French. cage caisse campagne cap coche cendre cerise chaire chaloupe char cheminée cle or clef coin comté corps couple duel empire epouse etain fable face faim fardel fibre figue foi front fruit gai geai gingembre golfe goût Disqualify for, Dissatisfied with, Dissent from, Distinct from Distinguish from, English. cage case champagne eape coach cinders oherry chair } sloop chariot chimney clef coigne county corpse couple duel empire spouse tin fable face famine fardeau fibre fig flute faith front fruit gay jay ginger gulf gout Latin. capsa campánia caput cinis cerasus carrus camínus clavis cuneus comitatus corpus copula duellum imperium sponsa stannum fabula facies fames phortos (Greek) fibra ficus Many French terms are employed in English either in their native form or slightly altered, and of these some even in France are of modern origin. We have dragoon from the name of the soldiers with whom Louis XIV. carried on the war, which received the name of his dragonades, against his French Protestant subjects in order to compel them to become Catholics. From the noun dragoon we have the verb to dragoon into. A roué, in slang language, a black-leg, is literally a wretch who deserves to be broken on the wheel-metaphorically one who has the same manners, as the courtiers of the profligate Duke of Orleans, Regent of France, who is said to have given the name to his abandoned associates. Guillotine, a term which we derive from France, is the name of an instrument for decapitating political offenders; it received its name from the inventor, and was first used amidst the early horrors of the first revolution in that country. Translations from the French have led to the in-coming amongst us of many French terms and phrases, greatly to the corruption of our mother English. Formerly, translations were said to be done into English." The phrase is not inappropriate, for many translations from the French are miserably done, a large portion of every page 66 consisting of French words, and idioms in an English dress; resembling a Frenchman aiming to speak English by putting on an English costume. Common-place novels, too, have brought into vogue many Gallicisms. Most blame-worthy is this defacement and corruption of our language, when they are perpetrated by historians of whom better things might be expected. "This practice has been well taken off by the "Spectator" in No. 185 of that work, which is strongly recommended to the perusal of those who possess it or can readily borrow it. Having read the remarks in the "Spectator," and read also what I have written in this lesson, let the student proceed to write an essay on fides frons fructus gaudium THE FRENCH ELEMENT IN THE ENGLISH LANGUAGE. Words with their proper Prepositions. F. R. qualis, of what kind sentio, I feel tinctus, dyed, coloured zingiber kolpos (Greck) gustus between Distrustful of, Divested of, tínguo, I dye, colour trauen, to trust Divide between (two), Dote on, Dwell in, at, on, Embark in, on board of, Embellished with, Employ in, on, about, Enamoured of, Report the following extract: divido, I divide dubito, I doubt begierig, desirous of embarquer, to go into a barque COMPOSITION. bellus, beautiful employer, to put to use ON THE CHOICE OF AUTHORS. If we are to read, it is a very important rule in the conduct of the understanding, that we should accustom the mind to keep the best company, by introducing it only to the best books. But there is a sort of vanity some men have, of talking of, and reading, obscure, half-forgotten authors, because it passes as a matter of course that he who quotes authors which are so little read, must be completely and thoroughly acquainted with those authors which are in every man's mouth. For instance, it is very common to quote Shakspeare; but it makes a sort of stare to quote Massinger. I have very little credit for being well acquainted with Virgil; but if I quote Silius Italicus, I may stand some chance of being reckoned a great scholar. In short, whoever wishes to strike out of the great road, and to make a short cut to fame, let him neglect Homer, and Virgil, and Horace, and Ariosto, and Milton, and, instead of these, read and talk of Fracastorius, Sannazarius, Lorenzini, Pastorini, and the thirty-six primary sonnetteers of Bettinelli; let him neglect everything which the suffrage of ages has made venerable and grand, and dig out of their graves a set of decayed scribblers, whom the silent verdict of the public has fairly condemned to everlasting oblivion. If he complains of the injustice with which they have been treated, and call for a new trial with loud and imporin the estimation of men of sense, he will be sure to make some tunate clamour, though I am afraid he will not make much progress noise in the crowd, and to be dubbed a man of very curious and extraordinary erudition.-Sydney Smith. LESSONS IN GERMAN.-No. XXXIX. SECTION LXXXI. Berlassen, when used reflexively, signifies, "to depend upon to rely upon." Ex.: Ich verlasse mich auf ihr Wort; I depend upon your word (I leave myself upon your word.) I. Abhängen, likewise, signifies to depend upon, to be dependent upon. Ex.: & hängt von Umständen ab; it depends upon circumstances. Thence is derived the adjective abhängig, dependent. Ex.: Er führt ein abhängiges Leben; he leads a dependent life. Die Vereinigten Staaten erklärten sich als ein unab. hängiges Volf; the United States declared themselves (as) an independent people. EXERCISE 84. able, easy; Ab’hängen, (Sce above); 6'bangig, dependent; Diese Leute stellen sich als ob sie von Hinaus eilen, to hasten Kanarienvogel, m. Deffnen, to open; Umstand, m. circumstance; Un'abhängig, indepen- I cannot agree to it. These people act (place them- EXERCISE 85. Kern, m. kernel; Kutscher, m. coachman; Ordnung, f. order, regulation; Schale, f. shell; Schwächling, m. weakling, weakly per Anʼgeben, to give, specify; An'strengung, f. exertion, effort, labour; Bereit, ready; Beruf, m. calling, vocation; Beruhigen, to quiet; Bestim'men, to fix, determine; Dafür, therefore, for it; Dank, m. thanks, acknowledgment; Grret'ten, to save, rescue, deliver; Furcht, f. fear, dread; 21 son; Umschließen, to inclose, surround; Umwerfen, to upset; Un'ortentlich, disorderly, irregular, confused; Unterlass'en, to leave off, omit, fail; Un'würdigkeit, f. unworthiness, indignity; Teller, m.plate; Verter'ben, to spoil, corrupt, destroy; Verzich'ten, (auf Etwas), to resign, i. e. as a privilege or a claim on anything; Vor'gehen, to go before, go too fast; Wagen, m. carriage; Weisheit, f wisdom; Wesen, n. being, exis tence: Zerbrechen, to break (in pieces). Es wiederfährt' uns in unserm Leben Es wiederfährt Manchem mehr Der Vogel ist zum Fenster hinaus. Die Freunde entzwei'ten sich. Die Pflaume ist ein Steinobst. Ist es nicht, als ob dieses Volk mic There happens to us in our lives serves. The bird has flown out of the The friends quarrelled (sepa- One should never assent to a 1. Dieses Jahr ist das Obst, sowie alle Früchte, wohl gerathen. 2. Dieser Baum trägt jedes Jahr sehr vieles Obst. 3. Ist alle Frucht Obst? 4. Nein, nicht alle, sondern nur solche, die (§ 65. 2.) an Bäumen wächst. 5. Dieser junge Marn verläßt sich zu viel auf seine Verwandten und zu wenig auf seine eigenen Fähigkeiten. 6. Er verläßt sich darauf, daß wir ihn die nächsten Woche besuchen. 7. Er verließ sich darauf, daß ihm Gott helfen werde. 8. Wer sich zu viel auf Andere verläßt, kann leicht getäuscht werden. 9. Ich halte (Sect. 69. II.) viel auf meine Freunde. 10. Er hält viel auf ein gemächliches Leben. 11. Dieser Mann hält zu viel auf sich und seine Klugheit, weßhalb er den Nath wohl. meinenter Freunde verschmäht. 12. Nur unter dieser Bedingung kann ich darein willigen. 13. Ich willige darein, in so fern (Sect. 79. V.) es feine üblen Folgen hat. 14. Er willigte varein, ohne mit allen Schwierigkeiten bekannt zu sein. 15. Dieses Kind thut gerade, als ob es hier zu Hause wäre. 16. Der Matrose stellte sich, als ob er von Sinnen wäre. 17. Er geberdet sich, als ob ihm das größte Unrecht widerfahren sei. 18. Dieser Mann stellt sich, als ob er beleidigt wäre. 19. Er stellt fich wie ein Kind von fünf Jahren. 20. Der Nachbar warf den zu dringlichen zur Thüre hinaus. 21 Der Knabe cilte zur Thüre hinaus, als ich dieselbe öffnete. 22. Zur Thüre hinaus, wer sich entzweit! (Göthe.) 23. Es hängt ganz von Umständen ab, ob ich schon nächstes Jahr nach Amerika reise oder nicht. 24. Es hängt sehr von Umständen ab, was er thun wird. 25. Ein so abhängiges Leben die Bauern in Deutschland führen, ein eben so unabhängiges führen fie in Amerika. 26. Ganz unab, hängig vermag kein Mensch auf Erten zu werden. 1. Last year the fruit did not turn out well. 2. This tree yields fruit but seldom. 3. This young gentleman relies too much upon his abilities. 4. No, he does not rely too much upon his abilities, but he knows it is not well to be dependent upon those of others. 5. I rely upon you that you will visit me next week. 6. Do exactly as if you were at home. 7. The criminal acted as if he were out of his senses. 8. This man acts exactly as a child. 9. Where is your canary.bird? It is flown out of the window. 10. How can I assent to a thing which is against my inclination? 11. Whoever quarrels shall be expelled the house. 12. It depends upon circumstances, whether I shall go to my friends. 13. Every man strives to be independent. 14. Depend upon it that I shall not help you again. SECTION LXXXII Nichts or, nicht dafür können, signifies "not to be in fault, or to blame,'" &c.; as, ich kann nichts dafür; it is not my fault, or I cannot help it; literally, I cannot, or can nothing therefore. Er kann nichts dafür, daß er so arm ist; he cannot help it, i. e. he is not to blame that he is so poor. So also interrogatively; as, kann die Welt etwas dafür, daß sich ein großer Geist in ein schlech tes Kleid versteckt? (Rabener.) Is the world t blame, that a great soul conceals itself in a plain dress? That is, Die Welt kann nichts dafür. Ich kann nichts dafür, taß ich mein Diese Uhr geht vor (or, zu schnell), Hat man mein Zimmer in Ortnung In der Reihe seiner Schmeichler hat Leben Sie wohl, mein Herr, uut em It is not my fault, that I have This watch goes too fast and that (one) goes too slow. Has my room been put in order? In the ranks of his flatterers he has not a true friend. There are many who believe, that in (the) most cases, the fortune or misfortune of a man depends on chance. Farewell, sir, and please remember me kindly to your lady. 11. Leben Sie 1. Sie können nichts dafür, daß Sie so unglücklich sind. 2. Er konnte nichts dafür, dieses Glas zerbrochen zu haben. 3. Ich kann nichts dafür geben, als meinen Dank. 4. Die Gründe dafür werde ich angeben, wenn es verlangt werden sollte. 5. Können Sie mir sagen, (Sect. 83. I.) wie viel Uhr (Sect. 25. IX.) es ist? 6. Nein, tenn meine Uhr ist stehen geblieben. 7. Steht Ihre Uhr schon lange? 8. Ja, beinahe eine Stunte. 9. Meine Uhr geht zu schnell, sie geht beinahe eine halbe Stunte vor. 10. Die Uhr meines Freundes geht fünf Minuten vor. wohl, und vergessen Sie nicht, mich bald wieder zu besuchen. 12. Leben Sie wohl, mein Herr! 13. Wann wollen wir zusammen Herrn N. besuchen? 14. Es hängt ganz von Ihnen ab (Sect. 81. I.), welche 3eit Sie dazu bestimmen wollen, ich bin zu jeder Zeit bereit, mitzugehen. 15. G8 hängt von Ihnen ab, diese Familie zu erretten over zu verderben. 16. Der Nachbar arbeitet in seinem Garten und sucht denselben in Ortnung zu bringen. 17. Vei Aller Anstrengung bringt er diese Sache nicht in Ordnung. 18. Er suchte mich in tie Reihe seiner Kameraten zu bringen. 19. Es hält schwer (Sect. 46. II.), einen unordentlichen Menschen an Ortnung zu gewöhnen. 20. Nach vieler Mühe hat er die Rechnung in Crinung gebracht. 21. Wer an dem Fuße eines steilen Berges stehen bleibt und aus Furcht vor Anstrengung denselben zu er, klimmen unterläßt und lieber auf die schöne Aussicht verzichtet, der zeigt damit an, daß er ein Schwächling und eines solchen Genusses unwerth ist, und wer aus eigner Schuld in ter Mitte seiner geistigen Ausbildung stehen bleibt, und den süßen Kern der Weisheit entbehren will, weil eine rauhe und harte Schale denselben umschließt, der zeigt ebenfalls nicht nur seine Umwärtigkeit, denselben zu genießen, an, sondern auch, wie wenig er ten Beruf und die Pflicht des Menschen, als eines geistigen Wesens, erkannt hat. 1. It is not my fault that you have had the mishap. 2. You are not to blame that the servant has broken the plate. 3. He could not give me anything for it, except his thanks. 4. He could not help it, he only spoke the truth. 5. Is the coachman to blame, that the carriage was upset? 6. No, he is not to be blamed, for the horses could not be quieted. 7. Can you tell me what time it is? 8. No, my watch goes too slow. 9. To fix the hour of my departure copande upon my parents 19 Farewell, Madam; please do not forget, to remember me to your parents 11. It depends upon you. what time you will fix to visit your friend; I shall always be ready to accompany you. 12. Fortune and misfortune. He and death, poverty and riches, all depend on the will of God SECTION LXXXIII. Sich versieben, (to understand one's self with m2 signifies to be a judge of, to be skilled in. Ex: Grit 14 he is skilled in everything. d 2 ** Go veficht fl. (literally, it understands its:lf, i, e it is unien? 23. 34 war tret viettel Sturden and demilen 24. Sma derstood, is self-evident,) answers to our phrase, of course," Sie weit ven temselben entfernt * 25. 34 hr bes-le en Halle Str or, as a matter of course." Ex: Gewicht 5 cr c ver weit von demselben entfernt geweien 20. 36 defe cut midezutie, i steht sich von selbst, daß ich maken Eltern gebeten mut of course,, or as a matter of course, I must cher my parents. The word c nun auf dieser, ever sei es in jener Bea. 27. Det Gefangene mazte „natürli" naturally, is often used in the same manner; as, et ei nun lange geung ras a ten warmen. Schon sa Saar wat te fait. natürlich mus e fe fein, of cours:, it must be so. | ruft habe entbehren müssen. 28 36 fan magen side ja tir Latme 1. Cagen answers to the English say or tell to tell or es sei denn, daß mein Bruter bet raar wen gir; göl mine 9. narrate, however, is often expressed in German by craien | Ich kann heute unniglich tiefca 2: ef beertigen, då sa tezz, kaj ið tiến Ex.: Was sagte er? What did he say? Wat bat er Ihnen gefagt? Nachmittag weniger gekirt wette 90. G mut Numart in the Stai What has he told, or said to you? Der alte Motroje erzählte eingelassen, es sei denn, daß er einen Bok Hale eine rührente Gefichte; the old sailor told, or related a moving (affecting) story. 9 far ale nicht aber mein MILIE” Sie net gehen Sie from II. dert is often answered in English by "gone, off," &c Ginen wie langen (§ 120. 4.) Spa Isefer Italte'ner reifieht sich auf Meu Fer tort-me' tifa III. Es sei denn, tas unless, except, &c, Ex: Der Mensch kann nicht wahrhaft glücklich sein, es sei tenn, daß er tugent haft sei; man can not be truly happy, unless he be virtuous Wahrlich, wahrlich, ich sage dir: Es sei denn das Zemant von Neuem geboren werte, kann er das Reich Gottes nicht sehen: EXERCISE 86 Anwalt, m. attorney, Hin'femmen, to come, echer, m. jest, sport; weit er auch von hier wehnt art (o lanze ich auch zu geben Lole, to brim'd: 1h ihr tennoch effe que eren Sie so se me[[ ? Ich gebe zu zem 3-knʼatzte Do you know how far you have It is self-evident that a lazy This Italian is a judge of music Mr. M. left (is off) this morning I am going to the dentist. I. Tell me if that is your own horse? 2 That farmer told me many things about agriculture. 3. I shall not go out today unless necessity compels me. 4 You will not enter the kingdom of heaven, unless you acknowledge the blessings of God 5. My brother went off yesterday and we have heard nothing of him. 6. It is self-evident that without nourishment man, animals, and plants cannot exist 7. My knife is gone, and none of the children know where. 8. Our money is al gone. 9 I know very well how far I have to go in this matter 10. Where do you go to? 11. I am going to my brother. 12 How far have you to go? 15. Just to the park. 14. What distance have you to go? 15 About three quarters of a mile 16. He believed the time had now arrived to open his own path through life. ILLUSTRATION. PROB. 1. Suppose that a man divided 72 pounds among his three sons in the following manner :-To a be gave a certain number of pounds; to n he gave three times as many as to a; and to c be gave the remainder, which was half as many pounds as a and a received. How many pounds did he give to each ? 1. To solve this problem arithmetically, the pupil would resson thus: A had a certain part, i. e.. one share; a received three As far as he resides from here, times as much, or three shaves; but c had half as much as a and and so great a distance as IB; hence he must have received two shares. By adding their have to walk, I, nevertheless, respective shares, the sum is sir shares, which, by the conditions of visit him every day. the question, is equal to 72 pounds. If, then, & shares are equal Whither are you hastening so to 72 pounds, 1 share is equal to of 72, viz., 12 pounds, which rapidly? is A's share. E had three times as many, viz., 36 pounds, and c half as many pounds as both, viz., 24 pounds. 2. Now, to solve the same problem by algebra, he would use letters and signs, thus: LESSONS IN ALGEBRA.-No. I SECTION I ART. 1.-ALGEBRA is a general method of solving problems, and signs. Leta represent A's share; then by the conditions, 23 will represent B's share; and 42 will represent c's share. Then 6x=72, for the whole is equal to all its parts; and 1s=12 Add together the several shares, or z's; thus, +3x+25=6x pounds A's share; 3=36 pounds B's share; and 2r=24 pounds 1. Ter Lieb ist seines Verbrechens überführt werten unt es versteht sich von selbst taß er bestraft werden wirt. 2 Der Vater ist seit diesem Morgen fert und bis jept noch nicht wieder zurückgekehrt 3 Das Buch ift fert unt feiner tiefer Schüler will (§ 83. 8. Rem.) wissen, wo es hinc's share. gekommen ist. 4 Vicine Neffen find fortgegangen, ehne zu sagen wohin fie gehen würten. 5. Unser Cbft ist alle. (Sect, 41. III.) 6 Auch noch so vieles Gelt wirt all, wenn man verschwenterisch ist 7. Det tückisbe Kaiser Soliman II fagte furz vor seinem Lote: meine Säfte PROOF.-Add together the number of pounds received by each, and the sum will be equal to 72 pounds, the amount divided. In this algebraic solution it will be observed: First, that we represent the number of pounds which a received by z. Second, ta obtain B's share, we must multiply A's share by 3. This mathi cation is represented by two lines crossing each other like a capital X. Third, to find c's share, we must take half the sum of A's and B's share. This division is denoted by a line between two dots. Fourth, the addition of their respective shares is denoted by another cross formed by a horizontal and perpendicular line. Take another example: PROB. 2.-A boy wishes to lay out 96 pence for peaches and oranges, and wants to get an equal number of each. He finds that he must give 2 pence for a peach, and 4 pence for an orange. How many can he buy of each? of Let x denote the number of each. Now, since the price of one peach is 2 pence, the price of a peaches will be x x 2 pence, or 2x pence. For the same reason, 2 X 4, or 4x pence, will denote the price of oranges. Then will 2x-4x, that is, 6r, be equal to 96 pence by the conditions of that question, and 1x is equal to 96 pence, viz., 16 pence, which is the number he bought of each. 2. QUANTITIES in algebra are generally expressed by letters, as in the preceding problems. Thus may be put for 2 or 15, or any other number which we may wish to express. It must not be inferred, however, that the letter used has no determinate value. Its value is fixed for the occasion or problem on which it is employed; and remains unaltered throughout the solution of that problem. But on a different occasion, or in another problem, the same letter may be put for any other number. Thus, in prob. 1, x was put for A's share of the money. Its value was 12 pounds, and remained fixed through the operation. In prob. 2, was put for the number of each kind of fruit. Its value was 16, and it remained so through the calculation. 3. By the term quantity, we mean anything that can be multi-A plied, divided, or measured. Thus, a line, weight, time, number, &c., are called quantities. 4. The first letters of the alphabet, a, b, c, &c., are used to express known quantities; and the last, z, y, a, &c. letters, those which are unknown. 5. Known quantities are those whose values are given, or may be easily inferred from the conditions of the problem under con sideration. 6. Unknowr quantities are those whose values are not given but required. 7. Sometimes, however, the given quantities, instead of being expressed by letters, are given in figures 8. Besides letters and figures, it will also be seen that we use certain signs or characters in algebra to indicate the relations of the quantities, or the operations which are to be performed with them, instead of writing out these relations and operations in words. Among these is the sign of addition (+), subtraction (−), equality (=), &c. 9. Addition is represented by two lines (+), one horizontal, the other perpendicular, forming a cross, which is called plus.. It signifies "more," or "added to." Thus a + signifies that is to be added to a. It is read a plus b, or a added to b, or a and b. 10. Subtraction is represented by a short horizontal line (-) which is called minus. Thus, a--b signifies that b is to be "sub. tracted" from a; and is read a minus b, or a less b. 11. The sign + is prefixed to quantities which are considered as positive or affirmative; and the sign, to those which are supposed to be negative. For the nature of this distinction, see Arts. 36 and 37. is used; as If the first is less than the other, the character 16. A numeral figure is often prefixed to a letter. This is called The co-efficient may be either a whole number or a fraction. Thus fb is two-thirds of b. When the co-efficient is not expressed, 1 is always to be understood. Thus a is the same as la, i. e., once a, or one times. 17. The co-efficient may also be a letter, as well as a figure. In the quantity mb, m may be considered the co-efficient of b; because b is to be taken as many times as there are units in m. If m stands for 6, then mb is six times b. In 3abc, 3 may be considered as the co-efficient of abc; 3a the co-efficient of be; or 3ab the co-efficient of c. 12. The sign is generally omitted before the first or leading quantity, unless it is negative; then it must always be written. When no sign is prefixed to a quantity, + is always understood. Thus a+ is the same as +a+b. 13. Sometimes both and (the latter being put under the former, +) are prefixed to the same letter. The sign is then said to be ambiguous. Thus a+b signifies, that in certain cases, comprehended in a general solution, b is to be added to a, and in other cases subtracted from it. 18. A simple quantity is either a single letter or number, or several letters connected together without the signs and Thus a, ab, abd, and 86, are each of them simple quantities. 19. A compound quantity consists of a number of simple quantities connected by the signor. Thus a+b, d-y, b-d+ 34, are each compound quantities. The members of which each is composed are called terms. 20. If there are two terms in a compound quantity, it is called a binomial. Thus a+b and a-6 are binomials. The latter is also called a residual quantity, because it expresses the difference of two quantities, or the remainder, after one is taken from the other. compound quantity consisting of three terms, is sometimes called a trinomial; one of four terms, a quadrinomial, &c. 21. When the several members of a compound quantity are to be subjected to the same operation, they must be connected by a line () called a vinculum, or by a parenthesis (). Thus a-b+c, or a-(b+c), shows that the sum of b and c is to be subtracted from a. But a-be signifies that b and c is to be subtracted from a, while e is to be added. 22. A single letter, or a number of letters, representing any quantities with their relations, is called an algebraic expression, or formula. Thus a+b+3d is an algebraic expression. 23. Multiplication is usually denoted by two oblique lines And of them is called a factor. In the product ab, a is a factor, and 25. A quantity is said to be resolved into factors, when any 26. Division is expressed in two ways: 1st. By a horizontal Observ.-When all the signs are plus, or all minus, they are said to be alike; when some are plus and others minus, they are called unlike. a 2nd. Division is more commonly expressed in the form of a 14. The equality of two quantities, or sets of quantities, is expressed fraction, putting the dividend in the place of the numerator, and by two parallel lines. Thus a+b=d, signifies that a and b together are equal to d. So 8+4=16-4=10+2=7+ the divisor in that of the denominator. Thus - is a divided by b. 2 +3. 15. When the first of the two quantities compared is greater than the other, the character is placed between them. Thus a> b signifies that a is greater than b. b 27. When four quantities are proportional, the proportion is expressed by points, in the same manner as in the Rule of Proportion in arithmetic. Thus ab::c:d signifies that a has to b, the 356 same ratio which e has to d. And ab: cd :: a+m: b+n, means that ab is to cd, as the sum of a and m, to the sum of b and n. 28. Algebraic quantities are said to be alike, when they are expressed by the same letters, and are of the same power; and unlike, when the letters are different, or when the same letter is raised to different powers. Thus ab, 3ab, -ab, and -6ab, are like quantities, because the letters are the same in each, although the signs and co-efficients are different. But 3a, 3y, 3bx, are unlike quantities, because the letters are unlike, although there is no difference in the signs and co-efficients. So x, xx, and xxx, are unlike quantities, because they are different powers of the same quantity. (They are usually written x, x and x3.) And universally if any quantity is repeated as a factor a number of times in one instance, and a different number of times in another, the products will be unlike quantities; thus, cc, ecce, and e, are unlike quantities. But if the same quantity is repeated as a factor the same number of times in each instance, the products are like quantities. Thus, aaa, aaa, aar and aaa, are like quantities. 29. One quantity is said to be a multiple of another, when the former contains the latter a certain number of times without a remainder. Thns 10a is a multiple of 2a; and 24 is a multiple of 6. 30. One quantity is said to be a measure of another, when the former is contained in the latter any number of times, without a remainder. Thus 36 is a measure of 156; and 7 is a measure of 35. 31. The value of an expression, is the number or quantity for Thus the value of 3+4 is 7; of which the expression stands. 3X4 is 12 of 6 is 2. 32 The RECIPROCAL of a quantity, is the quotient arising from dividing a UNIT by that quantity. The reciprocal of a is 1 reciprocal of a+b is ; the reciprocal of 4 is a+b 14 33. What is the algebraic expression for the following statement, in which the letters a, b, c, &c., may be supposed to represent any given quantities? EXAMPLE. The product of a, b, and c, divided by the difference of c and d, is equal to the sum of b and c added to 15 times h. = =b+c+15h. abc Ans. a+b h EXERCISES. 2. The product of the difference of a and h into the sum of b, c and d, is equal to 37 times m, added to the quotient of divided by the sum of h and b. 3. The sum of a and b, is to the quotient of b divided by c, as the product of a into e, is to 12 times h. 4. The sum of a, b and c, divided by six times their product, is equal to four times their sum diminished by d. 5. The quotient of 6 divided by the sum of a and b, is equal to 7 times d, diminished by the quotient of b, divided by 36. 31. What will the following expressions become, when words are substituted for the signs? =abc6m+. a a+c. -=dxa+b+c c-Gd 10. 6. Ans: The sum of a and b divided by h, is equal to the product of a, b and c diminished by 6 times m, and increased by the quotient of a divided by the sum of a and c. EXERCISES. 6+6 d-4 9. a-bac:: : 3×(h+d+y). m ୯ =(a+h) (b—c.) = a bax (d+h) ; the beginning. In doing this the sign X must not be omitted between EXAMPLE.-11. +a+mn= +3+8x10=9+3+80= 2 92. Ans. a m cd 'a-h d+ab + 2m h+dm 3+ (b—c) 35. At the close of an algebraic process it is often necessary to tore the numbers for which letters have been substituted at the b+mn bc+n_4+80 4X2+10 + = =8. Ans. + cd 3d 2X6 3x6 EXERCISES. +bem 26 15. abm+ +2n. m ax (d+c) m 16. (a+c)x(nm)+ m (c+b)x(m-d) n-bc (4d-b)X(a-c) 12 -a. de POSITIVE AND NEGATIVE QUANTITIES. 36. A POSITIVE of AFFIRMATIVE quantity is one which is to be added, and has the sign + prefixed to it. (Art. 11.) 37. A NEGATIVE quantity is one which is required to be SUBTRACTED, and has the sign — prefixed to it. When several quantities enter into a calculation, it is frequently necessary that some of them should be added together, while others are subtracted. If, for instance, the profits of trade are the subject of calculation, and the gain is considered positive, the loss will be negative; because the latter must be subtracted from the former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debit and the credit are distinguished by opposite signs. 38. The terms positive and negative, as used in the mathematics, are merely relative. They imply that there is, either in the nature of the quantities, or in their circumstances, or in the purposes which they are to answer in calculation, some such opposition as requires that one should be subtracted from the other. But this opposition is not that of existence and non-existence, nor of one thing greater than nothing, and another less than nothing. For in many cases either of the signs may be, indifferently and at pleasure, applied to the very same quantity; that is, the two characters may change places. In determining the progress of a ship, for instance, her easting may be marked +, and her westing; or the westing may be, and the easting - All that is necessary is, that the two signs be prefixed to the quantities, in such a manner as to show which are to be added, and which subtracted. In different processes they may be differently applied. On one occasion, a downward motion may be called positive, and on another occasion negative. 39. In every algebraic calculation, some one of the quantities must be fixed upon to be considered positive. All other quantities which will increase this must be positive also. But those which will tend to diminish it, must be negative. In a mercantile concern, if the stock is supposed to be positive, the profits will be positive; for they increase the stock; they are to be added to it. But the losses will be negative; for they diminish the stock; they are to be subtracted from it. 40. A negative quantity is frequently greater than the positive one with which it is connected. But how, it may be asked, can the former be subtracted from the latter? The greater is certainly not contained in the less: how then can it be taken out of it? The answer to this is, that the greater may be supposed first to exhaust the less, and then to leave a remainder equal to the difference between the two. If a man has in his possession 1,000 pounds and has contracted a debt of 1,500; the latter subtracted from the former, not only exhausts the whole of it, but leaves a balance of 500 against him. In common language, he is 500 pounds worse than nothing. 41. In this way, it frequently happens, in the course of an |