the better deed. them. .121 קכא 27 In combining different numbers, the greater is put first, according to the Hebrew mode of writing; thus, , 11; and instead of the letters, because the latter commence the The number 15 is marked by the letters, name of God in Hebrew. The thousands are denoted by the units with two dots above-thus,, 1,000. Gesenius, in his occur in the text of the Old Testament, but was first found on grammar, says that this numeral use of the letters did not the coins of the Maccabees in the middle of the second century before the Christian era. The articles are sometimes placed before an adjective, when it precedes a noun, as, an admirable painting, or, The better day The definite article the is sometimes set before adverbs in the comparative and superlative degrees, as, The more the merrier; or, The oftener I look at Raffaelle's paintings, the more I admire Articles are sometimes found joined to proper names, for the purpose of giving them distinction or eminence; thus we say of some large town, it is quite a London, that is, a place as busy or as bustling as London; or, the Howards, that is, the The Greeks, in the same manner as the Hebrews, divided family of the Howards; or, he is a Wellington, that is, a man the letters of their alphabet into three classes of nine cha as distinguished for skill and bravery as the Duke of Wel-racters each, to denote units, tens, and hundreds, respectively; lington; or, the Caesars, that is, the Roman emperors of the and as this alphabet contained only 24 letters, they adopted three marks which were formerly used as letters in the more ancient Greek alphabet, introducing one into the class of units, one into that of tens, and one into that of hundreds. The whole of the characters thus arranged are shown in the following table: name of Cæsar. Some nouns are used without articles; such as proper names, Andrew, London, Paris; or names of attributed or mental qualities, as, beauty, goodnature, virtue, charity, &c.; or words in which nothing is implied but the mere existence of the thing, as, This is not silk, but cotton; or, This is not gold, but silver gilt. There are also nouns which will not admit the use of the article, as when words are to be taken in their largest and Signs. Names. Values. Signs. Names. Values. Signs. Names. Values. most general sense; thus we say, Man is a rational, an accountable creature; that is, all men, without exception, are rational and accountable. (Questions on the foregoing Lesson will be given in our next.) LESSONS IN ARITHMETIC.-No. II. GREEK SYSTEM OF NOTATION. THE difficulty of inventing names for all numbers even to a limited extent, and of remembering them after they were invented, evidently led to the classification and arrangement exhibited in our system of numeration, which was explained m the first lesson. The next difficulty would be that of performing calculations by the help of the mere names-a process which, in such a case, must either be done mentally, or with the assistance of the ten fingers. The use of small stones or pebbles (in Latin, calculi), for the purpose of making calculations, is indicated by the origin of the word itself. The necessity of inventing signs or characters to represent numbers, and to facilitate the process of computation, would become more and more obvious as society advanced in civilisation. At first, men would most naturally employ the letters of the alphabet, in every language, as the readiest marks for numbers. Hence, we find that this practice was adopted by the Hebrews, the Greeks, the Romans, and various other nations of antiquity. In the two oldest collections of writings in the world, the Bible, and the works of Homer-the one written in Hebrew and the other in Greek-the letters of the alphabets of these languages are respectively used to denote the whole or parts of these books, in their proper numerical order. The letters of the alphabet in any language, however, would go but a little way in expressing numbers by signs, unless some system of classification and arrangement were adopted, or some method of increasing their value, according to a fixed scale, introduced. For the purpose of expressing large numbers by signs, the Hebrews divided the letters of their alphabet into three classes of nine characters each, to denote units, tens, and hundreds, respectively; and as this alphabet contained only 22 letters, they adopted five of its letters, which had a final form (that is, a peculiar form when they terminated a word), to complete the class of hundreds. The whole collection of signs was then arranged, as in the following table: HEBREW SYSTEM OF NOTATION. Signs. Names. Values. Signs. Names. Values. Signs. Names. Values. א Beth ב Yodh Kaph כ 10 20 100 200 300 400 2 Resh Kaph .500 ף Tsadhe 90 } a M The thousands in the Greek system of notation were denoted by the same letters as the units, with a dash, or accentual mark under; thus, 4, 1,000. Myriads, or tens of thousands, were denoted by the same letters with the letter M under-as 10,000. Instead of placing the letter M under, sometimes the letters Mu were placed to the right of the number of myriads, and sometimes only a point, instead of either, was placed in the same position, to indicate the same thing. Archimedes extended this notation, by taking the square of the myriad as a new unit or period, and forming a series of periods containing eight figures in each; so that he was enabled to express a number sufficient to denote all the sands of the sea. This system of notation, in some respects, anticipated the modern systems; and in others, surpassed them; but, unfortunately, it was confined to the knowledge of the learned. The same mathematician, one of the mightiest geniuses of antiquity, anticipated the discovery of logarithms, by a few happy thoughts, which were allowed to lie dormant for 2,000 yeurs, until Napier promulgated his immortal invention, in 1614, and in his turn forestalled the discoveries of later times. The origin of the notable improvement in notation, by which the nine characters for units only were employed, and the eighteen for tens and hundreds, thrown aside, is still a doubtful question, although it has generally been attributed to the Arabs or Moors of Spain, and examples of a similar notation are to be found in India. This improvement consists in giving to these nine characters a relative value-that is, a value depending on their positions, as well as an absolute value depending on their names. Thus denoting the place of units by a certain fixed mark or character, and placing each of the nine primary characters to the left of it, their values are increased tenfold, so that no new characters are necessary to denote the tens; again, denoting the place of tens and units by two of the primary characters to the left of these, their values are same fixed marks or characters, and placing each of the nine increased a hundred-fold, so that no new characters are necessary to denote the hundreds. It is evident that this process may be extended indefinitely, and applied so as to denote not only all the numbers whose names are expressed or indicated in our system of numeration, but all numbers whatsoever, although far beyond the reach of our numerical nomenclature. The first nine letters of any alphabet would, of course, answer the purpose of denoting the units, and a dot or any new letter might be employed to denote the vacant places of units, tens, &c., as they occur. The nine characters, and the cipher (which stands for nothing by itself, and is only employed to increase the value of other numbers), now usually adopted in all civilised countries, are the following-viz., of the twelfth century, which treats of the abacus Pythagoricus, viz. : I E A BILA 8 6 VULGAR SYSTEM OF ROMAN NOTATION. 0 1 2 3 4 5 6 7 8 9 cipher, one, two, three, four, five, six, seven, eight, nine. M. Chasles, a French writer who has recently made some 1 2 3 4 5 6 7 8 9 curious researches in the history of the mathematics, maintains that these characters have descended to us from the From the preceding observations, it is evident that the Greeks and the Romans, with the whole system of decimal Romans, especially those who were initiated into the doctrines notation. He asserts that a very obscure passage in the of Pythagoras, employed in their mathematical calculations a writings of Boethius, a Roman philosopher and senator, who system of notation and a set of characters very similar to our flourished at the end of the fifth century, has a direct reference own, and very different from those which they used in their to our decimal system of notation; that various manuscript ordinary writings, and which are denominated Roman figures. treatises written between the tenth and twelfth centuries As these are, to this day, very often used for various common relate to the same subject; that the celebrated Gerbert (after-purposes among ourselves, such as paging certain portions of wards Pope Sylvester II.) had greatly contributed to introduce a book, making dates, numbering chapters, &c., it will be and extend the knowledge of the decimal system, in the east, useful to give the following table of their values:towards the end of the tenth century; and that the apparatus employed to facilitate operations in this system, which was Signs. unquestionably constructed on the decimal scale, was universally denominated the abacus Pythagoricus, or Pythagoras's II board, until the beginning of the twelfth century. In the early part of this century, the cipher, which had been preceded in the Greek notation by a dot or period, was known at first by the names of rota, rotula, sipos, and afterwards by those of circulus and cifra, or cipkra. The term cipher, however, is applicable to all the characters used in our system of notation; and the art of arithmetic itself is hence called ciphering. This word is evidently derived from the Hebrew verb saphar, to number; and it is not improbable that even the art of numbering itself, as well as the symbols employed in it, may have spread over the east from the people who originally spake this language. At the revival of literature, the invention of the arithmetical characters was ascribed to the Indians, from whom it was pretended they came to us, through the Arabians; and the characters themselves were denominated figure Indorum. In this way all trace of the ancient system of notation, preserved in the abacus Pythagoricus, was insensibly lost in the writings of the moderns, while some ideas, no doubt taken from the Arabic literature, were introduced. Hence, at the present time, all remembrance of the abacus, and of the real origin of our system of notation, has disappeared, and their origin is referred to the Arabians and the Hindoos. Many passages, however, in the works of writers on the subject, even so late as the sixteenth century, show that the Greek and Latin origin of our decimal system was not then completely forgotten. It is also important to observe, that those who ascribe its origin to the Arabians, or the Asiatics, do not assert that we have preserved the characters employed by the inventors; and it may, indeed, be shown that the characters we now use are extremely analogous to those of Boethius, as well as to those which were employed in the treatises on the abacus written during the middle ages. The following are the characters which M. Chasles has discovered in a manuscript Values. 100 200 300 400 500 600 700 800 900 Combinations of these numbers are formed by placing those of highest value first in order; thus XI, 11; XII, 12; ČXXI, 121, &c. The thousands are denoted by the letter M or the combination CIO; the tens of thousands by CCIO; and so on. In this system, there is very considerable regularity and ingenuity, although it is not adapted for the purposes of calculation. The letter I, repeated any number of times denotes so many units; when placed to the right of another character, it adds a unit to the number represented by that character; when placed to the left, it takes away a unit. The letter X repeated any number of times, denotes so many tens; when placed to the right of a character of greater value, it adds ten to the number represented by that character; when placed to the left, it takes away ten. The letter C repeated any number of times, denotes so many hundreds; when placed to the right of a character of greater value, it adds a hundred; and when placed to the left, it takes away a hundred; and so on, with M &c. The letter C placed right and left, in the former case inverted, increases the value of CIO tenfold. A bar placed over any character or number increases its value a thousand fold. In some few cases, it has been ascertained that the Romans employed these literal characters even with values depending on their position; thus, in Pliny, we find XVI.XX.DCCC.XXIX used for the number 1,620,829. The following tables, especially the first, which combines the system of numeration with that of notation, will be found of the greatest utility. TABLE I.-ENGLISH SYSTEM OF NUMERATION AND NOTATION. TRILLIONS. Thousands of. Hundreds of.. • TRILLIONS. Tens of..... TABLE II.-FRENCH SYSTEM OF NUMERATION AND NOTATION. names that are applied to each place in the given number, according as they stand in the table, and fill up the blank places in this line of figures with ciphers; then the proposed number will be properly expressed in figures. TRILLIONS. THOUSANDS. SEPTILLIONS. Tens of QUINTILLIONS. TRILLIONS. Hundreds of. Hundreds of. Hundreds of SEXTILLIONS. QUINTILLIONS. QUADRILLIONS. UNITS. Tens of UNITS. 1, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 0. The use of these tables is to enable the learner to read any number that may be placed before him, or that may occur in calculation, or to write any number that may be proposed in words, with ease and certainty, according to the particular system he chooses to adopt, or that may be recommended to him. As we are partial to the English system, on account of its greater ability to grasp large numbers, we add the following rules, which have special reference to Table I., and which must be committed to memory : PROBLEM 1.-To read or express in words any proposed number expressed in figures.-Rule: Point the proposed number off by commas into periods of six figures each, from right to left, and then into half periods of three figures each. Apply to the number thus divided the names which belong to the respective figures and places of figures in each period, reading them from left to right that is, from the highest name to the lowest-taking care to avoid repetitions of the same word, and observing that wherever a cipher occurs, the name which belongs to that place must be omitted. It will be of importance to remember, that in the application of this table, the name hundred is applied to that of every third figure, and the additional name thousand to that of every sixth figure, reckoned from right to left. After the name hundred, in reading from left to right, the name ten immediately follows; and after the name hundred thousand, that of ten thousand, in the same order. After the name ten, in reading from left to right, the name of the period in which it occurs is applied; and after the name ten thousand, that of thousand is applied. The following examples will show the application of this rule. EXAMPLE 1.-Read, or express in words, the number 146385297831276543 Here, the number pointed according to the rule will stand thus: 146,385;297,831;276,543. In this number there are three complete periods, and six half periods. By referring to the table, we see that the name of the first period, reckoning from left to right, is billions, the next millions, and the next units. Hence, remembering the above rule, it is read or expressed in words, thus: One hundred and forty-six thousand three hundred and eighty-five billions, two hundred and ninety-seven thousand eight hundred and thirty-one millions, two hundred and seventy-six thousand five hundred and forty-three. The name units is generally omitted at the end of a number when read in the preceding manner. The number in the table itself is read thus: one septillion. EXAMPLE 2.-Read or express in words the number 101000230200001. This number, pointed according to the rule, will stand thus: 101;000,230;200,001. In this number there are two periods and a half; hence, it is read or expressed in words as follows: One hundred and one billions, two hundred and thirty millions, two hundred thousand and one. PROBLEM 2.-To write or express in figures any proposed num ber expressed in words.-Rule: Write in one line as many ciphers, from right to left, as will extend from units to the highest name in the proposed number, and point them off according to the table, as before. Write below these ciphers the figures which express the EXAMPLE 1.-Write or express in figures the number; Three hundred and forty-five thousand six hundred and seventy-two bil lions, one hundred and thirty-eight thousand seven hundred and ninety-two millions, five hundred and eighty-three thousand six hundred and forty-one. Here, the highest name in the proposed number being hundreds of thousands of billions, we write eighteen ciphers, as follows; and then write below them the figures which express the names belonging to each place, according to rule; thus: 000,000,000,000,000,000. 345,672;138,792;583,641. EXAMPLE 2.-Write or express in figures, the number; One hundred billions, two thousand and thirty-two millions, one hundred and one. Here, as before, the highest name being hundreds of billions, them the figures which express the names belonging to each we write fifteen ciphers as follows; and then write below place, according to rule, filling up the blank places with ciphers; thus: 000;000,000;000,000. QUESTIONS ON THE PRECEDING LESSON. 1. Write out the names of all the numbers from one to a hundred, and express them in figures. 2. Write out the names of the numbers which immediately follow one hundred; one hundred and ninety-nine; four hundred and ninety-nine; nine thousand nine hundred and ninety-nine; and a million. 6. Write or express the following numbers in figures — Three thousand and thirty-two. Six hundred and five thousand, and nineteen. Three hundred and forty-one thousand, seven hundred and eighty-two. Eighty millions, two hundred and three thousand and two. Two hundred and two millions, twenty thousand two hundred and two. Nine thousand nine hundred and ninety-nine millions, nine hundred and ninety-nine thousand, nine hundred and ninety-nine. Write also the number which follows this last one in order Twenty thousand millions. Two hundred thousand and twenty millions, two thousand. The next number to thirty thousand billions, nine hundred and ninety-nine thousand. (6) pré, meadow; allé, gone; donné, given; passé, passée, See (è) très, very; après, after; achète, buy; mère, mother; (i) lit, bed; dit, said; dire, to say; lire, to read; lime, (i) dîner, to dine; île, isle; dîme, tithe; abîme, abyss; épître, epistle; gîte, lodging. (o) mot, word; cachot, dungeon; repos, rest; trot, trot; globe, globe; carrosse, coach. (o) tôt, soon; plutôt, rather; rôti, roast meat; dépôt, deposit; prévôt, provost. (u) bu, drank; cru, believed; du, of the; élu, elected; menu, minute; prévu, foreseen; ecu, crown. (a) chûte, fall; brûlot, fireship; mûr, ripe; dûment, duly ; il fût, he might be; nous fûmes, we were. THE DIPHTHONGS. (ai) je donnai, I gave; je parlerai, I will speak; j'allai, I (au, eau) maux, evils; chameau, camel; chevaux, horses; (eu) peu, little; peur, fear; leur, their; il meurt, he dies; (eu) like u. J'eus, I had; tu eus, thou hadst ; il eut, he had; (ia) liant, binding; il lia, he bound; partial, partial; il cria, (oi) loi, law; moi, me; il voit, he sees; il boit, he drinks ; roi, (ou) bout, end; il coud, he sews; il moud, he grinds; loup, wolf; coup, blow; croute, crust. (ua) nuage, cloud; nuance, shade; il salua, he saluted; il remua, he moved; il contribua, he contributed. (ue) nue, naked; lue, f. read; reque, f. received; il salue, he salutes; il remue, he moves. (ui) lui, him; luisant, shining; cuire, to bake; cuit, baked; nuire, to injure; produire, to produce. (y) thyrse, thyrsus; type, type; style, style; yeux, eyes; Yvetot, Yvetot; payer, to pay; envoyer, to send; abbaye, abbey; essayer, to try; pays, country; paysage, landscape; paysan, peasant. See 30. EXERCISE 2.-THE NASAL SOUNDS. Innocent, innocent; inutile, useless; dîner, dinner; im- 31. am ambre, amber; chambre, chamber; Adam, Adam: ram- an tante, aunt; manteau, cloak; plan, plan; plancher, em remplir, to fill; temple, temple; temps, weather; assem- examination. 32. en amen, amen; specimen, specimen; examen, 34. im simple, simple; timbre, stamp; daim, deer; faim, hun- 25. om sombre, dark; nom, name; ombre, shadow; tombe, on pont, bridge; honte, shame; montre, watch; raison, reason; maison, house; fondre, to melt; non, no. 36. um humble, humble; parfum, perfume; humblement, humbly. 36. un lundi, Monday; brun, brown; alun, ahum; emprunter, to borrow; importun, importunate; un, one. EXERCISE 3.-THE CONSONANTS. (b) baum, balsam; blessure, wound; brun, brown; absolution, absolution; abstrait, abstract; abbaye, abbey ; Jacob, Jacob. (c) cacher, to conceal; coin, corner; décuple, decuple; cire, waz; cinq, five; chercher, to seek; je cache, I conecal; patriarche, patriarch; patriarchat, patriarchate; chambre, chamber; arche, arch; changer, to change; orchestre, orchestra; charbon, coal; sac, bag; suc, juice; elerc, clerk; banc, bench; flanc, flank; second, second; fécund, fruitful; façon, fashion; reçu, received. (d) daim, deer; don, gift; admirer, to admire; bord, border; (f) foin, hay; faim, hunger; froid, cold; bref, short; soif, (h) (j) hâte, haste; honte, shame; haut, high; herbe, herbage; almanach, almanac. jujube, jujube; jeune, young; juger, to judge; jurer, to swear; joindre, to join; jonc, rush; déjeûner, to breakfast Juif, Jew; jeu, play. (1) lame, blade; loi, law; illégal, illegal; illicite, unlawful paille, straw; soleil, sun; pareil, similar; bail, lease; railler, to rail; souiller, to soil; caille, quail; canaille, rabble; ville, town; village, village; mille, mile, thousand; péril, peril; pointilleux, punctilious; baril, barrel; fusil, gun; gentilhomme, nobleman; gentilshommes, noblemen ; bouteille, bottle. EXERCISE 4.-THE CONSONANTS CONTINUEd. (mn) mon, my; marge, margin; nom, name; champ, field; moine, monk; prompt, quick; condamner, to condemn; faim, hunger; son argent, his money; bon appétit, good appetite; lien étroit, close connexion. (P) partir, to go away; coup, blow; temps, weather; drap, cloth; sept, seven; baptême, baptism; cap, cape; Alep, Aleppo. (q) quérir, to fetch ; quitter, to leave; musique, music; logique, logic; quarante, forty; quoi, what; aquatique, aquatic, Quintilién, Quintilian; cinq, five; cinq livres, five books. (r) ranger, to arrange; errer, to err; arriver, to arrive, verser, to pour; je courrai, I will run ; je courais, I was running; jouir, to enjoy; car, for; plaisir, pleasure; amer, bitter; parler, to speak; changer, to change; fer, iron; hiver, winter. (s) silence, silence; soin, care; sans, without; base, base; (t) tiers, third; tiare, tiara; tort, wrong; portion,* portion; (w) Westphalie, Wiemar, Worms, Wurtemberg. this letter is pronounced like e in cedar, has generally the sound of sh, as in T, in the English words corresponding to the French words, in which motion, partial, &c. EXERCISE 6. (x) xylon, cotton plant; Xénophon; exiler, to exile; excuser, Non, Madame, vous avez seulement le vinaigre et la viande. to excuse; luxe, luxury; Alexandre, Alexander; maxime, 19. Avez vous la table? 20. Oui, Madame, j'ai la table. maxim; soixantième, sixtieth; six, six; sixième, sixth; six livres, six books; Bruxelles, Brussels; Aix-la-Chapelle; -dix, ten; Phénix; Ajax; deux hommes, two men ; dix amis, ten friends. (z) zèle, zeal; zone, zone; zoologie, zoology; vous lisez, you read; nez, nose; Metz; allez-y, go there; venez ici, come here. SECTION II. 3. TO BE TRANSLATED INTO FRENCH. 1. Have you the wheat? 2. Yes, Sir, I have the wheat. Who has the meat? 4. The butcher has the meat and the salt. 5. Has he the oats? 6. No, Madam, the horse has the oats. 7. Have we the wheat? 8. You have the wheat and the flour. 9. Who has the salt? 10. I have the salt and the meat. 11. Have we the vinegar, the tea, and the coffee? 1. In French the article [§ 13, (2.)* has, in the singular, a 12. No, Sir, the brother has the vinegar. 13. Who has the distinct form for each gender. Le fils, the son. Ex. La fille, the daughter, the girl. Le frère, the brother. 2. Before a word commencing with a vowel or an h mute [Section I., 11 h], the article is the same for both genders [§ 18, (7)]. Ex. 5. The e of the pronoun je is elided when that pronoun comes before a vowel or an h mute [§ 146]. horse? 14. The baker has the horse. 15. Have we the book and the pen? 16. No, Miss, the girl has the pen, and the miller has the book. 17. Have you the table, Sir? 18. No, Sir, I have only the book. 19. Who has the table? 20. We have the table, the pen, and the book. RESUME OF 6. In interrogative sentences, when the third person singular of a verb ends with a vowel, and is immediately followed by a Le tailleur a l'habit de drap du pronoun, a t, called euphonic, must be placed between the verb and the pronoun. A-t-il ? A-t-elle ? Has he? Le père a la viande, vous avez le café, et j'ai l'eau. L'homme a le pain, l'enfant a le sel, et nous avons le poivre. Avoine, f. oats. Blé, m. wheat. Boucher, m. butcher. Boulanger, m. baker. Farine, f. flour. Livre, m. book. Has she? The father has the meat, you have the coffee, and I have the water. TO BE TRANSLATED INTO ENGLISH. médecin. Vous avez la lettre de la sœur du A-t-il le livre de la dame. The cloth coat. The silk dress. EXAMPLES. The tailor has the physician's cloth coat. You have the baker's sister's letter. (the letter of the sister of the baker). Has he the lady's book. 1. Qui a le pain? Le boulanger à le pain. 3. A-t-il lager et la farine du meunier. 7. Avez vous le porte-crayon farine? 4. Oui, Monsieur, il a la farine. 5. Avons nous la viande? 6. Oui, Monsieur, vous avez la viande et le pain. 7. Le meunier a la farine. 8. Le boulanger a la farine et le blé. 9. Avons nous le livre et la plume? 10. Oui, Mademoiselle, vous avez le livre et la plume. 11. Le boucher a la viande. 12. Le meunier a la viande et j'ai le café. 13. Avez vous l'eau et le sel? 14. Oui, Monsieur, nous avons l'eau, le sel, et l'avoine. 15. Avons nous le thé? 16. Non, Monsieur, la fille a le thé, le vinaigre, et le sel. 17. Ai-je le vin? 18. Once for all, references of this kind are made to the SECOND PART OF THIS COURSE, of the commencement of which duei ntimation shall be given. d'argent? 8. Oui, Monsieur, nous avons le porte-crayon d'argent. 9. Avons nous l'avoine du cheval? 10. Vous avez l'avoine et le foin du cheval. 11. Qui a l'habit de drap du charpentier? 12. Le cordonnier a le chapeau de soie du tailteur. 13. Le tailleur a le soulier de cuir du cordonnier. 14. Avez vous la table de bois ? 15. Oui, Monsieur, j'ai la table de bois du charpentier. 16. Ont ils le couteau d'argent? 17. Ils ont le couteau d'argent. 18. Le frère du médecin a la montre d'argent. 19. La sœur du cordonnier a la robe de soie. 20. A-t-elle le soulier de cuir? 21. Non, Madame, elle a le soulier de satin. 22. Avons nous le bas de laine? 23. Non, Monsieur, vous avez le bas de soie du tailleur. 24. Qui |