124 



THE POPULAR EDUCATOR. 



4. You had pleased your teachers, because you had always obeyed 

 their commands. 5. Scarcely had our soldiers fortified the camp 

 when Csesar formed a line of battle. 6. We shall not sleep until 

 (before tliat) we have finished our business. 7. When the soldiers 

 have fortified the camp, they will prepare for the fight. 8. Take care, 

 boys, that you do not chatter. 9. The laws of the Spartans have tins 

 object, to instruct (that they may instruct) the youth in labours. 



10. No one doubted that you had always taken care of the boys. 



11. Tell me, by what consolation you have soothed the troubled mind 

 of your friend. 12. I know not why you have punished the boy. 

 13. I did not doubt that you had kept my precepts in memory. 14. 

 Do not chatter, daughters. 15. I come to ask you to walk (tJwt you 

 may walk) with me. 16. The soldiers ought to guard the city. 17. 

 Wisdom is the art of seeing. 18. We must obay the precepts of 

 virtue. 19. The art of sailing is most useful. 



EXERCISE 101. ENGLISH-LATIN. 



1. Sitierunt. 2. Esuriam. 3. Prseceptis meis non obedies. 4. 

 Pueri garriunt. 5. Patri non obedierunt. 6. Nescio cur patri non 

 obediverint. 7. Nemo dubitat quin boni pueri patri obediant. 8. Verba 

 mea memoria custodivit. 9. Curse mihi erit ut verba mea memoria 

 custodias. 10. Veniunt urbem muniturn. 11. Ars scribendi utilis est. 



12. Domus suee parietes coronis ornant. 13. Non prius doriniam 

 quam negotia finivero. 14. Negotia finivistine ? 15. Puerum punie- 

 bat, quum scholam intrabam. 



EXEKCISE 102. LATIN-ENGLISH. 



1. The father takes care that his son is well instructed. 2. 

 The father took care that his son was well instructed. 3. The citi- 

 zens fear that the camp is fortified by the enemies before the city. 

 4. The eyes have been clothed with very thin membranes. 5. When 

 the king was entering the city, the houses of all the citizens were 

 clothed and adorned with chaplets and flowers. 6. We shall not sleep 

 until (before that) your business is (shall have been) finished. 7. As 

 soon as the camp is (shall have been) fortified, the soldiars will pre- 

 pare for the fight. 8. We feared that the city had been blockaded by 

 the enemies. 9. Let the wicked be punished. 10. A good scholar 

 strives to be instructed in the knowledge of letters. 11. The city, 

 having been blockaded, ia punished by many evils. 12. A cultivated 

 man not only benefits himself but others also. 13. Boys must bo 

 diligently instructed. 



EXEKCISE 103. ENGLISH-LATIN. 



1. Custodiuntur. 2. Urbs custoditur. 3. Urbs custodietur. 4. 

 Urbs custodita est. 5. Curo ut urbs custodiatur 6. Nemo dubi- 

 tat quin urbs bene custodiatur. 7. Gives urbem custodire debent. 

 8. Cur cives non urbem custodiunt ? 9. Nescio cur cives non urbem 

 custodiant. 10. Metuo ut cives urbam custodiant. 11. Ad pugnain 

 se expedierunt. 12. Domus parietes floribus vestiti sunt. 



MECHANICS. IV. 



FORCES APPLIED TO TWO POINTS PARALLEL FORCES. 



THREE cases of forces applied to two points present themselves 

 for consideration. 1. When the lines of direction of the two 

 forces meet within the body. 2. When they meet without. 3. 

 When the two forces are parallel to each other. 



First Case. This is easily disposed of. When two forces 

 meet within a body, the point of meeting may be taken as the 

 point of application of both forces, which can there be com- 

 pounded into one ; and the case thus becomes that of a single 

 force applied to a single point. 



Second Case. Here also the two forces may be reduced to 

 one ; but as their directions meet outside the body, it is neces- 

 sary to show that their effect is the same as though the point 

 of meeting was a real point of application. This, in a future 

 lesson, can be demonstrated by a perfect proof ; but, in the mean- 

 time, the following considerations will satisfy you that it is true. 

 Let A p and B Q be the two forces applied to the points A and 

 B (as in Fig. 10), and o the outside point in which their direc- 

 tions meet. Also, let o R be the direc- 

 tion which their resultant would take were 

 the body extended to o and the forces 

 there applied. Suppose now that, in order 

 to extend it, a round bar of iron of uni- 

 form thickness is firmly soldered to it, so 

 as to include the line o R within its sub- 

 stance. The body being thus extended, 

 o may be considered a point of applica- 



rig. 10. 



tion of both forces, which we may conceive to be transferred 

 to it by two thin but strong wires, O A, o B, the mass of 

 which is so small that it may be neglected in comparison 

 with that of the body. The forces A p and B Q then evidently 



become one force, acting along O R on rod and body together, 

 and producing the same effect on both as though they acted 

 at A and B. But the effect taken separately of the resultant on 

 o K, and therefore of A r and B Q, is evidently the same 

 namely, a pressure along its length. Their effects, therefore, 

 on the body itself taken separately must be the same ; and o, 

 although outside, may be considered a point of application. 

 The two forces are reducible to one applied to the body at any 

 point on the line O u within the body. 



TWO PARALLEL FORCES. 



TJiird Case. The resultant single force can be determined in 

 this case also by the parallelogram of forces, but the proof given 

 by the greatest mechanician of antiquity -Archimedes of Syra- 

 cuse is, with a slight alteration, much preferable, on account 

 of its simplicity. I shall first take r _,__ o. 



two equal parallel forces, which act 

 in the same direction. Let A and B 

 (Fig. 11) be the points of applica- 

 tion, and their directions those of 

 the arrow-heads P and Q. Suppose, 

 moreover, that in magnitude they are 

 each one pound, or ounce, or ton say 

 one pound. Now, in the first place, the resultant, whatever 

 it be, must pass through the middle point of A B. The best 

 reason I can give you for this is, that the resultant cannot, 

 since the forces be equal, be nearer to one tha'j to the other. 

 If it were a tenth of an inch nearer to A, it should be also a 

 tenth nearer to B. 



Now, in order to find its magnitude and direction, let us sup- 

 pose that two other forces, A c, B D, each equal to a pound, are 

 applied to the body along the line A B in opposite directions. 

 These being equal, and therefore of themselves balancing each 

 other, can neither add to nor take from the effect of A P and 

 B Q, which may consequently be considered equivalent to the 

 four forces, A p, B Q, A c, B D. Let the two at A be now 

 compounded into one, acting in some direction between them 

 (I care not which), and let the same be done with the two at B. 

 Now produce these resultant directions backwards, until they 

 meet at c, and transfer the resultants themselves to that point. 

 Now resolve them back into their original components, and 

 you have two pounds, o c, and o D, acting against each other 

 parallel to A B, and two separate pounds pulling from o down- 

 wards parallel to A p and B Q. The two former cancel each 

 other, and there remain two pounds acting parallel to A P. 

 Hence we can say, that 



1. If two equal parallel forces act on a body in the same 

 direction, their resultant is parallel to either, and bisects, or 

 divides equally, the line joining their points of application. 



2. The resultant is in magnitude equal to their sum, or to 

 twice either force. 



As an example to illustrate, take two equally strong horses 

 pulling a carriage ; two equal forces are applied to the splinter- 

 bar, which give one force equal to double the strength of either 

 horse acting at its middle point. When the carriage is backed, 

 these forces are applied in the opposite direction directly to the 

 centre through the pole. 



We are now in a position to find the resultart of any two 

 parallel forces, the first step towards which is to determine the 

 resultant of any number of equal ones applied to a body at 

 equal distances along a line. The number may be either odd or 

 even. We shall consider each separately. First, take odd ; 

 and let it be seven, as in Fig. 12. Now, supposing each to be 

 one pound, if we take the middle one, which is evidently at the 

 middle of the line A B, we find that 

 there are three pounds on either 

 side of it acting in pairs at equal 

 distances from M. The resultant 

 of the nearest pair gives, as proved 

 above, two pounds at M ; the next i_. 



pair also give two, and so does the 



third. These make six pounds of resultant at M, which, with the 

 single one already there, are seven pounds the sum of all the 

 forces for resultant. Were the number thirteen the conclusion 

 would be the same. There would be six on either side of the 

 middle one, and you would have a resultant of thirteen pounds ; 

 and the same holds good of any other odd number you select, 

 be it large or small. 



13) 



