50 



THE POPULAR EDUCATOR. 



And this is true, whatever be the magnitude of the ball. It 

 may be as small as we please, even so small as an atom, or 

 what is called a " material particle," and yet there will be this 

 internal compression or straining. Thus we see that even the 

 "material particle," acted on by two equal and opposite forces, 

 cannot be said to be in the same condition before and after their 

 application. 



The case of equal and opposite forces presents some other 

 points of interest, which may well occupy your attention in this 

 lesson. Suppose, for example, two men pull against each other 

 with equal strength at the opposite ends of a rope. What will 

 be the strain on the rope? What will be its amount, considering 

 that both are pulling ? Most persons at first incline to say that 

 it is strained by the united strength of both, or by double the 

 strength of either man. Such is not the case ; the strain is 

 only equal to the strength of one of the men. What is the 

 reason of this ? A moment's reflection makes it evident. 

 Suppose one man only to pull ; the rope follows him, and there 

 is no strain on it. But the instant the other seizes his end and 

 pulls, strain begins, caused by his resistance. If he gives a 

 strong pull, it is great; if a weak, it is slight. But, to put this 

 in another way, suppose the first man leads, pulling with all his 

 might, while the other, holding on with less strength, is dragged 

 after. The rope is strained in this case also. By how much ? 

 By the less of the two forces. The stronger pull becomes 

 divided into two parts, one putting both the rope and the 

 second man in motion, and the other balancing the latter's pull. 

 It is this second portion which strains the rope, and must be 

 equal to the strength of the hinder man, while the other, which 

 causes motion, is the difference of the two pulls or forces. Sup- 

 pose, lastly, that the two pulls become equal, their difference 

 becomes nothing, motion ceases, and the men come to a stand- 

 still. But the strain remains, as before, equal to the hinder 

 force, which, being equal to that of the leading man, we can 

 eay it is equal to either of the forces. 



Let us next suppose that for one of the men an iron ring, 

 fastened on a wall, is substituted, to which one end of the rope 

 is attached. So long as the rope hangs loosely from the ring 

 there is no strain on it. Let the other man now pull at the far 

 end, the rope at once is strained, evidently not by the wall, but 

 by the man's pull. The wall puts forth no more effort to strain 

 it than it did before ; but simply resists the force communicated 

 to it through the rope. It is, in fact, a case of a force applied 

 to the wall through the rope, every point of which may be con- 

 sidered a point of its application. 



Again, take two equal weights attached to the ends of a 

 cord which passes over a pulley. The strain on the cord which 

 hangs down at either side is evidently equal to the weight on 

 that side ; and, since the weights are equal, the strains on both 

 sides, and therefore all through the cord, are equal to that 

 weight. 



In all these cases the forces were of the nature of a pull, 

 causing a stretching strain. But the conclusions hold equally 

 good of pushing forces. If two such, equal to each other, be 

 applied to a ball at opposite sides in opposite directions, the 

 compressing strain within the ball will bo equal to only one of 

 the forces. Or if the ball be pushed against a wall by only 

 one of them, though the wall resists, the strain will still be the 

 same equal to the single force. The resistance counts for 

 nothing. Also, when the two forces are unequal, and motion 

 ensues, there is a compressing strain equal to the smaller force, 

 while the motion produced is due to the difference of the forces. 

 When a man ascends a ladder with a hod of mortar, there are 

 two such compressing forces acting on his shoulder at the spot 

 on which the hod rests namely, his own muscular power push- 

 ing his shoulder upwards, and the weight of the hod and mortar 

 pushing it down. His ascent is effected by the difference of 

 these forces, the muscular being the greater ; while the com- 

 pressing strain is evidently the weight of the loaded hod. 

 These examples will make clear to yon the principle I have 

 been explaining; and you will find no difficulty in multiplying 

 them by thinking of others yourselves. 



We now pass to the case of three forces, whose directions are 

 all different, applied to a point, and producing equilibrium. 

 Now it is evident, first of all, that the three must pull or push 

 in the same plane or flat, such as, for instance, the flat surface 

 of a table ; for if two of them pulled along that surface, while 

 the third pulled in a slanting direction upwards, this latter f orca 



should lift the body off the table. Try the experiment with 

 three strings attached to a ring which lies flat on a table, two 

 of which are pulled horizontally along the table, and the third 

 in any direction upwards. The ring will be lifted, and soon the 

 three strings will come into one plane. I am not here taking 

 into account the weight of the ring and strings, which are a 

 fourth force applied to the body. For the sake of simplification, 

 to enable you to understand the principle, I suppose these to be 

 so small in comparison to the others as to count for nothing. 



Secondly, when three forces applied to a point are in equili- 

 brium, the resultant of any two of them is equal and opposite 

 to the third force. This is also evident ; for if it were not, the 

 resultant of the two and the third force, to which the three are 

 equivalent, would not be two forces equal, and opposite to each 

 other, and therefore could not make equilibrium. In the case 

 of the ring on the table, to which the three strings are attached, 

 if the direction of the effect of the pulls on two of the strings were 

 not opposite to that of the third pull, the three would make the 

 ring move to the side of the table, towards which these two 

 directions incline. And, furthermore, even if the directions 

 were opposite, the ring would move, if the effect of the two, or 

 their resultant, were not equal to the third force. These two 

 principles may be definitely stated as follows : 



1. When three forces applied to a point are in equilibrium, 

 they are in the same plane. 



2. The resultant of any two of three forces in equilibrium at 

 a point is equal and opposite to the third force. 



LESSONS IN GREEK. XV. 



EXERCISES FROM THE BOOK OF PROVERBS. 

 1. Tios o~o<pos tvcppaivfL iraTfpa, vlos 8e afyptaif AUTTTJ TTJ 

 2. Ilcpta avSpa rairfipo't, xetpes 8e avoptiwv ir\ovTiovffiv. 3. 

 EuAo-yia Kvpiov eirt Kf<pa\fj SIKO.IOV. 4. Mi/rj/x?} SIKCUUIV fj.fr' 

 fyKu/jituv (understand tffTi), ovop.a, 5e afff&ovs crfievvvTai. 5. 

 Micros tytipft vtiKos. 6. 'Os /c x'^ecoj' irpocr<J>fpei ffotpiav, pa/SSip 

 Tinrrei avSpa aKapSiov. 7. AJ/TJO Sij\uff<ros aTro/caAuirrei /JouAas 

 tv ffvvtSptia, iriffTos 8e irvori /COUTTTSJ Trpayfiara. 8. Tvvrj ffirovSaia. 

 ffrt<pavos T(f avSpt. 9. Aoyov aSi/coi/ u.ifffi Si/caios, aae^Tjs t 

 Sf aio~xwfTai. 10. 2i5?7pos ffi5i\pov owft, avt)p 8e irapo^wei 

 irpoffaairov tratpov. 11. 'CLcnrtp Spoffos (v a/uijT<j>, /cat itiffirtp ueros 



(V Oeptl, OUTWS OVK fffTIV CUppOVl TI/U9J. 12. A.KO.vQo.1 tyVOVTOU fi> 



X e 'P' neOvo~fj.ov, SouAeta Se tv x f 'P l " t<av atppovcuv. 13. 2o<ia KM 

 tvvoia ayadr) tv irv\ais ffoty<av (understand fifftv)- ffo<poi OVK 

 (KK\tvovfftv tK ffTouMTos Kvptov. 14. ATrotfj/rjcr/cei cuppcav tv 

 afj.apTia.is. 15. Mrj x a 'P f 7r ' KCLKOTTOIOIS, /njSe t]\ov a,fj,apTo>\ovs. 

 16. 4>o^ou TOV &tov, vif, /ecu jSacriAea. 17. Aoyois o~o(piav Tropa^aAAe 

 ffov ovs, /cat aKovf fu.ov \oyov. 18. EAeTjyUOcri'j'Tj /cat a\rj6tta 

 <pv\a.Kf) $ariAej. 19. Koa/Mos vfa.via.is o~o<pia, Soa Se Trpfcr^vrfptav 

 TroAiai. 20. lias avr\p (paivtrai tawrcp Si/caios, KaTfvOvvft 5e 

 /capSicts Kvptos. 21. A.Ko\affTov oivos, KO.I vfipicrriKov /ueflrj, TTOS 3t 

 cuppcav Towvrots o~vp.ir\fKfTO.i. 



VOCABULARY TO THE PASSAGES FROM THE PROVERBS. 



1. EvQpaivia, I rejoice (transitively) ; Auir>7, -ns, rj, grief. 



2. riei/ia, -as, y, poverty ; ra-rrftvow, I lower, degrade ; avopttos, 

 -a, -ov, manly, excellent ; irAourt^w, f make rich (from what noun 

 is the verb derived ?) 



3. Ev\oyia, -as, ij, a Uessing (what are the components of 

 the noun ?) Kupjor, -on, 6, lord, master, the Lord that is, the 

 Almighty, in the Old Testament ; SIKO.IOV for TOV SIKUWV. The 

 article is often omitted in the Greek version of the Hebrew 

 Scriptures. This version is called the Septuagint, sometimes 

 " the Seventy," because said to have been made by that number 

 of learned Jews in Alexandria in Egypt ; the translation was 

 completed in the second century before Christ. 



4. MVTJ/XTJ, -TJS, fi, memory, the memory ; fyKcau.iov, -ov, TO, praise, 

 eulogy, our word encomium; affffirjs, -ovs, impious, compare 

 o-f&o/j.ai, I ivorship ; ff$tvvvu.i, I extinguish; fffifvvvTai, is extin- 

 guished, that is, destroyed. 



5. Micros, -oDs, TO, hatred, connected with /xiereeo, I hate ; 

 vftKos, -ovs, TO, strife; here is exemplified the remark that the 

 Seventy are given to the omission of the article, for in Attic 

 Greek this proposition would be TO piffos tyeipei TO vetKos. 



6. 'Os, the relative pronoun he who; xeiAos, -ovs, TO, a lip; 

 pafiSos, -ov, TJ, a stick, staff; a/cap8jos, -ov (from a, not, and 

 KapSia, the heart), heartless, senseless. 



7. Aiy\uo-ffos (from Sis, twice, and yAwrTo, -ijs, r;, a tongue), 



