; that the one side of the equation, and all the known quantities to the 2. Divide 39 into four parts, such that if the first be increased other, taking care to change the signs of the terms transposed, and by 1, the second diminished by 2, the third multiplied by 3, and incorporate the terms that are alike. the fourth divided by 4, the results may be all equal. 4. Remove the co-efficient of the unknown quantity, by dividing 3. If a certain number is divided by 12, the quotient, all the terms in the equation by it; the result will be the solution dividend, and divisor, added together, will amount to 64. What required. is the number? PROOF.-Substitute the value of the unknown quantity for the 4. An estate is divided among four children, in such a manner letter which stands for it in the equation; and if the number that the first has £200 more than of the whole, the second satisfies the conditions of the question, it is the answer sought. has £340 more than of the whole, the third has £300 more PROBLEM 1.-A man being asked how much he gave for his than 4 of the whole, the fourth has £400 more than of the watch, replied : If you multiply the price by 4, to the product whole. What is the value of the estate ? add 70, and from this sum subtract 50, the remainder will be 5. What is that number which is as much less than 500, as a equal to 220 pounds. fifth part of it is greater than 40 ? In order to solve this question, we must first translate the 6. There are two numbers whose difference is 40, and which conditions of the problem into such an algebraic expression as are to each other as 6 to 5. What are the numbers ? will form an equation. 7. Suppose two coaches to start at the same hour, one from Let « be the price of the watch. London for Glasgow, and the other from Glasgow for London, This price is to be multiplied by 4, which makes 4x; to the the former travelling 10} and the latter 94 miles per hour. product 70 is to be added, making 4x + 70; from this, 50 is to Where will they meet, the distance between the two cities being be subtracted, making 4x + 70 — 50. 400 miles ? Here we have a number of the conditions, expressed in 8. Suppose everything to be as in the last question, except algebraic terms; but we have as yet no equation. We must that the coach from Glasgow starts two hours earlier than the observe, then, that by the last condition of the problem, the other. Where will they meet ? preceding terms are said to be equal to 220. 9. A dealer purchases 60 yards of cloth for £30; and by We have, therefore, this equation, 4x + 70 — 50 = 220; selling one part of it at 12s., another, twice as great, at 14, which reduced, gives æ= 50. Ans. and the rest at 10s. per yard, he gains £8. How many yards Here the value of x is found to be 50 pounds, which is the were in the several lots ? price of the watch. 10. Suppose two dealers each annually to double his capital PROOF.-The original equation is 4x + 70-50=220; sub. except an expenditure of £100; and that at the end of three stituting 50 for x, it becomes 4 x 50 + 70 — 50 = 220; that years the capital of one is found to be doubled, while the other is, 220 = 220. has only half what he had at first. How much had each to com PROBLEM 2.—What number is that to which, if its half be mence with ? added, and from the sum 20 be subtracted, the remainder will 11. If a person each year double his capital, except an expen be a fourth of the number itself ? diture of £300 the first year, 4400 the next year, and £500 th In stating questions of this kind, where fractions are con- third, and at the end of three years be found to be worth £5,500 cerned, it should be recollected that fx is the same as what was his original capital? 2 x 12. A father's age is now treble of his son's, while five year eto. ago it was quadruple. What are their present ages ? Let æ be the number required. 13. Divide £1,000 between A, B, and C, giving A £100 more and B £50 less, than C. Then by the conditions, we have r + - 20 = , and re 14. A spirit merchant finds that if he add 10 gallons to a cas of brandy, the mixture will be worth 21s. per gallon; but the ducing the equation, we have x=16. Ans. if he had ten gallons more, the value will be reduced to 18 16 PROOF.--Thus 16 + 20 How many gallons were in the cask ? 2 15. Find a number, such that if it be divided successively PROBLEM 3.—A father divides his estate among his three sons 2, 3, 4, 5, 6, 7, 8, 9, and 10, half the sum of the first for in such a manner, that the first has £1,000 less than the quotients increased by 20 shall be equal to the sum of th whole; the second has £800 less than one-third of the whole; remaining five. the third has £600 less than one-fourth of the whole. What is 16. Find two numbers differing by 6, and such that thre the value of the estate ? Ans. £4,1144. times the less may exceed twice the greater by 7. PROBLEM 4.-Divide 48 into two such parts, that if the less be 17. Find a number, such that if it be increased successive divided by 4, and the greater by 6, the sum of the quotients by 1, 2, and 3, the sum of one-half of the first result and on will be 9. third of the second shall exceed one-fourth of the third by 8. Leta be the smaller part; then 48 - « is the greater part; and, 48 by the conditions of the problem, we have + 6 LESSONS IN ITALIAN.-XIV Whence x=12; therefore, 12 is the less part, and 36 the EXERCISES FOR PRACTICE. greater part. We resume in this lesson our series of exercises which wil 172. Letters may be employed to express the known quantities afford the student sufficient practice in translating simple se in an equation, as well as the unknown. A particular value is tences in Italian into English, and turning English into Italia assigned to the letters, when they are introduced into the calcu- The copious vocabularies will afford the learner a useful oppor lation ; and at its close, the numbers are restored. tunity of storing his mind and memory with Italian words. EXAMPLE.-If to a certain number 720 be added, and the sum be divided by 125, the quotient will be equal to 7392 VOCABULARY. divided by 462. What is the number? Anche, also. E's-80, m., ès-sa, f., L' o-10-16-gio, the wste he, she, it (of per or clock. Let e be the number required; and let a= 720, 6=125, Che, who, whom, that, d=7392, and h= Ma, but. 462. sons and things). Do-te, where, n cap-pal-lo, the hat. Mól-to, very. Then, by the conditions of the problem, we have + a d I ca-val-lo, the horse. Per, for. bd - ah and reducing, we have æ= ñ E'i-la, she, it (in refe. Il fan-ciál-lo, the child. Per-de-to, lost. rence to a feminine I tem-pe-ri-no, the pen. Tro-ed-to, found. h (125 x 7392)--(720 x 462) noun). kaife. Restoring the numbers, we have x = =1280. 462 EXERCISE 8.--COLLOQUIAL. 1. Mí-o pá-dre è buô-no; é-gli ha án-che un buôn fra-télk EXERCISE 29.-MISCELLANEOUS PROBLEMS IN SIMPLE 2. Mi-a má-dre è buô-na; él-la ha án-che ú-na buð-na so-rål-1 EQUATIONS. 3. Ab-bia-mo ve-du-to vô-stro zi-o; e-gli a com-pra-to un gT 1. Divide 11 into two parts, such that the sum of twice the li-bro. 4. A-vé-te voi ve-du-to il no-stro giar-dí-no? és-s0 half the second may be 16. mól-to grán-de. 5. Hð com-prá-to ú-za pén-na; és-sa è molt 16 sur an ca, f. buô-na. 6. Il tú-o li-bro è píc-co-lo, ma és-so è buô-no. 7. Ab- Fiore, flower, bloom, Medico, physician. Sassonia, Saxony. biá-mo un pá-dre che è buô-no. 8. A-vé-te ú-na ma-dre che è priine; the most ex-Mese, month. Sciocco, fool, blockhead. buô-na. 9. Hỗ tn lí-bro che è mól-to pic-co-lo. 10. Mi-a 80 cellent or valuable Mezzo, m., mezza, f., Servo, servant. rėl-la ha ú-na pén-na che è mol-to grán-de. 11. Il lí-bro che part of anything; (ds),t middle. Sette, seven. model, standard. Miniera, mine. *-76-te com-prá-to è buô-no. 12. IT giar-dí-no che ab-biá-mo Settimana, week. Fe-du-to è molto grán-de. 13. Hai tu ve-dú-to il li-bro che Francoforte, Frankfort. Nome, name. Fonderia, foundry. Mondo, world. Si, so. Soldato, soldier. mí-o zí-o ha com-prá-to ? 14. Il lí-bro che vô-stro zí-o ha com- Galante, polite, civil; Nomina, designation to Soprannome, surname, prá-to è mól-to pío-co-lo, ma és-so è buð-no. 15. Hồ án-che obliging, kind; gen. office, appointment, family name. com-prá-to un lí-bro, ma és-so è grán-de. tleman-like; love nomination (decreto Sopranominato, making, VOCABULARY. amorous, di nomina, diploma, named. gallant (gallant 1 commission). Spada, sword. April, A-prí-le. January, Gen-ná-jo. Send for, man-dá-te a omo, upright, Occhio, eye (colpo d' Squillo, sound. Asks for, do-man-da. Loaf, pá-ne, m. prể-de-fe. honest man; a man occhio, sight, view, Stato, state. Ball, bil-lo. London, Lón-dra. Silk, sé-ta, f. of honour, a perfect prospect, instead of Stiria, Styria. Bread, pá-ne, m. Map, cár-ta geo-grá-fi- Small crust, cro-sti. gentleman). ve-dú-ta). Tabacchiera, snuff-box, Button, bot-to-no. no, m. Taddeo, Thaddeus. By, per. Master, pa-dró-ne, m. Sugar, zúc-che-rosts), m. dress, gentility, po- Oggi giorno, non-&-days. Taglia, size, stature, Changeable,ta-ria-bi-le. | May, Mag-gio. Tailor, sar-tó-re, m. liteness, (uomo di Origine, origin, do shape, figure, waist. City, cit-tà. Month, mé-se, m. Take, pren-dé-te. garbo, a polite man ; scent, birth. Talento, talent, inclina. Cloth, pán-no, m. Nine, mô-te. That are to be made, also an honest man). Oro, gold. tion, propensity, Coffee, caj-fi, m. On the contrary, all' che si fác-cia-no. Giona, Jonas. bent, bias, will (mal December, De-cém-bro. in-con-tro. To-morrow, giór-no di Giorno, day. Paglia, straw. talento, malignity, Dozen, doz-zf-na (ds), t. Order, ór-di-ne, m. do-ma-ni. Giuda, Judas. Pelle, 1., skin, hide, polt, maliciousness, maDress, 4-bi-to. Paris, Pa-ri-gi. To-morrow, do-má-ni. Grande, great. leather. lice, malevolence; Eat, man-gid-te. Performance (i..., To tell you, a dir-vi. Grido, cry, reputation, Permosso, congedo, per uomo di mal talento, Find me, cer-cá-te-mi. comedy), com-mé. Two, dú-e. report (uomo, medico mission, leave (of ill-natured man), Finish drinking, fi-ní. dia, f. Very, mól-to. di grido, celebrated absence), discharge. Tanto, so much. te di bé-re. Pleasant, a-me-no. Were you, sid-te stá-ti. man, physician), Teatro, theatre. Glass, bic-chit-re, m. Pound, lib-bra, f. What, che. Guanto, glove. Piacere, pleasure (cam- Tempra, temper. Half an ounce, méz-za Preparation, pre-pa-ra. Will you put on, met. Guerra, f., war. po di piacere, mili. Termine, space or point On-cia, t. zi-one, f. te-ré-te. Guglielmo, William. tary encampment of time, period, He had given him, gli Quarter, qudr-to. Wine, vi-no, m. Imperatore, emperor, for the diversion of term. fú-ro-no as-se gná-te. Room, cá-me-ra, t., sót- Yard, brác-cio, m. (pl. Importanza, import the prince). Testa, 1., head. Hour, 6-ra, f. to, below, under. le brác-cia, f.) anco. Poco, m., poca, f., little, Tintore, dyer. neath. Tocco, piece. I Inspettore, inspector. Ponto, bridge, Torniajo, turner. vò (prom, ni-tor-me-rô) do. Yesterday, je-ri. talia, Italy. Poverino, poor, unfor- Tre, three. Jeri, yesterday. tunate. Trenta, thirty. 1. The tailor asks for nine yards of cloth, two dozen of Lettera, letter (uomo di Legge, law. Prodigio, prodigy, Tromba, trumpet. miracle. Unito, united. battons, and half an ounce of silk. 2. Send for a loaf of sugar lettere, learned man, Progetto, project, plan. Uno, one. and two pounds of coffee. 3. I shall return in a quarter of scholar). Quattro, four, Uomo, mnn. an hour. 4. Finish this glass of wine, and eat this small Luna, moon. Quello, that. Vaso, vessel, vase. crust of bread. 5. Take the map and find me the city of Macigno, sandstone, Questo, this. Velluto, velvet. Paris and the city of London. 6. I come by order of the master mill-stone, stone. Reputazione, reputa- Venti, twenty. to tell you that the preparations for to-morrow are to be made. Malattia, disease, in tion, fame. Vestito, dress, 7. The month of April is changeable; the month of May, on the ness, malady. Russia, Russia. Vino, wine. Malo, ill, badly. Salute, 1., health. Vostro, your. contrary, is very pleasant. 8. The months of December and Jannary are the roughest in the year. 9. What dress will you EXERCISE 10. put on for the ball of to-morrow? 10. Were you at the performance of yesterday? 11. He had given him the lower stito di vel-lú-to. 4. Ví-no d' I-tá-lia. 5. Un cuộr di ma-cí. 1. Ta-bac-chiê-r3 do ô-ro. 2. Un vá-So do ar-gôn-to. 3. Verooms. VOCABULARY. gno. 6. Il fí-lo di fér-ro. 7. Guán-ti di pêl-le fi-na. 8. Abbigliamento, orna- Campo, camp. Cap-pôl-lo di pa-glia. 9. U-na mi-niê-ra do ô-ro, d ar-gen-to. Decreto, decree. ment, dress, fitting Cannone, cannon.* 12. Fih. Diavolo, devil. 10. Ac-ciá-jo d' In-ghil-tər-ra. 11. Fôr-ro di Stí-ria. out, equipment. Cappello, hat. Dolce, sweet, gentle, ra di Fran-co-för-te. 13. La fê-sta di do-má-ni. 14. Il gior.no Carattere, character; soft. đ 6g-gi. 15. La com-mê-dia di sê-ri. 16. I teá-tro d' 6g-gi Acciajo, steel handwriting, hand Domani, to-morrow. giorno. 17. Uo-na ma-lat-ti-a đi quát-tro sot-ti-ma-ne. 18. 11 Acqua, water. (caratteri, pl., types, Drittaccio (for dirittac- ví-no di Ot-to, di vén-ti án-ni. 19. La guêr-ra di set-te án-ni. 20. Afare, affair, business; letters) cio), arrant knave or Un bel cól-po d' do-chio. 21. Lo squí-lo dél-la tróm-ba. 22. U'-na station of life, con Cattivo, m., cattiva, L., sly fox. per-só-na di fé-de. 23. E'-gli è di tô-sta dú-ra. 24. Ud-mo di dition, mnk (uomo bad, wicked. Duro, m, đura, £, hard, cóc-te, di món-do. 25. Uô-mo di lêt-te-ro, đi đól-eo tôm-pra. di grand affare, a Certificato, certificate. obstinate. man of consequence Chiaro, light, bright- E, is. 26. Uô-mo di grand' af-fá-re, di gár-bo. 27. Ud-mo di cat-ti-va or importanoe; a con-dót-ta. 28. Ud-mo di grán-de a-bi-li-tà, di gran re-pu-taness, shining. Entro, within. very able or clever Colpo, blow, stroke. Estrazione, f., extrac zió-ne. 29. Ud-mo di mêz-za tá-glia. 30. Ud-mo di mal ta. man; a man of su- Commedia, a comedy, tion, descent. lên-to, di spá-da, di guêr-ra. 31. Ud-mo di bás-sa e-stra-zió-ne. perior genius play. Fa, it is, there is. 32. Ud-mo di pô-ca sa-lú-te. 33. La cô-sa è di grán-de im-por. Condotta, f., conduct, Fede, faith, fidelity. tán-za. 34. Un mê-di-co di grí-do. 35. L' ár-te del tor-ni-ā-jo, Altrettanto, as much behaviour. Abilità, ability. or Femmina, female. del tin-to-re. 36. La fon-de-ri-a de ca-rát-te-ri. 37. Cám-po di Confine, confino, con Ferro, iron, pia-cé-re. 38. Fi-la-tó-jo di co-tó-ne. 39. Pón-te di bár-che. fines, frontier. Festa, feast, festival, 40. In-spet-tó-re dél-la fon-de-rí-a de' can-nó-ni. 41. L' ab-biCorte, court (uomo di festivity. glia-men-to dei sol-dá-ti. 42. Pro-gêt-to di lég-ge. 43. Il decorte, courtier, for. Fiera, fair (for mer. Barca, barge, boat cré-to di nô-mi-na. 44. Cer-ti-fi-cá-to d' o-ri-gi-ne. 45. Sta-ti merly the court's chants). fool). Filatojo, spinning u-ní-ti d' A-mê-ri-ca. 46. L' Im-pe-ra-tó-re dél-le Rús-sie. 47. I Cosa, f., thing, matter. wheel, spinning-mill Basso, m., bassa, I., low. Cotone, cotton ( filatojo or manufactory. * Lég-ge, law, and låg-ge, he rends. + The student must bear in mind the difference of pronunciation and Fino, m., fina, f., fine, meaning between these two words : méz-ro (ts), dough-like, over-ripe, Cuore, heart. thin, delicate. shrivelled (of fruit), and mêz-so (ds) middle, hall, the centre, tho middle, means, mediation. * Can-no-ne, cannon, piece of ordnance, and ca-no-ne, rule, precept; Tóc-co, touch ; blow with a hammer, stroke of a clock; and tóc-co, canon (in ecclesiastical affairs and in music). toque, a kind of bonnet; piece, bit. talents). again. (ponte di barche, pontoon). Dello, beantiful, fine. drel. con-fi-ni dél-la Sas-s8-nia. 48. En-tro il têr-mi-ne di tre mé-si. has also a good book. 19. We have a large book and a small pen. 49. Un pro-di-gio di uô-mo. 50. Un uo-mo di trén-ta. 51. II 20. You have a good father and a good mother. 21. Have you also a fior di ga-lánt uô-mi-ni. 52. Quél-lo sciôc-co di vô-stro sêr-vo. brother? 22. I have a book. 23. I have bought a good book 24. We 53. Qué-sto diá-vo-lo di fém-mi-na. 54. Quel drit-tác-cio di have seen a large garden. 25. My brother has also seen a large garden. Gu-gli-el-mo. 55. Tôc-co di bric-cô-ne! 56. Quel po-ve-ri-no di 26. I have bought a pen. 27. Hast thou bought a good pen? 28. Hast mí-o fra-têl-lo! 57. Tán-to di vi-no ed al-tret-tán-to d' á-cqua. thou seen my book? 29. I have seen thy book and thy pen. 30. Hara you seen my little sister? 31. My father has bought a garden. 58. Fa un sì bêl chiá-ro di lú-na. 59. U'-no di no-me Gio-na. Thy sister has bought a little book. 60. Giú-da di So-pran-nó-me (so-pran-no-mi-na-to) Tad-dê-0. 61. Per-més-so (con-gê-do) di tre mé-si. EXERCISE 6. 1. L' ipocrosia è un omaggio che il vizio rende alla virtu. 2. LA VOCABULARY. natura non domanda che il necessario. 3. La ragione vuolo l' utile, Ba-l, m., bél-la, f., La car-róz-za (ts), L'o-sto-ri-a, inn, hotel, l' amor proprio cerca il dilettevole, la passione esige il superlido. * beautiful, handsome. coach, carriage. tavern, public-house. Gli alberi grandi danno più ombra che frutto 5. Iddio è il Padre Gran-dis-si-mo, m., La cúf-fia, cap, coif, Questo, m., qué-sta, f., cielo, gli uccelli dell' aria, i pesci del mare, le piante, gli animali sono degli uomini e il Conservatore delle creature. 6. le stelle del gran-dis-si-ma, f., hood (particularly this. very great. for women). Ri-ce-pu-to, received, le opere del Signore. 7. Lo scopo della creazione è infinito, l'ingegno Il fi-glio, the son. La ft-glia, the daughter. got. dell'uomo è debile. 8. La sapienza di Dio è come la luce del cielo, Il man-tel-lo, the cloak. La lat-to-ra, the letter. Scrit-to, written. 9. L'ordine, la bellezza e la giocondità del mondo sono le prove manifeste I re-gá-lo, the present, La ta-bac-chiê-ra, snuff- Sú-o, m., sú-a, f., his, d'un Essere supremo. 10. L'eccesso delle passioni & ordinariamente la gift. box. her, its. cagione dell' infelicità degli uomini. 11. Le agitazioni dell'ira, dell' L'a-na-lo, ring. L'om-brél-la, umbrella. Ven-du-to, sold. invidia e dell'orgoglio sconcertano violentemente l' equilibrio de' findi, il sistema de' nervi, e per fine danneggiano anche spesso il mecanismo EXERCISE 11.-COLLOQUIAL. del corpo. 12. Il piacere dell' intemperanza e dell' incontinenza è il 1. Qué-sto ca-vál-lo è bêl-lo. 2. Qué-sta ta-bao-chiê-ra è nemico che reca all ' uomo il più gran danno; esso indebolisce la su mól-to píc-co-la. 3. Quest' 0-ste-rí-a è grán-de. 4. Qué-sto forza, lo priva delle ricchezze e guasta il suo miglior bene, la salute. fan-ciúl-lo è mi-o fra-têl-lo. 5. Qué-sto lí-bro è per mi-o pá-dre. EXERCISE 7. 6. Qué-sto tem-pe-rí-no è per mi-o fra-têl-lo. 7. Hồ tro-vá-to 1. The uncle's cloak. 2. John's coat. 3. My sister's house. 4. un'a-nel-lo. 8. Dó-ve a-vé-te vói tro-vá-to quest' a-nel-lo? 9. The rising, the setting of the sun, 5. The name of Just, of Great La lô-stra pic-co-la So-r-l-la ha un bên lí-bro. 10. Mi-a ma-dre 6. Sheep's wool. 7. Point of view, 8. The house of correction. 2. ha com-prá-to qué-sto cap-pêl-lo. 11. Tú-o fra-têl-lo ha ve-dú- One hears a pistol-shot. 10. Stone and marble quarries. 11. His to qué-sta bêl-la car-rôz-za. 12. Il vô-stro píc-co-lo fra-tel-lo è masterpiece. 12. The body-guard. 13. With a single stroke of a pen . un buôn fan-ciúl-lo. 13. Dó-ve hai tu com-prá-to qué-sta ta- flowers. 18. What a blockhead thou art! 19. The knife's point. 2). 14. A bell stroke. 15. Window-pane. 16. Cream. 17. A garland of bac-chiê-ra? 14. Quest' o-ro-18-gio è mól-to buô-no. 15. Qué- | A silver vein. 21. To-morrow is post-day. 22. A master of drawing, sto bell' a-nêl-lo è per qué-sto fan-ciúl-lo. 16. Mí-o zí-o ha un of fencing. 23. Court of Appeal. 24. Lottery ticket, pawnbroker's fi-glio ed ú-na fí-glia. 17. Hỏ ve-dú-to tú-o fra-têl-lo e tú-a so ticket. 25. The post-horses. 26. Office certificate. 27. Austrian emrel-la. 18. Ab-bia-mo ri-ce-vu-to un re-ga-lo. 19. A-vé-te vói pire. 28. Kingdom of England, of Scotland, of Ireland. 29. The city serít-to ú-na lêt-te-ra? 20. Mi-a so-rel-la ha ri-co-vú-to ú-na of London, of Edinburgh, of Dublin, of Manchester, of Liverpool, of bêl-la cúf-fia. 21. Hố ven-dú-to la mi-a car-rôz-za. 22. Hai Birmingham, of Glasgow. 30. The month of January, of May. 31. tu án-che ven-dú-to la túa car-rôz-za? 23. Qué-sto re-gá-lo à The name of Joseph, of Francis. 32. The island of Sicily, of Sardinia. per VÔ-stra zi-a. 24. V8-stro fí-glio è mól-to píc-co-lo, ma é-gli 33. A quarter of an hour. 34. A kind of dogs. 35. A horse-race. 36. è buô-no. 25. Mí-a fí-glia è gran-dís-si-ma. 26. Qué-sto pá-dre Stamp tax. Garrison troops. 37. The road of Trieste. 38. Tonnage duty, *. 40. A game at cards. ha ú-na bêl-la fí-glia. 27. Qué-sto fan-cíul-lo è mí-o fí-glio. 28. head-dress. 43. The order of the day. 44. Ten yards of linen, of cloth, 41. An ostrich feather. 42. The Il giar-dí-no che hồ ve-dú-to è gran-dís-si-mo. 29. Mí-o pá-dre 45. A barrel of oil, of vinegar. 46. A pound of ment, of cheese. 47 ha per-du-to il sú-o cap-pêl-lo e la sú-a om-brel-la. 30. Nô-stro A hundredweight of sugar, of coffee. 48. A bushel of corn. 49. A zí-o ha vend-dú-to la sú-a bêl-la car-rôz-za. 31. Mi-a 80-rêl-la piece of bread, a piece of roast meat. 50. A quarter of a pound ! ha tro-vá-to il sú-o a-nel-lo. 32. Qué-sto pá-dre ha per-dú-to butter. 51. A glass of wine, of beer. 52. I have bought ten bottles sú-a fí-glia. 33. Qué-sta má-dre ha per-dú-to sú-o fi-glio. 34. of Burgundy and six of champagne. 53. A box of pipes. 54. A great Mí-o zi-o ha com-prá-to ú-na cúf-fia per la sú-a píc-co-la fí-glia. numerable multitude of people. 57. A pair of old shoes. number of wolves. 55. A quantity of sheep, of oxen. 58. Two 35. Qué-sto re-gá-lo è per mi-a so-rel-la. ciúl-lo ha scrít-to ú-na gran-dís-si-ma lêt-tera per sú-a má-dre. pairs of boots, of trousers, of stockings. 59. A score of sequins. . 61. A cup of coffee. 37. Nô-stra zi-a ha com-prá-to un bel-lís-simo man-têl-lo per sú-o pinch of snuff. 62. A cup of tea. 63. A 64. Measure me for a cloak and a pair of trousers. tí-glio. 38. A-re-te vói tro-va-to un a-nêl-lo ? 39. Mí-o gío ha | 65. A team of horses. per-dú-to il sú-o man-têl-lo. LESSONS IN ASTRONOMY.-III. KEY TO EXERCISES IN LESSONS IN ITALIAN-XIII, EXERCISE 4. HISTORY OF THE SCIENCE (continued)-KEPLER'S SECOND AND 1. The memory. 2. Of the face. THIRD LAWS-GALILEO-INVENTION OF THE TELESCOPE 3. To the hill. 4. From the esplanade. 5. The slaughter-houses. NEWTON-LAW OF UNIVERSAL GRAVITATION. 6. Of the inns. 7. To the doors. 8. From the streets. 9. To one's face. 10. In the vineyard. BEFORE proceeding to consider the other laws discovered by 11. In the forests. 12. With straw. 13. With the vine, 14. With Kepler which govern the motions of the planets, we mast just 15. Through misfortune. 16. By the valley. 17. For the pause to explain one or two astronomical terms which we shall follies. 18. Upon the carriage. 19. On the rocks... 20. The dawn. frequently meet with. The 21. Of the joy. 22. To the opinion, 23. From the tavern. The ideas. 25. Of the herbs. 26. To the arts. 27. From the sun, as we have seen, is city. 28. In sledges. 29. In the imagination. 30. In the minds. situated in one of the foci 31. With water. 32. With the nail, 33. With the standards. 34. of the ellipse in which any Let s (Fig. 2) represent the position of the sun, and 1. I have a book and a pen. 2. Thou hast a good book and a good | ABCI the orbit in which pen. 3. I have a good brother. 4. Thou hast a good sister. 5. I the planet moves ; it evihave a large book; my sister has also a large book. 6. My brother dently is much nearer to the has a little pen. 7. Hast thou a sister? 8. I have a sister and a Fig. 3. brother. 9. Hast thou my pen? 10. I have thy book and thy pen. sun when at i than when it 11. We have a good father and a good mother. 12. We have also has moved on in its course to m. From this cause we find good brother and a good sister. 13. The garden is large. 14. I have that Mercury and Venus sometimes recede to a much greater a little book. 15. Hast thou also a book? 16. We have a large distance from the sun before they begin to retrace their steps garden. 17. My little brother has a good book, 18. My little sister than they do at other times. the pens. 24. D B II • 365 Now when any planet is in that part of its orbit which is whole work, and with intense joy found the result agree in the Dearest to the sun, it is said to be in perihelion, a term composed most marked way. The same calculations were then made for of two Greek words signifying "near the sun;" and when at the the other planets, and still found to hold good, and thus the other extremity of its course, it is said to be in aphelion, or away remaining law which governs our universe was at last disfrom the sun. covered. Only those who have themselves toiled on year after The distance o A, or half the major axis, is called the mean year in a search which often appeared hopeless, can realise distance, and this it is which is given in all tables of the solar the triumph of the philosopher as he thus discovered the laws system as the distance of the planets. of the heavens, and reduced all the apparently irregular motions Now it had long been noticed that the apparent daily motion of the heavenly orbs to three great but simple laws. of the sun was greater at some times than at others; and the The third law may be stated thus :-The squares of the planets, too, were found not to move exactly at a uniform rate, periodic times of any two planets bear the same proportion to their motion being more rapid when in perihelion than when in each other as the cubes of their mean distances. other parts of their orbits. Kepler accordingly determined to Thus, if we know the distance of a planet, we can calculate try and discover the law which established the relation between approximately its time of revolution round the sun ; and, on their distances and their speeds. This link was manifestly the other hand, if we know its time we can ascertain its diswanting to enable him to foretell the exact place the sun or a tance. When the distances and periods of the different planet would occupy at any given moment, for though he had members of our system have been given, the student will be mapped out their paths, yet, as their speeds varied, he could not able to check this rule for himself, and will find that the proportell in what part of those paths they would be unless he knew tion holds true to within a comparatively small amount. As an the amount of this variation. illustration, however, he may take the distances and periods of This work he accomplished with far less difficulty than he had Venus and the Earth, which may be set down in round numbers experienced in the discovery of his first great law, and he soon as follows:announced his second law, which teaches that Period. Distance. The velocity of any planet at every point of its orbit is such that Venus 224 days 68,000,000 miles. the line drawn from it to the sun always describes equal spaces in Earth 95,000,000 » equal times. This imaginary line, joining the centres of the sun and planet, is called the radius vector. The proportion between the periods here is has, and that beThis law may be easily understood by reference to Fig. 2, tween the distances of If now we take the square of the where s may be taken to represent the position of the sun, and former quantity, we shall find that it is nearly equal to the cube ABCDEFG the successive positions occupied by the planet of the latter. It must, however, be remembered that the at equal intervals, so that it passes from A to B in the same numbers above given are mere approximations; the exact figures period as from B to c or from c to D. will be given in a subsequent lesson. Now join these points with s, and we shall find that the space Almost contemporary with Kepler there lived another great A8 B passed over by the radius vector in one period is exactly philosopher and astronomer, Galileo by name, chiefly memorable equal to the space BSc passed over in the next, and to each now as having been the first to construct the astronomical of the other spaces CSD, D 8E, etc. We see thus that the telescope, though his powers were such as would have ensured rate of the planet's motion when near o is greater than as it his renown, even had this great discovery not been made by approaches A. The same would be found to hold good if we him. He was born in 1564, and became a philosophical teacher had divided the orbit into spaces passed over in one day, or in at Pisa. Here he soon rendered himself remarkable by his any other period. This law, then, together with the former violent opposition to some of the teachings of Aristotle, which he one, enables us exactly to calculate the place of any planet at proved by experiment to be incorrect. This brought upon him any time. Kepler, however, was not content with this : as he much odium, and even persecution; but though he thus opgazed on the different members of our system, he found so many posed the received views on mechanical subjects, he continued points of resemblance in their movements that he conceived for some time a firm believer in the Ptolemaio system, and there must be some general law connecting the whole into one even refused to hear any explanation of the views and theories grand unity. They all had the sun as one focus of their orbits, of Copernicus. After a time, however, he saw the folly of this, they all moved in elliptical orbits around that body and in the and commenced a careful inquiry, the result of which was that same direction, and all obeyed the same law as to rate of he became an ardent upholder of the new system. motion; he fancied, therefore, that some intimate bond of union In the early part of the seventeenth century, Galileo heard must exist between them, and set himself to discover it. of a discovery which had been made by an instrument-maker This was a more difficult task than either of the former had in Holland, by which distant objects could be made distinctly been, for there he had certain facts given, and had merely to visible. He therefore made every inquiry, and, after several form a theory that would explain them all; now he had to grope trials, at last succeeded in manufacturing a telescope which about for some law the very existence of which he only sus- possessed a magnifying power of 30. pected, nor could he tell whether it connected their distances, This he first directed towards the moon, and here ho at once their sizes, their periods, or their densities. He conjectured, detected many points of resemblance to the earth : there were however , that if there was any such relationship, it would pro- rugged mountainous parts and lofty elevations ; level plains bably be between their distances and their periods of revolution, likewise, which were at first called seas, were observed. A more especially as his second law showed that in the case of each greater discovery was, however, made when on the 7th of January, individual planet these quantities varied in a definite proportion. i610, he directed his magic tube towards the planet Jupiter. Accordingly, he commenced to compare their distances with their Not only did it present to him a brilliant disc streaked across periods, but not the faintest clue could he find in this way to with dark bands, but close to it were three small stars almost guide him; he then tried various multiples of these quantities, in a straight line. These were at first supposed to be merely but with no better success. So firmly, however, was he con- fixed stars ; on the following evening, however, when the televinced that some such law did exist, that he tried fresh com- scope was again directed to the planet, it was observed that binations and multiples; yet could find no relation whatever . they had moved along with it , and had also changed their He then compared the squares and the cubes, but, though positions with respect to each other. Here , then, was evidently trifling approximations were at times found, they were far from some new discovery, and most anxiously did Galileo await the complete, and he had still to try again. It now occurred to him recurrence of a clear evening to enable him to decide the to compare the squares of the periods with the cubes of the matter. The next view satisfied him that they were in reality distances, and these calculations ho accordingly worked out, moons accompanying the planet, and further, he found that but failed to detect any resemblance, owing, as afterwards ap. there were four of them. peared, to an error in some of his figures. The question now Intense excitement was created among astronomers by this began to appear hopeless, and was for a time laid aside, many discovery, some arging the absurdity of increasing the number anxious years having been spent in vain attempts to solve it; of the heavenly bodies beyond the sacred number seven, and bat still at times Kepler would almost instinctively look back others angry at the man who attempted to depose the earth over his old calculations, and in doing so one day he detected from its position of dignity, by asserting that Jupiter had the error above referred to. He accordingly went again over the four satellites while the earth had only one. Some even ro fused to look through the instrument which made such un- this attraction he called gravitation. The question then arose, heard-of revelations. The followers of Copernicus, on the whether this action was confined to the surface of the earth, or other hand, welcomed the discovery as presenting a miniature whether distant bodies were attracted in a similar way. The model of the solar system, and thus upholding their theory. intensity of this force was also believed to diminish with the The telescope soon made other discoveries. By its aid square of the distance; but the difficulty arose, how was this to Galileo found that Venus presented the same phases as the be tested. Even if a body could have been raised several miles moon, appearing at times as a narrow crescent, and then from the earth's surface, this amount would have been so slight, gradually becoming more and more illuminated, till at last it when compared with the radius of the earth, that no appreciable shone with an almost circular disc. It could not, however, be difference would have been manifested. seen with a complete diso, as at such a time the earth must be No way, therefore, appeared practicable of putting this theory in the part of its orbit exactly opposite to Venus, which would to the test, till at last the thought occurred to him, why not use therefore appear in conjunction with the sun and be lost in its the moon as the falling body, and ascertain the distance through brightness. which it falls in any given space of time-say, for instance, in This discovery was a very important one, as it afforded a one minute. This idea at first sight appears absurd, but the strong confirmation of the truth of the Copernican system. In annexed figure will enable us fact, an objection had been raised to this system on the ground to understand it. that these phases were not seen, as they ought to be if the We know that the moon retheory were true. The telescope, however, soon settled this volves around the earth in an difficulty and silenced these objections. orbit almost circular, as MBN. Another discovery was made when the planet Saturn was Now, suppose the moon to be examined. Instead of appearing with a circular disc, like the atm, its tendency at that moother heavenly bodies, it was found to be elongated, as if ears ment is to move along in the were affixed to each side of it. Owing to imperfections in the direction of the tangent mc, construction of his telescope, Galileo failed to discover that and in this direction it would this appearance was caused by a large ring which completely certainly move did not some Fig. 3. encircled it, and imagined that the planet was in reality composed other force bend it out of its of three smaller ones. Both these discoveries were, according course. This force Newton supposed to be the attraction of the to the practice of scientific men in those days, made known in earth, and determined to calculate whether or not the amount it anagrams only intelligible to those who possessed the key. deviated from a straight line was that which would arise from the We see even thus early what an important instrument the tele-earth's attraction. We can easily see that when the moon has scope proved, and we shall shortly find that almost all discoveries moved into the position B, the distance which it has deviated from since this time have been made by its use, and that now nearly its true path is equal to A B. He acoordingly calculated how great all our astronomical instruments consist, wholly or in part, of a this distance would become after the lapse of one minute-that telescope. We see thus to what important results the accident is, how far the moon would fall towards the earth in that time; of a child playing with two spectacle-glasses has led. he next computed the space through which a body removed The whole career of Galileo was a splendid one; it was, how to the distance of the moon ought to fall in the same period ever, somewhat marred near its close. The prominent position under the action of the earth's gravitation, and compared these he had taken as an upholder and defender of the new doctrines results together. had attracted the attention of the papal authorities, who re Though this calculation seems to be simple enough, it was in garded his views as heretical, and demanded of him a public reality the work of many years; and when at length it was recantation of his belief in the motion of the earth. This he completed, he found a considerable resemblance between the reluctantly gave, though he is related to have said immediately amounts, but not a sufficiently close one to satisfy him, and afterwards, “It moves for all that.” It seems probable, how his work was therefore, for a time, laid aside. After some time, ever, that he considered this act as one which he was called however, he heard that a more aocurate determination of the upon to perform by the Church, and that therefore it was his earth's diameter had been effected, and he accordingly repeated duty to obey ; still, it was, in several ways, a sad scene. Not his caleulations, substituting the new figures, and when at very long after this, in 1642, he died. In the same year there length the bewildering task was completed, the results were was born Newton, & man even more celebrated than Galileo or found to agree most accurately. Kepler. In order to fully satisfy himself of the accuracy of his theory, From this time onward we come across the names of so many the same calculations were gone through with reference to other great astronomers that we can but refer to a few of the more planets, and with the same results, and Newton then announced distinguished. Huyghens discovered that the appendage to this general law:Saturn was in reality a ring surrounding it, and further, he Every particle of matter in the universe attracts every other found one of the satellites of that planet. Napier had some particle with a force proportional to the quantity of matter in each, forty years before this invented logarithms, and thus reduced and decreasing inversely as the squares of their distances. the work of weeks to days or even hours; and a little later, re The motion of the planets is thus seen to be compounded flecting telescopes were introduced by Gregory. of two--the one, the original motion or impetus given them The name of Newton, however, stands foremost amidst all by the Creator, and remaining anchecked by any counteracting these names as the discoverer of the one great law on which all force; the other, the attraction of the body round which they those of Kepler depend. Kepler seems to have suspected that rotate. some such general law did exist, but failed to discover it; he Having attained this result, Newton set himself one more seems likewise to have been aware of the fact that the tides task, and that was to ascertain on mathematical principles the were in some way influenced by the moon, and that the other curve in which the planets onght under these conditions to more. heavenly bodies were in some way connected so as to influence This was a calculation requiring the utmost amount of matheone another; but he could not find what this mysterious bond matical skill. Newton, however, possessed this, and set about of union was, and therefore with him it was a mere conjecture. the work, fully expecting to find that the curve must be ap Newton, however, applied himself to clear up this difficulty. ellipse. He found, however, when the result was complete, one It is said that his attention was first directed to the subject by which represented not only this, but any of the "conic sections," observing an apple fall one day while he was sitting in a summer- that is, of the curves which may be obtained by cutting a cone. house in his garden. There was nothing remarkable in the circum- These are—the circle, which is the curve obtained when it is cut stance in itself , for it was an event that might be seen any day; parallel to the base ; the ellipse, when it is out a little inclined it set him thinking, however, and he began to inquire why the to this; the parabola, when the line passes parallel to one side apple should fall downwards or towards the ground, instead of of the cone; and the hyperbola, when it is parallel to the axis. apwards or to one side. To most men such a question would In any one of these curves, then, a planet may more under the have appeared utterly vain and frivolous ; to him, however, it influence of these general laws; and, as we shall find, the appeared an important step towards great and vast results, and satellites of Saturn move in the first, the planets generally in such in reality it became. After careful inquiry, he found that the second, while the comets career onwards in parabolas or all bodies were attracted towards the centre of the earth, and hyperbolas. |