Hollinger Corp. pH8.5 LC 6301 .U5 P4 Copy 1 m/^ fIrit¥ersitY and SchsQl Extensien. GEOMETRY— Course B. A. W. PHILLIPS, 1889. Yale University. u Copyright. 1889, By a. W. Phillips. Press of J. J. Little & Co. Aster Place, New York. Geometry.-Course B. solid geometry. OBJECTS OF THE COURSE. First— 1:0 enable the student to make the constructions and solve the problems of Mensuration, which depend upon the princi- ples of elementary Solid Geometry. Second— 1:0 enable the student to demonstrate the principles upon which these constructions and solutions depend. METHODS OF STUDY. The student is advised to study the definitions and demonstra- tions found in the text-book of solid geometry which he has used in plane geometry. He should be able to draw the figures required for the problems of the syllabus in accordance with the methods sug- gested. He should then solve the problems in the syllabus, and write out an outline of the demonstrations of the propositions in- volved. MODELS. It is recommended that in the instruction of a class, or where several persons study together, that, especially in the early part of the work, models of the figures be made of wood, pasteboard, metal. UNIVERSITY EXTENSION. wire, strings, etc., so that a clear conception may be formed of the figure in space. DRAWINGS. Each student should then learn how to make the drawings mathematically from measurements taken from the models. The simplest means of drawing the figures is in what is called " cabinet projection," several examples of which are given in the following pages. In the particular system which is most fully explained here, the drawings may be regarded as the shadows of the skeleton outlines of geometrical models, as they would be cast by the sun on a vertical plane. The position of the sun is assumed to be such that all lines in the model perpendicular to the vertical plane are represented in the drawing by lines of one half their length, and inclined at an angle of sixty degrees to horizontal lines in the drawing. In this system : 1°. All vertical lines in the model are vertical lines in the drawing. 2°. All horizontal lines in the model parallel to the vertical plane are horizontal lines in the drawing. 3°. All lines in the model pei^pendicular to the vertical plane are inclined at an angle of 6o° to horizontal lines in the drawing. 4°. All lines parallel to the vertical plane in the model have the same length in the drawing. 5°. All lines perpendicular to the vertical plane in the model are one half as long in the drawing. (In making any drawing, proportional dividers may be used GEOMETRY.— COURSE B. B / / / K Fig. I. to lay off the lines perpendicular to the vertical plane.) Figure i illustrates the method of drawing a cube whose edges are either parallel or perpendicular to the vertical plane on which the shadow would be cast. From any point O, draw the vertical line O B and the horizontal line O A, each equal in length to the edge of the cube. Make the angle A O C equal to 60°, and O C equal to half the length of the cube. Complete the figure by drawing the remaining edges of the cube parallel respectively to those already drawn. Figs. 2 and 3 illlustrate the method of drawing a pyramid whose base is in the horizontal plane, but the sides of the base are neither parallel nor perpendicular to the vertical plane. The quadrilateral in Fig. 2 rep- resents the actual size and position of the base. O/ is a line drawn in the plane of this base parallel to the vertical plane, and the several perpendiculars drawn to the line 0/ from the corners of the base, and meeting it in ^ ^^ d f, show what kind of lines must be drawn in any horizontal figure in order to lay it off in the projection here de- FiG. 2. UNIVERSITY EXTENSION. scribed. A is the position of the foot of a perpendicular drawn from the vertex of the pyramid to the base. £ M H i ■"7" / / / «- 5 Fig. d f 3- Fig. 3 is the pro- jection of the pyramid. O/is a horizontal line with points a b c d f marked upon it in the same scale as the cor- responding line of Fig. 2. The perpendiculars (S ' / / / f- / / in Fig. 2, drawn from abed, etc., to the cor- ners of the base, are drawn in Fig. 3 at an angle of 60° to the line O /, and on a scale of one half. The extremities of these lines are joined to make the base of the figure. At A the vertical line A B is drawn of the same height as the pyramid, and the vertex B is joined to each one of the corners of the base. The large parallelogram represents the projection of a rectangular board on which the pyramid stands. Figs. 4 and 5 illustrate how the horizontal circular base of a figure may be drawn in this pro- jection. Let the circle in Fig. GEOMETRY.— COURSE B. this 4 be the base of a figure, as it stands, with respect to a horizontal line of reference, as O g, parallel to the vertical plane of projection. Draw a horizontal diameter to the circle. Divide diameter into a equal parts, and throu< several points of division A X c ^ draw chords to the circle, pj^ ^ which shall be perpendicular to O g, and meet that line in ^ <^ ^, etc. Fig. 5 is the projection of Fig. 4. In Fig. 5, O^ is a horizontal line of the same length and divided in the same way as the corresponding line of Fig. 4. At Z r / / /J ,' ^^ — » ^ f S each of the points of division a d c, etc., a line is drawn, making an angle of 60° with O g. The segments on each of these lines between O g and the curve and its horizontal diameter are half the corresponding segments in Fig. 4. After laying off these segments the points are joined. Fig. 6 illustrates how any simple model may be drawn in this projection. The large parallelogram is the projection of a rect- angular horizontal board. 8 UNIVERSITY EXTENSION. AB is a rod perpendicular to ttiis board through a given point; D C, a Hne of the board parallel to one of its edges. O E is per- C/ pendicular to D C, and A C B, A E B, and A D B are strings join- ing A and B, and passing through the board at D, E, and C. The lines A B and D C, and the sides of the parallelogram parallel to D C, are the same length as in the model. O E is inclined to D C 60°, and is drawn on the scale one half. The other lines, being neither parallel nor perpendicular to the vertical plane of pro- jection, are not drawn to a scale, but sim- ply connect points established by lines parallel or perpendicular to the vertical plane of projection. Figs. 7, 8, 9 represent the outlines of a cylinder, cone, and sphere in this pro- jection. The bases of the cylinder and Fig. 9. GEOMETRY. — COURSE B. cone and the horizontal section of the sphere may be drawn on the same principle as the projection of the circle in Fig. 5. \ \ p \ B A y "A Fig. 10. Fig. II. Figs. 10 and 11 are constructed on the principle that all lines either parallel or perpendicular to the vertical plane of projection are of the same length in the drawings as in the models. The lines CO are inclined 45° or 135" to the line O A. It maybe said in general that the lines perpendicular to the vertical plane may be inclined at any angle whatever to the horizontal lines in the draw- ing, and may be made on any scale whatever. One very convenient arrangement is to make the angle 45° and draw the lines so inclined on a scale of one-half. Fig. 12 is an example of what is called isometric projection. In this system it is assumed that the projection of a cube is made by a shadow cast on a plane perpendicular to the direction of the rays of light, and that the rays are in the direction of one of the longest diagonals of the cube. The lines, therefore, to be measured and lO UNIVERSITY EXTENSION. laid off for any figure are the vertical lines, the lines perpendic- ular to some fixed vertical plane, and the lines parallel to horizon- tal lines in that vertical plane. The scale of all these lines is the same, and they are laid off respectively parallel to O B, O A, and O C in the figure. The angles A O B and B O C are each 60°. BOOK I. LINES AND PLANES IN SPACE 1. Can more than one plane be passed through (a) A line ? (c) Two intersecting lines ? (d) A line and a point? {d) Tv/o parallel lines? What is the intersection of two planes ? 2. Why will a three-legged stool always stand firm on a plane floor ? 3. Can a person determine by a straight-edge whether a surface is a plane or not ? In what ways ? 4. How would a person determine whether a post in a room was perpendicular to the floor ? Could he determine it by two carpenter's squares ? GEOMETRY.— COURSE B. II 5. How would one find where a perpendicular from a point in the ceiling would strike the floor by the use of a ten-foot pole, the height of the ceiling being less than 10 feet ? 6. Suppose a straight line A B is marked on the floor : At its two extremities, A and B, a string is fastened, the string being longer than A B. How could a plane perpendicular to A B at its middle point be determined ? 7. Suppose a rectangular door is swung on a vertical post : How can one tell whether the floor is level or not ? 8. When the door in the above example swings, is it parallel in all positions to every vertical line in the room ? 9. A table whose top is level stands on a level floor. Why is the edge of a ruler lying anywhere on the top of the table parallel to the floor ? 10. Suppose any straight line drawn obliquely across a room. From a point in the hinge of a desk-lid in the room how would one draw a line parallel to the first line ? How would the desk-lid be made parallel to the first line ? 11. The floor and ceiling of a room are parallel planes. Why is a post perpendicular to one perpendicular to the other ? 12. The floor and ceiling of a room are both perpendicular to a post. Why are they parallel to each other ? 13. Why are the floor and ceiling of a room (parallel planes) every- where equidistant ? 14. Two parallel lines intersect two parallel planes. Why are they equal ? 15. A line A B cuts three parallel planes in A, F, and B, and a line 12 UNIVERSITY EXTENSION. C D cuts the same planes in C, G, and D. If A F = 3, F B == 4, and C D = 6, compute C G. 16. Prove that two angles not in the same plane, whose sides are re- spectively parallel and lie in the same direction, are equal, and that their planes are parallel. 17. How may the diedral angle between two planes be measured ? For example, the angle between any two faces of a pyramid : the angle between two faces of a crystal. 18. How may a plane surface be set up so as to make an angle of 50° with the floors ? How may two plane surfaces be set at right angles to each other ? How made parallel? 19. How shall the angle which a line makes with a plane be measured? How shall a line be drawn which shall be a perpendicular common to any two non-parallel lines in space ? 20. When are two triedral angles equal to each other ? 21. Prove that the sum of the face angles about the vertex of any pyramid is less than four right angles. GEOMETRY. — COURSE B. 1 3 BOOK II. SOLIDS BOUNDED BY PLANES. THE COVERINGS. 22. Find the amount of paper needed to cover the walls and ceiling of a room 18 feet long, 16 feet wide, and to feet high. 23. A building 16 feet square has a roof in the form of a square pyramid. The altitude of this pyramid is 10 feet. Required the surface of the roof. 24. The side of the base of a square pyramid measures 10 feet, and the distance from each corner of the base to the vertex is 15 feet. Required the surface of the roof. 25. The base of a pyramid is a regular hexagon inscribed in a circle of 5 feet radius. The slant height of this pyramid is 8 feet. Find the total surface of the pyramid including the base. 26. A rectangular building 12 feet wide by 20 feet long has a roof in the form of a wedge. The altitude of this roof is 10 feet. The ridge-pole of the roof is 8 feet in length. Required the surface of the roof. 27. A bin is in the form of the frustum of a square pyramid in- verted. The bottom is 3 feet square, the top 9 feet square, the depth 4 feet. What is the number of square feet of cov- ering required for the bottom and sides ? 28. An oblique pyramid has a square base the side of which is 12 inches. The altitude of the pyramid is 8 feet, and the vertex UNIVERSITY EXTENSION. is in the same vertical line with one corner of the base. Re- quired the surface of the sides of the pyramid. 29. Cut out from pasteboard the covering of the following regular polyedrons : (a) When an edge is equal to unity, (l>) When the radius of the circumscribed sphere is unity, for as many as can be found by geometry. Tetraedron. ■ Octaedron. Icosaedron. Cube. Dodecaedron. 30. How would the diedral angle of the edge be measured ? THE VOLUMES. 31. Find the contents of a right parallelopiped. 32. When are two solids similar ? ^^. Find the contents of a right triangular prism, the sides of the base being 3, 4, and 5 inches respectively, and the altitude being 8 inches. 34. Prisms having equivalent bases are to each other as their alti- tudes. 35. Two prisms which have the same height are to each other as their bases. 36. Find the contents of an oblique prism whose base is a rectangle, whose sides are 3 and 6 inches, and whose altitude is 3 inches — the length of the oblique edge is 4 inches. Show, by taking a package of paper cut this size, that it may be deformed from a right parallelopiped into an oblique one in two direc- tions. The altitude and base will of course remain the same. GEOMETRY. — COURSE B. 15 37. Prove that two pyramids having equal bases and the same alti- tude have the same volume, by the " method of limits." Illustrate this proposition by packages of paper. ^8. Show the same also when the bases are equivalent^ but of differ- ent shape. 39. Show that when two pyramids have the same altitude they are to each other as their bases. 40. Show that, when a pyramid is cut off by a plane parallel to its base, the part cut off is similar to the whole. 41. Two similar prisms or pyramids are to each other as the cubes of their homologous sides. 42. A cube can be cut into six equal square pyramids having their bases the faces of the cube and their vertices in the center. Show by this that the volume of the pyramid is equal to the base multiplied by one-third the altitude. 43. Show that the volume of any pyramid is equal to the product of the base by one-third the altitude. 44. Show that the volume of the frustum of a regular square pyra- mid is equal to 1 [B^ + b'^ + Bb) h, when B and b and h are respectively the sides of the lower base, the side of the upper base, and the altitude. [This may be worked by algebra, using the principle of similar pyramids. Find first the altitude of the complete pyramid. Then subtract the volume of what was cut off from the volume of the complete pyramid, and reduce the expres- sion for the remainder to the above form,] 45. Make of wood a triangular prism of any convenient size, and then cut it into three triangular pyramids, Show that these pyramids are equivalent to each other. l6 UNIVERSITY EXTENSION. Note. — The student can find numerous examples in the arithmetics as exer- cises in the rules of practical mensuration. Especial attention should be given to examples involving the flooring, plastering, and papering of rooms, the amount of brick or stone needed in constructing the walls, the lumber used in the roof, etc.; also, the surface and capacity of cisterns and bins, and problems involving what is called "board measui-e." BOOK III. THE CONE AND CYLINDER. 46. The convex covering of the right cylinder is equal to a rectangle whose length is the length of the cylinder, and whose width is the circumference of the cylinder. 47. The surfaces of two cylinders are to each other as 4 to 16 ; the radius of the base of the first is 6 inches. Required, the con- vex surfaces of both cylinders. 48. The convex covering of the right cone is a sector of a circle whose radius is the slant height of the cone, and the angle at the center is less than a circumference. 49. Compute the surface of a right cone, the radius of the base being i inch, and the slant height 4 inches, and cut out the covering from paper ; also, compute the angle of the sector of the circle and cut out the covering, supposing the altitude of the cone to be 2 inches, and the radius of the base ij inches. 50. Calculate the surface of the frustum of a right cone, the radius of whose upper base is i inch, of the lower base 2^ inches, and the slant height 2^ inches. GEOMETRY. — COURSE B. 1 7 51. Prove that the volume of a cylinder is equal to the area of the base multiplied by the altitude. 52. Prove that the volume of a cone is equal to one-third the product of the base by the altitude. 53. Prove that the volume of the frustum of a cone is equal to ^ {A -}- a + \/Aa) h where A and a are respectively the areas of the lower and upper bases, and h the altitude of the frustum^. [The same method may be employed as in No. 44.] Note. — The student should apply these rules to the problems of mensuration. 54. The area of the base of a right cone is 3 square feet, and its height 30 inches. Find the height of a cone the solid con- tents of which is four times as great, but the diameter of whose base is only one-third of the given one. BOOK IV. THE SPHERE. [In studying the sphere it is of advantage to use a slated globe, on which the figures which relate to the surface may be constructed. The globe should be fitted to a cylindrical cup such that the globe can be turned easily in the cup, and of a depth equal to the radius of the sphere. The rim of the cup may then be used as a ruler for drawing the arcs of great circles on the sphere.] PROBLEMS TO BE CONSTRUCTED ON THE GLOBE. 55. {a) Bisect a given arc. [b) Erect a perpendicular to a given great circle from a point on the great circle, and also from a point outside. UNIVERSITY EXTENSION. 56. (a) Construct an angle equal to a given angle, (l?) Construct a triangle given the three sides, (c) Construct a triangle given two sides and the included angle, (d) Construct for all pos- sible cases a triangle given two sides and an angle opposite one of them. 57. On the sphere, determine by the dividers the length of the diam- eter. 58. Explain the methods of measuring the angle of any spherical triangle. 59. What are the methods of proving two spherical triangles equal ? 60. Derive the rule for finding the surface of a sphere. 61. Compute the surface of a sphere whose radius is 6 inches. What is the radius of a sphere whose surface is one half that of the above ? 62. Find the surface of the earth, supposing its radius to be 4,000 miles. Derive the rule for finding the convex surface of a zone of the earth given the altitude. 6^. Prove that the area of a spherical triangle is equal to its spherical excess, multiplied by one-eighth of the surface of the sphere. [We compute the area of a spherical triangle by adding the angles together, subtracting from their sum 180°, and dividing the remainder by 90°. Then multiply the resulting fraction hjjTt R'-^ for the area.] 64. Compute the volume of the earth, and also that of a zone. LIBRARY OF CONGRESS 029 944 928 2 LIBRARY OF CONGRESS 029 944 928 2