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to? or,

sage come?

n.

LESSONS IN GERMAN.—XV.

auf den Berg. 13. Wo geht unser alter Nachbar hin? oder, Wohin geht SECTION XXVII.-SEPARABLE PARTICLES—(continued).

unser alter Nachbar? ($ 89. 1.) 14. Gr ist jeßt in dem kleinen Garten,

aber er geht bald in den großen Garten. 15. Seine Frau ist in diesem o refers to the place where anything may be supposed to exist Hause, aber fein Vetter geht in jene Bildergallerie. 16. Ich stehe an dem or transpire, as :-Wo ist mein Messer ? Where is my knife ? Wo (am) Fenster, und sie fommen ans (8 4. 2) Fenster. 17. Der Ritter sißt laufen die Kinder ? Where (in what place) are the children run schon auf seinem guten Pferde, und der Knecht springt auch so eben auf sein ning?

gutes Pferd. 18. Der Mann sißt am ($ 4. 2) Tische, und das Buch liegt Da is used in answer to wo; that is, to designate some parti. auf dem Tische. 19. Ich habe keinen Hut auf dem Kopfe. 20. Wo geht cular place, as :-Da ist es, here it is. Da laufen fie, they are der Soldat hin? 21. Die Soldaten gehen aufs ($ 4. 2) Feld ; fie find running here.

schon auf dem Felde. 22. Der Frosch springt in den Fluß und schwimmt Vin denotes direction, or motion from the speaker, as : Warum in tem Flusse, und die Gans schwimmt auf dem Teiche. 23. Ich habe diese laufen die Kinder hin? Why are the children running thither? Worte irgendwo gelesen. 24. Ich kann meine Müße nirgends finden, ob

Her is the opposite, in signification, to hin; denoting motion or gleich sie irgendwo in diesem Zimmer sein muß. direction toward the speaker, as :-Warum laufen die Kinder her?

EXERCISE 45. Why are the children running hither ?

Hier signifies“ in this place," as :-Warum bleiben die Kinder hier? 1. Where is the picture-gallery of this town? 2. Where was Why do the children remain here?

that gentleman born (geboren]? 3. He was born in Bohemia. 4. These words are frequently compounded, one with the other ; Where does your friend, the actor, reside ? 5. He resides in the thus, from wo and bin, we have the compound wohin ; from wo city. 6. Whither do these emigrants go? 7. Whence do these and her, woher; from da and bin, dahin; from da and her, daher; from immigrants come ? 8. They come from France. 9. Where bier and hin, hierhin ; and from hier and her, hierher (sometimes con- much is given, much is required. 10. Here the revenge [Rache] tracted to þicher). ($ 103. 3.)

and whetted dagger (gewekte Dolch] of a traitor enter not (oringt Examples of the use of wo, da, hin, her, and hier compounded. nicht? ;-beneath (unter) the shade of this tree comes no king.

11. He threw down the book before me. 12. Whither art thou Wo reisen unsere Freunde hin? Where do our friends travel going? 13. I am going to my brother-in-law. 14. Will these ober,

emigrants go to America ? 15. No, they will stop here. 16. Wonin reisen unsere Freunde? Whither do our friends travel ? | There is water in the pond. 17. Where does she come from? Sie reisen tahin, wo ihre Ver- They travel thither, where their 18. She comes from Germany, wandten wohnen.

relatives reside. B. fommen diese Zugvögel her? Where do these birds of pas

VOCABULARY. det,

sage come from? Or,

Aus'wanderer, m. emi-| Dahin'ziehen, to pro- Italie'ner, m. Italian. Woher kommen diese Zugvögel ? Whence do these birds of pas- grant.

ceed thither. Italie'nisch, adj. Ita

Bedien'te, m. servant. Ein'wanderer, m. im- lian. Sie fommen daher, wo es ießt zu They come from (there) where Befann'te, m.acquain. migrant.

Milwaufee,

Milfalt für sie ift. it is now too cold for them.

tance.

Englisch, English. waukee.

Pöhmen, n. Bohemia. Euro'pa, n. Europe. Nie, never.
VOCABULARY.

Bremen, n. Bremen. Feldherr, m. comman- Spa'nien, n. Spain. Badstube, f. bake-; Kopf, m. head. Schwimmen, to swim. Dahin'gehen, to

go

der-in-chief. Spanisch, Spanish. house. Liegen, to lie. Seiler,m. rope-maker. thither.

Französisch, French. Theil, m. part. Balt, soon. Müße, f. cap. Sißen, to sit.

Dahin'reisen, to travel Fremde, m. stranger. Venedig, n. Venice. Bildergallerie, f. pic- Nirgends, nowhere. Springen, to spring, thither.

Havre, n. Havre. Wie viel ? how much? ture-gallery. Obgleich', although, leap.

Dahin'schiden, to send Hierher'fommen, to

how many ? broích, m. frog. notwithstanding. Stehen, to stand. thither.

come hither. Wohl, well. Cani, f. goose.

D'pernhaus, n. opera- Teich, m. pond.
Hin'gehen, to go to. house.
Werkstatt, f. work.

EXERCISE 46.
Hirte, m. shepherd. Ritter, m. knight. shop.

1. Dic Soldaten sind hier, und der Feldherr fommt auch hierher. 2. Stgenowo, some Soau'spieler, m. actor. Wohin'? whither ? Der Feind ist schon da, und unsere tapfern Brüder müssen dahin zichen. 3. where. Schon, already.

what way?

Wann gehen sie nach Spanien ? 4. Ich will gar (Sect. XIV. 3) nicht 3est, now.

Schwager, m. brother. Zud'erbäder, m. con- dahin gehen, aber mein Vater will in nächster Woche dahin reisen. 5. Sind in-law.

fectioner.

Sie schon da gewesen ? 6. Nein, aber einer meiner Bekannten war ta und

will nie wieder dahin gehen. 7. Wir gehen auf den Berg, wollen Sie mit RÉSUMÉ OF EXAMPLES.

und gehen? 8. Will der Russe seinen Bedienten in die Stadt schiđen? 9. Bo ist tas größte Glück: an dem Where is the greatest happi- Gr hat ihn schon dahin geschidt. 10. Werten die Truppen hierher fom

Hofe eines tyran'nischen Könige, ness: at the court of a tyran- men ? 11. Sie werden nicht hierher kommen. 12. Wo fommen riese cter in der Hütte eines zufriede. nical king, or in the cottage Fremden her? 13. Sie sind Einwanderer und fommen aus Böhmen. 14. nen Tagʻlöhners ?

of a contented day-labourer ? Ist dieses Schiff von Bremen over Havre ? 15. Es ist weder von Bremen, Do geben Sie hin? an den Hof oder Whither do you go? to the noch von Savre, es ist von Venedig. 16. Gehen diese französischen Eins in die Hütte ?

court or into the cottage ? wanderer nach Milwaufee? 17. Ein Theil von ihnen geht tahin, die Der Feltherr fißt auf dem Pferte The commander-in-chief upon antern bleiben in New - Yorf. 18. Die Einwanderer in Amerika sind

und reitet ruhig längs den Reihen the horse rides tranquilly Auswanderer aus Europa und aus andern Theilen der alten Welt. 19. der Sola'ten hin und her.

along the ranks of the sol. Wann wollen Sie auf das Feld gehen? 20. Ich bin schon auf dem Felde diers to and fro.

gewesen, und kann nicht wieder dahin gehen, aber ich muß ießt bald in den Mergen reitet er mit seinen Schaaren To-morrow he rides with his Garten gehen, denn mein Lehrer ist da und will mich sehen. 21. Warum auf das Schlachtfeld.

troops to the battle-field. will dieser Italiener nicht englisch sprechen? 22. Er möchte es wohl In te Hoffnung finden die Un'glüc. The unfortunate find consola- (Sect. XLIII. 4) sprechen, aber er kann es noch nicht ; er spricht nur lichen Troft.

tion in hope.

italienisch und spanisch. 23. Wie viel Sprachen fönnen Sie sprechen? 24. Der Bater ist da, aber ter Bruder The father is here, but the Icy spreche nur zwei, aber ich will noch andere lernen. ist in ter Start. brother is in the city.

EXERCISE 47. Ich gebe heute dahin, wohin ich schon I go to-day (thither) where I gestern gehen wollte.

wished (already) to go yester- 1. When did he live? 2. He lived in the fourteenth century day.

[im vierzehnten Jahrhuntert]. 3. My friend told me he would never

go there again (wieter]. 4. Do you go to Spain ? 5. No, I shall EXERCISE 44.

not go thither. 6. The commander-in-chief has sent his troops 1. Wo ist der Schwager? 2. Or fikt an dem (am) Tische. 3. Wo geht where the danger was most [die meiste Gefahr]. 7. Is this ship der Zuderbäder hin? 4. Er geht in die Badstube. 5. Wo ist sein Freund, from Spain or from Havre ? 8. No, it is neither [weder] from wa Shauspieler? 6. Er ist in dem Opernhause. 7. Wo geht sein Freund, Spain nor (noch) from Havre; it comes from Hamburg. 9. These ter Sailer, hin? 8. Er geht in seine Werkstatt. 9. Wo ist der Hirte? immigrants are going to Milwaukee, and are emigrants from 10. Er ist auf dem Berge. 11. Wo geht der Hirte hin? 12. Er geht | Bohemia and Venice. 10. Can you leap over that gate [Thor] ? LESSONS IN PENMANSHIP.-XVI.

turned in an easy and flowing stroke. To show the necessity of

this, the learner has only to turn the loop before reaching the In our last lesson, in Copy-slip No. 52, we gave an example of line h h, when he will find that this imparts a stunted appearthe elementary looped stroke which enters into the composition ance to the stroke, or to any letter into whose composition it of the letters j, y, and g, and, with a little modification, into enters, which is far from satisfactory. the formation of the letter Z. To make this new elementary To form the letter j, it is only necessary to place a dot above stroke, a thick down-stroke must be commenced at the line a a, the elementary looped stroke that has just been described, on as in Copy-slip No. 57, for example, and carried downwards in a the line d d, which is, as it has been stated in a previous lesson slanting direction towards the left. As the stroke approaches (pago 61), three-sixteenths of an inch above the line a a. In the line b b, the pressure on the pen must be lessened and Copy-slip No. 54 the elementary strokes entering into the com. gradually reduced until the thick stroke narrows into a hair- ! position of the letters y and g are shown, the first of these

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line, which is turned at the line h h, and brought upwards over letters consisting of the top-and-bottom-turn and the elementary the line b b, in a direction slanting upwards towards the right, looped stroke, while the second is formed by a combination of crossing the down-stroke in a graceful curve a little below the this stroke and the letter 0. In Copy-slips No. 55 and 56, the last-named line.

letters y and g are given, showing how the elementary strokes The distance between the lines b b and h h should be of which they are composed are joined together, while in Copsexactly nine-sixteenths of an inch. The learner, on referring slip No. 57 an example is given of the method in which the to Copy-slips No. 30 (page 133) and No. 39 (page 173) will see letter j is joined to any letter that follows it, and the letter y that letters carried below the line b b terminate on a line at the to a letter that precedes it. distance of seven-sixteenths of an inch below it, when the stroke The learner has now been taught how to make nineteen out of below b b is of uniform thickness throughout, as in the letter p, the twenty-six letters of the writing alphabet, and these we shall or has a bottom-turn to the right, as in the letter q. In the bring under his notice in a single lesson, after giving a few rore formation, however, of looped letters, an eighth of an inch more examples for practice in writing letters looped below the line bb, is required to give space enough to admit of the loop being i and combining them with others.

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to ? or,

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LESSONS IN GERMAN.-XV.

auf den Berg. 13. Wo geht unser alter Nachbar hin ? oder, Wohin geht

unser alter Nachbar? (8 89. 1.) 14. Er ist jeßt in dem kleinen Garten, SECTION XXVII.-SEPARABLE PARTICLES—(continued).

aber er geht bald in den großen Garten. 15. Seine Frau ist in diesem Bo refers to the place where anything may be supposed to exist Hause, aber fein Vetter geht in jene Bildergallerie. 16. Ich stehe an dem or transpire, as :-Wo ist mein Messer? Where is my knife ? Wo (am) Fenster, und fie fommen ans ($ 4. 2) Fenster. 17. Der Ritter sißt laufen die Sinter? Where (in what place) are the children run- schon auf seinem guten Pferde, und der Knecht springt auch so eben auf sein ning ?

gutes Pferb. 18. Der Mann sißt am ($ 4. 2) Tische, und tas Buch liegt Da is used in answer to wo; that is, to designate some parti- auf tem Tische. 19. Ich habe keinen Hut auf dem Kupfe. 20. Wo geht cular place, as :-Da ist ce, here it is. Da laufen fie, they are der Solrat hin? 21. Die Soldaten gehen aufs ($ 4. 2) Feld; fie sind running here.

schon auf dem Felde. 22. Der Frosch springt in den Fluß und schwimmt Vin denotes direction, or motion from the speaker, as :- Warum in tem Flusse, und die Gans schwimmt auf dem Teiche. 23. Ich habe diese laufen die Kinder hin? Why are the children running thither? Worte irgendwo gelesen. 24. Ich fann meine Müge nirgende finden, ob

Her is the opposite, in signification, to hin; denoting motion or gleich sie irgendwo in diesem Zimmer sein muß. direction toward the speaker, as :-Warum laufen die Kinder her?

EXERCISE 45. Why are the children running hither ?

Hier signifies " in this place," as :-Warum bleiben die Kinder hier? 1. Where is the picture gallery of this town? 2. Where was Why do the children remain here?

that gentleman born (geboren]? 3. He was born in Bohemia. 4. These words are frequently compounded, one with the other; Where does your friend, the actor, reside ? 5. He resides in the thus, from wo and hin, we have the compound wohin ; from wo city. 6. Whither do these emigrants go? 7. Whence do these and ber, woher; from da and hin, dahin; from da and her, daher ; from immigrants come ? 8. They come from France. 9. Where hier and hin, hierhin ; and from hier and her, hierher (sometimes con- much is given, much is required. 10. Here the revenge [Rache] tracted to hicher). ($ 103. 3.)

and whetted dagger (geweßte Dolch] of a traitor enter not (bringt Examples of the use of mo, ba, bin, her, and hier compounded. night) ; beneath [unter) the shade of this tree comes no king.

11. He threw down the book before me. 12. Whither art thou We reisen unsere Freunde hin? Where do our friends travel going? 13. I am going to my brother-in-law. 14. Will these cter,

emigrants go to America ? 15. No, they will stop here. 16. Wohin reisen unsere Freunde ? Whither do our friends travel ? There is water in the pond. 17. Where does she come from ? Sie reisen tahin, wo ihre Ver. They travel thither, where their 18. She comes from Germany. wandten wohnen.

relatives reside. Wo fommen diese Zugvögel her? Where do these birds of pas

VOCABULARY. cer, sage come from ? or,

Aug'wanderer, m. emi- Dahin'ziehen, to pro- | Italie-ner, m. Italian. Wober fommen diese Zugvögel ? Whence do these birds of pas- grant.

ceed thither. Italie'nisch, adj. Itabage come?

Bedien-te, m. servant. Gin'wanderer, m. im. lian. Sie tommen daher, wo es jeßt zu They come from (there) where Bekann'te, m. acquain- migrant.

Milwaufee,

Milfalt für fie ift. it is now too cold for them.

tance.

Englisch, English. waukee.

Pöhmen, n. Bohemia. Euro'pa, n. Europe. Nie, never.
VOCABULARY.

Bremen, n. Bremen. Feltherr, m. comman- Spa'nien, n. Spain.
Bad-ftube, f. bake-, Kopf, m. head. Schwimmen, to swim. Dahin'gehen, to go der-in-chief. Spanisch, Spanish.
house.
Liegen, to lie.
Seiler,m.rope-maker. thither.

Französisch, French. | Theil, m. part. Bald, soon. Müße, f. cap. Sißen, to sit.

Dahin'reisen, to travel | Fremde, m. stranger. Venedig, n. Venice. Biltergallerie, f. pic. Nirgends, nowhere. Springen, to spring, thither.

Havre, n. Havre. Wie viel ? how much? ture-gallery. Obgleich', although, leap.

Dahin'schiden, to send Hierher'kommen, to

how many ? Brody, m. frog. notwithstanding Stehen, to stand.

thither.

come hither.

Wohl, well.
Gans, f. goose.

D'pernhaus, n, opera- Teich, m. pond.
Hin'gehen, to go to. house.
Werkstatt, f. work.

EXERCISE 46.
Hitte, m. shepherd. Ritter, m. knight. shop.

1. Dic Soldaten sind hier, und der Feldherr fommt auch hierher. 2. Srgentmo, some Schau'spieler, m. actor. Wohin'? whither ? Der Feind ist schon da, und unsere tapfern Brüder müssen dahin ziehen. 3. where. Schon, already.

what way? Wann gehen sie nach Spanien ? 4. Ich will gar (Sect. XIV. 3) nicht Jest, now.

Schwager, m. brother-Zuc'erbäder, m. con- tahin gehen, aber mein Vater will in nächster Woche tahin reisen. 5. Sind in-law.

fectioner.

Sie schon da gewesen? 6. Nein, aber einer meiner Bekannten war ta und

will nie wieder dahin gehen. 7. Wir gehen auf den Berg, wollen Sie mit RÉSUMÉ OF EXAMPLES.

ung gehen? 8. Will der Russe seinen Bedienten in die Stadt schiden? 9. Be ift tas größte Glüd: an dem Where is the greatest happi- Er hat ihn schon tahin geschi&t. 10. Werten die Truppen hierher fom

Hofe eines tyran'nischen Könige, ness: at the court of a tyran- men ? 11. Sic werden nicht hierher fommen. 12. Wo fommen diese ste ir ter Hütte eines zufrie'tes nical king, or in the cottage Fremden her ? 13. Sie sind Einwanderer und fommen aus Böhmen. 14. nen Tagʻlöhners ?

of a contented day-labourer ? Ist dieses Schiff von Bremen oder Havre ? 15. && ist weder von Bremen, Bo gehen Sie hin? an den Hof oder Whither do you go? to the noch von Havre, es ist von Venedig. 16. Gehen diese französischen Ein. in die Hütte ?

court or into the cottage ? wanderer nach Milwaufee? 17. Ein Theil von ihnen geht tahin, die Der Feltherr Rißt auf dem Pferde The commander-in-chief upon andern bleiben in New York. 18. Die Einwanderer in Amerita sind

unb reitet ruhig längs den Reihen the horse rides tranquilly Auswanderer aus Europa und aus andern Theilen der alten Welt. 19. ter Solba'ten hin und her.

along the ranks of the sol. Wann wollen Sie auf das Feld gehen? 20. Ich bin schon auf dem Felde diers to and fro.

gewesen, und fann nicht wieder dahin gehen, aber ich muß ießt bald in den Mergen reitet er mit seinen Schaaren To-morrow he rides with his Garten gehen, denn mein Lehrer ist da und will mich sehen. 21. Warum auf das Sdlachtfeld.

troops to the battle-field. will dieser Italiener nicht englisch sprechen? 22. Er möchte es wohl ÍR ta Hoffnung finden die Un'glüd. The unfortunate find consola- (Sect. XLIII. 4) sprechen, aber er kann es noch nicht; er spricht nur lisen Trost.

tion in hope.

italienisch und spanisch. 23. Wie viel Sprachen fönnen Sie sprechen? 24. Der Bater ist da, aber der Bruder The father is here, but the Ich spreche nur zwei, aber ich will noch andere lernen. ift in der Stadt. brother is in the city.

EXERCISE 47. 34 gebe heute tahin, wohin ich schon I go to-day (thither) where I gestern geben wollte.

wished (already) to go yester- 1. When did he live? 2. He lived in the fourteenth century day.

[im vierzehnten Jahrhundert). 3. My friend told me he would never

go there again (wieder). 4. Do you go to Spain ? 5. No, I shall EXERCISE 44.

not go thither. 6. The commander-in-chief has sent his troops 1. Bo ist der Schwager? 2. Or fikt an dem (am) Tische. 3. Wo geht where the danger was most [die meiste Vefahr]. 7. Is this ship ta Zuderbåder hin? 4. Er geht in die Badstube. 5. Wo ist sein Freund, from Spain or from Havre ? 8. No, it is neither [weber] from bec Stauspieler ? 6. Er ist in dem Opernhause. 7. Wo geht sein Freund, Spain nor (noch) from Havre; it comes from Hamburg. 9. These te Seiler, bin? 8. &r geht in seine Werfftatt. 9. Wo ist der Hirte? immigrants are going to Milwaukee, and are emigrants from 10. Gr ift auf dem Berge. 11. Wo geht der Hirte hin? 12. Er geht | Bohemia and Venice. 10. Can you leap over that gate [Thor] ?

come over.

11. I could when I was young. 12. He bade (bat] me go thither, ab, hinauf geht unser Lauf. 8. Das Reh sprang den Berg hinab, während that he might speak to me about it.

ber Hase den Hügel hinauflief. 9. Die Soldaten ftürzten aus der Caserne

heraus, als der Feind in die Stabt hineinstürmie. 10. Als die Nachtwuche SECTION XXVIII.-SEPARABLE PARTICLES-(continued). in das Haus trat, eilte der erschrođene Dieb die Treppe herunter. 11. 3ch

Wo, ba, hin, etc., besides being compounded one with another fann nicht aus den Kreuzwegen dieses Gartens hinausfominen. 12. Wiljen (Sect. XXVII.) are also united with prepositions; thus pro- Sie nicht, wie dieser Vogel hereingekommen ift? 13. Ja, aber er weiß ducing a separate class of adverbs, as :-Wovon sprechen Sie? Of nicht, wo er wieder hinauskommen hann. 14. Der junge Schweiger schaute what (whereof) are you speaking ? Ich spreche von meinen Büchern; hinüber nach den blauen Bergen seines Vaterlantes. 15. Kommen Sie wollen Sie eins davon haben? I am speaking of my books; will you heute nicht herunter? 16. Ja, wenn der Oheim heraufkommt, werde ich have one of them ? (one thereof ?) Ich bin auf dem Dache ; foms hinabgeben. 17. Haben Sie diesen Mann schon gesehen? 18. Ja, er lam men Sie herauf! I am on the roof; come up! Ich kann nicht zur Thüre berein, als ich hinausging. 19. Der Freund fuhr in einer Stunte hinauf gehen; kommen Sie berab! I cannot go up; you come

den Fluß hinüber und herüber. 20. Der Strom stürzt mit großem Geräusch down!

den Felsen herab. Hinab, hinauf, hinaus, herab, etc., when used with nouns, are

EXERCISE 49. translated by prepositions ; and the adverb, unlike its English

1. The son hastened down to receive his father. 2. His equivalent, is placed after the noun, as :-Id gebe den Berg speech lasted over two hours. 3. The roe sprang out from his hinauf, I go up the mountain. Kommen Sie den Berg herab, hiding place. 4. Will you go over to Frankfort to-day by. the come down the mountain.

steamboat? 5. No, I shall go over by the railroad and return 1. The verb tommen frequently answers to our “get," as :-Wie ist er in diesen Garten getommen? How did he "get" into this by the steamboat. 6. Do not go beyond the crossway. 7. I

saw your friend come in as your uncle went out. 8. These men garden? Er weiß nicht, wie er heraus tommen soll, he does not who go over that bridge are in danger of their lives. 9. Will know how to "get" out. Id tomme mit diesem Manne sehr gut you go out to-day with your friend ? 10. From this hill we can fort, I "get" along with this man very well.

look over our country. 11. How did the thief get into your VOCABULARY.

house ? 12. Edward precipitated himself from the rock. 13. I

shall pass your house this morning, and shall come in, without Brau’sen, to roar. Berü'berfommen, to, Hügel, m. hill, hillock. your asking me to do so. Cajū'te, f. cabin.

Kreuzweg, m.crossway. Caser'ne, f. barrack. Heruntereilen, to has- Lauf, m. course, cur. Dampfboot, n. steam- ten down.

rent.

LESSONS IN ARITHMETIC.-XVI. boat.

Herun'terfommen, to Nachtwache, f. watch.
Dieb, m. thief.

come down.
Reh, n. roe.

DECIMALS (continued).
Eisenbahn, f. railroad. Sinab'gehen, to

go Schiffsbrūđe, f. bridge 15. Terminating and Circulating Decimals. Reducing Fractions Erschrect'en, to terrify. down.

of boats.

to Decimals. Felsen, m, rock. Hinab'springen, to leap Schweizer, m. Swiss. It is evident, from what has been said, that vulgar fractions Hase, m. hare.

down.
Straße, f. street.

can be reduced to decimals by the process of the division of Herabstürzen, to pre- Hinaufʻlaufen, to run Strom, m. stream. decimals. For we have only to write down the dividend with a cipitate.

up.
Stunde, f. hour.

decimal point, followed by a series of ciphers, and then divide Herauf'fommen, to Hinausgehen, to go out. Thüre, or Thür, f. door. by the divisor, according to the rule already given for the division come up. Sinaus fommen, to Treppe, f. stair.

of decimals. Thus, to may be reduced to a decimal as follows:Heraudíformen, to

come out.
Treten, to tread, step.

40 ) 7.000 ( "175 come out.

Hinein'gehen, to go in. über, over, beyond. Heraus'stürzen, to rush, Hinü'berfahren, to pass Vaterland, n. native spring out. country.

300 Herein'fommen, to Ginü'berkommen, to go Während, during, come in.

while. Herein'stürmen, to rush Hinü'berschauen, to Wenn, if. in.

look over.

Wieder, again. Herú'berfahren, to come Hinun'tergeben, to go Zwi'schendect, n. deck over (in a vehicle). down.

(between deck).

Therefore 16 = 175
RÉSUMÉ OF EXAMPLES.

Decimals which, after continuing the division of the fractions Sie sehen hinab' in das wilde Meer. They look down into the wild from which they arise far enough, at last give a result without a

remainder, are called terminating decimals. Da gießt unend'licher Regen herab'. There pours down interminable 16. To determine whether a Fraction will produce a Terminat

rain.

ing Decimal or not. Die Knaben eilten den Berg hinauf'. The boys hastened up the

Since a decimal is a fraction with 10 or power of 10 for its mountain.

denominator, it is evident that if a given fraction will produce a Der Bergmann steigt herauf aus The miner comes up out of the terminating decimal, it must be capable of being expressed in der Liefe des Schachtel.

depth of the shaft.

the form of an equivalent fraction, which shall have a power of 10 Petrus ging hinaus' und weinte bit'. Peter went out and wept bit- for its denominator. terlich. terly.

Now 10 is composed of the prime factors 2 and 5. Hence, if Und hincin' mit bebåch'tigern Schritt And thither (therein) with con- the denominator of the given fraction, when in its lowest terms, cin Löwe tritt.

siderate step a lion strides. contains any factor besides 2 and 5, it cannot produce a terEr wirft sich in die brau'sende Fluth. He throws himself into the minating decimal. But if the denominator contains only 2's and

roaring flood.

5's as its factors, then, by multiplying numerator and denomiDer Richter, rief den Bauer herein'. The judge called the peasant in. nator of the fraction by a factor, we can always transform Das Leben des Menschen schwanft, The life of man, like a skiff, the fraction into an equivalent one, having a power of 10 for its

wie ein Nachen, hinüber und fluctuates hither and thither. denominator-that is, into a terminating decimal. herüber.

For example: Der Dachveder fiel vom Hause The tiler fell down from the 4 will produce a terminating decimal, because 250 is comherun'ter.

honse.

posed only of factors 2 and 5.

250 = 5 x 5 x 5 x 2. EXERCISE 48. 1. Saben Sie meinen Freund geschen? 2. Ja, er ist die Straße hinab. 250, the denominator, a power of 10.

Hence, if we multiply this by 2 X 2, 12., 4, we shall make gegangen. 3. Wollen Sie in die Cajute hineingchen? 4. Nein, ich gehe in das Zwischenscd hinunter. 5. fahren Sie heute mit (§ 112. 7) dem

4 x 217 Therefore 233

= 888.868. Dampfboote nach Mainz hinüber? 6. 3a, und diesen Abend werde ich mit

Ersenbahn über die neue Schiffebrüde wieder herüber fommen. 7. Hin. Similarly, I gives a terminating decimal, for 8 is 2 x 2 x 2,

40

over.

280

over.

200
200

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4 X 250

1000

1000

60

40 35

50
49

7

*|**

30
28

* 20

14

60

and therefore by multiplying by 5 X 5 X 5, we make it 21. This will be made plainer by examples.
10 X 10 X 10, or 1000.

Reduce to a decimal.
3 x 5 x 5 x 5
Hence i
375 = 375.

7) 2.0000 (-28571428 ...
10 x 10 x 10

14 17. We see from the preceding remarks the truth of the following

56 Rule for determining whether a given Vulgar Fraction will produce a Terminating Decimal.

Reduce the given fraction to its lowest terms, and split the denominator into its prime factors.

If the denominator have for its factors 2's or 5's, or both, and no other factors, the fraction will give a terminating decimal, but not otherwise.

10 18. To determine in this case the Decimal without actually dividing.

If one of the factors 2 and 5 occur fewer times than the other, multiply numerator and denominator of the fraction by that power of the factor which occurs the fewest times in the denominator, which will make the number of times it occurs equal to the number of times the other occurs.

Thus, in the instance already given, 250 is made up of three 5's and one 2 as factors. We therefore multiply numerator and denominator by the second power of 2.

Here it will be seen that at the point indicated by the star Similarly, in 1, 8 being the third power of 2, we multiply the remainder 2 occurs, and therefore the division will, after numerator and denominator by the third power of 5.

this point, be identical in every respect with that already perObs.-It will be perceived that the number of decimal places formed. Hence the figures in the quotient, 285714, will conin the terminating decimal which is equivalent to a vulgar frac- tinually recur, or the quotieuc is the pure circulating decimal, tion, will be the same as the greatest number of times that either 285714. of the factors 2 or 5 is repeated in its denominator, when the It will be observed that the period here is as large as it could fraction is reduced to its lowest terms.

possibly be, for the greatest possible remainder is 6, and all the EXAMPLE.—Determine the decimal which is equal to 1990. remainders from 1 up to 6 inclusive occur.

382 382 Reduced to its lowest terms this is

[The process has been exhibited in the form of Long Division, 125 or 55. Multiplying

to allow of the remainders appearing in the operation.] numerator and denominator by 23, or 8, the fraction becomes

22. Reduce to a decimal. se, or 3.056.

We see at once that the quotient will be a circulating decimal, 19. Circulating or Recurring Decimals.

since W being in its lowest terms, 3 is a factor of the denominator. Decimals in which the same series of figures is repeated in. definitely, are called circulating or recurring decimals; and the

30) 17.0000 ( -566 ...

15 0 series of figures thus repeated is called the period. Thus, 3.21737373, etc., . ...., where 73 is continually repeated

200 ad infinitum, is a circulating decimal.

180 Similarly, .01342342342 ... and .6666 ., are recurring decimals.

* 20 A recurring decimal is indicated by writing a dot over each figure of the period, or, sometimes, where the period is long, by

Here the remainder 20 is at once repeated, and therefore the writing a dot over the first and last figures only of the period. quotient after the first figure 5 will consist of 6 continually roThus, the decimals we have given before as examples

would be peated, or it will be the mixed circulating •56. written

23. Reduce 12 to a decimal. 3.2173, 01342, or .01342, and ...

55 ) 129-00 (2-345 Answer. Decimals in which the period commences immediately after

190 the decimal point, are sometimes called pure circulating or recurring decimals; others being entitled mixed circulating decimals.

250 Thus above 6 is a pure, while the other two are mixed circulating decimals. 20. Fractions producing Circulating Decimals.

300 We have seen that all vulgar fractions in their lowest terms, which have any other factors besides 2 and 5 in their denomina

• 25 tors, will not produce terminating decimals; that is to say, in performing the division we shall never arrive at a remainder Here the remainder 25 occurs again, and therefore the which is zero. We shall, however, arrive at a remainder which periodical part of the quotient will, after this point, consist of is the same as one of the remainders which has already occurred. the figures 45 continually repeated. This is evident from the following considerations :

EXERCISE 34. The largest possible remainder in any division is the divisor diminished by unity, and therefore there cannot possibly be more 1. Determine which of the following fractions will produce than this number of different remainders. Hence, at the very terminating decimals, and find the equivalent decimals without farthest, after this number of remainders have occurred, a re- executing the division :mainder will occur which is the same as one of the preceding re

Bo38, 36, 6, gó, vio allo sta, 92, 11:14. mainders. Now it is plain that when this is the case, the whole

2. Reduce the following fractions to decimals :-
of the operation which has been performed since that remainder
last occurred will be repeated, and that the same remainder will

3. a.
5. .

7. it. occur again after exactly the same interval, and so on ad infini

8. 27 x 14 9. as of 31. 10. 24 of 1253. tum. Now to every remainder there will correspond a figure in

34

12. the quotient, and therefore the figures in the quotient corres

27}

316

153

. ponding to the interval between two remainders which are the

273+

21

13. same will continually recur.

375 113

110

165

220

275

1. s.

6. .

6

X

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12

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