350 



THE POPULAR EDUCATOR. 



We now give our readers the continuation of the words that 

 may be sang to the Exercises in the preceding pages, those to 

 " Cyprus," Exercise 33, being from the pen of Mrs. Mary 

 Howitt : 



2. To him they are but as the stones 



Beneath his feet that lie, 

 It entereth not his thoughts that they 



From him claim sympathy ; 

 It entereth. not his thoughts that God 



Heareth the sufferer's groan, 

 That, in his righteous eye, their life 



Is precious as his own. 



The following are the second and third verses of the words to 

 the " Spanish Chant," Exercise 34 : 



2. See to the harvest field 

 Gleaners have hasted, 

 Gathering the scattered ears, 

 None should be wasted. 

 FREELY WE ALL RECEIVE, 

 FREELY THEN WE SHOULD GIVE, 

 On Him " in whom we live" 

 All our care casting. 



3. Spring came and passed away, 

 Summer is ending ; 

 Autumn will soon decay, 

 With winter blending. 

 "While time is given us here, 

 Oh, may we prize it dear, 



In love and godly fear 

 Each moment spending. 



The following additional verses are to be sung to the air 

 called " Clifton Grove," Exercise 35. The words to this Exer- 

 cise were written by Joanna Baillie : 



2. The lady in her curtained bed, 

 The herdsman in his wattled shed, 

 The clansman in the heathered hall, 

 Sweet sleep be with you, one and all ! 

 We part in hopes of days as bright 



As this gone by ; good night, good night 1 



3. Sweet sleep be with us, one and all! 

 And if upon its stillness fall 



The visions of a busy brain, 



We'll have our pleasure o'er again, 



To warm the heart ! to cheer the sight ! 



Gay dreams to all ! good night, good night I 



MECHANICS. XI. 



THE MOMENTS IN THE LEVEE OF FORCES. THE WHEEL 

 AND AXLE. THE COMPOUND WHEEL AND AXLE. 



WE now return to Mechanics pure and simple, applyirg the 

 geometrical principles enunciated in previous lessons to 

 determine 



THE MOMENTS IN THE LEVER OP FORCES NOT PARALLEL. 



Two such forces, A p, A Q (Fig. 60), being supposed to meet at 

 some point, A, to which they are transferred, and there com- 

 pounded into a resultant A R, represented by the diagonal of the 



Fig. 60. 



parallelogram, A p R Q, and o being a point taken at random 

 on that diagonal, we can prove the following very important 

 proposition: 



The moments of two intersecting forces in reference to any point 

 on their resultant are equal to each other Now the moment of a 



Fig. 61. 



force in reference to a point, as has been already explained, is 

 the product of the force by the perpendicular dropped on it 

 from that point. In Pig. 60, therefore, the moment of A p in 

 reference to o, a point on the resultant, is A p multiplied into 

 O x, the perpendicular from O on A p. So likewise is the 

 moment of A Q in reference to O equal to A Q multiplied into 

 o T, the corresponding perpendicular. What I have then to 

 prove is that these products are equal. But they are equal ; 

 for, from the second geometrical principle above, tho areas of 

 the triangles A o p, A o Q, are -half these products ; and, by the 

 third, since these triangles stand on the common base A o, and 

 the line p Q joining their vertices, being a diagonal, is bisected 

 by A R, that is, by that base, their areas are equal. Tho 

 moments of A p and A Q, therefore, in reference to o are equal, 

 as I undertook to prove. 



Now, to apply this to the lever, using the same figure, let us 

 suppose the two forces to be A p, A Q, meeting, as I have stated 

 to be necessary, at some point A. Then it is evident, since there 

 is but one point fixed in the body, that there cannot bo equi- 

 librium unless the resultant of A p and A Q passes through that 

 point, and is there resisted by the supports that fix it. Tho 

 fulcrum, therefore, you see, must be on the resultant, and there- 

 fore taking o to be the fulcrum, we must have A p multiplied into 

 o x equal to A Q multiplied into O Y, that is, 

 the moments of the forces in reference to 

 the fulcrum must be equal. We arrive 

 thus at the two following modes of stating 

 the condition of equilibrium in a lever, 

 either of which may be selected for use as 

 the occasion requires : 



1 . In a lever, the forces not being parallel, 

 the power multiplied by the perpendicular 

 from tho fulcrum on its direction is equal 

 to the resistance multiplied by the perpen- 

 dicular on its direction. 



2. The power and resistance are to each other inversely as the 

 perpendiculars dropped from the fulcrum on their respective 

 directions. 



THE WHEEL AND AXLE. 



This useful mechanism, of which several forms are given in 

 Figs. 61, 62, and 63, is a kind of lever, or succession of levers, 

 revolving round an axis, from which they project at right angles. 

 Corresponding to this central axle 

 line is a cylindrical axle of some 

 thickness, round which winds the 

 rope which bears the resistance, or 

 weight, to be raised. In Fig. 61 is 

 the simplest form of the instrument, 

 consisting of an horizontal axle and 

 four levers, which are worked in 

 succession by the power. In the 

 ship's capstan for raising the anchor 

 (Fig. 62), the resistance acts horizon- 

 tally, a man pushing also horizontally 

 at the end of each lever, the power being multiplied in the pro- 

 portion of the number of levers and men. Wo have in Fig. 63 

 another form, where the levers are the spokes of a wheel, and the 

 power A works in succession on them along 

 the tire as they come round. 



The principle in all is the same, whether 

 tho resistance and power be parallel or not, 

 and may be understood from Fig. 64, which 

 represents a transverse section, the outer 

 circle being the wheel and the inner the 

 axle. The central line of the axle, which 

 you must conceive perpendicular to the 

 paper at the centre of these circles, is the 

 fulcrum, represented by the point o. The 

 line A B thus is seen to be the lever, at the 

 ends of which the power, p, and resistance, 

 W, act ; and, as already proved, these forces 

 must be inversely as O A to o B, which lines 

 are the radii of the wheel and axle respectively. When the power 

 and resistance act parallel to each other this is evident ; but the 

 same holds good were they not so to act, as in the capstan, where 

 the power is continually changing direction as the sailors go round; 

 for, referring again to Fig. 64, if the power were to act not in the 



Fig. 62. 



