236 



THE POPULAR EDUCATOR. 



lead a dependent one. 26. No man on earth can become quite 

 independent. 



EXERCISE 155 (Vol. III., page 139). 



1. 9Sorige3 Satjr fint tie 5rud;te md;t gut geratl)en. 2. 3)tefcr 93aum 

 tragt nut fetten grtid;te. 3. JBiefcr junge Jperr oerlApt ftd; $u met auf feine 

 SA[;tgfetten. 4. Sftein, er ttertagt fid; nid;t ju tiel auf feine Safyigfeiten, 

 tenn er toeip, bap eg nidfjt gut ift, ftd) auf bieienigen 2lnbercr ju oerlaffen. 



5. 3d) sertaffe mid) auf ie, bap ie mid) nacbfte 2Bud;e befudjen rcerten. 



6. Styun @te gerate, ali ob Sie $u Jjaufe toArcn. 7. 3)et 9Serbred;ct 

 ftcttte ftd;, at ob er walmfinnig toare. 8. !iefer 9Kann ftdlt ftd; gerate 

 nnc ein Jtinb. 9. S03o ift 3f)r J?aitariciwogc( ? @r ift jum Senfter IjinauS 

 gefUigen. 10. SBie faun id; in eine ad;e etimnKigcn, tie gegen mcine 

 JTleigungift? 11. (Sin jeter, ter fid; entjroett, mtrb au bent aufe ge 

 trteben, 12. 3 fyangt von llmftanben ab, ob id; jit meinen Sreunten 

 gefyen toerbe. 13. Setev SWenfd; ftrebt unabb.a'ngig ju roerten. 14. 33er- 

 laffe bid; barauf, tap icf> bir nicbt micter 6,elfctt werte 



EXERCISE 156 (Vol. III., page 179). 



1. It is not your fault that you are so unhappy. 2. It was not his 

 fault that he broke this glass. 3. I can give nothing for it, except 

 my thanks. 4. I shall state the reasons for it, if it be requested. 

 5. Can you tell me what o'clock it is ? 6. No, for my watch h^,s 

 stopped. 7. Has your watch stopped long ? 8. Yes, nearly an hour. 

 9. My watch goes too fast, it has gained nearly half an hour. 10. My 

 friend's watch is five minutes too fast. 11. Good-bye, and do not 

 forget to visit me soon again. 12. Good-bye, sir. 13. When shall 

 we both visit Mr. N. ? 14. It depends entirely upon you what time 

 you wish to appoint for it; I am ready at any time to accompany you. 

 15. It depends upon you to save or to ruin this family. 16. The 

 neighbour works in his garden, and tries to put it in order. 17. With 

 all his exertions he never settles this matter. 18. He tried to get 

 me into the ranks of his comrades. 19. It is difficult to accustom 

 a disorderly man to regularity. 20. After great trouble he lias 

 cleired up the account. 21. He who stops at the foot of a steep 

 mountain, and through fear of exertion omits to climb up it, and 

 prefers resigning a beautiful view, shows thereby that he is a weak 

 person, and unworthy of such enjoyment; and he who stops in the 

 midst of his intellectual cultivation through his own fault, and is 

 willing to dispense with the sweet kernel of wisdom because a rough 

 and hard shell surrounds it, likewise shows not only his unworthiness 

 to enjoy it, but also how little he has understood the vocation and 

 the duty of man as a spiritual being. 



EXERCISE 157 (Vol. III., page 179). 



1. 3d; fann nicfjt tafur, tap @ic bag llnglucf getyabt Ijaben. 2. ie 

 fonnten nid;t tafur, tap bie 3JZagb ten Setter jcrbrod;en fyat. . 3. @r fonnte 

 mir mcfytg tafur geben, al8 feinen 3)anf. 4. Sr fonnte md;ts tafur, er 

 fprad; nur tie SBafyrljeit. 5. JJann ter ,Rutfd)er etroa tafur, tajj bet 

 2Bajen umgeroorfen luurte? 6. Stein, er fonnte md;t tafur, benn bie 

 SJfcrte roarcn nid;t ju bcrnfyigcn. 7. .Ronnen ie mir fagen, toelcfye 3eit eg 

 ift? 8. Stein, meine ll(;r gc(;t nad;. 9. te tunbe meiner 2lbretfe ju 

 ceftimmen 6,angt con meinen (Sltern ab. 10. Seben ie tcob,!, 93tabam ; 

 unt sergeffen ie ntctyt, tn\b Sfiren @(tern ju empfe^fen. 11. @3 IjSngt 

 con Sfmen ab, te(d;e 3eit ie beftimmen hjotlen, Styre Sreunte ju 6efud;en; 

 id; luerte ftetS bcrctt fein, ie ju begkiten. 12. lurf unt Ung(ucf, Seben 

 unb Sob, 2lrmutb, unb 9icid;tt;um, ( Mc3 ^flngt son bem SKSittcn otte ab. 



LESSONS IN MENSURATION, IV. 



AREAS OF IRREGULAR FIGURES AND FIGURES BOUNDED BY 

 CURVED LINES. 



PROBLEM XII. To find the area of a regular polygon, the 

 length of the side being given. 



Eule : Find the radius of the inscribed circle by previous 

 rule ; then multiply the length of the side by the number of sides, 

 and this by the radius, and half the product will give the area. 



EXAMPLE. The length of the side of a regular pentagon is 

 3 ; what is its area ? 



Find the radius of the circumscribed circle, thus : 



ofAo 79 



The Z_ (angle) at the centre = *?- = 72 ; _ = 36 = /. 



A o c (Fig. 15, page 77). g 



Then half the side or 7, = 1'5 =base of right-angled triangle- 

 base 1'5 



A o c, and hypothenuse A o = 



5878 =2<55 ' 



EXERCISE 158 (Vol. III., page 179). 



1. The thief was convicted of his crime, and of course he will be 

 punished. 2. The father went away this morning, and has not yet 

 returned. 3. The book has been lost, and all these scholars pretend 

 not to know where it has gone to. 4. My nephews went away with- 

 out saying where they were going. 5. Our fruit is all gone. 6. Any 

 amount of money will go if one is wasteful. 7. The Turkish emperor, 

 Soliman II., said, shortly before his death, " My strength is gone, but 

 not my courage." 8. How far are you going to walk ? 9. I walk till 

 I get tired, generally as far as the park. 10. My friend knows very 

 tvell how far he has to go in this affair. 11. Even in joke one ought 

 to know how far one can go, because even in jest one may offend. 

 12. Where are you going ? 13. I am going to my attorney. 14. 

 How fur have you to go ? 15. To the end of the town. 16. How 

 long will it take you to walk ? 17. More than an hour. 18. How 

 far have you walked? 19. I have been as far as the river. 20. 

 How long have you been walking ? 21. I have been walking above 

 half an hour. 22. How long have you been from home ? 23. I have 

 been away three-quarters of an hour. 24. Have you been far away 

 from it ? 25. I hive been nearly half an hour's walk from home. 

 26. I hope to see you again, whether it be in this world or in the 

 next. 27. The prisoner thought it was now long enough for him to 

 have been obliged to dispense with the warm light of the sun t:nd 

 the fresh air. 28. I cannot come to your house to-morrow, unless 

 my brother should quite recover till then. 29. I cannot possibly 

 finish this letter to-day, unless 'L am less disturbed this after- 

 noon. 30. Nobody will be admitted into the town, unless he has a 



Then 



nat. sine ^_ 36 



Again, perpendicular o c = hypothenuse X nat. sine 

 \ or 2-55 X -3090 = 2'06 = radius of inscribed circle. 



3 X 5 XJLgg = 15-45, area of pentagon. 

 2 



EXERCISE 16. 



1. What is the area of a pentagon whose side is 3'82 ? 



2. The side of a hexagon measures 20 poles ; what is its area P 



3. The side of an octagon measures 20 yards ; what is its area ? 



4. The side of an equilateral triangle is 389 links ; required 

 its area. 



5. The side of an octagon is 156 feet ; what is its area ? 



PROBLEM XIII. To find the area of any irregular figure, the 

 boundary sides of which are straight. 



Eule : Divide the figure into separate triangles. If the- 

 diagonals are given, find at once the area of the respective 

 triangles, as explained in Problem XI., and their sum will be the 

 area. If the diagonals are not iven, they must be obtained by 

 actual measurement. 



EXERCISE 17. 



1. The four sides of an irregular figure are as follow: 

 BA = 12; AC = 20; CD = 18; and D B = 10 ; and the diagonal 

 from A to D measures 6. What is the area of the figure ? 



2. The four sides being as above, but the diagonal being 12, 

 what is the area ? 



3. A figure has five sides, as follows : A B = 22 ; B c = 18 ; 

 c D = 32 ; D E = 18 ; and E A = 20 ; and the diagonals E B and 

 BD measure respectively 23'25 and 16'75. What is the area of 

 the figure ? 



We shall now consider the superficial area of surfaces bounded 

 by curved lines ; and we request our reader to refresh his memory 

 by a reference to our remarks upon the proportion which, exists 

 between the diameter and the circumference of a circle. 



PROBLEM XIV. The radius of a circle being given, to find 

 its area. 



Eule 1 : Multiply the radius by the circumference, and halve 

 the product. 



Note. The circumference of a circle being to its diameter in 

 the proportion of 3'1416 (approximately) to 1, it follows that 

 its proportion to the radius is as 3'1416 (a number we shall 

 designate generally by IT) to J, and hence the truth of the 

 above rule. 



EXAMPLE 1. Eequired tlio area of a circle whose diameter 

 (D) is 1. 



D 1 . 1 IT 3-1416 _. 



Here R = O r - ; and * ne area 1S <r X~^ or r = '7854. 

 22 224 



This number may be with great advantage borne in mind by 

 the student, it being the area of a circle whose diameter is 

 unity. It is often used in estimating circular areas. 



EXAMPLE 2. The radius of a circle is 1. What is its area ? 



In this oase, R being 1, D is 2, and the circumference becomes 

 2* or 6-2832. 



Therefore the area is R X K = 31416. 



13 



EXERCISE 18. 



1. The diameter of a circle is 3 ; what is its area? 



2. The circumference of a circle is 3-1416; what is its area? 



3. The diameter of a circle is 4ft. 11 in. ; what is its area? 



