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Prop. XXVII.-The area of a segment is equal to the area of a sector with the same arc, minus the area of the triangle formed by the chord and the two radii.

(In Fig. 29 the segment formed by the arc a c b and the chord a b is equal to the area of the sector formed by the two radii o a and o b and the arc a c b, less the triangle formed by the chord a b and the two radii o a and o b.)

135.-Problems Involving Circles and Areas of Circles.
Prob. 24. From a given arc form a complete circumference.

b

FIG. 30.

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From Prop. XXIII: The diameter 60 inches X3.1416188.496 inches.

Prob. 26.—If an engine has drivers 62 in. in diameter and truck wheels 30 in. in diameter, how many times will the truck wheels turn while the drivers are turning 100 times? Ans. 206.667 times.

From Prop. XXIII: The circumference of the drivers is 62 in. X 3.1416194.7792 in. The circumference of the truck wheels is 30 in. X 3.141694.248 in. In 100 revolutions the drivers will travel 194.7792 in. X 100 19.477.92 in., which divided by 94.248 in. (the distance traveled by the truck wheels in 1 revolution) = 206.667+ times.

Prob. 27. If a driver rolls 233.569 ft. in 14 revolutions, what is its diameter? Ans. 63.7261 in.

The distance traveled in 14 revolutions is equal to 2,802.828 in., or in 1 revolution 200.202 in., which is the circumference of the driver. From Prop. XXIV: 200.202 3.141663.7261 in.

in.

Prob. 28.-If there be a pressure of 160 pounds of steam in the front end of an 18 in. x 26 in. cylinder what is the total pressure on the piston?

Ans. 40,715.136 lbs.

From Prop. XXIV: The area of the piston is equal to the square of the diameter multiplied by .7854, therefore, 182324 sq. in., and 324 sq. in. X.7854254.4696 sq. in., the area of the piston. As the steam pressure is 160 lbs. per square inch the total pressure on the piston is 254.4696 X 16040,715.136 lbs.

Prob. 29. Suppose the sector a b o in Fig. 28 has the following dimenThe angle a ob measures 90° and the radius o b measures 8 in What is the area of the sector? Ans. 50.2656 sq. in.

sions:

The sector is from a circle having a radius of 8 in. and a diameter of 16 in. The area of a circle 16 in. in diameter is, 162.7854201.0624 sq. in. From Prop. XXVI : The measurement of the angle of the sector is 90°, which is 14 of 360°, the measurement of the entire circumference of the circle. The area of the circle, 201.0624 sq. in. ÷450.2656 sq. in., which is the area of the sector a b o.

Prob. 30.-In Fig. 29 suppose the chord a b equals 1 inch, and the radii o a and ob each equals 1 inch, and the perpendicular o d equals .866 inch, what is the area of the segment formed by the chord a b and the subtended arc a c b? Ans. .0606 sq. in.

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If the radius is 1 in., the diameter of the circle of which the segment is a part is 2 in. and the area of the circle is 22.78543.1416 sq. in. From Prop. XXVI: By measurement we find that the angle a o c is 60°, which is 1/6 of the entire circumference, and, therefore, the area of the sector a c bo is 1 of 3.1416 sq. in., or .5236 sq. in. From Prop. XX (January Magazine): The area of the triangle a bo is found by multiplying the altitude by half the base, therefore .866 in. X.5 in..433 sq. in. From Prop. XXVII: The area of the sector .5236 sq. in., minus the area of the triangle .433 sq. in.=.0906 sq. in., which is the area of the segment formed by the chord a b and the subtended arc a c b.

An Incident of Industrial Life

By W. L. French

B

ERNARD JOYCE was a brass finisher employed at a great car works. His history was not different from many another man who labors for some corporation that he and his family might have bread. From 7 a. m. until 6 p. m., barring the noon hour, he toiled for two dollars per day, and he was one of the best paid employes in the shop; making hand rails, mountings and ornaments for the coaches and putting them in place, to please the public eye and for its convenience. Out of his pay each month he endeavored to lay aside a few dollars against the rainy day that comes to all, and fortunate are those who are prepared for its coming. But try as he did, how could he lay aside much when the company owned the house, the lights, and the water, and charged roundly for each, and kept wages as low as possible.

Now and then Bernard lost a few days through illness, and work as steady as he could he saved but little, and at times he felt discouraged and discontented. Holidays there were none. Such days and Sundays he rested, to be ready for the morrow's grind. Then it takes money to enjoy a holiday, and the workmen at the car works had none to spend that way. He had married a nice girl with domestic tastes when he was twenty-three and she was twenty, and they had entered on the battle of life full of zeal and hope for the future.

There was now a little girl of seven, and there had been a little boy who had come and stayed three years with them, but his short life had been full of pain and sickness, and one day he yielded it

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employes receiving the best pay could hardly strike a balance between expense and income at the end of the week. The first reduction in the hours of labor had come immediately after a committee had waited on the manager, asking for better pay. The manager had claimed that the company could not pay more, that they would have to shorten the hours of labor because of so little work ahead. The committee knew it was untrue, and they still recognized it as such when the hours of labor were cut. The discontent grew, and the mutterings of dissatisfaction became louder, and the rules were enforced more rigidly by the company. It was evident that open warfare was only a question of a little time.

The employes as a whole were getting behind daily and they felt that a position that would not give them a living was not very valuable and might better be sacrificed in an endeavor to better themselves than to be held to under the conditions which they labored. This feeling was the stronger from the fact that the men knew the company could do reasonably well by them but was seeking to put them in a condition of absolute helplessness by involving them hopelessly in debt.

Another committee was sent to the manager with a demand for certain concessions. The manager told them he was powerless to grant their request and he advised them to go to the president who owned the controlling interest in the works, and they did so. At first the president tried to put them off with a flow of smooth talk, but when this failed he laughed at them and told them to go on and do their worst. Why should he care? He had an abundance if the works never operated another day. He could starve them all to death and laugh at their misery, and when it was all over he could start the works with new men.

The next morning the works were closed. The papers printed accounts of "the strike," some giving the real facts of the case, saying it was a "lock-out," but the majority coloring their accounts to suit the car company, for the men were not rich. The special police came with

their guns and clubs and in a few days that human vulture, the "scab," was in evidence, as he always is, no matter what conditions or compensation is given him. Without him, the employer would be helpless to break any strike; he seems to know his power to do evil and gloats over it.

The car company was spending far more money in the effort to defeat their old employes than would have paid the increase in wages asked by the men for several years. But then there is nothing like "running one's own business," even though it does cost lots of money. Some of the old employes went elsewhere while some went back to work worse off than before. All were starved out.

Bernard Joyce had been one of those who had gone away in search of work after the strike was lost, and he secured it. His old employers found him out and he was discharged after three monthsservice without any sufficient reason being given, but in his own mind Bernard was certain of the reason. He found other positions but they went the same way as the first one. It seemed impossible for him to secure permanent employment because of the "blacklist."

His family had reached his new home and when they came Joyce had, as he supposed, steady employment and there was no misgivings for the future in their minds. With loss of work their money was soon gone, and day after day Bernard tramped the streets along with the ever-growing army of unemployed. He found an odd job here and another there, barely enough to keep soul and body together without seeking charity, yet they were in actual want. His neat tidy appearance was gone and this was against him in his search for work, and day by day he grew more wretched and discour aged. At such times he brooded over the iniquities of the blacklist which kept men out of work at the will of their former employers, until that feeling came up in his heart which men have when they commit murder.

To add to the burden of his woes his wife grew ill. Poor food, and sometimes none at all; two miserable rooms in an unsanitary, rickety tenement house at the top of two flights of stairs were conditions which, with the worry over their misfortunes, had broken her health at last. A doctor was found who came to see her. He knew there was but little chance that he would ever receive his pay for coming, but he had gone hundreds of times before under just such conditions

and why should he not go again? There is so much misery and want in the world that one should do all he can to relieve it. This was his belief and he practiced it fully.

There was nothing he could do for Mrs. Joyce except to ease her days of pain before she sank into an eternal sleep, and that he did. There was no sorrow in her heart for the life she was leaving. Why should she sorrow for it? The happy days were gone. There was nothing ahead but sorrow, suffering and poverty. Only for her parting with husband and daughter she mourned. One day she closed her eyes and went away forever, leaving two sorrowing hearts behind. Bernard Joyce sold enough trinkets and household goods to give his wife proper burial. The thought or a pauper's grave for her was as death to him, and she was laid away quietly where the morning sun would shine on her unmarked grave.

Joyce decided that he and his little daughter, Mercy, would live where they were until the blacklist was forgotten, then they would move elsewhere, and Mercy should be properly educated and looked after. There was an old German woman living on the same floor who had been very kind to his wife. She promised to look after Mercy as much as she could, and although as poor as they she kept her word and often gave Mercy food she needed herself. The poor are kinder to the poor than are the rich, for they share their sufferings, know their needs, and feel the touch of human sorrows that warms the heart toward mankind.

Bernard Joyce's ambition was gone; his hope died with his wife. He commenced to drink and spent more time in and about saloons than he did looking for work and more of the little money he obtained went that way than at home. In fact, he had come to believe that for him the search for work was worse than useless. A few months of this kind of life made such a wreck of Joyce that his friends of former years would have deemed impossible. Through it all Mercy loved him, although at times when he was drunk she was afraid of his strange actions and would go away to bed and cry herself to sleep. Joyce endeavored always to be kind to the child, and he was except when liquor had dulled his sensibilities. He had formed a comradeship with one Thomas Black, a thief when he could be without fear of detection, a gambler and confidence man all the time. Joyce had been mixed with him in the doing of some deeds to secure money that were all but

criminal and his moral nature had been hardened up for something worse. Black had never earned an honest dollar and never expected to if he could avoid it. He had long dreamed of some grand stroke in crime that should yield him big returns and make him independent of petty crimes for all time and he believed the time had arrived for the carrying out of his project with the assistance of Joyce.

Because of an industrial depression many people drew their money from the banks and concealed it about their homes. Black, sneaking about different banks, had observed an old couple draw out a large sum of money, and he had followed them until they entered their home where he was certain they would conceal it. Believing there would be little danger in robbing people so old as they, he determined to secure Joyce's assistance and try and steal it that same night. Liquor and bad associations had done their work for Joyce and he readily fell in with the plot as laid before him by Black. If they were successful in getting the money-and there was no doubt they would be they I could quit the life they were leading, go elsewhere and live as honest men do. Their plans laid, they separated to meet at a certain saloon from whence they would go to the scene of their crime at an hour deemed most advisable.

Joyce went to his miserable home. Ordinarily he would not have gone there but the old German woman who usually looked after Mercy's welfare was away caring for a poor sick woman who lived across the street, and a sense of duty drew him home despite his drunken condition. Mercy had been much alone during the old woman's absence. She had grown thin and pale since her mother's death and she showed plainly the marks of neglect and want. Of late she had taken to lying down much of the time in a half stupor and if her father had been sober and the man of other days he would quickly have realized her needs and condition and given her the care she required. All the afternoon and evening she had lain, hoping her papa would come home and stay. Her only companion as the hours dragged slowly by was an old doll, as ragged and forlorn looking as the child herself, yet wonderfully dear and comforting to Mercy. At last her father had come but more under the influence of drink than usual, to brace him up for the crime ahead. She welcomed him gladly and for a few minutes played about but soon lay down again. He petted her and tried to prepare her some food from the

scanty store they had, but she would not eat, for it was medicine and care more than food the child needed.

Joyce grew restless. He had hoped that Mercy would fall asleep and that he could steal quietly away without her knowledge but she started uneasily at every sound and apparently could not sleep. At length the time came when he must go if he would keep his appointment with Black and he arose. Mercy looked at him pleadingly and begged him to stay with her. If he had been himself he would have realized her condition and not have left her. As it was he almost determined to break his agreement with Black and stay at home, but the thought of how much might be gained by the night's work overpowered his other reasonings and he decided to go. Alas for the reasoning of a mind bent upon crime. He kissed her good-bye and went away with her childish pleadings still ringing in his ears.

After he was gone Mercy lay and cried softly to herself; starting at every sound and shrinking in terror when a flickering street light cast fantastic shadows on the wall. Worn out at last she went to sleep with the angel of death hovering over her.

Soon after midnight Joyce and Black stood in front of the house they intended to rob. They were going to search for the money and if they could not find it they were then going to force the old couple to tell where it was hidden, secure it and make their escape. Black easily forced the lock and soon they were inside the house and the search was begun. The old man was a light sleeper and fancying that he heard someone moving about he quietly arose and securing a gun went on a still hunt for the intruders and came upon them unawares. There was a cannon-like roar from the old shot-gun and Black, with a scream of agony fell dying on the floor with a great hole torn in his chest.

Joyce made a dash out of the room before the old man could fire again, but as he ran across the lawn the owner of the house fired at him but failed to hit him. Out of the yard and down the street he ran. Now he heard someone coming on a run and he dodged into the nearest alley and stood quiet while the policeman on the beat ran by on his way to the scene of the disturbance. Residents of the neighborhood were making their appearance at doors and windows and Joyce slipped away as quietly as he could.

Before long Black's body would be recognized and the police would be looking for him as his companion in crime, and

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