circles on a sphere which have the same poles ? QUERY 3. Can two great circles have the same poles ? EXERCISES 69. Prove that every circle of a sphere through the poles of a given circle is a great circle. 70. Every point on a circle of a sphere is equidistant from a pole of that circle. Theorem 11 638. The arcs of the great circles joining any point of a small circle to one of its poles are equal. HINT. Apply 25, 179. QUERY 4. What kind of a spherical triangle is PAB? 639. Corollary. The points on a sphere ivhich are a constant distance from a fixed point on the sphere lie on a circle of the sphere of which the fixed point is the pole. HINTS. Let P be the fixed point and PA be the constant distance. Pass the plane through A which is perpendicular to the radius PO, and apply 425 (1), 595. 470 SOLID GEOMETRY 640. Constructions on the sphere. It follows from 639 that if the point of a pair of compasses is placed on a point of a sphere and a continuous curve is drawn on the sphere with the aid of the compasses, this curve will be a circle of the sphere. If it is intended to perform the constructions of spherical geometry by operations that can be carried out on the surface of the sphere, this method of drawing circles must replace their determination by the passing of planes. A method of drawing a great circle deter- mined by two given points is found in 654. Construction 2 641. Construct the diameter of a sphere from measure- ments on its surface. . 2 FIG. 3 Given a sphere S. To construct its diameter from measurements on its surface. Analysis. If K is a small circle on S and its poles are P and T, one observes that the triangle PA T is a right triangle, and that AR is the altitude on the hypotenuse PT. If we can construct AP and AR, the triangle APR and hence APT can be constructed. The diameter PT will then be found. BOOK VIII 471 Construction. Set the compasses with the distance AP between the points and construct K, a small circle of the sphere (Fig. 1). 639 With the compasses determine the lengths of the chords AB, BC, and CA, which join any three points of K, as A, B, and C. In a plane construct the triangle A^B^C^ whose sides are equal respectively to AB, BC, and CA (Fig. 2). 232 Circumscribe the circle K about A 1 B 1 C 1 and denote its center Construct a plane right triangle P^A^T^ (Fig. 3) having one leg, v equal to AP, and the altitude on the hypotenuse, A^R^ equal to Afl^ Proof. Therefore Also Now Therefore But and Therefore and Hence P 2 T 2 is the required diameter. PT is J_ to plane of circle K. AR is the altitude on PT. PA T is a rt. Z. AA 1 B 1 C 1 is congruent to A ABC. 2 R 2 is congruent to A PAR, A P 2 A 2 T 2 is congruent to A PA T. T is equal to PT, the diameter of S. 637 414 217 33 27 Why? 97 Why ? 642. Polar distance. The length of the arc of a great cir- cle joining any point of a great or a small circle of a sphere to the nearer pole of the circle is called p the polar distance of the circle. Thus, in the figure the arc AP is the ^ polar distance of the circle. 643. Corollary. Two equal circles on a sphere have equal polar distances. HINT. Prove by superposition. 472 SOLID GEOMETRY Theorem 12 644. Two symmetric spherical triangles are equal in area. A A' Given the two symmetric spherical triangles ABC and A 'B' C', that is, two spherical triangles whose parts are equal each to each but arranged in opposite orders. To prove AABC = AA'B'C f . Proof. Pass the planes determined by the vertices of ABC and A'B'C'j respectively, forming small circles S and S', in which the given spherical A are inscribed. Since AB=A'B' > AC=A'C', BC = B'C', Hyp. the sides of the plane A ABC and A'B'C' are equal. 178 Hence the two small circles are equal. Why ? Let P and P' be the poles of S and S' respectively. Then the spherical APAB, PAC, PCS, P'A'B', P'A'C', P'C'B', are all isosceles. 638 Therefore APAC=AP'A'C', APAB=AP'A'B', PBC=AP'B'C'. 635 Adding, APAC+APAB+APBC=AP'A'C'+AP'A'B' or AABC=AA'B'C'. BOOK VIII 4T3 645. Relation between plane and spherical geometry. The subject of plane geometry consists in proving propositions which follow from the assumptions and definitions there laid down. We have also a spherical geometry, in which the fig- ures are drawn not on a plane but on a sphere. In order to understand the similarities and the contrasts between plane geometry and spherical geometry, it is necessary to determine what figures on the sphere correspond to the line and to the circle on the plane, and to find out how closely the assump- tions made regarding the properties and relations of these fundamental elements in the plane may be carried over to spherical geometry. We have already seen ( 640) that a small circle on a sphere can be drawn with compasses and therefore corre- sponds to a circle in the plane. In fact, a small circle is the curve on the sphere such that the distances to any of its points from the pole of the circle are equal. The two most important properties of the line in plane geometry are the following: 1. A line is determined by any two of its points. 2. The shortest path between two points of a plane is along the line joining them. From 598 it follows that a great circle is determined by any two of its points unless those points lie at the extremi- ties of a diameter. With the exception noted the great circle has property 1 of the line in a plane. The same exception appears when we make the statement regarding great circles corresponding to the fact that two lines in a plane never have two points in common, for two great circles never have two points in common except those points which are the extremities of a diameter. We shall now show that (2) corresponds to a property of the great circle. 474 SOLID GEOMETRY Theorem 13 646. The minor arc of the great circle joining two points on a sphere is the shortest curve on the sphere connecting the two points. Given two points, A and 5, on the sphere, and AB the minor arc of the great circle joining them. To prove that AB is the shortest curve on the sphere connect- ing A and B. Proof. Select any point C on the arc AB. With A as a center and with AC as radius construct the small circle S. Similarly, with B as a center and EC as radius construct T. 639 The circles S and T have only the point C in common. For, take D, any point on S other than C. In the spherical AABD, AD + DB > A C + CB. 632 But AD=AC. 638 Therefore DB > CB. Why ? Hence D is not on T, and consequently is not common to S and T. Now let AFB be any curve on the sphere connecting A and B which does not contain C. It must cut S and T in distinct points K and L, by the first part of the proof. Consider the curve which might be drawn connecting A and C, which is congruent to A A'; and the curve similarly connecting B and C, which is congruent to LB. The sum of the lengths of these curves is less than AKLB by the length of KL. BOOK VIII 475 Hence there is a curve on the sphere through C joining A and B, which is shorter than AFB. But since C was any point on the minor arc AB, the shortest curve on the sphere connecting A and B must contain all points of AB, and hence coincide with it. 647. Distances on a sphere. We are now justified in call- ing the minor arc of the great circle joining two points on the sphere the shortest distance between them, and in taking the great circle as the figure in spherical geometry which corresponds to the straight line in plane geometry. QUERY 1. How many points are required to determine a circle in the plane ? Mention any exception. QUERY 2. How many points are required to determine a small circle on the sphere ? Mention any exception. QUERY 3. What use is made in navigation of the fact stated in 646 ? 648. Assumption of free motion on a sphere. Since the sphere has the same curvature throughout, it follows that figures on the sphere may be moved from place to place upon it with- out altering their size or shape. This corresponds to the important assumption of plane geometry contained in 20. 649. Restrictions on spherical geometry. In two important respects geometry on the sphere differs from geometry in the plane. In the plane, triangles whose parts are equal each to each but arranged in opposite order, like those in the adjacent figure, can be brought into coincidence by turning one of them over in space and applying the equal parts to each other. Symmetric tri- angles on the sphere cannot be brought into coincidence in this way, as one can easily see by cutting out the symmetric triangles from the peel 476 SOLID GEOMETKY of an orange. Although symmetric spherical triangles are not congruent, they are equal in area ( 644). Since every pair of great circles on a sphere meet, there is nothing on the sphere corresponding to parallel lines in the plane. For parallel lines never meet however far they are produced. Hence theorems in plane geometry which depend either on the existence of parallel lines or on the parallel assumption ( 45) cannot be carried over into spherical geom- try. If, however, we are careful to avoid such theorems, we may state a large number of theorems from plane geometry which are true on the sphere. The following theorems of spherical geometrv may be stated without proof, since the corresponding theorems in plane geom- etry do not depend on the notion of parallels. 1. At a point in a great circle, one and only one great cir- cle can be drawn perpendicular to it. 2. Vertical angles are equal. 3. Two right spherical triangles are congruent if the hypot- enuse and an adjacant angle of one are equal respectively to the hypotenuse and an adjacent angle of the other, and if the corresponding parts are arranged in the same order. 4. Two spherical triangles on the same sphere are con- gruent if two sides and the included angle are equal respec- tively to two sides and the included angle of the other and if the corresponding parts are arranged in the same order in the two triangles. 5. Two spherical triangles on the same sphere are congru- ent if a side and the adjacent angles of one are equal to an angle and two adjacent sides of the other, and if the corre- sponding parts are arranged in the same order. It should be noted that 4 and 5 may be proved by superposition in precisely the same manner as the corresponding theorems in plane geometry. BOOK VIII 477 EXERCISES 71. Two spherical triangles on the same sphere are symmetric if two sides and the included angle in one are equal to two sides and the included angle in the other, and if the corresponding parts are arranged in opposite orders. HINTS. Denote the given triangles by T l and 7 T 2 . The triangle sym- metric to 7\ has its parts arranged in the same order as the correspond- ing parts of T z , and by 4 above is congruent to T 2 . Hence 7\ and 7' 2 are symmetric. 72. Two spherical triangles on the same sphere are symmetric if a side and two adjacent angles of one are equal to a side and two adjacent angles of the other, and if the corresponding parts are arranged in opposite orders. 73. Find four theorems other than those given on page 476 which correspond to theorems in plane geometry and whose truth can be inferred without proof. 74. Find four theorems of plane geometry which correspond to propositions in spherical geometry which are not true. 650. Sum of angles in a spherical triangle. Since parallels do not exist on the sphere, there is no such figure as a spher- ical parallelogram, trapezoid, or square. The theorem of plane geometry that the sum of the angles in a triangle equals two right angles necessarily depends upon the parallel assumption. Since this assumption does not hold upon the sphere, one would not expect the sum of the three angles of a spherical triangle to equal 180 degrees. We now proceed to prove the theorems which will lead us to the facts in the case of spherical triangles. 651. Quadrant. The arc of a great circle subtended by a right angle at the center of a sphere is called a quadrant. 652. Corollary. The polar distance of a great circle is a quadrant. 478 SOLID GEOMETRY Theorem 14 653. If two points are taken a quadrant's distance from a given point, they determine the great circle of which the given point is the pole. Given the point P on a sphere, and two other points of the sphere, A and B, such that PA and PB are both quadrants. To prove that the great circle of which AB is an arc has P for its pole. Proof. Pass the planes of the great circles determined by PA, PB, and AB respectively. These planes intersect at 0, the center of the sphere. 596 Z.POA =^POB= 90. Why ? Therefore PO is _L to the plane of AB, 415 and P is the pole of the great circle of which AB is an arc. 637 654. Corollary. Construct on the sphere a great circle determined by two given points. HINTS. Let the given points be A and B. Con- struct the point of intersection of the great circles of which A and B are poles ( 640), and apply 653. EXERCISE 75. If two great circles on the same sphere are both perpendicular to a given great circle on that sphere, they meet at the poles of the given great circle. BOOK VIII 479 Theorem 15 655. A spherical angle is measured by the arc of the great circle of which its vertex is a pole, and which is included between its sides, produced if necessary. P Given any spherical angle XPY, and A and B the points where the sides of this angle, produced if necessary, meet the great circle of which P is the pole. To prove that Z.XPY is measured ly arc AB. Proof. Draw the tangents EP and SP to the great circles AP and EP respectively. Then Z.XPY is measured by ^.RPS. 626 But RP and SP are both _L to PO. 192 Also A O and BO are both _L to PO. Why ? Therefore Z A OB = Z RPS. Why ? But /.AGE is measured by the arc AB. Why ? Therefore /.RPS or Z.XPY is measured by the arc AB. 656. Corollary. If one great circle passes through a pole of another, the circles are perpendicular to each other. QUERY. If the angle XPYin the figure for Theorem 15 is 40, how many degrees are there in the sum of the angles of the triangle PA B ? EXERCISE 76. If one vertex of a spherical triangle is the pole of the opposite side, prove that the sum of the angles of the triangle equals the sum of its sides, each measured in degrees. 480 SOLID GEOMETRY NOTE. Since the measure of both the dihedral angle and the spher- ical angle are denned in terms of certain plane angles, it follows that they are both expressed numerically in terms of the units which meas- ure plane angles, namely, degrees, minutes, and seconds. The arc of a circle is also measured in terms of these units. But this fact does not imply that these magnitudes are of the same kind, any more than the measure of the height of houses, trees, and mountains in terms of feet implies any similarity in their geometric form. QUERY 1. What is the locus of points a quadrant's distance from a given point on a sphere? QUERY 2. Is each angle of any spherical triangle measured by the side opposite it ? QUERY 3. Can a spherical triangle be constructed so that each angle has the same measure as its opposite side? QUERY 4! Can a spherical triangle be constructed so that two, but not three, angles are measured by the sides opposite them ? 657. Polar triangle. Let ABC be a spherical triangle. Let A' be the pole of the great circle of which BC is an arc, which is no more than a quadrant's distance from A, Let B' and C' be similarly chosen with respect to the other sides of ABC. Then A'B'C' is called the polar triangle of ABC. An inspection of the above figures will show that A'B'C' will lie entirely outside, or entirely inside, ABC, according as the sides of ABC are all less than a quadrant or all greater than a quadrant. If at least one side of ABC is less than a quadrant, while at least one side is greater than a quadrant, its polar triangle will overlap it. BOOK VIII 481 Theorem 16 658. If A'B'O' is the polar triangle of ABC, then ABC is the polar triangle of A'B'C'. Given any triangle ABC and its polar triangle A'B'C'. To prove that ABC is also the polar A of A'B'C'. Proof. C" is the pole of AB, and B' is the pole of AC. 657 Hence A is a quadrant's distance from B' and from C'. 652 Hence A is the pole of B'C'. 653 Since A is in the same hemisphere with A', by hypothesis, it is one vertex of the polar A of A'B'C 1 . 657 Similarly, B and C are vertices of the polar A of A'B'C'. Therefore ABC is the polar A of A'B'C'. QUERY 1. Is there any triangle on the sphere which is its own polar triangle ? QUERY 2. If two sides of a triangle are quadrants, does it bear any particular relation to its polar triangle ? QUERY 3. If one side of a spherical triangle is a quadrant, does it bear any particular relation to its polar triangle ? QUERY 4. If two angles of a spherical triangle are right angles, does the triangle bear any particular relation to its polar triangle ? EXERCISE 77. Construct the polar triangle of a given triangle, giving a reason for each step. 482 SOLID GEOMETKY Theorem 17 659. Each angle of a spherical triangle is the supple- ment of the side lying opposite it in its polar triangle. Given any spherical triangle ABC and its polar triangle A'B'C 1 . To prove that Z.A + B'C'=180, ZB+A'C r =180, Z C + B'A' = 180. Proof. Let F and // be the intersections of B'C' with AB and A C produced, respectively. Now C'F = 90 and HB' = 90. 657 C'F + HB' = C'H + HF + HF + FB' = 180. Why ? But HF measures Z.A. 655 Hence (C'H + HF + FB') + HF = C'B' + Z.A= 180. Why ? Similarly, Z.B +A'C' = 180 and Z C + .4 'B' = 180. QUERY 1. If a spherical triangle is equilateral, what can be said of its polar? QUERY 2. If a spherical triangle is equiangular, what can be said of its polar ? QUERY 3. If a spherical triangle is isosceles, what can be said of its polar? QUERY 4. If a spherical triangle has two angles equal to each other, what can be said of its polar ? QUERY 5. If all of the angles of a spherical triangle are right angles, what can be said of its polar ? BOOK VIII 483 Theorem 18 660. Two spherical triangles on the same sphere which have three angles of one equal to three angles of the other are congruent or symmetric, according as the correspond- ing parts are arranged in the same or in opposite orders. X Case L When the parts are arranged in the same order. Given the spherical triangles ABC and XYZ, in which angle A equals angle X, angle B equals angle F, angle C equals angle Z, and the parts A, B, and C follow each other in the same order as the parts JJT, 7, and Z. To prove that A ABC = AXYZ. Proof. Construct the polar triangles A'B'C' and X'Y'Z'. A 'B' = 180 - C, X ' ' = 180 - Z, etc. 659 Therefore A'B' = X'Y', B'C 1 = Y'Z', and C'A' = Z'X'. Why? Hence AA'B'C' is congruent to AX'Y'Z', 635 and Z.4 ' = ZZ', Z' = Z Y', and Z C ' = ZZ r . 27 Hence EC = YZ, CA = ZX, and AB = XY. 659 Therefore AABC = AXYZ. 635 Case II. When the parts are arranged in opposite orders. Denote the given triangles by T^ and T 2 . Any triangle, such as S, which is symmetric to T l has its angles equal to those of T^ and the corresponding parts arranged in the same order as those in jT 2 . Hence the triangle S is congruent to T f Case I. Therefore, since T l and S are symmetric, it follows that T l and T^ are symmetric. 484 SOLID GEOMETRY NOTE. It is observed that this theorem is in striking contrast to the corresponding theorem in plane geometry. Since congruence (or symmetry) follows from equality of angles in two spherical triangles, there is no such thing as similar triangles in spherical geometry. Hence it is possible to construct only one triangle on a sphere whose angles are known. EXERCISES 78. If two angles of a spherical triangle are equal, the triangle is isosceles. 79. If the three angles of a spherical triangle are equal, the triangle is equilateral. 80. If two trihedral angles have the dihedrals of one equal to the dihedrals of the other, the corresponding face angles are equal and the trihedrals are either symmetric or congruent. Theorem 19 661. The sum of the angles of a spherical triangle is greater than 180 and less than o40. Given the spherical triangle ABC. To prove that 180 4 and < 8 right angles. 137. The sides of a spherical triangle ABC are 60, 100, and 80 respectively. How many degrees in the vertex angle of a birec- tangular triangle of the same area on the same sphere ? 138. If the diameter of the earth is taken as 8000 miles, and the distance of the sun as 93,000,000 miles, what per cent of the total light and heat of the sun is received by the earth ? INDEX Angle, between line and plane, 352 ; dihedral, 339; plane, 339; poly- hedral, 419 ; trihedral, 420 Bisector of dihedral angle, 345 Cavalieri's theory, 369 Circumscribed prism, 385 Circumscribed pyramid, 418 Circumscribed sphere, 449 Cone, 405; axis of, 408; circular, 408 ; lateral area of, 412 ; of revo- lution, 416 ; right circular, 408 ; slant height of, 408 ; spherical, 460 ; volume of, 410 ; volume of spherical, 461 Cones, similar, 417 Conical surface, 405 Construction, operations of, 314 ; on the sphere, 470 Coplanar, 308 Cube, 362 Cylinder, 375 ; axis of, 381 ; circular, 375 ; of revolution, 381 ; volume of, 378 Cylinders, similar, 381 Cylindrical surface, 375 Diagonal, 363 Dihedral angle, 339 Distance, to plane, 337 ; on a sphere, 475 Element, of cone, 405 ; of cylinder, 375 Ellipse, 407 Foot of a line, 324 Given, meaning of, 306 (footnote) Great circle, 446 Hemisphere, 448 Hyperbola, 407 Inscribed prism, 378 Inscribed pyramid, 409 Inscribed sphere, 449 Intersection, 309 Line, 305 Line-segment, 305 Lime, 485 Pappus, Theorem of, 383 Parabola, 407 Parallel lines, 313 Parallel planes, 313 Parallelepiped, 362 ; right, 362 Perpendicular, 325 Perpendicular planes, 341 Perspective, 308 Plane, 305 Plane angle, 339 Polar distance, 471 493 494 SOLID GEOMETRY Polar triangle, 480 Poll-, 469 Polyhedral angle, 419 Polyhedron, 358 ; convex, 358 ; diag- onal of, 363 ; regular, 424 Prism, 358; oblique, 360; regular, 361 ; right, 360 ; right section of, 360 ; truncated, 361 Prismatoid, 435 Projection, of area, 352; of line, 349 ; of point, 349 Proposition, converse, 334 ; direct, 334 ; opposite, 334 Pyramid, 387 ; altitude of, 387 ; frustum of, 395; lateral area of, 387; regular, 399; slant height of, 399 Pyramidal surface, 388 Quadrant, 477 Rectangular solid, 362 Regular polyhedron, 424 Regular prism, 361 Regular pyramid, 399 Right dihedral, 341 Similar cones, 417 Similar cylinders, 381 Similar figures, 435 Similar polyhedrons, 432 Skew lines, 354 Skew quadrilateral, '354 Slant height, of cone, 408 ; of frustum, 401 ; of pyramid, 399 Small circle, 447 Sphere, 443 ; radius of, 443 ; tangent plane, 445 ; volume of, 443 Spherical angle, 465 Spherical cone, 460 Spherical degree, 485 Spherical excess, 486 Spherical geometry, 464 ; restrictions on, 475 Spherical polygon, 464 Spherical sector, 460 Spherical segment, 460 Symmetric spherical triangles, 466 Symmetric trihedrals, 431 Tetrahedron, 397 ; regular, 399 Triangle, bi-rectangular, 485 ; polar, 480 Trihedral angle, 420 Truncated, 361 Undefined terms, 305 Volume, of cone, 410; of cylinder, 378 ; of rectangular solid, 365 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL. 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