“Pure” tone is free from (1) the heavy and hollow note of and girls, men and women. Faithful advisers may be of much the chest; (2) the "guttural,” choked, stifled, or hard sound of service to young students in this particular. the swollen and compressed throat; (3) the hoarse, husky, 3.-Versatility or Pliancy of Voice * hash,” “ reedy;" and grating style, which comes from too signifies that power of easy and instant adaptation, by wlich forcible “expiration," and too wide opening of the throat; it takes on the appropriate utterance of every emotion which (4) the nasal twang, which is caused by forcing the breath occurs in the reading or speaking of a piece characterised by against the nasal passage, and, at the same time, partially varied feeling or intense passion. closing it; (5) the wiry, or false ring of the voice, which unites To acquire this invaluable property of voice, the most useful the guttural and the nasal tones; (6) the affected mincing course of practice is the repeated reading or reciting of passages voice of the mouth, which is cansed by not allowing the due marked by striking contrasts of tom, as loud or soft, high oz proportion of breath to escape through the nose. The natural, low, fast or slow. smooth, and pure tone of the voice, as exhibited in the vivid The following exercises should be repeated till the student utterance natural to healthy childhood, to good vocal music, or can give them in succession, with perfect adaptation of voico to appropriate priblic speaking, avoids every effect arising from in each case, and with instantaneous precision of effect. an undue preponderance, or excess, in the action of the muscles of the chest, of the throat, or of any other organ, and, at the Exercises for Versatility or Pliancy of Voice. same time, secures all the good qualities resulting from the just Very Loud. and well-proportioned exercise of each. A true and smooth And dar'st thou, then, atterance derives resonance from the chest, firmness from the To beard the lion in his den, throat, and clearness from the head and mouth. The Douglas in his hail ? And hop'st thou hence unscathed to go ? Without these qualities, it is impossible to give right effect No! by St. Bride of Bothwell, no !to the beauty and grandeur of noble sentiments, whether ex Up, drawbridge, groom! What! warder, ho! pressed in prose or in verse. Let the poitcullis fall! Childhood and youth are the favourable seasons for acquiring Very Soft. and fixing, in permanent possession, the good qualities of I've seen the moon climb the mountain's brow, agreeable and effective utterance. The self-taught cannot exert I've watched the mists o'er the river stealing,too much vigilance, nor take too much pains, to avoid the But ne'er did I feel in my breast till now, encroachments of faulty habit in this important requisite to a So deep, so calm, and so holy a feeling :gcod elocution. 'Tis soft as the thrill which memory throws The subjoined exercise should be frequently and attentively Athwart the soul in the hour of repose. practised, with a view to avoid every sound which mars the Very Low, purity of the tone, or hinders a perfect smoothness of voice. I had a dream, which was not all a dream, The bright sun was extinguished; and the stars Did wander darkling in the eternal space, Rayless, and pathless; and the icy earth Swung blind and blackening in the moonless air. Very High. I awoke :-where was I?-Do I see A human face look down on me? And doth a roof above me close ? Do these limbs on a couch repose ? Is this a chamber where I lie ? And is it mortal, yon bright eye, That watches me with gentle glance ? Very Slow. of old hast Thou laid the foundation of the earth; and the Of charming symphony they introduce heavens are the work of hy hands. They shall perish, but Thou Their sacred song, and waken raptures high. shalt endure; yea, all of them shall wax old, like a garinent; as a The various passions and emotions of the soul are, to a great vesture shalt thou change them, and they shall be changed: but Thou extent, indicated by the “quality” of the voice. Thus, the art the same; and Thy years shall have no end. malignant and all excessive emotions, as, anger, hatred, revenge, Very Quick. fear, and horror, are remarkable for "guttural quality," and I am the Rider of the wind, The Stirrer of the storm! strong “aspiration,” or “ expiration,” accompanying the vocal The hurricane I left behind sound, and forming “impuro” tone; substituting a “harsh,” Is yet with lightning warm ;husky, aspirated utterance, for the “orotund,” or the “ pure To speed to thee, o'er shore and sea tone; while pathos, serenity, love, joy, courage, take a soft and I swept upon the blast. smooth "oral,” or head tone, perfectly pure, or swelling into 4.--True Pitch of Voice. "orotund.” Ave, solemnity, reverence, and melancholy, take a The proper pitch of the voice, when no peculiar emotion deep“ pectoral” murmur; the voice resounding, as it were, in demands high or low notes, is—for the purposes of ordinary the cavity of the chest, but still keeping perfectly “pure ” in reading or speaking-a little below the habitual note of contone, or expanding into full “orotund.” versation, for the person who reads or speaks. Public discourse, The young student cannot bo too deeply impressed with the being usually on graver subjects and occasions than mero private importance of cultivating early a pure and smooth utterance. communication, naturally and properly adopts this level. The excessively deep “ pectoral” tone sounds hollow and sepul But, through mistake or inadvertency, we sometimes hear chral ; the “guttural” tone is coarse, and harsh, and grating to persons read and speak on too low a key for the easy and the ear ; the “ nasal” tone is ludicrous ; and the combination expressive use of the voice, and sometimes, on the other hand, of “ guttural” and “nasal” tone is repulsive and extremely on a key too high for convenient or agreeable utterance. disagreeable. Some speakers, through excessive negligence, The following sentences should be repeated till the note on allow thernselves to combine the “ pectoral," " guttural,” and which they are pitched is distinctly recognised, and perfectly "nasal” tones in one sound, for which the word grunt is the remembered, so as to become a key to all similar passages. only approximate designation that can be found. Affectation Erercise on Middle Pitch. or false taste, on the other hand, induces some speakers to aerung an extra fine, or double-distilled, In every period of life, the acquisition of knowledge is one of the “oral” tone, which most pleasing employments of the human mind. But in youth, there minces every word in the mouth, as if the breast had no part to are circumstances which make it productive of higher enjoyment. It perform in human utterance. is then that everything has the charm of novelty; that curiosity arci The tones of serious, serene, cheerful, and kindly feeling, are fancy are awake, and that the heart swells with the anticipations of nature's genuine standard of agreeable voice, as is evinced in future eminence and utility. the utterance of healthy and happy childhood. But prevalent Contrast this pitch with that of the pieces before quoted as neglect permits these to be lost in the habitual tones of boys examples of “high” and “low." 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 appear. the 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 (page 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 line, which is turned at the line hh, 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 socond is formed by a combination of crossing the down-stroke in a graceful curve a little below the this stroke and the letter o. 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 Copy exactly 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 more formation, however, of looped letters, an eighth of an inch moro examples for practice in writing letters looped below the line bb, is required to give space enough to admit of the loop being and combining them with others. to ? or, n. LESSONS IN GERMAN.—XV. auf den Berg. 13. Wo geht unser alter Nachbar hin? oder, Wohin geht unser alter Nacıbar? ($ 89. 1.) 14. Er ist jeßt in dem fleinen Garten, SECTION XXVII.-SEPARABLE PARTICLES—(continued). aber er geht bald in den großen Garten. 15. Scine 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 ste fommen ane ($ 4. 2) Fenster. 17. Der Ritter figt laufen die Kinder? Where (in what place) are the children run-ichon auf seinem guten Pferde, und der Knecht springt auch so eben auf sein ning ? gutes Pferd. 18. Der Mann lißt am (8 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 feinen Hut auf dem Kopfe. 20. Wo geht cular place, as :-Da ist es, here it is. Da laufen fie, they are ter Solrat hin? 21. Die Soltaten gehen aufs (8 4. 2) peld ; sie sind running here. schon auf dem Felte. 22. Der Frosch springt in den Fluß und schwimmt Vin denotes direction, or motion from the speaker, as: Warum in dem 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. It fann meine Müße nirgends finden, ob Her is the opposite, in signification, to hin; denoting motion or gleich sie irgendwo in diesem Zimmer sein muß. EXERCISE 45. :-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 þin, we have the compound wohin; from wo city. 6. Whither do these emigrants go? 7. Whence do these and her, woher; from ba and hin, dahin; from ba 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 (gewezte Dolch] of a traitor enter not (dringt 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 Ho reisen unsere Freunde hin? Where do our friends travel going? 13. I am going to my brother-in-law. 14. Will these otet, 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 P Sie reisen bahin, wo ihre Ver. They travel thither, where their 18. She comes from Germany. wandten wohnen. relatives reside. VOCABULARY. Wo kommen diese Zugvögel her? Where do these birds of pasober, sage come from? or, Aus-wanderer, m. emi. Dahin'ziehen, to pro- i Italie'ner, m. Italian. Woher kommen diese Zugvögel ? Whence do these birds of pas- grant. ceed thither. Italie'nisch, adj. Itasage come ? Bedien'te, m. servant. Ein'wanderer, m. im- lian. Sie kommen daher, wo es jeßt zu They come from (there) where Bekann'te, m.acquain- migrant. Milwaukee, Mil. falt für fie ift. it is now too cold for them. tance. Englisch, English. waukee. Nie, never. Bremen, n. Bremen. Feldherr, m. comman. Spa'nien, n. Spain. Bad'stube, 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 | Fremte, m. stranger. Venedig, n. Venice. Biltergallerie, f. pic- Nirgends, nowhere. Springen, to spring, thither. Havre, n. Havre. Wie viel ? how much? ture-gallery. Dbgleid)', although, leap. Dahin'ichiden, to send Hierher'kommen, to how many ? Frosch, m. frog. notwithstanding. Stehen, to stand. thither. come hither. Wohl, well. D'pernhaus, n. opera- Teich, m. pond. þin'gehen, to go to. Werkstatt, f. work EXERCISE 46. Hirte, m, shepherd. Ritter, m. knight. shop. 1. Die Soldaten sind hier, und der Feldherr kommt auch hierher. 2. Irgentwe, some Schau'spieler, m. actor. Wohin' ? whither ? Der Feind ist schon da, und unsere tapfern Brüter müssen dahin ziehen. 3. where. Schon, already. what way? Wann gehen sie nach Spanien? 4. Ich will gar (Sect. XIV. 3) nicht 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 da und will nie wieder babin gehen. 7. Wir gehen auf den Berg, wollen Sie mit RÉSUMÉ OF EXAMPLES. uns geben? 8. Will der Russe seinen Bedienten in die Stadt schiden? 9. Wo it tas größte Glüd: an dem Where is the greatest happi. Er hat ihn schon dahin geschidt. 10. Werden die Truppen hierher fom Hofe eines tyran'nischen Könige, ness: at the court of a tyran- men? 11. Sie werden nicht hierher fommen. 12. Wo fommen diese ober in der Hütte eines zufrie'te. 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. Gå ist weder von Bremen, Wo gehen Sie hin? an ben Hof oder Whither do you go to the noch von Havre, es ist von Venetig. 16. Gehen diese französischen Ein: in die Hütte ? court or into the cottage ? wanderer nach Milwaufee? 17. Gin Theil von ihnen geht dahin, die Der Feldherr fißt auf dem Pferde The commander-in-chief upon andern bleiben in New York. 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 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 bako in den Morgen reitet er mit seinen Schaaren To-morrow he rides with his Garten gehen, denn mein Lehrer ist ta und will mich sehen. 21. Warum auf tas Schlachtfeld. troops to the battle-field. will dieser Italiener nicht englisch sprechen? 22. Er möchte es wohl In ter Hoffnung finden die Un-glück. The unfortunate find consola- (Sect. XLIII. 4) sprechen, aber er kann es noch nicht; er spricht nur lichen Trost. tion in hope. italienisch und spanisch. 23. Wie viel Sprachen können Sie sprechen? 24. Der Vater ist da, aber der Bruder The father is here, but the Ich spreche nur zwei, aber ich will noch andere lernen. ist in der Stadt. brother is in the city. EXERCISE 47. Ich gehe 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 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. Wo ist der Schwager ? 2. Er ligt an dem (am) Tische. 3. Wo geht where the danger was most [tie meiste Gefahr). 7. Is this ship ter Zuckerbäder hin 4. Er geht in die Badstube. 5. Wo ist sein Freund. from Spain or from Havre ? 8. No, it is neither [weder) from ter Schauspieler ? 6. &r ist in dem Opernhause. 7. Wo geht sein Freund, Spain nor (noch) from Havre; it comes from Hamburg. 9. These fier Seiler, hin? 8. Er geht in seine Werfftatt. 9. Wo ist der Hirte? immigrants are going to Milwaukee, and are emigrants from 10. Gr oft auf dem Berge.' 11. Wo geht der Hitte hin? 12. Er geht | Bohemia and Venice. 10. Can you leap over that gate [Thor] ? Gand, f. goose. house. Jeßt, now. 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, währeno that he might speak to me about it. ter Hase ten Hügel hinauslief. 9. Die Soldaten stürzten aus der Caserne heraus, als der Feind in die Stadt bineinstürmte. 10. Als rie Nachtwache SECTION XXVIII.-SEPARABLE PARTICLES-(continued). in das Haus trat, eilte der evschrockene Dieb die Treppe herunter. 11. 3o Wo, da, hin, etc., besides being compounded one with another fann nicht aus den Kreuzwegen dieses Gartens hinauskommen. 12. Wissen (Sect. XXVII.) are also united with prepositions; thus pro- Sie nicht, wie dieser Vogel hereingetommen ist? 13. Ja, aber er weiß ducing a separate class of adverbs, as :-Wovon sprechen Sie? Of nicht, wo er wieder binausfommen finn. 14. Der junge Schweizer (daute what (whereof) are you speaking ? Ich sprecise von meinen Büchern ; hinüber nach ten blauen Bergen seines Vaterlantes. 15. Stommen Sie wollen Sie eins davon haben? I am speaking of my books; will you heute nicht herunter ? 16. Ja, wenn der Dheim herauffommt, werde ich have one of them ? (one thereof ?) Ich bin auf dem Dadie ; kom hinabgehen. 17. Haben Sie diesen Mann schon gesehen? 18. Ja, er fam men Sie herauf! I am on the roof ; come up! Ich kann nicht zur Thüre herein, als ich hinausging. 19. Der Freund fuhr in einer Stunte hinauf gehen; fommen Sie herab I cannot go up; you come den Flug hinüber und herüber. 20. Der Strom stürzt mit großen Geräusch down! den Felsen herab. Hinab, hinauf, þinaus, 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 :--Ich gehe den Berg speech lasted over two hours. 3. The roo 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 tho come down the mountain. steamboat ? 5. No, I shall go over by the railroad and return 1. The verb fommen frequently answers to our "get," as :Wie ist er in diesen Warten gekommen? 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 kommen soll, he does not who go over that bridge are in danger of their lives. 9. Will know how to “get” out. Ich fomme 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. houso ? 12. Edward precipitated himself from the rock. 13. I shall pass your house this morning, and shall come in, without Brau'sen, to roar. Herü'berfommen, to Gugel, m. hill, hillock. your asking me to do so. Cajü'te, f. cabin. Kreuzweg, m.crossway. Caser'ne, f. barrack. Herun'tereilen, to has- Lauf, m. course, cur. Dampfboot, n. steam- ten down. rent. LESSONS IN ARITHMETIC.—XVI. boat. Herun'terfommen, to Nachtwache, f. watch. come down. DECIMALS (continued). of . ats. to Decimals. Feisen, 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 Herab'stürzen, to pre- Hinauflaufen, to run Strom, m. stream. decimals. For we have only to write down the dividend with a cipitate. np. decimal point, followed by a series of ciphers, and then divide Herauf'kommen, to Hinaus'geben, to go out. Thüre, or Thür, f. door. by the divisor, according to the rule already given for the division come up. Hinaus-kommen, to Treppe, f. stair. of decimals. Thus, o may be reduced to a decimal as follows:Herausófommen, to come out. Treten, to tread, step. come out. Mineinógehen, to go in. über, over, beyond. 40) 7.000 ( 175 40 Beraud'stürzen, to rush, Hinü'berfahren, to pass Ba'terlant, n. native spring out. country. Herein'tommen, to Hinu'berfommen, to go Während, during, 280 come in. while. Herein'stürmen, to rush | Hinūóberschauen, to Wenn, if. in. look over. Wieter, again. Heri berfahren, to come Sinun'tergehen, to gol Zwischendeck, n, deck over (in a vehicle). down. (between deck). Therefore 10 = Decimals which, after continuing the division of the fractions Sie sehen hinab' in bag wilde Mcer. 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 deterinine whether a Fraction will produce a Terminat. rain. ing Decimal or not. Die Knaben eilten den Berg hinauf. The boys hastened up tho 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 & Der Bergmann fteigt herauft aus The miner comes up out of the terminating decimal, it must be capable of being expressed in te: Tiefe des Schachtes. depth of the shaft. the form of an equivalent fraction, which shall have a power of 10 Petrug ging hinaus' und weinte bit'. Peter went out and wept bit. for its denominator. terli.t. terly. Now 10 is composed of the prime factors 2 and 5. Hence, if Und bincin' mit bedåch'tigem Schritt And thither (therein) with con- the denominator of the given fraction, when in its lowest terms, ein Löwe tritt. siderate step a lion strides. contains any factor besides 2 and 5, it cannot produce a terGr wirst sich in die brau'sende Fluthy. 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 denomi. Der Richter rief den Bauer herein'. The judge called the peasant in. nator of the fraction by a factor, we can always transform Oas leben des Menschen schwankt, The life of man, like a skiff, the fraction into an equivalent one, having a power of 10 for its mrie ein Nachen, hinăóber und fluctuates hither and thither. denominator--that is, into a terminating decimal. bcrüber. For example: T r Dach'teder fiel vom Hause The tiler fell down from the will produce a terminating decimal, because 250 is com. herun'ter. house. posed only of factors 2 and 5. 250 = 5 x 5 x 5 x 2. EXERCISE 48. 1. Saben Sie meinen Freund geschen? 2. 3.1, er ist tie Straße hinab. 250, the denominator, a power of 10. Hence, if we multiply this by 2 X 2, i.e., 4, we shall make degangen. 3. Wollen Sie in die Gajüte bineingeben ? 4. Nein, ich gehe 4 x 217 in tag zwischented hinunter. 5. Fahren Sie heute mit (§ 112. 7) dem Therefore i 868.868. Dampfboote nach Mainz hinüber? 6. Ja, und diesen Abend werde ich mit Der Gisenbahn über die neue Schilfebrüde wieder herüber kommen. 7. Hin. Similarly, I gives a terminating decimal, for 8 is 2 x 2 x 2, a over. 300 over. 200 200 sea. 4 X 250 100 1000 60 40 35 50 5349 or 53: 10009 15 0 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 1C, or 1000. Reduce to a decimal. 7) 20000 ( .28571428 ... = 375. 14 17. We see from the preceding remarks the truth of the following 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. 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 sta: Similarly, in ), 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 quotient 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 148. remainders from 1 up to 6 inclusive occur. 382 382 Reduced to its lowest terms this is 125 [The process has been exhibited in the form of Long Division, Multiplying to allow of the remainders appearing in the operation.] numerator and denominator by 23, or 8, the fraction becomes 22. Reduce 1 to a decimal. , or 3.056. We see at once that the quotient will be a circulating decimal, 19. Circulating or Recurring Decimals. since being in its lowest terms, 3 is a factor of the denominator. Decimals in which the same series of figures is repeated in. 30 ) 17.0000 ( -566 . . definitely, are called circulating or recurring decimals; and the 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. Similarly, .01342342342 . . . ., and .6666 . . are recurring decimals. 20 A recurring decimal is indicated by writing a dot over each Here the remainder 20 is at once repeated, and therefore the figure of the period, or, sometimes, where the period is long, by quotient after the first figure 5 will consist of 6 continually rewriting a dot over the first and last figures only of the period. Thus, the decimals we have given before as examples would be peated, or it will be the mixed circulating •56. written 23. Reduce to a decimal. 3-2173, 01342, or 01342, and .i. 55 ) 129-00 (2-345 Answer. Decimals in which the period commences immediately after the decimal point, are sometimes called pure circulating or recurring decimals; others being entitled mixed circulating decimals. Thus above •6 is a pure, while the other two are mixed circu. lating decimals. 20. Fractions producing Circulating Decimals. 275 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 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 7, 25, 6, 5, 6, sia stato a 41, 41811, 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 1. }. 5. . 7. 18. occur again after exactly the same interval, and so on ad infini 8. 23 X I. 9. a's of 36 10. 210 of 1258 tum. Now to every remainder there will correspond a figure in 871 2 11. the quotient, and therefore the figures in the quotient corres 27} 3,3 15. ponding to the interval between two remainders which are the 2734 12 214 13. same will continually recur. 180 110 190 250 800 which is zero. 31 01 2. §. 3. &. 6. .. |