s;o 



THE POPULAR EDUCATOE. 



Fig. 44. 



of the opposing sets, and their centres of application; and there- 

 lore, by the aid of tlie principle above established, learn that 



1. The Eesultant of a system of Parallel Forces, which act, 

 some in one direction others in the opposite, is in magnitude 

 the Difference of the Sums of the Opposing sets of Forces. 



2. Its Point of Application is had by finding the parallel 

 centre of each opposing set, and taking a point on the side of 

 the greater sum, on the production of the line joining these 

 centres whose distances from these points are inversely as the 

 sums of the opposing forces. 



For example : Suppose eight parallel forces are applied to the 

 eight corners of a box, five of 2, 4, 6, 7, and 9 pounds directed to 

 the east, and three of 10, 11, and 15 pounds to the west; the 

 resultant will be 8 pounds, acting towards the west and at a 

 point on the line joining the parallel centres of the two sets, 

 and outside the greater, whose distances from these centres are 

 inversely as 36 to 28. 



These principles, with others previously established, we now 

 apply to the Lever; first taking the cases in which the 

 forces, usually termed the " Power " and the " Resistance," or 

 " Weight," are parallel. The principle of leverage may be 

 understood by the aid of Fig. 44. Two balls, say of iron, 



connected by a 

 thin bar, are sup- 

 ported by a cord 

 at a point o. 

 How is this point 

 to be selected so 

 that the balls 

 may equally ba- 

 lance each other, 

 the weight of 

 the rod not being 

 taken into con- 

 sideration ? Again, having recourse to numbers, let the balls 

 be 13 pounds and 4 pounds, and their centres the points A and B ; 

 how is o to be found ? Evidently by cutting A B so that A o 

 be to B o inversely as 13 to 4 ; or, on dividing that line into 

 seventeen equal parts, so that four of them be in A O and 

 thirteen in B o. If the bar be supported by the cord from 

 above, or by a prop from below, at this point there is equilibrium. 

 This is the principle of the Lever, of which the ball, B, may be con- 

 sidered the Power, and the ball, A, the Resistance. We say, there- 

 fore, that the support, or prop, commonly called the " fulcrum," 

 must be so placed that the arms A o, B o of the lever on each 

 side of it be to one another inversely as the Power and Resist- 

 ance. 



But, as inverse ratio puzzles some persons, I shall put the 

 matter in another light. You observe that at the end, A, 

 of this lever, there are only 4 equal parts in the arm, but 

 13 pounds in the resistance, while in the arm, B O, the parts are 

 13, and the pounds only 4. Now, suppose the parts were all inches, 

 then if you at either end multiply the number of indies in an 

 arm by the number of pounds on that arm, you get the same 

 number namely, 52, for product. Choose any other numbers 

 different for 13 and 4, and the result is the same ; the numbers 

 at either end multiplied together give the same product. There- 

 fore another way of stating the Condition of Equilibrium in a 

 lever is, that the product of the Power and arm on one side 

 should be equal to that of the Resistance and arm on the 

 other. 



But here be careful to be clear as to what is meant by " the 

 product of Power and arm, Resistance and arm." This 

 puzzles some persons extremely, from its never being clearly 

 explained to them. Strictly speaking, the product of a force 

 and a line, or of a resistance and an arm, is nonsense. Multiply 

 a bag of flour by the Iron beam from the end of which it hangs, 

 and who can divine what the result of the operation is to be ? 

 neither flour nor iron, but something between ! Well, then, to 

 remove every possibility of confusion on this point, keep in mind 

 (as the example above shows) that we multiply numbers only, not 

 the Power and its arm, or the Resistance and its arm, but the 

 NUMBER which denotes the units of FORCE in one, by the NUM- 

 BER which denotes the units of LENGTH in the other. Then you 

 can make no mistake, there will be no confusion ; and you can still 

 say, knowing the meaning of your words, that the Power multi- 

 plied by its arm is equal to the Resistance multiplied by the 

 other arm. This product is commonly termed the " Moment" 



of the Power or Resistance, and the Condition of Equilibrium 

 is stated as follows : 



For Equilibrium in a Lever the Moments of the Power, with 

 reference to the fulcrum, and Resistance should be equal. 



ANSWERS TO QUESTIONS IN LESSON V. 



1. To prevent the perpendicular from his centre of gravity falling 

 outside his base as he springs on the fore-foot to advance. On coining 

 down to counterpoise the centre of gravity's falling forward. 



2. He draws his feet under the chair, in order to get a base over 

 which, by leaning forward, he brings his centre of gravity, aud lifts 

 that centre upwards by his muscular strength. 



3. He leans to the opposite side in order to keep the Common centre 

 of gravity of himself and bucket over the base of support. 



4. Else the perpendicular from his centre of gravity would meet the 

 ground in advance of his feet. 



5. Because the resultant of the forward motion, and the weight of 

 horse and rider acting at their common centre of gravity, is then more 

 apt to meet the ground outside the base of support of the horse's legs. 



6. Because in that case the perpendicular from the centre of gravity, 

 being lower down, is less apt to meet the ground outside the base 

 when the road slopes to one side. 



[It will be noticed that some of the figures which have been 

 employed in Lesson VI. in Mechanics, have been introduced a 

 second time in the present lesson. This has been done to spare 

 the reader the trouble and annoyance of having to turn' from 

 one page to another when reference has been made in the course 

 of a lesson to any figure which has been used before as a means 

 of illustrating the text. Whenever, therefore, any figure is re- 

 peated, it must be understood that this is the reason for its 

 repetition.] 



LESSONS IN FRENCH. XVI. 



SECTION I. FBENCH PEONUNCIATION (continued). 

 75. WE proceed with our illustrations of the nasal vowel sounds 

 im and in, om and on : 



The nh in the pronunciation of anh and orih must have a short 

 stopped sound, as in the im of timbre, and the on of bon-bon. 

 The full sound of n, which would give vann for vin, and bonn 

 for 6011, should be studiously avoided. 



IM. 



FRENCH. PRONUNCIATION. ENGLISH. 



Imbecile Auh-bay-seel Foolish. 



Impenitence Anh-pay-uee-tahns Impenitenct. 



Impdratoire Anh-pay-rat-oahr or t'wahr Mastcr-u-ort. 



Impossible Auh-po-sibl' Impossible. 



Limbo Lanhb Limb. 



Limpide Lanh-peed Limpid. 



FRENCH. 

 Cinq 

 Cheinin 

 Fin 



Instant 

 Medecin 

 Vin 



FRENCH. 

 Bombe 

 Comble 

 Lombard 

 N ombre 

 T'loinb 

 Trompette 



IN. 



PRONUNCIATION. 

 Sanhk 

 Sh'manh 

 Fanh 



Anh-staunh 

 Mayd'-sahn 

 Vanh 



OM. 



PRONUNCIATION. 

 Bonhb 

 Konhbl' 

 Lonh-bar 

 Nonh-br' 

 Plonh 

 Tronh-pett 



ENGLISH. 

 Five. 

 Road. 

 End. 

 Instant. 

 Physician. 

 Wine. 



ENGLISH. 

 Shell. 



Consummation. 

 Lombard. 

 Nvmler. 

 Lead (a. metal). 

 Trumpet. 



ON. 



FRENCH. PRONUNCIATION. ENGLISH. 



Bon Bonh Good. 



Canton Kahn-tonh Canton. 



Done Donh Tlwn. 



Long-temps Lonb-taunh A great while. 



Maison May-zouh House. 



Mon . Monh Mine. 



liaison Eay-zonh Reason. 



Eepondit Kay-ponh-dee Replied. 



76. The French word monsieur is pronounced by foreigners 

 all sorts of ways, except the right way, in common conversation. 

 The author knows of no one French word so much in use by 

 those who speak the English language as this, and yet pro- 

 nounced so variously and incorrectly. Let us analyse this word, 

 and, if possible, set forth its correct sound. 



Remember, then, that the n and r of the word monsieur tiro 



