LESSONS IN 



alaiibt, fr fci a.ef.illcn. 



'tj 

 iiibte, id; ware nif t,i ^t- 



: t, cr (vertc nif UMeter gliiif 



:v;rte b.ilt tie Dber- 



Mben. 



VOCABULARY. 



Th-y i I been nick. 



iit I had never been* 



!>y again. 



It in said *he will soon have tho 

 ascendancy. 



RI':SUMK OF EXAMPLES. 



Cfr bebaiiF'tet, tap cS n>ar fei. Ho asserts that it is true. 



3d) will, tap" 11 frar'famcr I'eiefl. I will that thou be more frugal. 



< fetyeint mir, tap cr traxrig ifl. It appears to mo that he ia 



sorrowful. 



5Kan glaubt, tap n?ir rcid) fcicn. It is supposed that we are rich. 



vrcmte fcit, fo fcit ibr Although you are strangers you 

 ted; irillfenrmcit. are nevertheless welcome. 



(f febeint mir, tap fie llmcrifa'ncr It appears to mo that they are 



6* fiefct au, a(8 cb er nicbt gefunb' He appears aa though he were 



njare. not healthy. 



3d; glaubc, tap cr franf gcroc'fcn ifl. I think that he has been pick. 



2)fan fagt, tap er fcben b,icr gcnje'fcn They say that he has already 



fci. been here. 



3d; fecffe, tflji tu glutflid; gerce'fen I hope that you will have been 



fein rcivft. fortunate. 



Cr fagte mir, tap tu ba 33ud; bcS Ho told me yon had the 



vi-r babeft. teacher's book. 



3d? be$roeifle, tap ter 3agcr bic ffh'nte I doubt that tho hunter has tho 



^at. gun. 



3JJan wrmut^ct, tap ib> ict @clb It is supposed that you have 



Ijabt. much money. 



*Dian ivcip, tap ftc Srcute an ticfcr It is known that they have 



<gad;c babcn pleasure in this affair. 



3cf) bortc, tap cr cin grcpcJ S5cr I heard that he had a large 



me'gcn b/atte. fortune. 



cr Dnfcl enaM'te, tap cr cine an'. The uncle said (narrated) he 



gcneb.me Wcife ge^abt' ^abe. had had a pleasant journey. 



EXERCISE 80. 



1. J&abcn Sic aud; geljcrt, icb fci vem $fcrbe gefallen? 2. Srtein, id; 

 Ijorte, @ic fcien au tent 21>agen gcfaHen. 3. Die @efd;id;te meltct, tap 

 StU^, tte(d;er 'JJJagtcburg im trcifngiaftrigen JJricge ercberte, fcbr barbarifd; 

 tcr'abren fci. 4. lljein a?ruter fagte, Sic fcien fcbr gclebt iverten. 5. 

 !Eie 8ran;cfcn bebaurten, ftc fcicn tie (^ebiltetflcn in ber SBett. 6. 3^re 

 @cfiu;cflcr glaubtc, Sic waren in tcr 2tatt gcmefen. 7. Die SngJanter 

 ftnt tcr flWeinung, fie feien tic erren te SWcere*. 8. iefer SRcifcnte n> 

 jtlblte, cr fci ju'cimat in JRcnt getocfcn. 9. Sr ^efft, cr wcrtc in ad;t Sagen 

 in retcn fcin. 10. @ic furctjten, 2ic feien ju langfam im Jjantcln gcwefcn. 

 11. Sflir glaubten, ie warcn auf tent Vante. 12. 3d> glaubc, nsir roArcn 

 gcflcrn ^u (ud) gefcmmen, irtenn tn^ iJBctter fdjoncr getccfen a'Jre. 13. 3d; 

 glaubtc, cr ware tcr u\irnenten Stimmc fcincr (5'ltern eingetenf gcroefcn. 

 14. Cr fagte <irar, cr fei franf, abcr i ! iclc bebaurten, c? fci ^crflcllung 

 vru ibm gnr-et'cn. 15. Seine '-l ; cnuantten fagen, fcin (SMucf babe fcin 

 llnglucf bcrbeigcfubrt. 16. 3d; bertc mit i'etauern, Sic b.ltten ba 

 SJcrccnnebcr i^cbabt. 17. Ta irb in tern ebcrn Simmer irar, birte id; <ic 

 nicbt rufen. 18. SBJan er^lblt, tcr linear babe bi* in ten 'let fcin H i'ater 

 lanb trcu rcrtbciEigt. 19. 3d? lu-rtc, ticfcr jun^c Aranjefc tvcrte ein grepc* 

 ^ermogcn crben. 20. 3d; glaubc, tap vide i \ . auf Grten ibr 



Outca gcbabt baben iverteit. 



EXERCISE 81. 



1. People say the?o gentlemen have neen tipsy, but they are 

 mistaken. 2. They say that residence in Paris is more agree- 

 able than in London. 3. We could not believe that this was 

 true. 4. It ia universally believed that tho enemy has crossed 



tho frontier. 5. Ha MMrtod that it WM better to nUj * 1 

 than to go oat. 6. I wuh tbst b* mmjr bo trmtt'4 with .BOM 

 kindaen. 7. He telb every one that yon * ft ? ty nob man 5 

 von were, yoo would not be o paurkMU. 8. Have JOB 

 beard, too, tbat your friend DM fallen from hi* bone r 



:iat ho ha* fallen oat of the coach. 10. I hope 

 that you will be with your parent* in a fortnight. 11.1 <i-,-Aji 

 that ho can bo no ungrateful. 12. Thu iranger eayg that he 

 hoi been twice to India, and wa* very nick on hui lait voyage. 



LESSONS IN GEOMETRY. XII. 



As the next baton will pat the learner in poeeeation of the laet 

 of the problems tbat we intend to five on the eonetmotion of 

 figure* contained by three and four trai ( rht fee* namely, the 

 triangle, the square, the rectangle, and the parallelogram we 

 would recommend him to go carefully over the whole of the pre- 

 sent series of problems from the commencement, eonrtrocting M 

 many figures aa he possibly can, to meet the requirements of the) 

 data in each case. And in doing this we advise him to try to 

 construct figures different in form to tbose which we have given 

 in these pages, as, if he can do this, he may be sure that he has 

 gained a thorough knowledge of tho various methods of con- 

 struction set forth in the different problems. 



The problem in practical geometry that was brought before the 

 notice of the student in the last lesson, showing him how to 

 construct a square that shall be equal in superficial area to the 

 sum of two squares described on two given straight lines, has 

 given him tho key to the construction of squares, rectangles, 

 and parallelograms, equal in superficial area to the sum or 

 difference of any two or more squares, rectangles, or parallelo- 

 grams, as the case may bo ; and it has also shown him that the 

 main principle on which their construction depends, is the relation 

 between tho triangle, the figure contained by the least possible 

 number of straight lines (since two straight linos cannot CTK>!OS 

 a space, although one curved line can, as in the case of the circle), 

 and all regular figures contained by straight lines namely, the 

 square, tho rectangle, and the parallelogram. It may be as well 

 to repeat that this principle is, that when a square, rectangle, 

 or parallelogram is upon the same base and between the same 

 parallels, tho area of the square, rectangle, or parallelogram 

 (as the case may be), is double the area of the triangle. 



Now supposing we have a square, rectangle, or parallelogram 

 before us, and we wish to construct a triangle equal in area to 

 either of these figures, what have we to do ? Manifestly nothing 

 more than to draw one of the diagonals of the figure in question, 

 produce the base indefinitely in the necessary direction, and, 

 after setting off on it a straight line equal in length to the aide 

 of tho square, rectangle, or parallelogram, that serves as its 

 base, to join the extremity of the line thus set off with the 

 upper end of the diagonal. This will be evident on an inspection 

 of Fig. 42, where, in the square (rectangle or parallelogram) 

 A B c D, the diagonal A c is drawn ; the base c D, on which the 

 square (rectangle or parallelogram) A B c D stands, is produced 

 indefinitely in the direction of r ; a straight line, D X, set off 

 along it from the point D, equal to D c ; and the straight line 

 E A drawn, joining the points >: and A, and completing the triangle 

 A E c, which is equal in superficial area to the square (rectangle 

 or parallelogram) A B c D. 



And, conversely, when we wish to draw a rectangle or parallelo- 

 gram equal to a given triangle, all we have to do is to bisect 

 the base of the triangle, and on either half of the base construct 

 the required rectangle or parallelogram, after drawing through 

 the apex of the triangle a straight line parallel to the base. In 

 the case of the rectangle, after bisecting the base of the triangle 

 as, for example, in Fig. 43, where the base of the triangle ABO 

 is bisected in D and drawing a straight line, r Q, of indefinite 

 length, through the apex A of the triangle A BC parallel to its 

 base BC, a rectangle equal in superficial area to the hiangl" 

 A B c is formed by drawing the straight lines c B, D through 

 the extremities c and D of c D, one-half of the base B c, perpen- 

 dicular to B c, and meeting r g in K and r ; or by drawing the 

 piTpfiidicnlars D F, B a, through the extremities D and B ef B D, 

 tho other half of the base meeting F q i in v and a. 



In tho case of the parallelogram, if it be required to make two 

 of its opposite Hides equal to a given straight line, as the straight 

 lino x in Fig. 43, or two of its opposite angles equal to a given 

 angle, as the angle T, we must from the extremity of one-half 



