dressing-gown. Schleier, m. 8, pl. -, veil. Sporen, m. -8, pl. -, spur. Taffet, m. -es, -8, pl. -c, taffeta. brush. Bahnstocher, m. 5, pl. -, tooth- Die Stadt und das Haus. Balken, m. -8, pl. -, beam. orchard. Baumschule, f., pl. -n, nursery. " spout Decke, f., pl. -n, ceiling. Dorf, n. -es, -8, pl. Dörfer, village. Erdgeschoß, n.-sses, pl. -sse, groundfloor. Fensterladen, m. -8, pl. -låten, window-shutters. Flecken, m. -8, pl/-, borough. Fußboten, m. -8, pl. -beten, floor. Gasse, ƒ. -, pl. -n, lane. Gefängniß, n. -e8, pl. -, prison. Gewächshaus, n. -es, pl. -häuser, green-house. Gewölbe, n. -8, pl. -, vault. Glockenspiel, n. -ce, 8, pl. -c, chime. Geffe, f. pl. -n, kennel. Hede, f, pl. -n, hedge. Hütte, f., pl. -n, cottage, hut. Kamin, n. -es, -8, pl. -e, chimney. cellar. Klofter, n. -8, pl. Klöster, convent, Küche, ƒ. -, pl. -n, kitchen. Landstraße, f. -, pl. -n, highway. town-house, council-house. Schelle, f. -, pl. -n, bell, (small.) room. Schloß, n. -πes pl. Schlösser, Schiefer, m. -, pl. -, slate. Speicher, m. -8, pl. -, loft, garret. Treppengeländer, n. -8, pl. -, stair- Thurm, m. -e8, -8, pl. Thürme, Vorstadt, f. -, pl. -ftätte, suburb. Weinberg, m. -es, -s, pl. -e, vineyard. Wiese, f. -, pl. -n, meadow. Ziegel, m. -8, pl. -, tile. LESSONS IN ALGEBRA.—No. III. THE following are the answers to the Exercises in Addition ANSWERS TO EXERCISES FOR PRACTICE. (1.) 6abcd4m+7; (2.) 3y — dx+hm+1; (3.) abm+bm -5y+x+16; (4.) 8am+3xy-11; (5.) 9ahy+16; (6.) 11ad+ 'xy; (8.) by +3(b − a)x+3a; (9.) 6ax+xy; (10.) 6b+44cdf3ay; (11.) 18a+4ax-5bx+63cx+3bz-17xy; (12.) 8ab6bc (14.) 3df+4ax+74y+30; (15.) 55a+686; (16.) 7(a+b); (17.) +4cd7xy+17mn+18fg 2ax; (13.) 8abc+25abd+5xyz; 2xy(a+b); (18.) 2ax+5aa+3x+3xxx; (19.) 7y+9yy+5xу· 6xx; (20.) 9aaa; (21.)7yyyy+12xx; (22.) — 6(a+b) — 12 (x -y)-13; (23.) 4(ab)+9a(x+y) — 6y; (24.) 15axy +3 bed; (25.) 17(x+y) — 9x (a—b); (26.) 15abc+2(x+y); (27.) 24xy-6abc+4mn-25a; (28.) 4a(x+y)+76(x+y)=(4a+7b) (x+y). ANSWERS TO ADDITIONAL EXERCISES. (1.) ax2+a2x+xy2+3by2+y3; (2.) 11a3-10a2b-14ab2+16 63; (3.) 10x32x2+3x-2; (4.) 2a+2b+2c+2d; (5.) a— 6f; (6.) 6y3; (7.) (a+b+c)x3+(b−c+d)x2; (8.) (m+n+2)z2 — (n+p+1)z. SECTION III. ART, 57.-SUBTRACTION is the finding of the difference between any two quantities or collections of quantities. EXERCISE 1.-Charles has 5a pears, and James has 3a pears. How many more pears has Charles than James? In this example, we wish to take 3a pears from 5a pears. Hence the exBut subtraction is denoted by the sign But 5a-3a=2a pression 5a-3a pears represents the answer. pears; which is the answer. EXERCISE 2.-A gentleman owns a house valued at £4,500; but he is in debt £800. How much is he worth? Here we have £4,500-£800 £3700. Ans. 58. Let us now attend to the principle upon which these operations are performed. Let us suppose that you open a book account with your neighbour, and that when cast up, the debtor side, which is considered positive, is £500, and the credit side, which is considered negative, is £300. On balancing the account, you find that he owes you £500–£300= £200. Now, if you take £50 from the positive or debtor side, it will have the same effect on the balance, as if you added £50 to the negative or credit side; on the other hand, if you take £50 from the negative or credit side, it will have the same effect on the balance, as if you added £50 to the positive or debtor side. 59. In like manner, if, in the expression 12a- 5a, you take 3a from 12a, it will have the same effect on the expression, as if you added 3a to 5a and retained the negative sign in the sum; thus, 9a-5a is the same as 12a- 8a. Again, if in the expression 12a-5a, you take 3a from 5a, and retain the negative sign in the difference, it will have the same effect on the expres-c8, -8, pl. Ställe,sion, as if you added 3a to 12a; thus, 12a-2a, is the same as 15a-5a. Statthor, n. -8, -8, pl. -e, gate, 60. Hence universally, taking away a positive quantity from an Treibhaus n. -es, pl. -häuser, hot-algebraic expression is the same in effect as adding an equal negative quantity; and taking away a negative quantia is the same as adding an equal positive one. house. Geiftliche, m. -r, pl. -n, clergy man. Gerber, m. -8, pl. -, currier. Hirt, m. en, pl. -en, herdman. Juwelier, m. -es, -8, pl. - brazier. Kupferstecher, m. -8, pl. -, engraver. Mäher, m. -8, pl. mower. Marktschreier, m. -6, pl. quack. Maurer, m. -8, pl. mason. Messerschmied, m. -es, -8, pl. -e, cutler. Megger, m. -8, pl. -,(See Fleischer.) Musikant, m. -en, pl. -en, musician, fiddler. Nachtwächter, m. -8, pl. -, watch man. Näherin, f., pl. -nen, seams tress. Naturforscher, m. -8, pl. -, natur alist. Obsthändlerin, f. -, pl. -nen, fruit woman. Papst, m. -es, pl. Päpste, pope. Berrücken'macher, m. -8, pl. -, hairdresser. Pfar'er, m. -8. pl. -, vicar, parson. Pferdehändler, m. -8, pl. -, horse- | Vaden, m -8, pl. -, cheek. dealer. Philosoph'. m. -en, pl. -en, philo- Pre'diger, m. -8, pl. -, preacher. Backenbart, m. -es, -8, pl. -bärte, whiskers. Bart, m. -es. -8, pl. Bärte, beard. Gaumen, m. -8, pl. -, palate. Glied, n. -es, -8, pl. -er, limb, member. Hats, m. -es, pl. Hälse, neck. Leber, f., pl. -n, liver. Muskel, f., pl. -n, muscle. Schädel, m. -8, pl. -, skull. Schlag'ader, f. pl-, . -n, artery. Wäscherin, f. -, -nen, washer- IV. MALADIES AND INFIRMITIES. Krankheiten und woman. Weber, m. -8, pl. -. weaver. Wechsler, m. -8, pl. -, money-Anfall, m. -es, -8, pl. -fälle, fit. changer. Wundarzt, m. -es, pl. ärzte, (See Chirurg.) Zahnarzt, m. -es, pl. -ärzte, dentist. Zuckerbäcker, m. -8, pl. -, confectioner. brechen. Blattern, pl. the small pox. Fieber, n. 8, fever. Geschwür, n. -es, -8, pl. -e, ulcer. cramp. Krebs, m. -c8, pl. -e, cancer. Bocken, pl. (See Blättern.) Quetschung, f., pl. -en, contu sion. Recept, n. -es, -8, pl. -e. prescrip tion. Salbe, f., pl. -n, salve. Verrenkung, f., pl. -ex, disloca- Wassersucht, f. -, dropsy. V. ARTICLES OF DRESS. Kleidungsstück c. Urenkel, m. -8, pl. -, great-grand-Hermel, m. -8, pl. -, sleeve. Ur'großvater, m -8, pl. -väter, great-grand-father. Verlobung, f., pl. -en, betroth ment. Augenlich, n. -e8, pl. -er, eye-lid. Atlas, m. -sses, pl. -sse, satin. Flor, m. -e, -8, pl. Flore, crape. Frack, m. -es, -8, pl. Fráce, dress coat. Franse, f. pl. -e, fringe. Halstuch, n. -es, -6, pl. -tücher, neck-cloth. Haube, f, pl. -n, cap. Kapre, f., pl. -n, cap. Kleid, n. -es, -8, pl. -er, dress, Kopfput, m. -es, head-dress. cushion. dressing-gown. Schleier, m. 8, pl. -, veil. Sporen, m. -8, pl. -, spur. Stiefelknecht, m. -es, -8, pl. -THE following are the answers to the Exercises in Addition Taffet, m. -es, -8, pl. -e, taffeta. ANSWERS TO EXERCISES FOR PRACTICE. (1.) 6abcd-4m+7 ; (2.) 3y — dx+hm+1; (3.) abm+bm −5y+x+16; (4.) 8am+3xy — 11; (5.) 9ahy+16; (6.) 11ad+ xy; (8.) by +3(b − a)x+3a; (9.) 6ax+xy; (10.) 66+44cdf3xy; (11.) 18a4ax 5bx+63cx+3bz-17xy; (12.) 8ab-6bc tooth-+4cd7xy+17mn+18fg2ax; (13.) 8abc+25abd+5xyz; (14.) 3df+4ax+74y+30; (15.) 55a+68b; (16.) 7(a+b); (17.) 2xy(a+b); (18.) 2ax+5aa+3x+3xxx; (19.) 7y+9yy+5xy. Zahnstscher, m. -5, pl. -, tooth-6xx; (20.) daaa; (21.)7yyyy+12xx; (22.) — 6(a+b) — 12 (x -y)-13; (23.) 4(a—b)+9a(x+y)—by; (24.) 15axy +3 bed; (25.) 17(x+y) — 9x(a — b); (26.) 15abc+2(x+y); (27.) 24xy-6abc4mn-25a; (28.) 4a(x+y)+76(x+y)=(4a+7b) (x+y). brush. pick. Die Stadt und das Haus. Balken, m. -8, pl. -, beam. Baumschule, f., pl. -n, nursery. Dede, f., pl. -n, ceiling. Dorf, n. -es, -8, pl. Dörfer, village. Erdgeschoß, n.-sses, pl. -sse, groundfloor. Fensterlaten, m. -8, pl. -låten, window-shutters. Flecken, m. -8, pl/-, borough. Fußboten, m. -8, pl. -böcen, floor. Gasse, f. -, pl. -n, lane. Gefängniß, n. -c8, pl. -e, prison. Gewächshaus, n. -es, pl. -häuser, green-house. Gewölbe, n. -8, pl. -, vault. Glockenspiel, n. -es, 8, pl. -e, chime. Goffe, f. pl. -n, kennel. Hecke, f, pl. -n, hedge. Hütte, f., pl. -n, cottage, hut. Kloster, n. 4, pl. Klöster, convent, -häuser, Küche, f. -, pl. -n, kitchen. theatre. Schelle, f., pl. -n, bell, (small) room. Schloß, n. -sses pl. Schlösser, Schiefer, m. -8, pl. -, slate. Sreicher, m. -8, pl. loft, garret. ANSWERS TO ADDITIONAL EXERCISES. (1.) ax2+a2x+xy2+3by2+y3; (2.) 11a3-10ab14ab2+16 63; (3.) 10x32x2+3x-2; (4.) 2a+26+2c+2d; (5.) a6f; (6.) 6y3; (7.) (a+b+c)x3+(b−c+d)x2; (8.) (m+n+2)z2 — (n+p+1)z. SECTION III. ART. 57.-SUBTRACTION is the finding of the difference between any two quantities or collections of quantities. EXERCISE 1.-Charles has 5a pears, and James has 3a pears. How many more pears has Charles than James? In this example, we wish to take 3a pears from 5a pears. Hence the exBut subtraction is denoted by the sign pression 5a-3a pears represents the answer. But 5a-3a=2a pears; which is the answer. EXERCISE 2.-A gentleman owns a house valued at £4,500; but he is in debt £800. How much is he worth? Here we have £4,500-£800 £3700. Ans. 58. Let us now attend to the principle upon which these operations are performed. Let us suppose that you open a book account with your neighbour, and that when cast up, the debtor side, which is considered positive, is £500, and the credit side. which is considered negative, is £300. On balancing the account, you find that he owes you £500-£300=£200. Now, if you take £50 from the positive or debtor side, it will have the same effect on the balance, as if you added £50 to the negative or credit side; on the other hand, if you take £50 from the negative or credit side, it will have the same effect on the balance, as if you added £50 to the positive or debtor side. 59. In like manner, if, in the expression 12a- 5a, you take 3a from 12a, it will have the same effect on the expression, as if you added 3a to 5a and retained the negative sign in the sum; thus, 9a-5a is the same as 12a-8a. Again, if in the expression 12a-5a, you take 3a from 5a, and retain the negative sign in the difference, it will have the same effect on the expres-es, -8, pl. Ställe,sion, as if you added 3a to 12a; thus, 12a-2a, is the same as 15a-5a. Statither, n. -es, -8, pl. -e, gate, Stockwerk, n. -es, -, pl. -e, story. 61. Upon this principle is founded the following GENERAL RULE FOR SUBTRACTION. Change the signs of all the quantities to be subtracted, i. e., of - and the subtrahend, or suppose them to be changed from to from-to; then if the quantities are ALIKE, unite the terms as in addition (Arts. 50, 51); but, if the quantities are UNLIKE, write the terms of the subtrahend after those of the minuend. (Art. 55.) OTHERWISE :-Put the quantity to be subtracted in brackets, and write it after the quantity from which it is to be subtracted, with the sign minus between them; then apply the Rules of Addition. Here, xy + 35. From xy+d, take 7ad-xy+d+hm. d-(7ad-xy+d+hm) =—7ad+2xy. hm. Answer. 66. On the other hand, when a number of quantities are to be introduced within the marks of parenthesis, with immediately preceding it, their signs must be changed. Thus, ―m+bdx+3h =— (m − b+dx —3h). EXAMPLES FOR PRACTICE. 1. From 6ab+7xy+18dfg, take 3xy+4ab8dfg. 2. From 35ax-21ab-37m, take 30m-15ab10ax. 3. From 9ay+19bx+22bc, take 12ay+316c+50bx. 4. From 8xy-10ab6d, take-12ab+10d+24xy. 6. From 18bc-xy+22gh, take 41xy —gh+bc. 7. From 21ax+y+ac―ay, take 4a-bc+x-yz —dc. (6ab12xy+ad). 13. Introduce the following quantities within a parenthesis 6da with immediately preceding, without altering their value; viz.-a+b-c-d+f+gh. -14da 3. 14da 4.-28 6da -16 5.-166 -126 46 6.-14da 6da 8da -166 +8da14. Also, abcdx+dƒ— x −y+ghf—be+xyz. 17.-14da 15. From 4xx+6bbb, take 3xx+4bbb. + 6da -20da 6dd+3d-4ddd-10de2dddd — 4dy 62. From these examples, it will be seen that the difference between a positive and a negative quantity may be greater than either of the two quantities. In a thermometer, the difference between 28 degrees above zero, and 16 degrees below, is 44 degrees. The difference between gaining 1000 pounds in trade, and losing 500 pounds, is equivalent to 1500 pounds. 63. PROOF.-Subtraction may be proved, as in arithmetic, by adding the remainder to the subtrahend. The sum ought to be equal to the minuend, upon the obvious principle, that the difference of two quantities added to one of them, is equal to the other. 23. From 2xy-1 Subtract -xy+7. 16. From 20yy-2y+12aaa, take 15yy-2y-12aaa. 17. From 8(a+b)+10(x+y), take 2(a+b)—6(x+y). 18. From 4(a+b)—16(x—y), take 17(a+b)+36 (x−y). 19. From 2a - aa+ba, take a — -4aa6ba. 20. From xx+3x-xxx, take 2x+3xx+10xxx. 21. From 18-25ab+20x+3y, take 3x+3y-25ab+1. 22. From 6(a-y)—17(a+y), take 3(a+y) — 7 (a—y). 23. From ax-xy-my-6, take 6ax-6xy-ay+46—7df. 24. From 66a-4b, take 20a-b-30a-16a-3b+5a. 2. From x+4x3y+6x¢y2+4xy3+y1, take x1 — 4x3y+6x°y2 — 4xy3+y. 3. From 4a3-8a'+16a5, take 3a3—4a1+5a3. 4. From a+b+c, take —a+b+c. 5. From 4a3-3a2x+ax2—2x3, take 2a3 — 4a2x — ax3 — x3. 6. Take x3-3x2+3x-1 from 2+3x2+3x+1. 7. From ax+by2 take ca2-dy2. LESSONS IN FRENCH.-No. LVII. By Professor LOUIS FASQUELLE, LL.D. § 49.-REMARKS ON THE PECULIARITIES OF SOME VERES OF THE FIRST CONJUGATION. (1.) In verbs ending in ger, in order to retain the soft pro-7ah+axy nunciation of the g, the e of the infinitive is preserved, whenever the g would come before a, or o: +10ah 64. When there are several terms alike in the subtrahend, they may be united and their sum be used. Thus, 31. From ab, subtract 3am+am+7am+2am+6am. Here ab-3am-am-7am-2am-6am-ab-19am. Answer, 32. From y, subtract a—a—a-a. Answer y — 4a. 33. From az- -bc+3ax+7bc, subtract 4bc—2ax+bc+4ax. 34. From ad+3dc—bx, subtract 3ad+7bx-de+ad. 65. The sign, placed before the marks of parenthesis which include a number of quantities, requires, that when these marks are removed, the signs of all the quantities thus included should be changed. Thus a- -(b-c+d) signifies that the quantities b-c and +d, are to be subtracted from a. Remove the parenthesis and the expression will then become a—b+c—d, an expression which has exactly the same meaning as the former. |