LESSON'S IN' rj KRMA.V. 





Fig. 23. 



EXERCIBt 



1 . lua count survey, ob*erratiou* are uiade from two rocks. A and 

 B, at **., ou a lighthouse and a headland, which are 2857 yard* apart. 

 At A the lighthouse bean 78 32' and the headland 30 2V : 

 At B the headland bean 65 15' and the lighthouie 2s 431 from A. 

 Bequired the distance between the rocks, and their bearing* from t!..- 

 lighthouse and tbe headland. 



PKOBUUI VL To find the height of an inaccessible object 



situated above the plane 

 of observation, and ita 

 height abovo that piano 



123). 



Take two stations, A 

 and B, in line with the 

 object, and measure A B ; 

 observe the angles B A <:, 

 F B c, F B D. From the 

 first two the total height, 

 F c, and the distance, F B, 

 may be obtained, as in 

 Problem III. F B and 

 FED being known, F D is 



obtained as in Problem I. F D = height above plane of observa- 

 tion, and F c - F D = height of object. 

 EXERCISE 14. 



1. In order to find the dimensions of a window of a cathedral, 

 which is inaccessible, I select two stations in front of it, 61 ft. apart. 

 From the far station the elevation of the top of the window is 31 30' ; 

 and from the near station the elevations of the top and bottom are 48 

 and 36 30' respectively. What is its height, and how high above the 

 ground ? 



2. A castle standing on the top of a cliff is observed from two 

 stations at sea, which ore in a line with it, and a quarter of a mile 

 apart. The elevation of the top of the castle, seen from the remote 

 station, is 16 P 28' ; the elevations of the top and bottom, seen from 

 the near station, are 52 14' and 48 38' respectively. What is its 

 height, and what its elevation above the sea ? 



A variety of useful miscellaneous examples will be found in 

 " Galbraith and Haughton's Trigonometry," from which th.3 

 foregoing exercises have been taken. 



KEY TO EXERCISES IN LESSONS IN PLANE TRIGO- 

 NOMETRY. III., V. 



EXERCISE 2.* 



1. A=61; B = 29; c = 35-065.: 11. 605 yds. and 718 yds. respec- 

 tively (omitting fractions). As 

 these distances were traversed in 

 equal times, the speeds ware pro- 



2. A= 61; 0=297-5; 1 = 164-815. 



3. A = 36; o=3-2715; c =5-56. 



4. B = 29; a = 39-71; c = 45'38. 



5. A=15; B=75; c = 2588-2 ft. portional to the distances; hence 



6. A= 36; B = 54 a ; b = 212.5-5. tho speed of the faster train was 



7. B=15; o= -00483; b=-001295. nearly 21f miles per hour. 



8. A = 29 ; B = 61 ; o ='485. 12. 67 yds. ; 60. 



9. A = 40; B = 50.t 13. 386'4 yds. ; 103'4 yds. 

 10. 34; 60ft. 



(No Key is required to Exercise 3.) 

 EXERCISE 4. 



1. A = 36 49' 36" ; B = 4537'; C= 97 33' 21". 



2. A = 48 38' 53" ; B = 45 9' 17" ; C = 86 11 ' 55". 



3. A = 45 33' 34" ; B = 59 40' '13" ; C = 75 45' 40". 



4. A = 96" 53" 16"; B = 48 46' 21"; C = 31 2<X 17". 



EXERCISE 5. 



1. A = 96 18' 21' ; B = 45" 20' 16" ; c => 136'1. 

 3. A =31 8' 53"; B = 19 15' 1" ; c = 7614. 



3. B = 119 31' 58"; C = 38 11' 32" ; a = 75'24. 



4. B = 54 2' 20", or 125 57' 40" ; C = 83 32' 40'', or 11 37' 20" ; 

 = 191-5, or 38-83. 



5. A = 42 2^ 49"; B = 100 54' 41" ; b = 77'208. 



6. A = 101 53' 31"; C = 12 42' 6" ; c = 49 '37. 



7. A = 90; B = 5141'; b= 78-46. 



EXERCISE 6. 



1. A =- 60 57' 10" ; b = 206-65 ; c = 220'55. 



2. A =22 52' 45"; b = 2221-8; c= 1513'1. 



EXERCISE 7. 



1. 929-54452 sq. ft, I 3. 1-335. I 5. 692'337. 7. -03818. 



8. -47523 *q. miles. | 4. 40535. | 6. 150769. 8. 1'710. 



* The above answers are only approximately correct, ths table of 

 ratios in Section X. having been purposely restricted to three places 

 ot decimals, to render calculations less difficult. 



t Certain ratios of these angles were accidentally omitted from the 

 table. 



l.KSSONS IN GERMAN. -LXXIV. 



| 132. It: 



A noon or pronoun which U the immediate object of an actm 

 transitive verb is pat in the acctwative ; as, JXcr frit taa4: 

 fcat Sail*, the dog guard* the hooM. 



OBSERVATIONS. (1.) The accusative, M before said, frtfaf 

 the ease of the direct or immediate object ( 129. 1) u Btiri 

 with all verbs, whatever their classification in other respect*, 

 that have a transitive signification. Accordingly, under **< 

 rule come all those impersonal and reflective verb* that take 

 after them tho accusative ; all those verbs having a eautative 

 signification; as, fatten, to fell, i.e., to caute to fall; as also 

 nearly all verbs compounded with the prefix be. The excep- 

 tions are begegnen, be6.aa.en, berteben, berub.en, bebarren, and Jxnxiebfen. 



(2.) Seljreii, to teach ; ncnntn, to name ; $eien, to call ; faelter, 

 to reproach (with vile names) ; tauftn, to baptise (christen), 

 take after them two accusatives: as, tr lebrt mu$ lit tcutfct 

 Srrac^e, he teaches me the German language ; cr ncnnt ibn fetnen 

 Wetter, he calls him his deliverer. (See Section LII.) 



(3.) The accusative is used with such terms as roiegen, to 



f often, to cost; gelten, to pass for; irertb, worth; 



fttyroer, heavy; reidj, rich; tang, long; n?eit, wide, to mark 



definitely the measure or distance indicated by these words; as, 



:ocf tfl einen 3uf. long, this stick is a foot long ; er 

 SJionate alt, he is four months old. In the earlier German, 

 these words of measure or distance were put in the genitive ; 

 as, enter panne n>eit, a span wide. 



(4.) As words expressing time indefinitely are put in the geni- 

 tive ( 128. Obs.), so those denoting a particular point, or dura' 

 tion of time, are put in the accusative ; as, i<$ trartete jiret Sage, I 

 waited two days. 



133.-EULE. 



A noun or pronoun, used merely to explain or specify that 

 which is signified by a preceding noun or pronoun, is said to be 

 in apposition, and must be in the same case ; as, (iuere, tin grcjec 

 yJctncr, Cicero, a great orator; ter JRatli meine* sfirutert, ttf 9ta$t<> 

 gelefyrten, the advice of my brother, the lawyer. 



OBSERVATIONS. The proper names of months, countries, 

 towns, and the like appellatives, ore put in apposition with 

 their common names, where, in English, the two words stand 

 connected, for the most part, by the preposition of; as, let 

 aKonat "luguft, the month (of) August; ti; Statt fionten, the 

 city (of) London ; tic llmerftt<5t Orfcrt, the University (of) 

 Oxford. 



134.-THE PRONOUNS. 



RULE. A pronoun must agree with the noun or pronoun 

 which it represents, in person, number, and gender ; as, !et 

 SDJann, n>elcf>er wetfe t|t, the man who is wise ; tie grau, el($e ffetBio. 

 iff, the woman who is diligent ; fca JZir.t, nxld;c flctn ift, the child 

 that is small. 



OBSERVATIONS. (1.) The neuter pronoun e is used in a 

 general and indefinite way to represent words of all ganders 

 and numbers ; as, c ijl let QNann, it is the man ; ti tjl kit Stau, 

 it is the woman ; ti tft tat Sim, it is the child; tl fmt tie 9N<inntr, 

 they are the men, etc. In like manner, also, often are used the 

 pronouns ba, (that) ; ttt, (this); a, (what) ; as also the 

 neuter adjective allt8, (all) ; as, ta ftnt tneine 9Ji^tet, these are 

 my judges. 



(2.) When tho antecedent is a personal appellation formed by 

 one of the diminutive (neuter) terminations d; e n and 1 1 i n, the 

 pronoun, instead of being in the neuter, takes generally the 

 gender natural to tho person represented ; as, too ift 3$r c$n- 

 Qtn 3ft et (not e) im arten Where is your little son ? Is 

 fie in the garden ? The same remark applies to 3Bfifc (iconwn) 

 and 8rautn$immtr (lady). When, however, a child or servant is 

 referred to, the neuter is often employed. 



(3.) A collective noun may in German, as in English, be re- 

 presented by a pronoun in the plural number : as fcit GkifUt$ftil 

 n?ar fflr tyre Ste^tc fcbc beforgt, the clergy were very anxious about 

 their rights. 



(4.) The relative in German can never, as in English, be sup- 

 pressed; thus in English we say, the letter (which) you wrote; 

 but in German it must be, ter firitf, nxtyen tu f<$riebft. 



(5.) The neuter pronoun tt. at the beginning of a sentence, 

 ia often merely expletive, and answers to the English word 



