in Arithmetic Copyrighted 1901 by the Author. W. S. Carter Mensuration 136.-Solids have length, breadth and thickness, and are bounded by plane surfaces, curved surfaces, or by both plane and curved surfaces. The Volume of a solid is the number of cubical units it contains; therefore solids are measured with cubic measure. (Cubic measure was explained in Arts. 81 and 84 in the February, 1902 Magazine.) The Altitude of a solid is the perpendicular distance from its base to its highest point. (In Fig. 31 e a is the altitude.) A Plane Surface is a straight or even surface; a surface with which a straight line would coincide if applied in any direction. (In Figs. 31, 32, 33 and 34 all surfaces are plane surfaces.) NOTE. The lateral plane sides or faces of a solid when taken collectively is called its "convex surface." as one surface is called the "convex surface" of the solid. As will be learned later, however, this is a misnomer. 137.-A Prism is a solid which has two faces equal and parallel to each other and polygon in form, and with any number of lateral faces which are parallelograms in form. (Figs. 31, 32, 33 and 34 are prisms. The top and bottom faces are the bases and are equal and parallel with each other, and the perpendicular faces are the lateral faces.) A Right Prism is one all of whose lateral faces are perpendicular to its base. (Figs. 31, 32, 33 and 34 are right prisms.) An Oblique Prism is one whose lateral faces are not perpendicular to its base. (Fig. 35 is an oblique prism.) A Regular Prism is one whose bases are regular polygons. (Figs. 31, 32 and 34 are regular prisms.) A Parallelopiped is a prism all of whose faces are parallelograms. (Figs. 31 and 34 are parallelopipeds.) A Triangular Prism is one whose bases are triangles. (Fig. 32 is a triangular prism.) A Quadrilateral Prism is one whose bases are quadrilaterals. (Figs. 31 and 34 are quadrilateral prisms. In the same manner all prisms may be described by the number of sides in the perimeter of the base.) A Cube is a prism all of whose faces are equal squares. (Fig. 34 is a cube.) A Truncated Prism, or as it is usually called, the Frustum of a Prism, is one from which a portion has been cut off by a plane that is not parallel to the bases. (In Fig. 36 the right prism has been truncated by 138.-Propositions Relating to Prisms. FIG. 36. k Prop. XXVIII.-The convex surface of a prism is equal to the perimeter of the base multiplied by the altitude. (In Fig. 31 let us presume that the four lateral sides are each 10 in. wide and 21.25 in. long, that is, let us suppose the perimeter of the base to be 40 in. and the altitude of the prism 21.25 in. Then, 40 in. X21.25 in. 850 sq, in., which is the area of the convex surface of the prism.) Prop. XXIX.-The volume of a prism is equal to the area of the base multiplied by the altitude. (In Fig. 31 if the altitude be 21.25 in. and the base 10 in. square, the area of the base is 10 in. x 10 in. 100 sq. in., and the volume of the prism is 100 sq. in. X 21.25 in. 2125 cu. in.) NOTE. In an oblique prism the altitude is not the length of the lateral sides, but is the perpendicular distance between the two base lines. (In Fig. 35 ij and not a e is the altitude.) Prop. XXX. The volume of the frustum of a prism is equal to the area of the base multiplied by the average altitude of the vertical edges or lines of intersection between the lateral faces. 4 (In Fig. 36 suppose the vertical edges to be as follows: ae 10 in.; bf 10 in.; dh=7in.; cg=7in. The average altitude is 10 in. +10 in. +7 in. +7 in.—8.5 in. Let us suppose the area of the base to be 100 sq. in., then the volume of the frustum of the prism is 100 sq. in. X 8.5 in. 850 cu. in.) 139.-Problems Involving Volumes of Prisms. Prob. 31. Suppose Fig. 31 to be a regular prism, with base 7 in. square and altitude 15 in. (a) What is the area of the entire surface of the prism? (b) What is the volume of the prism? (a) The area of one base is 7 in. X7 in.49 sq. in. The area of the two bases is 49 sq. in. x 2=98 sq. in. The perimeter of the base is 7 in. X 4 28 in. The altitude being 15 in. the area of the convex surface is 28 in. X 15 in. = 420 sq. in. The area of the entire surface is the area of the base (98 sq. in.) + the area of the convex surface (420 sq. in.) 518 s 8 sq. in. (b) The area of the base is 7 in. X 7 in. 49 sq. in., and the altitude is given as 15 inches. The volume of the prism is 49 sq. in. X 15 in. 735 cu. in. Prob. 32. Suppose Fig. 32 to be a triangular prism with the following dimensions: ac 10 in.; g b 8.66 in.; ad: 21 in. (a) What is the area of the entire surface of the prism? (b) What is the volume of the prism? Ans. (a) 716.6 sq. in. (b) 909.3 cu. in. From Prop. XIX (February Magazine): The area of one base is 10 8.66 in. (a) in. X 43.3 sq. in. The area of both bases is 43.3 sq. in. X 286.6 sq. in. The perimeter of the base is 10 in. X 330 in. The area of the convex surface is 30 in. X the altitude (21 in.) 630 sq. in. The area of the entire surface is the area of the two bases (86.6 sq. in.) + the area of the convex surface (630 sq. in.) = 716.6 sq. in. 8.66 in. (b) The area of the base is 10 in. X 43.3 sq. in. The volume of the prism is the area of the base (43.3 sq. in.) X the altitude of the prism (21 in.)=9:9.3 cu. in. Prob. 33. Suppose Fig. 34 to be a cube, each edge of which measures 100 in. (a) What is the area of the entire surface? (b) What is the volume of the cube? area of the two bases is (a) The area of the base of the cube is 100 in. X 100 in. : 10.000 sq. in. The 10,000 sq. in. X 2=20,000 sq. in. The perimeter of the base The area of the convex surface is the perimeter (400 in.) X the is 100 in. 4 = 400 in altitude (100 in.): : 40,000 sq. in. The area of the entire surface is the area of the two bases (20,000 sq. in.) + the area of the convex surface (40,000 sq. in.) = 60,000 sq. in. (b) The area of the base is 100 in. X 100 in. 10,000 sq. in. cube is the area of the base (10,000 sq. in.) X the altitude (100 in.) The volume of the = 1,000,000 cu. in. with the following Prob. 34. Suppose Fig. 36 to be a frustum of a prism dimensions: bf= 10 in.; a e= 9 in.; cg 7 in.; dh 6 in.; e h=9 in.; hg= 10 in. What is the volume of the frustum? The average altitude is X 10 in. Ans. 720 cu. in. 10 in. +9 in. +7 in. +6 in.—8 in. The area of the base is 9 in. 4 90 sq. in. The volume of the frustum of the prism is the area of the base (90 sq. in.) X the average altitude (8 in.)=720 cu. in. tex of the triangle a b d and the perpendicular bf from the vertex of the triangle dbc. The area of the triangle a b d is the base da (12 in.) X half the altitude e b (4.5 in.) 54 sq. in. The area of the triangle d b c is the base dc (15 in.) X half the altitude bf (3.5 in.) 52.5 sq. in. The area of the entire base is 54 sq. in. +52.5 sq. in. = 106.5 sq. in. The volume of the prism is the area of the base (106.5 sq. in.) X the altitude (80 in.) 8520 cu. in. By W. DeKeith Why Bob Quit the Road at Tom Kelsay was boss of the pit side in the roundhouse back shop and had the history of every old man in the company's employ his memory's finger ends, so to speak, and all who heard his remark knew that just back of it was a good story. Tom had his own peculiar way of letting them know when he had a story on hand, and to have asked him for one before he indicated his desire to relate it would have met about the same success as opening the throttle on an engine that had not steam enough to move herself. However, as he was about to begin his narrative intended to interest his companions, who had been discussing transfers from one branch of service to another, the subject of Tom's story entered a door in the rear of the shop and approached the group. "Here's Bob, now. Let him tell it himself. I don't say that he got skeered out a-runnin' his train, for everybody knows Bob has got narves like a steel trap, but he's here, so let him tell it." After a greeting all around, Pat Kelly said: "Say, Bobby, you used to run a train once I mean you was conductor, wasn't you?" boys, when this thought came to me I'd watch the job pretty close, for I had passenger fever as bad as any of 'em ever get it. One day we were called for 84 which as lots of you remember left Elkhart at 9.10 a. m. Maybe you've heard, too, what a corker it was for work. Then, Fullerton made up for us by bad dispatchin' what it lacked to make it one of the hardest jobs out of doors. It would have been tolerable if the pit hadn't opened about then, but when Storrs began to stack his gravel at Kendallville, which was the outlet at the main line from the pit, we just felt that we wanted to tie up right there. The trip I am talking about in particular had for us one of the toughest days I had ever seen, and already upon my arrival at Kendallville I had filled my third page in the train book. You know this means better than 120 cars handled, and in less than fifty miles at that. We had just nicely anchored there when Schumm comes out with both hands full of card bills, and a message between his teeth which I knew was for me. ""Take thirty loads of gravel from Kendallville,' it said. "I could not repress an uncomplimentary expression, which arose at the order. "Does that d-n leatherhead imagine they are 'straw hats? We have fortyfive cars now, with orders to take all eastbounds. We'll not get in in time for Sun "Yep," replied Bobby with his usual day-school,' I said considerably worked up nonchalance. "How was it you broke off on it and went into the Old Man's office?" "Guess I've time to tell you about it if you want to hear it," he said, looking at his watch. "Tom has heard the yarn and could tell it if he would but, since you've asked me, here goes: "Let's see; it's six years ago this summer aint it, that we were taking the first gravel out of the pit?" "Seven, Bobby, since the pit opened, and it was then that-" "That's right, Pat; it is seven. Well, at any rate, that summer I was running freight and was standing pretty well ahead for passenger. I often wondered if I should be lucky enough to live to see myself punchin' tickets and, honestly over the prospect of hot boxes and numerous other troubles, incident to handling long trains whose chief tonnage was next to the caboose. "Notwithstanding all our kicks and complaints we were soon making excellent headway, and in less than an hour were sitting in the cupola watching the serpentine course our engine was taking us as she nosed around the curves east of bers on my report, when my brakeman town. I had just finished noting the num leaned over and said: "What's at Waterloo, Bob?' "I held up nine fingers, too weary in body and mind to say nine cars. ""Take 'em all?' he asked, giving me a cynical look which plainly said, 'If I |