346 



THE POPULAR EDUCATOR. 



line A p, but along any other tangent to the large circle, the per- 

 pendicular from the fulcrum o on its direction would still be the 

 radius of the wheel ; and, by the general principle of the lever 

 established in this lesson, the power and 

 resistance would be still inversely as the 

 radii of wheel and axle. 



A treadmill, used for punishment in prisons, 

 is another instrument of this kind, the power 

 being the weight of the prisoners ascending 

 the steps placed on the outside of the wheel, 

 and the resistance the weight of the water 

 pumped, the corn ground, or other work done. 

 The windlass is another, turned generally by 

 a winch handle, and used to raise water from 

 wells, or lift goods into stores. In Fig. 21 



Pig. 6i. 



(page 188) the reader will find an example of the utility of the 

 wheel and axle as a mechanical power in the crane, by which two 

 men, by turning the winch-handle attached to the axle, are able 

 to lift a horse out of the steamer alongside of the quay. 



A particular form of the windlass, which was first invented in 

 China, and which may therefore be called the "Chinese windlass," 

 is given in Fig. 65, where only the axle is represented, consisting 

 of two parts, one thicker than the other, but both forming one 

 solid piece. The winch handle, 

 or wheel, is to the right con- ^ 

 nected with the larger axle. The 

 weight to be raised is suspended 

 from a hook attached to a pul- 

 ley, round which the lifting rope 

 passes, one part winding round 

 the thick axle while the other 

 unwinds from the thin. The 

 weight with each turn of the 

 wheel ascends by the difference 

 between the length of the rope 

 that winds and unwinds, that 

 is, by the difference between 

 the circumferences of the 

 two axles. Moreover, since 

 the weight is equally divided 

 between the two ropes which 

 ascend from the pulley, the 



Fig. 65. 



force acting at the circumference of each axle is half (he weight. 

 It is evident, moreover, that the power applied to the winch 

 handle has to balance the difference of the actions of these 

 forces at the axle, or the moment of the power must be equal to 

 the difference of the moments of these forces. But each force 

 being half the weight, its moment is half the weight multiplied 

 by the radius of the axle at which it acts ; and therefore their 

 difference is equal to half the weight multiplied by the difference 

 of the radii of the axles, or, which comes to the same thing-, 

 to the weight into half the difference of these radii. But the 

 moment of the power being that force into the radius of the 

 wheel, we immediately learn that 



In the Chinese windlass the power multiplied 

 by the radius of the wheel is equal to the resist- 

 ance multiplied by the difference of the radii 

 of the axles. 



THE COMPOUND WHEEL AND AXLE. 



This is a combination of wheels and axles, of 

 the kind already explained, made for the same 

 purpose as. the similar combination of levers in 

 Lesson VIII., namely, the mechanical advan- 

 tage of a multiplication of the effect of the 

 power. The wheel and axle being once clearly 

 understood to be a lever, there can be no 

 difficulty in. extending the rule which holds 

 good of the compound lever to this combina- 

 tion. In Fig. 66 is such a combination. By 

 cogged teeth the axle of each wheel works on the circumference 

 of the next succeeding, the power, p, being applied by a rope to 

 the circumference of the first wheel, which does not require teeth. 

 It is evident that, as explained of the compound lever, the condi- 

 tion of equilibrium must be that 



In the compound wheel and axle, the power is to the resistance 

 as the product of the radii of the axles is to the product of the 

 radii of the wheels. 



LESSONS IN GERMAN. XXI. 



SECTION XL. PECULIAR IDIOMS (contwwed). 



2Ba8 fur em ( 66. 5), literally, what for a, answers to the English 

 " what kind of," or simply " what ;" as : 2Ba8 fur em '-Bud) l;afcen 

 @ie ? what kind of a book have you ? 28ag fur etn 3fteffer ifl ka * 

 what kind of a knife is that ? gitr, in this connection, loses its 

 prepositional character, and may precede any case, as : 2Ba3 fur 

 SSudKt ftnb tte8 ? what kind of books are these ? SSaS fur SSudjer 

 l;akn @te ? what kind of books have you ? 2ftit n>a8 fur etnent 

 S3ttd;e fint <Sie 6efrf;afttgt? with what.kind of (a) book are you em- 

 ployed ? 



1. 3Ba3 fur is likewise used in the way of exclamation, corre- 

 sponding to " what,"* as : 2Ba3 fitr !EI)orl;eit ! what folly ! 28a 

 fur etn 9Wann ! what a man ! SOeld;, abbreviated from njelcfyer, is 

 used in the same manner, as : SBeld; etn 3ftann ! what a man ! 



2. 3eter and iegttclja- are often preceded by the indefinite article, 

 and are then, accordingly, inflected after the Mixed Declension. 

 (Sect. X.) They are never used in the plural, as : Dcr ob iete 

 9Jhnfd;en, or eineS jeten 2Jicnfd)en ifl gcnnfi, the death of every man is 

 certain. (Sin 3eter mujj ftevkn, every one must die. 



3. 2l(ler, unlike the English " all," is joined directly to its 

 noun without any article intervening, as:- 2lller SBetn, all the 

 wine. 2lllc3 SBaffer, all the water, etc. 



Our word "all," when connected with the names of countries, 

 towns, etc., as also in such phrases as "all day, all the time, 

 all my life," etc., is not expressed in German by all, but by 

 gan$, as : an$ Suropa, all Europe. auj SSo^men, all Bohemia. 

 3)tc ganjc djnjctj, all Switzerland. !Den gonjcn Sag, all the day, or 

 the whole day. J>ie gaitje 3ett, mcin ganjcS Seben, etc. 



2U(e or all, in some elliptical phrases, is equivalent to our " all 

 gone," "no more," and the like, as: cm elb if! afle, his money 

 is all gone. 



4. dUanrfKr answers to "many a," as: 2Wattd;cr 3Wd;e ifl im 

 gtucf (tdt), many a rich man is unhappy. 



5. olcfjcr is often preceded by the indefinite article, as also 

 by fetn, and is then, like jefcer and iegltdjer, inflected after the 

 Mixed Declension, as : @r ifl etneS fefctyen ScfeenS nictjt tuurtitg, he is 

 not worthy of such a (a such) life. 3d; I;ci6e fetn fetches 33ud;, I 

 have no such book. 



6. 2U(er, manner, fold;er (and ttelcfyer, see E. 1) often drop the 

 last syllable, and are then undeclined. Thus, aller, when it pre- 

 cedes a pronoun, is often abbreviated to all; manner, when it 

 precedes an adjective, often becomes ma net); foWjer (as also 

 ledger) is always thus abbreviated when it precedes the indefi- 

 nite article, as also, sometimes, when it precedes an adjective, 

 as: 3d) fyabe all me in elb Dtrfcttn, I have lost all my money. 

 3d; fyabc all btefe 58ud;er gcfauft, I have bought all these books. 

 Hi a nd; el)rltd;cr SDJann tft arm, many an honest man is poor. 

 old; etn Sag tft angcnefym, such a day is agreeable. old; 

 fd;onc3 Javier ifl rlieucr, such beautiful paper is dear, etc. It 

 should however be noted, that, as in the above examples, when 

 the abbreviated form is followed by an adjective, this latter, in- 

 stead of being inflected after the New Declension (Sect. IX. 2), 

 follows that of the Old ( 29). 



7. G'inigcr and etlid;er are regularly declined. They are 

 nearly synonymous, and answer to our words " some, a few," 

 etc., as : @r farad; nur etnige SBorte, he spoke only a few words. 

 (r Jjrtt nod; ctltcfje greunbe in S)eutfd;lanb, he has still some friends 

 in Germany. @r luotynt in cimgcr ntfcrnung son bcr tabt, he re- 

 sides at some, or a little distance from the city. Siarf; ctntger 3eit 

 fam er, after some time he came. 3d) I;abe nod; ct(id;c6 aftefyl, I still 

 have got some flour. @tlid;c5 fiel an ten 2Beg, some fell by the way- 

 side (Mark iv. 4). 



8. GtnjaS, besides the signification noticed in Sect. XIV. 2, has 

 also an adverbial use, and answers to " somewhat," as : @r tft 

 efiuaS alter, af3 id;, he is somewhat (or something) older than I. 

 (5-3 ift ctiuaS falter, al8 sorgeftcrn, it is somewhat colder than the day 

 before yesterday. 



VOCABULARY. 



916'legen, to lay aside. 



SlnMtdf, m. aspect. 



21n'nal)cnmg, /. ap- 

 proach. 



SSegeljen, to commit. 



S3cfcfyaf'ttgcn, to em- 

 ploy. 



a3efd;aier'te, /. hard- 

 ship. 



a3eft$'en, to possess. 



2?en.ntn'terung, /. ad- 

 miration. 



Stlten, to form, con- 

 stitute. 



it, 



SBltnr, blind. 



Savii'dcr, about 

 thereon. 



5>acn', of it, there- 

 of. 



Scnncd;, still, not- 

 withstanding. 



