A Truncated Pyramid is one from which a portion has been cut off by a plane that is not parallel to the base. (In Fig. 43 the plane a b c is not parallel to the base de f. It cuts off the upper portion of the pyramid, leaving a truncated pyramid.) The Altitude of a pyramid is the perpendicular distance from its base line to its vertex. (In Figs. 38, 39, 40 and 41 c e is the altitude of each pyramid.) The Slant Height of a pyramid is the altitude of the triangle that forms one of its lateral sides. The slant height of a frustum of a pyramid or of a truncated pyramid is that part of the slant height of the pyramid remaining between the two bases. In Figs. 38, 39 and 41 c d is the slant height. In Fig. 42 g h is the slant height of the frustum of the pyramid. 141.-Propositions Relating to Pyramids. Prop. XXXI.-The area of the convex surface of a pyramid is equal to the perimeter of the base multiplied by one-half of the slant height. = In Fig. 39 let us presume the base to be 1 inch square and the slant height c d to be 2 in.; then, the perimeter of the base (4 in.) X one-half the slant height (1 in.) 4 sq. in., which is the area of the convex surface of the pyramid. Prop. XXXII.-The volume of a pyramid is equal to the area of the base multiplied by one-third of its altitude. In Fig. 41 let us presume the area of the base to be 4 sq. In. and the altitude ce to be 3 in.; then, the area of the base (4 sq. in.) X one-third of the altitude (1 in.) 4 cu. in., which is the volume of the pyramid. Prop. XXXIII.-The area of the convex surface of the frustum of a pyramid is equal to one-half the sum of the perimeters of the two bases multiplied by the slant height. In Fig. 42 let us presume that each of the sides of the base a b c to be 5 in. and each of the sides of the base def to be 10 in, aud the slant height gh to be 9 in.; then, one-half the sum of the perimeters ( 15 in. 30 in. 22.5 in.) X the slant height (9 in.) 202.5 sq. in., which is the area of the convex surface of the frustum of the pyramid. = Prop. XXXIV. The volume of a frustum of a pyramid is equal to the sum of the areas of the two bases, plus the square root of their product, multiplied by one-third the altitude. In Fig. 42 let us presume that the area of the base a b c = 6 sq. in., the area of the base de ƒ 24 sq. in., the altitude a i - : 3 in.; then, the sum of the areas of the two bases is 30 sq. in. The product of the two areas = 144 sq. in., and the square root of 144 is 12, which, added to the sum of the two bases, is 42 sq. in, Then, 42 sq. in. X one-third of the altitude (1 in.) 42 cu. in.. which is the volume of the frustum of the pyramid. Prop. XXXV. The volume of a truncated pyramid is equal to the sum of the areas of the two bases, plus the square root of their product, multiplied by one-third of the average altitude. It will be noted that Prop. XXXV is the same as Prop. XXXIV. except that the 10 in. 8 in. + 6 in. 3 142.-Problems Involving Areas and Volumes of Pyramids. = = Prob. 35. Suppose Fig. 38 to be a pyramid with the following dimensions: af 10 in.; ƒ b 12 in.; a b 11 in.; c d 18 in.; ƒg 10.66 in. (a) What is the area of the convex surface of the pyramid? (b) What is the area of the entire surface of the pyramid ? From Prop. XXXI. (a) The perimeter of the base is 10 in. † 12 in. + 11 in. 33 in. One-half the slant height is 18 in. 29 in. The perimeter of the base (33 in.) X one-half the slant height (9 in.) = 297 sq. in., which is the area of the convex surface. (b) The area of the base of the pyramid is as follows: a b (11 in.) X one-half of fg (5.33 in.) = 58.63 sq. in., which added to the area of the convex surface (297 sq. in.) makes the entire surface of the pyramid 355.63 sq. in. Suppose Fig. 38 to be a pyramid of the following dimensions: 100 sq. in.; altitude ce= 15 in. What is the volume of the Ans. 500 cu. in. From Prop. XXXII. The area of the base (100 sq. in.) X one-third the altitude (5 in.) 500 cu. in., which is the volume of the pyramid. = Prob. 37. Suppose Fig. 42 to be a frustum of a pyramid with the following dimensions: Perimeter of the base a b c 10 in.; perimeter of the base What is the convex surface of the Ans. 300 sq. in. def 10 in. 50 in.; slant height & h frustum of the pyramid ? From Prop. XXXIII. The sum of the perimeters of the two bases is 10 in. +50 in. 60 in., one-half of which is 30 in., which multiplied by the slant height (10 in.) gives for the area of the convex surface of the frustum of the pyramid 300 sq. in. Prob. 38. Suppose Fig. 43 to be a truncated pyramid with the following. dimensions : Area of the base a b c 25 sq. in.; area of the base def= 500 sq. in.; altitude at a in. 33 in.; altitude at b 28 in.; altitude at c = 20 What is the volume of the truncated pyramid ? Ans. 17,193.6 cu. in. From Prop. XXXV. The sum of the areas of the two bases is 25 sq. in. -- 500 sq. in. 525 sq. in. The product of the two bases is 500 × 25 = 12,500, of which the square root is 111.8 sq. in. The sum of the areas of the two bases (525 sq. in.) + the square root of their product (111.8 sq. in.) 636.8 sq. in. The average altitude is =27 in. Then, 636.8 sq. in. X the average altitude (27 in.) ⇒ 17,193.6 cu. in.. which is the volume of the truncated pyramid. 33 in.28 in. + 20 in. 3 Labor's Critics "I believe in labor unions," said the College President, “But I think I could improve upon their rules. If with what the masters chose to give, the members were content; If they handed the non-unionist their tools, And patted him, like brothers, where his backbone ought to be, And said, 'Pray take our job for what you'll get, For our rights are nonconflicting in this country of the free We're just as free to starve as you to sweat!' If they strained their every nerve to turn out piece-work by the heap, If they welcomed in apprentices to do work on the cheap— Why, then I think trades unions would be great! With these few and slight restrictions, which are well and wisely meant, "I believe in labor unions," said the Bishop to his flock, For the violence that boycotts and does injury to stock Let nation threaten nation (if the last's of smaller size)! But unionists should never let their angry passions rise, For that is such a naughty thing to do! They should strike in white kid gloves and patent-leather dancing-shoes, If they'll behave like gentlemen, of course I'll not refuse To be among the stanchest of their friends. If there's nothing in their actions that our Christian nerves will shock, "We believe in labor unions," say the Editors of tact, For the workingman's an angel; like an angel he must act, Although his sick wife freeze, he must be silent as a clam- He must bear the worst injustice with the meekness of a lamb, If the widow's mite be grabbed from her, she must not make a fuss, She must show an equanimity such as you see in us, If they never make mistakes, and will always take a hint, If the President had logic, and the Bishop had more sense, They might help to solve the problem as to how, in self-defense, Still, they are but fellow-mortals, and no doubt they've done their best. -Annie C. Muirhead, in the Outlook. Lame Johnny's Gratitude By W. L. French L hour that the law had dallied with for years and, while the committee did not get them all in the first "round up," as it was termed, the strength and perfect organization of the gang was broken, and gradually the rest of its members were "wiped out." For those who really desired law and order better times were coming. AME JOHNNY lived on the edge ing, trying and executing thieves in an of a creek that wound its way along the edge of the Black Hills, twisting through the canyons down to the Cheyenne River and, by virtue of his being the oldest settler, the creek now bears his name. Lame Johnny was the only name anyone in the Western country knew that he ever possessed; doubtless he had another name back in the Eastern land and either he had lost it in his wanderings or, for reasons of his own, allowed it to drop into the oblivion of the past along with many other things concerning himself. Lame Johnny was not of a very prepossessing of manhood. He was gray and grizzled and thin and bent, and he limped as he walked from an injured ankle that had been splintered by a bullet some time during his career. He put one in mind of a bit of seasoned hickory that would bend and twist much before breaking. Long miles of tramping through the Hills or of sleeping out had no terrors for him. He was not the kind of a man one would select for a heroic act, yet from the very nature of the life he lived of exposure, hardship and danger he would do such an act without realizing that it was anything out of the ordinary. Lame Johnny preferred horseback riding when possible and, as he never bought a horse in his life, it was usually one of some cattleman's that did him the service of carrying him through the sand hills, and unless pushed hard by the owner or his men the horse in the end brought up in some eastern market to be sold. Lame Johnny was one of a gang of "rustlers" scattered through the cattle country and, while posing as honest settlers, stole horses and rustled cattle and, if business was dull in those lines, robbed a stage coach to liven things up. The fact of the pretended honesty of them, and their being scattered so widely apart in a thinly settled country made their detection and punishment an almost impossible matter until the vigilance committee, that grim, unswerving administrator of Western justice, was organized and swept through the cattle country arrest Rumors of a coming railroad were spreading through the sand hills, a forerunner of peace and law. It was coming as rapidly as the work could be hurried along, and with it, as its constant companion, traveled the telegraph with the power of quick communication, and in their wake would come the people to settle this new country, and coming from old settled communities where law abided would plant the standard of peace and justice in their new homes, and the lawless element would be forced into living orderly lives or else living none at all. Still the lawless ones hardly realized the change that was coming in their old haunts. Some of them scorned the thought that "two streaks of rust," as they contemptuously termed the railroad, could make any change in the existing order of things in the far stretching sand hills. They had yet to learn of the civilizing power of the railroads. The railroad came and crossed the creek a short distance below Lame Johnny's cabin and pushed on into the Northwest. Lame Johnny watched the men at their toil and mingled much with them, and for a time he became dissatisfied with the life he was living, but after the men had gone on with the railroad he dropped back into the monotony and wickedness of his old life. In fact, he was worse than ordinary after this quiet spell. Stealing horses, rustling cattle, and robbing stage coaches grew to be an active industry, and the cattlemen grew more and more enraged until finally, driven to desperation, they became openly hostile, and bands of angry cattlemen rode here and there through the sand hills hanging every man known to be a suspect, and it was an exceedingly poor time to be caught in bad company. From away to the east and south they came spreading out like a fan as they advanced to the northwest up into the edge of the somber Black Hills. They hanged them on the pine trees on the mountain side, on the whispering cottonwoods along the streams and, where trees were scarce, on telegraph poles and road crossing sign posts. Lame Johnny was one of the marked men and he was well aware of it, and realized his danger when rumor reached him of the events transpiring among the sand hills. He finally decided that it was best for him to get out of the country for a few months if he desired to live, and he was making preparations to that end late one afternoon when, on looking across the prairie, he saw a group of some half dozen horsemen riding toward his cabin. He could not determine at the distance they were away whether they were friends or enemies. Long exposure to danger had taught him caution, and picking up his rifle he stole out of the cabin down to the edge of the creek whose high banks hid him from view and worked his way along the stream to where he had kept his horse concealed since the first rumors of danger reached him. He crept up to the top of the creek bank and looked cautiously toward his cabin about which the horsemen were gathered. It was evident by their actions that they were strangers about the place. After satisfying themselves that he was not in the building they commenced to search about for him, and one of them discovering some scattered footprints in the sand forming a broken trail toward the creek they all started in that direction and Lame Johnny realized that he must run for it and trust that his fresh horse would distance their jaded ones. To decide was to act, and a moment later Lame Johnny rode his horse up the opposite bank and struck out boldly to the north. He was observed at once and his enemies immediately rode forward in full pursuit. Lame Johnny had not only the advantage of a fresh horse and of a good lead but also of knowing the lay of the country perfectly. He soon discovered that he needed all of these for his pursuers were the best mounted and were gaining on him steadily. Evidently they had come forth prepared for an occasion like the present. His only hope lay in night coming on and aiding him in his escape, but even that was denied him, for as the sun faded from sight behind the western hills the big white moon came smilingly up in the eastern sky and there was no darkness to help him. Some of his pursuers were within gunshot and an occasional bullet sang spitefully past in Lame Johnny's neighborhood, but he could not reply without turning about in his saddle, and he could not spare time from guiding his horse for that, and there was small chance of making a successful shot at the pace they were riding, for he was riding for his life and so were they. He determined to ride out across the creek and the railroad and deep into the sand hills where, perhaps, he could elude his enemies; if not he would surrender. Down the creek bank he rode and in another instant his horse was wallowing in a bed of quick sand, mired hard and fact, and steadily sinking deeper. There was no time for lingering. Lame Johnny balanced himself on the animal's back and leaped safely to the other bank up which he ran and there, standing on the railroad track with its rear lights twinkling in the moonlight stood a freight train, the engine down by the water tank getting water. Straight toward the way car Lame Johnny ran. The conductor standing out on the rear platform had witnessed the whole drama out under the moonlight and, knowing something of what was then going on in the cattle country, and not wishing to be involved in any of their troubles, when he saw the figure of the pursued running toward the car he swung his lantern high in a signal to the engineer to leave town. As it raised the second time in the air the globe was shattered by a rifle ball and the light went out. The horseman who had stopped on the other bank of the creek in an endeavor to find a place to cross, that was free from quick sands, had observed the signal and, fearing their man would escape on the train before they could cross and secure him, had shot out the light. The engineer had seen the signal and when he heard the shot that put out the light he realized there was some trouble behind that the conductor wanted to escape, and he pulled out of town with a vengeance just as Lame Johnny scrambled up on the back platform of the way car. "What do you want here?" asked the conductor shortly. "A ride out of the country," said Lame Johnny, panting from his recent severe exertions. "So it would seem. Come inside; this is a poor place just now for a discussion," said the conductor, as a bullet buried it |