ally interrupted by deep cisterns in which the water was allowed to settle and deposit its sediment. This water, so carefully managed, was remarkable for its coolness and salubrity, and its clear green colour. The Aqua Tepula, built 126 B.C., and Aqua Julia, constructed by Agrippa, 34 B.C., were aqueducts which brought water to the city by two conduits passing one above and one under the channel of the same aqueduct just described. Sixthly, came the Aqua Virgo, also constructed by Agrippa, who lived under the Emperor Augustus, who laboured to improve and beautify Rome, and who, according to Pliny, constructed in one year 70 pools, 105 fountains, and 130 reservoirs. This aqueduct commenced in a very copious spring, which rose in the midst of a marsh about 14 miles from the city; it ran circuitously a distance of about 18 miles, and in its course passed through a tunnel four-fifths of a mile long. The Aqua Alsietina, now called the Aqua Paola, was built by Augustus to bring water to the Naumachia, a sheet of water formed by the same emperor for the representation of sea-fights. The eighth aqueduct was the Aqua Claudia, which was begun by Caligula and finished by Claudius, 51 A.D. It took its rise 30 miles from Rome, forming a subterranean stream 36 miles long, and running 10 miles along the surface of the ground. This aqueduct was vaulted for the space of 3 miles, and supported on arcades for the space of 7 miles, being carried along such a high level as to be able to supply all the hills of ancient Rome. It was built of hewn stone, and the ruins furnished the materials for the Aqua Felice, a modern watercourse built by Pope Sixtus V. The Anio Novus or New Anio, and its branch the Aqua Trajana, were built at a later date, with the Aqua Antoniana and some others. The Anio Novus was 62 miles long, and with the Aqua Claudia doubled the quantity of water hitherto supplied to Rome by the older aqueducts. the same method be adopted to bring water into London that was done in ancient Rome-namely, by aqueducts to conduct it from pure sources at a great distance, and at a very considerable elevation-the great desideratum of a constant and sufficient supply of pure water to the metropolis will never be obtained. Aqueducts constructed for the purpose of irrigation, or for the supply of towns with water, often require architectural and engineering works as difficult and as important in their construction as canals for inland navigation, or railways for internal communication. Specimens of Roman architecture in these departments were given in our last lesson. Many are to be found in France. The aqueduct which supplies Nismes with water is of this description. Such also is the celebrated Pont du Gard, which is stili in good preservation. As a specimen of modern skill, in that country, may be named the canal which brings the waters of the Durance down from the Pont de Pertius, and conveys them to Marseilles after a passage of about 60 miles, of which nearly 11 are under ground. This canal or aqueduct passes across several deep valleys, over splendid aqueducts, such as the Jancourelle, the Valbonette, and the Valmousse. But the most remarkable specimen of modern canal architecture, of any that now exists, is the aqueduct of Roquefavour on the same canal. This wonderful structure is about 266 feet above the bottom of the valley over which it is built. It is composed of three rows of arches placed one above another, and is about 1,312 feet long. The beauty alone of this construction is not the most striking feature. The simplicity and elegance of the methods employed in its erection are especially worthy of admiration. We should like to see the skill of modern English architects and engineers employed in bringing the waters of some vast reservoir collected on some lofty eminence at a distance from the metropolis. Such water, filtered previous to transmission, and carried by simple gravitation along some splendid aqueduct, would pour into London a river of pure water, and render the office of the Sanitary Commissioners almost a sinecure. LESSONS IN LATIN.-XXVII. REGULAR VERBS.-THE SECOND CONJUGATION. PASSIVE VOICE. EXAMPLE. Moneor, 2, I am reminded. Indicative. Moneris. Plu. Monemur. Sing. PRESENT TENSE. Subjunctive. Monear. Moneāris. Moneamur. Characteristic letter, E long. Imperative. Infinitive. Participle. [ētor. Monēri, Monĕre or mon- It was in the reign of the Emperor Augustus, who greatly extended the aqueducts of Rome, that the practice of tunnelling was commenced; and other emperors, who followed him, carried out the same important department of engineering. The Emperor Trajan particularly exerted himself in the improvement of these aqueducts. Such works were executed in a bold and original manner; nothing could damp the skill and enterprise of the Roman architects. They drained lakes, excavated mines in the mountains, and elevated valleys by rows of accumulated arcades. The water of the aqueducts was kept cool by covering it with vaults; and they were often so spacious that, according to Procopius, they admitted of a man riding Chief Parts: Moneor, menītus sum, monēri. through them on horseback. The supply of water in Rome from these wonderful erections was indeed so abundant, that Strabo says whole rivers of water flowed through the streets of Sing. Moncor. Rome. Pliny justly considered these aqueducts as the wonder of the world, for their grandeur, extent, and utility; and it seems very surprising to us, when we contemplate the high pitch to which civil engineering and architecture have risen among ourselves, that we have no such splendid aqueducts to supply our modern cities, such as London, with water. According to the enumeration of Frontinus, the nine earlier aqueducts of Rome delivered every day about 173,000,000 of our imperial gallons; and it is supposed that, when all the aqueducts Plu. together were in operation, upwards of 310,000,000 of imperial gallons of water were supplied to the ancient city. Now, reckoning the population of Rome to have been 1,000,000, which it probably never exceeded, no less than 300 gallons of water were allowed for the daily use of each inhabitant. Rarely, indeed, have cities, either ancient or modern, supplied their inhabitants with such quantities. According to the calculations of Prony, the French engineer, three aqueducts, the Aqua Felice, Juliana, and Paulina, with some additional sources, supply modern Rome with 33,000,000 of imperial gallons Sing. Monttus sum. Monitus sim. of water in 24 hours. This, divided among a supposed population of 150,000, gives about 220 imperial gallons of water for each inhabitant, being about one-third less than that which was furnished to the inhabitants when the city was the mistress of the nations, and at the height of her ancient splendour. We believe that London is at present far more plentifully supplied with water than even ancient Rome was, in proportion to her population, and at the period of her greatest prosperity; but we very much doubt whether the quality of Plu. Moniti eramus. Moniti essemus. the water supplied to the former would bear comparison with that supplied to the latter. It seems to us, indeed, that unless Monemini. Moneamini. Monemini. Monentor. IMPERFECT TENSE. Monebar. Monerer. Monebaris (e). Monereris. Monerctur. 9. 1. Exerceor. 2. Exerceris. 3. Exercetur. 4. Exercebar. 5. Exercebaris. 6. Exercebatur. 7. Exercebor. 8. Exercebere. Exercebitur. 10. Pater curat ut ego bene exercear. 11. Oppletur fossa. 12. Curo ut bene exercearis. 13. Curo ut puer bene exerceatur. 14. Pater curabat ut filius bene exerceretur. 15. Curabam ut bene exercereris. 16. Curabam ut filia tua bene exerceretur. 17. Quis nescit quam præclaris fructibus animi nostri in literarum studiis augeantur? 18. Timemus ne exercitus noster ab hostibus vincatur. 19. Omnes cives metuebant ne urbs ab hostibus obsidione cingeretur. 2). Quum in literis exercemur, animi nostri multarum rerum utilium cognitione augentur. 21. Quum subito periculo terremur, non debemus extemplo de salute desperare. 22. Virtutis honos nulla oblivione delebitur. 23. Pueri in literarum studiis strenue exerciti sunt. 24. Metuebamus ne urbs ab hostibus obsidione cincta esset. 25. Metuo ne milites subito periculo territi sint. 26. Strenue exercetor puer. 28. Boni discipuli 27. Ne rerum difficultatibus a proposito deterretor. student exerceri in literarum studiis. 29. Puer bene educatus omnibus placet. 30. Hostes territi in castris manent. 31. Pueri strenue exercendi sunt. EXERCISE 95.-ENGLISH-LATIN. Instances. SUPINES. 1. Lectum 2. Lectu. After this model write out fundo, fundĕre, fudi, fusum, I pour; tribuo, tribuere, tribui, tributum, I bestow; and scribo, scribere, scripsi, scriptum, I write. Acies, -ei, f., a line of struxi, instructum, 3, I draw up, form (E. R. instruct). Libenter, willingly. Pingo, pingere, pinxi, pinctum, 3, I paint (E. R. picture). Quoad, as long as. defendere, Gero, gerere, gessi, Simulatque, as soon as. EXERCISE 96.-LATIN-ENGLISH. 1. Duximus. 2. Duxisti. 3. Ducis. 4. Ducebam. 5. Ducet. 6. Ducat. 7. Dum ego pingebam, tu scribebas, et frater legebat. 8. Hostes aciem instruebant. 9. Quoad vives bene vives. 10. Si virtutem coletis, boni te diligent. 11. Hostes aciem instruxerunt. 12. Hostes aciem instruent. 13. Multas literas hodie scripsimus. 14. Bellum atrocissimum gesserunt hostes. 15. Cæsar aciem instruxerat. 16. Simulatque literas scripserimus ambulabimus. 17. Curo ut puerorum animos excolam. 18. Curabam ut filii mei preceptor animum excoleret. 19. Nemo dubitat quin ego puerum semper diligenter correxerim. 20. Metuimus ne hostes urbem combusserint. 21. Nemo dubitat quin hostes urbem obsidione cincturi sint. 22. Narrate nobis 1. The boys are earnestly exercised. 2. Let boys be earnestly exercised. 3. The boys must be strenuously exercised. 4. The boys will be strenuously exercised. 5. The boys are strenuously exercised. 6. The boys were being strenuously exercised. 7. The boys have been strenuously exercised. 8. The boys will have been strenuously exercised. 9. I take care that the boys are (may be, in Latin) strenuously exercised. 10. I took care the boys were (might be) strenuously exercised. 11. My sisters have been strenuously exercised. girl will have been strenuously exercised. 13. I fear the city will be quid parentes scripserint. 23. Scribito. 24. Disce, puer. 25. Boni surrounded with a blockade (blockaded). THE THIRD CONJUGATION. ACTIVE VOICE. EXAMPLE.-Lego, 3, I read. 12. The Chef Parts: Lego, legi, lectum, legĕre. Characteristic letter, I short. pueri libenter discunt. 26. Miles, se fortiter contra hostes defendens, laudatur. 27. Cupiditates coercere debemus. EXERCISE 97.-ENGLISH-LATIN. 1. I defended the city. 2. The soldiers defended the city. 3. They will defend the city. 4. They have defended the city. 5. They were writing. 6. He has written a letter. 7. No one doubts that you will write a good letter. 8. Take care to write a letter. 9. The teacher takes care that his scholars write good letters. 10. I have written a letter to-day. 11. The enemies will draw up (their) line of battle. 12. The soldiers have burnt the city. 13. I have read the letter which thou wrotest. 14. I fear that the enemies will blockade the city. 15. Correct that boy. 16. The master will take care to correct his scholars. 17. Tell (narro) me what thou saidst to thy father. 18. Restrain thy desires. 19. We ought to restrain our desires. 20. A boy (by) restraining his desires is loved. 21. Strenuously cultivate thy mind, my son. Lecturum Lecturus. Legisse, KEY TO EXERCISES IN LESSONS IN LATIN.-XXVI. 1. I exercise. 2. I was exercising. 3. He was exercising. 4. I will exercise. 5. I rejoice that thou art well. 6. The teacher was rejoicing that you were obeying his commands. 7. Thou wast pleasing thyself, (thou wast) displeasing others. 8. No forgetfulness will blot out the honour of virtue. 9. I exercised. 10. Greece flourished in all the arts. 11. I praise you because you have properly exercised your mind in study. 12. Why were you silent? 13. Thy boy was suddenly silent. 14. The mother was silent. 15. All are silent. 16. Unless you have obeyed the precepts of virtue, the entrance to heaven will not be open to you. 17. If thou hast restrained thy desires thou wilt be happy. 18. I take care to improve (that I may improve) the morals and to exercise the body of the boy. 19. I advise you to observe (that you may observe) the commands of your father. 20. I feared I was displeasing you. 21. Take care to improve the morals and to exercise the body of the boy. 22. I feared that an enemy was injuring me. 23. The boy feared that his mother was silent. 21. I took care to improve (that I might improre) the morals and to exercise the body of the boy. 25. I took care that you should improve the morals and (that you should) exercise the body of the boy. 26. I took care that the teacher should improve the morals and exercise the body of the boy. 27. I fear that you will (may) not come. 28. The husband fears that his wife will (may) die. 29. The teacher feared that the scholar would not obey his words. 30. The bad boy fears that the teacher will come. EXERCISE 93.-ENGLISH-LATIN. 1. Ille me monebat. 2. Illi regem monebant. 3. Ego vos monerem. 4. Vos me moneretis. 5. Illi puerum monuerunt. 6. Tu mulierem monebas. 7. Ego præceptorem monebo. 8. Tace. 9. Tacete. 10. Tacento. 11. Mulier repente tacuit. 12. Cura ut emendes. 13. Cura ut civium mores emendes. 14. Timeo ne tibi displiceat. 15. Pueri timebant ne patri displicerent. 16. Omnibus placet. 17. Bonus malis displicebit. 18. Cur taces? 19. Metuunt ne Cæsar patriam vincat. 2). Bona sorores timent ut fratres valeant. 21. Valesne? 22. Timeout valeas. 23. Si corpus exercueris valebis. 24. Mater timet ut mihi aditus in cœlum patent. LESSONS IN ALGEBRA.-I. ART. 1.—ALGEBRA is a general method of solving problems, and of investigating the relations of quantities by means of letters and signs. The following will afford illustrations of this method of arriving at the solutions of problems by the use of signs and letters instead of figures as in arithmetic. PROBLEM I.-Suppose that a man divided 72 pounds among his three sons in the following manner :--To A he gave a certain number of pounds; to B he gave three times as many as to A; and to C he gave the remainder, which was half as many pounds as A and B received. How many pounds did the donor give to each ? finds that he must give 2 pence for a peach, and 4 pence for an orange. How many can he buy of each? Let a denote the number of each. Now, since the price of one 2x pence. peach is 2 pence, the price of a peaches will be x X 2 pence, or For the same reason, ≈ × 4, or 4x pence, will denote Then will 2x + 4x, or 6x, be equal to the price of a oranges. 96 pence by the conditions of that question, and le ore (for when 1 is the co-efficient of a number [See Art. 16 below] it is always understood, and hever expressed) is equal to of 96 pence, namely, 16 pence, and 16 is therefore the number he bought of each. 2. Quantities in algebra are generally expressed by letters, as in the preceding problems. Thus b 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 Problem I., z was put for A's share of the money. Its value was 12 pounds, and remained fixed through the operation. In Problem II., was put for the number of each kind of fruit. Its value was 16, and it remained so throughout the whole of the calculation. 3. By the term quantity, we mean anything that can be multiplied, divided, or measured. Thus, length, weight, time, number, etc., are called quantities. 4. The first letters of the alphabet, a, b, c, etc., are used to express known quantities; and the last letters, z, y, z, etc., 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 consideration. 6. Unknown 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. To solve this problem arithmetically, the pupil would rea- 8. Besides letters and figures, it will also be seen that we use son thus:-A had a certain part, that is one share; B received certain signs or characters in algebra to indicate the relations of three times as much, or three shares; but C had half as much the quantities, or the operations which are to be performed with as A and B; hence he must have received two shares. By them, instead of writing out these relations and operations in adding their respective shares, the sum is six shares, which, by words. Among these are the signs of addition (+), subtraction the conditions of the question, is equal to 72 pounds. If, then, (-), equality (=), etc. 6 shares are equal to 72 pounds, 1 share is equal to of 72, namely, 12 pounds, which is A's share. B had three times as many, namely, 36 pounds; and C half as many pounds as both, namely, 24 pounds. Now, to solve the same problem by algebra, he would use letters and signs, thus: Let a represent A's share; then by the conditions, 20 a multiplied by 3, or x × 3 (when X, the sign of multiplication, is used instead of the words "multiplied by "), will represent B's share, and 4x, the sum of the shares of A and B divided by 2, or 4÷2 (when, the sign of division, is used instead of the words "divided by "), will represent C's share. Now, X3 may be written 3x, and 4r2 may be written 2x; so then adding together the several shares of A, B, and C, namely, z, 3, and 2x, and putting +, the sign of addition, between them, we get x + 3x + 2x, which is equal to 6x; or using, the sign of equality, for the words "is equal to," we get + 3x + 2x = 6. Then 6x 72, for the whole is equal to all its parts; and læ = 12 pounds, A's share; 3 = 36 pounds, B's share; and 2x = 21 pounds, C's share. = Proof.-Add together the number of pounds received by each, and the sum will be equal to 72 pounds, the amount divided between A, B, and C. In this algebraic solution it will be observed: First, that we represent the number of pounds which A received by a. Second, to obtain B's share, we must multiply A's share by 3. This multiplication 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 divi im is denoted by a line between two dots. Fourth, the addition of their respective shares is denoted by another cross formed by an horizontal and a perpendicular line. Take another example: PROBLEM II-A boy wishes to lay out 96 pence for peaches and oranges, and wants to get an equal number of each. He 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 ab signifies that b 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 "subtracted" from a; and the expression (see Art. 22 below) 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 Articles 36 and 37. 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 ab 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 ab signifies, that in certain cases, comprehended in a general solution, b is to be added to a, and in other cases subtracted from it. Observation. 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. 14. The equality of two quantities, or sets of quantities, iexpressed by two parallel lines, =. Thus abd signifie that a and b together are equal to d. So 84 16 — 4 = 10 +27 + 2 + 3. 15. When the first of the two quantities compared is greate than the other, the character > is placed between them. Thu ab signifies that a is greater than b. If the first is less than the other, the character is used; a ab, namely, a is less than b. In both cases, the quantity towards which the character opens is greater than the other. 16. A numeral figure is often prefixed to a letter. This is called a co-efficient. It shows how often the quantity expressed by the letter is to be taken. Thus 26 signifies twice b; and 96, 9 times b, or 9 multiplied into b. The co-efficient may be either a whole number or a fraction. Thas b 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, that is to say, 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 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. 18. A simple quantity is either a single letter or number, or several letters connected together without the signs + or -. Thus a, ab, abd, and 8b, are each of them simple quantities. 19. A compound quantity consists of a quantities connected by the sign+ or -. t-d+3h, are each compound quantities. which each is composed are called terms. number of simple Thus ab, d-y, The members of 20. A simple term is called a monomial; thus, a, b, -care monomials. If there are two terms in a compound quantity, it is called a binomial: thus a+b and a-b are binomials. The latter term (a - b) is also called a residual quantity, because it expresses the difference of two quantities, or the remainder after one is taken from the other. A compound quantity, consisting of three terms, is sometimes called a trinomial; one of four terms, a quadrinomial. A quantity consisting of several terms is, however, generally called a polynomial. 21. When the several members of a compound quantity are to be subjected to the same operation, they are 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-b+c signifies that b is to be subtracted from a, and c is to be added to the result. 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 crossing each other, thus X : hence, a x b is a multiplied into b; and 6 x 3 is 6 times 3, or 6 multiplied into 3. Sometimes a point is used to indicate multiplication: thus, a.b is the same 23 a x b. But the sign of multiplication is more commonly con omitted between simple quantities, and the letters are rected together in the form of a word or syllable: thus, ab is the same as a.b or a x b; and bede is the same as bx c x When a compound quantity is to be multiplied, a riculum or parenthesis is used, as in the case of subtraction. Thus the sum of a and b multiplied into the sum of c and d, is a + bx c +d, or (a + b) × (c + d). And (6+2) × 5 is 8 x 5, cr 40. But 6 + (2 x 5) is 6 + 10, or 16. When the marks of parenthesis are used, the sign of multiplication is frequently omitted. Thus (x+y) (x− y) is (x + y) × (x−y). 24. When two or more quantities are multiplied together, each of them is called a factor. In the product ab, a is a factor, and so is b. In the product xx (a + m), a is one of the factors, and (a + m) the other. Hence every co-efficient may be considered as a factor (Art. 17). In the product 3y, 3 is a factor as well as y. 25. A quantity is said to be resolved into factors, when any factors are taken which, being multiplied together, will produce the given quantity. Thus 3ab may be resolved into the two factors 3a and b, because 3a X b is 3eb. And 5amn may be resolved into the three factors 5a, and m, and n. And 48 may be resolved into the two factors 2 x 24, or 3 x 16, or 4 x 12, or 6 x 8; or into the three factors 2 x 3 x 8, or 4 X 6 x 2, etc. 26. Division is expressed in two ways: (1.) By an horizontal fize between two dots, which shows that the quantity preling it is to be divided by that which follows. Thus a ÷ c, = a divided by c. (2.) Division is more commonly expressed in the form of a action, putting the dividend in the place of the numerator, the divisor in that of the denominator. Thus is a divided by b. a ს READINGS IN GERMAN.-I. INTRODUCTION. THE object of learning a modern language is not simply, as in the case of one that is no longer spoken, to be able to read and write, but also to speak it. For this purpose it is obviously necessary to acquire a knowledge of the pronunciation as well as the meaning of the words. Hence we are not surprised at having received many applications from the readers of our lessons in German for some instruction on this subject; and it is our intention to publish in the pages of the POPULAR EDUCATOR a series of German Reading Lessons expressly prepared with a view to teach the proper pronunciation of the language. These lessons will be found much better adapted to answer the purpose than any mere collection of rules, however carefully drawn up, and however clearly expressed. In no case is the principle, that example is better than precept, more applicable than in that of pronunciation, a knowledge of which can only be acquired by frequent exemplification. We have no hesitation in saying that the study of our lessons will enable the reader to pronounce German, if not with absolute perfection, at least so as to be easily understood by a native, which is, after all, the only practical object in view. It is proper to observe, that whilst the lessons are especially intended to teach pronunciation, they are also calculated to be very useful to our readers as exercises in translation, being easy in construction, simple in style, rich in words, and adapted in substance to persons of all ages. A vocabulary will be appended to each lesson, containing an explanation of the meaning of every word in it which has not been previously explained. As few words will be explained more than once in the whole course of the vocabularies, it will be necessary for the learner to study each with great care on its first occurrence, that he may avoid the inconvenience of having to look through preceding pages for the meaning. DIRECTIONS FOR THE USE OF THE INTERLINEAR Pronounce every syllable as in English. To make a vowel long which otherwise would be short, or might be either short or long, an apostrophe has been placed immediately after that vowel. Thus vol will rhyme with doll, but vo'l will rhyme with whole. To make a vowel short which otherwise would be long, or might be either long or short, the short sign or breve has been placed over that vowel. Thus mild will rhyme with wild, but mild will rhyme with build. Boot will rhyme with root, but boot will rhyme with put. ah is long, and sounded as in father; a is short, and sounded as in castle. ey is to be pronounced as in obey. ai, representing the short sound of e, when unaccented and terminating a syllable, should be pronounced like cy in the noun sur'-vey. The letters ey could not well be used for this sound, as they have generally the sound of ee when unac cented. has no corresponding sound in English; place the organs as if to pronounce o long; keep them exactly in this position, and then try to pronounce the German e or English a. For the short sound of this vowel place the organs again as if to pronounce o, and without changing, try to pronounce ef, el, eck, em, en, ep, er, ess, et, and you will utter the sound required. The sound which comes nearest to it is the e in her. has no parallel in English. Pronounce oo in ooze, firmly maintain this position, and try to pronounce long e in eel; the sound uttered will be the one required. For the short sound, place the organs in a similar position, and without changing it try to say if, il, ick, im, in, ip, ir, iss, it. For those who have studied French it may be well to remark that the German i has the same sound as the French u. ou is always to be sounded as in out, our. gh before e and i must be pronounced like g in dy has different sounds, according to its posi In the interlinear pronunciation we shall ch, when it is pronounced like an aspi 1. you may observe) the commands of your father. 20. I feared I was displeasing you. 21. Take care to improve the morals and to exercise the body of the boy. 22. I feared that an enemy was injuring me. 23. The boy feared that his mother was silent. 21. I took care to improve (that I might improve) the morals and to exercise the body of the boy. 25. I took care that you should improve the morals and finds that he must give 2 pence for a peach, and 4 pence for an orange. How many can he buy of each? Let a 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 2. Quantities in algebra are generally expressed by letters, as in the preceding problems. Thus b 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 24. Mater timet ut mihi Problem I., a was put for A's share of the money. Its value was 12 pounds, and remained fixed through the operation. In Problem II., was put for the number of each kind of fruit. Its value was 16, and it remained so throughout the whole of the calculation. timebant ne patri displicerent. 16. Omnibus placet. 17. Bonus malis LESSONS IN ALGEBRA.-I. ART. 1.-ALGEBRA is a general method of solving problems, The following will afford illustrations of this method of arriving at the solutions of problems by the use of signs and letters instead of figures as in arithmetic. PROBLEM I. Suppose that a man divided 72 pounds among his three sons in the following manner :-To A he gave a certain number of pounds; to B he gave three times as many as to A; and to C he gave the remainder, which was half as many pounds as A and B received. How many pounds did the donor give to each ? 3. By the term quantity, we mean anything that can be multiplied, divided, or measured. Thus, length, weight, time, number, etc., are called quantities. 4. The first letters of the alphabet, a, b, c, etc., are used to express known quantities; and the last letters, z, y, z, etc., 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 consideration. 6. Unknown 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. To solve this problem arithmetically, the pupil would rea- 8. Besides letters and figures, it will also be seen that we use son thus:-A had a certain part, that is one share; B received certain signs or characters in algebra to indicate the relations of three times as much, or three shares; but C had half as much the quantities, or the operations which are to be performed with as A and B; hence he must have received two shares. By them, instead of writing out these relations and operations in adding their respective shares, the sum is six shares, which, by words. Among these are the signs of addition (+), subtraction the conditions of the question, is equal to 72 pounds. If, then, (−), equality (=), etc. 6 shares are equal to 72 pounds, 1 share is equal to of 72, namely, 12 pounds, which is A's share. B had three times as many, namely, 36 pounds; and C half as many pounds as both, namely, 24 pounds. Now, to solve the same problem by algebra, he would use letters and signs, thus: Let a represent A's share; then by the conditions, multiplied by 3, or a × 3 (when X, the sign of multiplication, is used instead of the words "multiplied by"), will represent B's share, and 4x, the sum of the shares of A and B divided by 2, or 4 ÷ 2 (when, the sign of division, is used instead of the words "divided by "), will represent C's share. Now, xX 3 may be written 3x, and 4x2 may be written 2x; so then adding together the several shares of A, B, and C, namely, x, 3x, and 2x, and putting +, the sign of addition, between them, we get x + 3x + 2x, which is equal to 6x; or using, the sign of equality, for the words "is equal to," we 6.x. get x+3x+2x Then 6x 72, for the whole is equal to all its parts; and 1x = 12 pounds, A's share; 3 = 36 pounds, B's share; and 2x = 24 pounds, C's share. = Proof-Add together the number of pounds received by each, and the sum will be equal to 72 pounds, the amount divided between A, B, and C. In this algebraic solution it will be observed: First, that we represent the number of pounds which A received by x. Second, to obtain E's share, we must multiply A's share by 3. This multiplication 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 an horizontal and a perpendicular line. Take another example: PROBLEM II-A boy wishes to lay out 96 pence for peaches and oranges, and wants to get an equal number of each. He 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+b signifies that b 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 "subtracted" from a; and the expression (see Art. 22 below) 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 Articles 36 and 37. 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 ab 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 ab signifies, that in certain cases, comprehended in a general solution, b is to be added to a, and in other cases subtracted from it. Observation. 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. 14. The equality of two quantities, or sets of quantities, is expressed by two parallel lines, =. Thus a+b= d signific that a and b together are equal to d. So 8416-4=10 +27 + 2 + 3. 15. When the first of the two quantities compared is greater than the other, the character is placed between them. Thus ab signifies that a is greater than b. |