8. Tace. 10. you may observes the commands of your father. 20. I feared I was finds that he must give 2 pence for a peach, and 4 pence for displeasing you. 21. Take care to improve the morals and to exercise an orange. How many can he buy of each ? the body of the boy. 22. I feared that an enemy was injuring me. Let a denote the number of each. Now, since the price of one 23. The boy feared that his mother was silent . 21. I took care to peach is 2 pence, the price of x peaches will be a x 2 pence, or of the boy. 25. I took care that you should improve the morals and 2x pence. For the same reason, æ X 4, or 4x pence, will denote (that you should) exercise the body of the boy. 26. I took care that the price of x oranges. Then will 2x + 4x, or 6x, be equal to the teacher should improve the morals and exercise the body of the 96 pence by the conditions of that question, and lx or 2 (for boy. 27. I fear that you will (may) not come. 23. The husband fears when 1 is the co-efficient of a number (See Art. 16 below) it is that his wife will (may) die. 29. The teacher feared that the scholar always understood, and hever expressed) is equal to 4 of 96 would not obey his words. 30. The bad boy fears that the teacher pence, namely, 16 pence, and 16 is therefore the number he will come. bought of each. 2. Quantities in algebra are generally expressed by letters, as 1. Ile me monebat. 2. Illi regem monebant. 3. Ego vos monerem. in the preceding problems. Thus b may be put for 2 or 15, or 4. Vos me moneretis. 5. Ii puerum monuerunt. 6. Tu mulierem any other number which we may wish to express. It must not be monebas. 7. Ego præceptorem monebo. 9. Tacete. inferred, however, that the letter used has no determinate value. Tacento. 11. Mulier repente tacuit. 12. Cura ut emendes. 13. Cura ut civium mores emendes. 14. Timeo ne tibi displiceat. 15. Pueri Its value is fixed for the occasion or problem on which it is timebant ne patri displicerent. 16. Omnibus placet. 17. Bonus malis employed, and remains unaltered throughout the solution of displicebit. 18. Cur taces ? 19. Metuunt ne Cæsar patriam vincat. that problem. But on a different occasion, or in another problem, 20. Bonæ sorores timent ut fratres valeant. 21. Valesne ? 22. Timeo ut the same letter may be put for any other number. Thus, in valeas. 23. Si corpus exercueris valebis. 24. Mater timet ut mihi Problem I., & was put for A's share of the money. Its value aditus in ccelum pateat. was 12 pounds, and remained fixed through the operation. In Problem II., a was put for the number of each kind of fruit. LESSONS IN ALGEBRA.-I. Its value was 16, and it remained so throughout the whole of the calculation. DEFINITIONS. 3. By the term quantity, we mean anything that can be mul. ART. 1.--ALGEBRA is a general method of solving problems, tiplied, divided, or measured. Thus, length, weight, time, number, and of investigating the relations of quantities by means of eto., are called quantities. letters and signs. 4. The first letters of the alphabet, a, b, c, etc., are used to The following will afford illustrations of this method of express known quantities; and the last letters, 2, 4, 2, etc., arriving at the solutions of problems by the use of signs and those which are unknown. letters instead of figures as in arithmetic. 5. Known quantities are those whose values are given, or PROBLEM I.–Suppose that a man divided 72 pounds among his may be easily inferred from the conditions of the problem three sons in the following manner :--To A he gave a certain under consideration. number of pounds; to B he gave three times as many as to A; 6. Unknown quantities are those whose values are not given, and to C he gave the remainder, which was half as many pounds but required. as A and B received. How many pounds did the donor give to 7. Sometimes, however, the given quantities, instead of being each? 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 nse son thus:-A had a certain part, that is one share; B received certain signs or characters in algebra to indicate the relations of ühree 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, 9. Addition is represented by two lines (+), one horizontal, namely, 12 pounds, which is A's share. B had three times as the other perpendicular, forming a cross, which is called plus. many, namely, 36 pounds; and C half as many pounds as both, It signifies "more," or " added to." Thus a + b signifies that namely, 24 pounds. b is to be added to a. It is read a plus b, or a added to b, or a Now, to solve the same problem by algebra, he would use and b. letters and sigs, thus : 10. Subtraction is represented by a short horizontal line (-) Let a represent A's share; then by the conditions, which is called minus. Thus, a - b signifies that b is to be * multiplied by 3, or < X 3 (when X, the sign of multipli. “subtracted" from a; and the expression (see Art. 22 below) cation, is used instead of the words “multiplied by"), will repre- is read a minus b, or a less b. sent B's share, and 11. The sign + is prefixed to quantities which are considered 4x, the sum of the shares of A and B divided by 2, or 4x + 2 as positive or affirmative; and the sign – to those which are (when “, the sign of division, is used instead of the words supposed to be negative. For the nature of this distinction, see “ divided by"), will represent C's share. Articles 36 and 37. Now, « X 3 may be written 3x, and 4x + 2 may be written 12. The sign is generally omitted before the first or leading 2x; so then adding together the several shares of A, B, and C, quantity, unless it is negative; then it must always be written. namely, 1, 3x, and 2x, and putting +, the sign of addition, When no sign is prefixed to a quantity, + is always understood. between them, we get x + 3x + 2x, which is equal to 6x; or Thus a + b is the same as + a + b. using = s, the sign of equality, for the words " is equal to," we 13. Sometimes both + and - (the latter being put under get & + 3x + 2x = 6t. Then 6x = 72, for the whole is equal the former, +) are prefixed to the same letter. The sign is then to all its parts; and lx = 12 pounds, A's share; 3x = 36 said to be ambiguous. Thus a + b signifies, that in certain pounds, B's share; and 2 = 24 pounds, C's share. cases, comprehended in a general solution, b is to be added to a, Proof.--Add together the number of pounds received by each, and in other cases subtracted from it. and the sum will be equal to 72 pounds, the amount divided Observation - When all the signs are plus, or all minus, they between A, B, and C. are said to be alike; when some are plus and others minus, they In this algebraic solution it will be observed : First, that we are called unlike. represent the numler of pounds which A received by a. Second, 14. The equality of two quantities, or sets of quantities, ia to obtain B's share, we must multiply A's share by 3. This expressed by two parallel lines, =. Thus a + b = d signifies multiplication is represented by two lines crossing each other like that a and 6 together are equal to d. So 8 + 4 = = 16 - 4 = 10 a capital X. Third, to find C's share, we must take half the + 2 = 7+ 2 + 3. sum of A's and B's share. Tjis division is denoted by a line 15. When the first of the two quantities compared is greater between two dots. Fourth, the addition of their respective than the other, the character > is placed between them. Thus shares is denoted by another cross formed by an horizontal and a > b signifies that a is greater than b. & perpendicular line. Take another example : If the first is less than the other, the character < is used; a PROBLEM II.-A boy wishes to lay out 96 pence for peaches a<b, namely, a is less than b. In both cases, the quantit and oranges, and wants to get an equal number of ench. He towards which the character opens is greater than the other. 16. A numeral figure is often prefixed to a letter. This is READINGS IN GERMAN.-I. called a co-efficient. It shows how often the quantity expressed INTRODUCTION. by the letter is to be taken. Thus 26 signifies twice b; and 96, 9 times b, or 9 multiplied into b. The object of learning a modern language is not simply, as in The co-efficient may be either a whole number or a fraction, the case of one that is no longer spoken, to be able to read and Thas jó is two-thirds of b. When the co-efficient is not ex- write, but also to speak it. For this purpose it is obviously pressed, 1 is always to be understood. Thus e is the same as necessary to acquire a knowledge of the pronunciation as well la, that is to say, once a, or one times. as the meaning of the words. Henco we are not surprised at 17. The co-efficient may also be a letter, as well as a figure. having received many applications from the readers of our lesIn the quantity mb, m may be considered the co-efficient of b; sons in German for some instruction on this subject; and it is because ) is to be taken as many times as there are units in m. our intention to publish in the pages of the POPULAR EDUCATOR If ok stands for 6, then mb is six times b. In 3abc, 3 may be con- a series of German Reading Lessons expressly prepared with a sidered as the co-efficient of abc; 3a the co-efficient of be; or view to teach the proper pronunciation of the language. These 3.sb the co-efficient of c. lessons will be found much better adapted to answer the pur18. A simple quantity is either a single letter or number, pose than any mere collection of rules, however carefully drawn of several letters connected together without the signs + or up, and however clearly expressed. In no case is the principle, - Thus a, ab, abd, and 86, are each of them simple quan- that example is better than precept, more applicable than in that of pronunciation, a knowledge of which can only be acquired 19. A compound quantity consists of a number of simple by frequent exemplification. We have no hesitation in saying guantities connected by the sign + or -. Thus a +b, d-y, that the study of our lessons will enable the reader to pronounce b-d +3h, are each compound quantities. The members of German, if not with absolute perfection, at least so as to be which each is composed are called terms. easily understood by a native, which is, after all, the only prac20. A simple term is called a monomial; thus, a, b, -c are tical object in view. moromials. If there are two terms in a compound quantity, it It is proper to observe, that whilst the lessons are especially is called a binomial : thus a + b and a-6 are binomials. intended to teach pronunciation, they are also calculated to be The latter term (a - b) is also called a residual quantity, very useful to our readers as exercises in translation, being easy becanse it expresses the difference of two quantities, or the in construction, simple in style, rich in words, and adapted in remainder after one is taken from the other. A compound substance to persons of all ages. A vocabulary will be appended quantity, consisting of three terms, is sometimes called a tri- to each lesson, containing an explanation of the meaning of omial; one of four terms, a quadrinomial. A quantity consist- every word in it which has not been previously explained. As ing of several terms is, however, generally called a polynomial. few words will be explained more than once in the whole course 21. When the several members of a compound quantity of the vocabularies, it will be necessary for the learner to study se to be subjected to the same operation, they are connected each with great care on its first occurrence, that he may avoid by a line called a vinculum (-), or by a parenthesis (). Thus the inconvenience of having to look through preceding pages 4-6+ or a-(b + c), shows that the sum of b and c is to be for the meaning. subtracted from a. But a-b+c signifies that b is to be sub- DIRECTIONS FOR THE USE OF THE INTERLINEAR tracted from a, and c is to be added to the result. PRONUNCIATION. 22. A single letter, or a number of letters, representing any Pronounce every syllable as in English. quantities with their relations, is called an algebraic expression To make a vowel long which otherwise would be short, or er formula. Thus a +b + 3d is an algebraic expression. 23. Multiplication is usually denoted by two oblique lines immediately after that vowel. Thus vol will rhyme with do, might be either short or long, an apostrophe has been placed crossing each other, thus X : hence, a x b is a multiplied into but vo’l will rhyme with whole. B; and 6 x 3 is 6 times 3, or 6 multiplied into 3. Sometimes To make a vowel short which otherwise would be long, or s point is nsed to indicate multiplication: thus, a .b is the same might be either long or short, the short sign or breve has been 23 a Xb. But the sign of multiplication is more commonly placed over that vowel. Thus mild will rhyme with wild, but smitted between simple quantities, and the letters are con mild will rhyme with build. Boot will rhyme with root, but boot rected together in the form of a word or syllable: thus, ab is will rhyme with put. the same as q.b or a xb; and bede is the same as b × c X Xt. When a compound quantity is to be multiplied, a ah is long, and sounded as in father ; a is short, and sounded poculum or parenthesis is used, as in the case of subtraction. as in castle. Thus the sum of a and b multiplied into the sum of c and d, is ey is to be pronounced as in obey. *+hxc+d, or (a + b) x (c+d). And (6+2) 5 is 8 X 5, ai, representing the short sound of e, when unaccented and 40. Bat 6 + (2 X 5) is 6 + 10, or 16. When the marks of terminating a syllable, should be pronounced like ey in the parenthesis are used, the sign of multiplication is frequently sound, as they have generally the sound of ee when unac noun sur'-vey. The letters cy could not well be used for this mmitted. Thus (x+y) (c-y) is (a + y) (—y). cented. 2. When two or more quantities are multiplied together, sast of them is called a factor. In the product ab, a is a factor, Ó has no corresponding sound in English ; plaee the organs as kad so is h. In the product a X (a + m), « is one of the factors, if to pronounce o long; keep them exactly in this position, ad (a + m) the other. Hence every co-efficient may be con- and then try to pronounce the German e or English a. For sidered a4 a factor (Art. 17). In the product 3y, 3 is a factor the short sound of this vowel place the organs again as if to pronounce o, and without changing, try to pronounce ef, 25. A quantity is said to be resolved into factors, when any el, eck, em, en, ep, er, ess, et, and you will utter the sound factors are taken which, being multiplied together, will pro- required. The sound which comes nearest to it is the e in dace the given quantity. Thus 3ab may be resolved into the her. two factors 3a and b, because 3a x b is 3&b. And 5amn may û has no parallel in English. Pronounce oo in ooze, firmly be resolved into the three factors 5a, and m, and n. And 48 maintain this position, and try to pronounce long e in eel; may be resolved into the two factors 2 x 24, or 3 x 16, or the sound uttered will be the one required. For the short 4 x 12, or 6 X 8; or into the three factors 2 X 3 X 8, or 4 x 6 sound, place the organs in a similar position, and without changing it try to say if, il, ick, im, in, ip, ir, iss, it. 26. Division is expressed in two ways: (1.) By an horizontal For those who have studied French it may be well to reSe between two dots +-, which shows that the quantity pre- mark that the German i has the same sound as the cling it is to be divided by that which follows. Thus a = C, French u. 2. Division is more commonly expressed in the form of a ou is always to be sounded as in out, our. rection, putting the dividend in the place of the numerator, gh before e and i must be pronounced like g. ch has different sounds, according to its po si the divisor in that of the denominator. Thus is a In the interlinear pronunciation we shall oh, when it is pronounced like an aspi vell as y. x 2, etc. divided by b. you may observe) the commands of your father. 20. I feared I was finds that he must give 2 pence for a peach, and 4 pence for displeasing you. 21. Take care to improve the morals and to exercise an orange. How many can he buy of each ? the body of the boy. 22. I feared that an enemy was injuring me. 23. The boy feared that his mother was silent. 24. I took care to peach is 2 pence, the price of a peaches will be x x 2 pence, or Let æ denote the number of each. Now, since the price of one 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 2x pence. For the same reason, æ x 4, or 4x pence, will denote (that you should) exercise the body of the boy. 26. I took care that the price of æ oranges. Then will 2x + 4e, or 6x, be equal to the teacher should improve the morals and exercise the body of the 96 pence by the conditions of that question, and lx or w (for boy. 27. I fear that you will (may) not come. 28. The husband fears when 1 is the co-efficient of a number (See Art. 16 below] it is that his wife will (may) die. 29. The teacher feared that the scholar always understood, and never expressed) is equal to of 96 would not obey his words. 30. The bad boy fears that the teacher pence, namely, 16 pence, and 16 is therefore the number he will come. bought of each. EXERCISE 93. ENGLISH-LATIN. 2. Quantities in algebra are generally expressed by letters, as 1. Ille me monebat. 2. Illi regem monebant. 3. Ego vos monerem. in the preceding problems. Thus b may be put for 2 or 15, or 4. Vos me moneretis. 5. Illi puerum monuerunt. 6. Tu mulierem any other number which we may wish to express. It must not be monebas. 7. Ego præceptorem monebo. 8. Tace. 9. Tacete. 10. inferrod, however, that the letter used has no determinate value. Tacento. 11. Mulier repente tacuit. 12. Cura ut emendes. 13. Cura Its value is fixed for the occasion or problem on which it is timebant ne patri displicerent. 16. Omnibus placet. 17. Bonus malis employed, and remains unaltered throughout the solution of displicebit. 18. Cur taces ? 19. Metuunt ne Cæsar patriam vincat. that problem. But on a different occasion, or in another problem, 20. Bonæ sorores timent ut fratres valeant. 21. Valesne ? 22. Timeo ut the same letter may be put for any other number. Thus, in valeas. 23. Si corpus exercueris valebis. 24. Mater timet ut mihi Problem I., & was put for A's share of the money. Its value aditus in cælum patent. was 12 pounds, and remained fixed through the operation. In Problem II., æ was put for the number of each kind of fruit. LESSONS IN ALGEBRA.-I. Its value was 16, and it remained so throughout the whole of the calculation. DEFINITIONS. 3. By the term quantity, we mean anything that can be mulART. 1.-ALGEBRA is a general method of solving problems, tiplied, divided, or measured. Thus, length, weight, time, number, and of investigating the relations of quantities by means of etc., are called quantities. letters and signs. 4. The first letters of the alphabet, a, b, c, etc., are used to The following will afford illustrations of this method of express known quantities; and the last letters, x, y, x, etc., arriving at the solutions of problems by the use of signs and those which are unknown. letters instead of figures as in arithmetic. 5. Known quantities are those whose values are given, or PROBLEM I.--Suppose that a man divided 72 pounds among his may be easily inferred from the conditions of the problem three sons in the following manner :--To A he gave a certain under consideration. number of pounds; to B he gave three times as many as to A; 6. Unknown quantities are those whose values are not given, and to C he gave the remainder, which was half as many pounds but required. as A and B received. How many pounds did the donor give to 7. Sometimes, however, the given quantities, instead of being each? 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 ihree 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, i share is equal to 4 of 72, 9. Addition is represented by two lines (+), one horizontal, namely, 12 pounds, which is A's share. B had three times as the other perpendicular, forming a cross, which is called plus. many, namely, 36 pounds; and C half as many pounds as both, It signifies "more," or " added to." Thus a + b signifies that namely, 24 pounds. b is to be added to a. It is read a plus b, or a added to b, or a Now, to solve the same problem by algebra, he would use and b. letters and signs, thus : 10. Subtraction is represented by a short horizontal line (-) Let æ represent A's share; then by the conditions, which is called minus. Thus, a - b signifies that b is to be a multiplied by 3, or X X 3 (when X, the sign of multipli- “ subtracted" from a; and the expression (see Art. 22 below) cation, is used instead of the words “multiplied by"), will repre- is read a minus b, or a less b. sent B's share, and 11. The sign + is prefixed to quantities which are considered 4x, the sum of the shares of A and B divided by 2, or 4x + 2 as positive or affirmative ; and the sign – to those which are (when +, the sign of division, is used instead of the words supposed to be negative. For the nature of this distinction, see “divided by"), will represent C's share. Articles 36 and 37. Now, * X 3 may be written 3x, and 4x + 2 may be written 12. The sign is generally omitted before the first or leading 2x; so then adding together the several shares of A, B, and c, quantity, unless it is negative; then it must always be written. namely, 2, 3x, and 2x, and putting t, the sign of addition, When no sign is prefixed to a quantity, + is always understood. between them, we get « + 3x + 2x, which is equal to 6x; or Thus a + b is the same as + a + b. using =, the sign of equality, for the wcrds " is equal to," we 13. Sometimes both + and -- (the latter being put under get & + 3x + 2x = 6x. Then 6x = 72, for the whole is equal the former, +) are prefixed to the same letter. The sign is then to all its parts; and lx = 12 pounds, A's share; 3x 36 said to be ambiguous. Thus a + b signifies, that in certain pounds, B's share; and 2x = 24 pounds, C's share. cases, comprehended in a general solution, b is to be added to a, Proof.-Add together the number of pounds received by each, and in other cases subtracted from it. and the sum will be equal to 72 pounds, the amount divided Observation. When all the signs are plus, or all minus, they between A, B, and C. are said to be alike; when some are plus and others minus, they In this algebraic solution it will be observed : First, that we are called unlike. represent the numler of pounds which A received by x. Second, 14. The equality of two quantities, or sets of quantities, is to obtain B's share, we must multiply A's share by 3. This expressed by two parallel lines, =. Thus a + b = d signifies multiplication is represented by two lines crossing each other like that a and b together are equal to d. So 8 + 4 = 16 - 4= 10 a capital X. Third, to find C's share, we must take half the + 2 = 7+ 2 + 3. sum of A's and B's share. Tis diri im is denoted by a line 15. When the first of the two quantities compared is greater between tuco dots. Fourth, the addition of their respective than the other, the character > is placed between them. Thus shares is denoted by another cross formed by an horizontal and a > b signifies that a is greater than b. a perpendicular line. Take another example : If the first is less than the other, the character < is used; as PROBLEM II.--A boy wishes to lay out 96 pence for peaches a<b, namely, a is less than b. In both cases, the quantity and oranges, and wants to get an equal number of cach. He towards which the character opens is greater than the other, -care 16. A numeral figure is often prefixed to a letter. This is READINGS IN GERMAN.-I. 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, INTRODUCTION. 9 times b, or 9 multiplied into b. The object of learning a modern language is not simply, as in The co-efficient may be either a whole number or a fraction. the case of one that is no longer spoken, to be able to read and Thus gb is two-thirds of b. When the co-efficient is not ex- write, but also to speak it. For this purpose it is obviously pressed, 1 is always to be understood. Thus a is the same as necessary to acquire a knowledge of the pronunciation as well le, that is to say, once a, or one times. as the meaning of the words. Hence we are not surprised at 17. The co-efficient may also be a letter, as well as a figure. having received many applications from the readers of our lesIn the quantity mb, m may be considered the co-efficient of b; sons in German for some instruction on this subject; and it is because b is to be taken as many times as there are units in m. our intention to publish in the pages of the POPULAR EDUCATOR If w stands for 6, then mb is six times b. In 3abc, 3 may be con- a series of German Reading Lessons expressly prepared with a sidered as the co-efficient of abe; 3a the co-efficient of bc; or view to teach the proper pronunciation of the language. These 3ab the co-efficient of c. lessons will be found much better adapted to answer the pur18. A simple quantity is either a single letter or number, pose than any mere collection of rules, however carefully drawn or several letters connected together without the signs + or up, and however clearly expressed. In no case is the principle, - Thas a, ab, abd, and 86, are each of them simple quan- that example is better than precept, more applicable than in tities. that of pronunciation, a knowledge of which can only be acquired 19. A compound quantity consists of a number of simple by frequent exemplification. We have no hesitation in saying quantities connected by the sign + or -, Thus a + b, d-y, that the study of our lessons will enable the reader to pronounce b-d +3h, are each compound quantities. The members of German, if not with absolute perfection, at least so as to be which each is composed are called terms. easily understood by a native, which is, after all, the only prac20. A simple term is called a monomial ; thus, a, b, tical object in view. monomials. If there are two terms in a compound quantity, it It is proper to observe, that whilst the lessons are especially is called a binomial: thus a + b and a--b are binomials. intended to teach pronunciation, they are also calculated to be The latter term (a - b) is also called a residual quantity, very useful to our readers as exercises in translation, being easy because it expresses the difference of two quantities, or the in construction, simple in style, rich in words, and adapted in remainder after one is taken from the other. A compound substance to persons of all ages. A vocabulary will be appended quantity, consisting of three terms, is sometimes called a tri- to each lesson, containing an explanation of the meaning of wonial; one of four terms, a quadrinomial. A quantity consist- every word in it which has not been previously explained. As ing of several terms is, however, generally called a polynomial. few words will be explained more than once in the whole course 21. When the several members of a compound quantity of the vocabularies, it will be necessary for the learner to study are to be subjected to the same operation, they are connected each with great care on its first occurrence, that he may avoid by a line called a vinculum (-), or by a parenthesis (). Thus the inconvenience of having to look through preceding pages 4-6+, or a-(6 + c), shows that the sum of b and c is to be for the meaning. subtracted from a. But a-b+c signifies that b is to be sub DIRECTIONS FOR THE USE OF THE INTERLINEAR tracted from a, and c is to be added to the result. PRONUNCIATION. 22. A single letter, or a number of letters, representing any Pronounce every syllable as in English. quantities with their relations, is called an algebraic expression To make a vowel long which otherwise would be short, or or formula. Thus a + b + 3d is an algebraic expression. might be either short or long, an apostrophe has been placed 23. Multiplication is usually denoted by two oblique lines immediately after that vowel. Thus vol will rhyme with doll, crossing each other, thus X : hence, a x b is a multiplied into but vo'l will rhyme with whole. b; and 6 x 3 is 6 times 3, or 6 multiplied into 3. Sometimes To make a vowel short which otherwise would be long, or a point is used to indicate multiplication: thus, a.b is the same might be either long or short, the short sign or breve has been az e xb. But the sign of multiplication is more commonly placed over that vowel. Thus mild will rhyme with wild, but omitted between simple quantities, and the letters are con mild will rhyme with build. Boot will rhyme with root, but boot nected together in the form of a word or syllable: thus, ab is will rhyme with put. the same as a.b or a xb; and bode is the same as b XcX dxe. When a compound quantity is to be multiplied, a ah is long, and sounded as in father; a is short, and sounded rinculum or parenthesis is used, as in the case of subtraction. as in castle. Thus the sum of a and b multiplied into the sum of c and d, is ey is to be pronounced as in obey. +bxe+d, or (a + b) (c+d). And (6 +2) x 5 is 8 x 5, ai, representing the short sound of ę, when unaccented and cr 40. But 6 + (2 X 5) is 6+ io, or 16. When the marks of terminating a syllable, should be pronounced like ey in the parenthesis are used, the sign of multiplication is frequently noun sur'-vey. The letters ey could not well be used for this omitted. Thus (x + y (x - y) is (x + y) X (—y). sound, as they have generally the sound of ee when unao cented. 24. When two or more quantities are multiplied together, each of them is called a factor. In the product ab, a is a factor, ó has no corresponding sound in English ; plaee the organs as and so is b. In the product a X (a + m), « is one of the factors, if to pronounce o long; keep them exactly in this position, and (a + m) the other. Hence every co-efficient may be con and then try to pronounce the German e or English a. For sidered as a factor (Art. 17). In the product 3y, 3 is a factor the short sound of this vowel place the organs again as if to pronounce o, and without changing, try to pronounce ef, 23. A quantity is said to be resolved into factors, when any el, eck, em, en, ep, er, ess, et, and you will utter the sound factors are taken which, being multiplied together, will pro required. The sound which comes nearest to it is the e in duce the given quantity. Thus 3ab may be resolved into the her. two factors 3a and b, because 3a x b is 3cb. And 5amn may ü has no parallel in English. Pronounce og in ooze, firmly be resolved into the three factors 5a, and m, and n. And 48 maintain this position, and try to pronounce long e in eel; may be resolved into the two factors 2 x 24, or 3 x 16, or the sound uttered will be the one required. For the short 4 X 12, or 6 X 8; or into the three factors 2 X 3 X 8, or 4 x 6 sound, place the organs in a similar position, and without X 2, etc. changing it try to say if, il, ick, im, in, ip, ir, iss, it. 26. Division is expressed in two ways: (1.) By an horizontal For those who have studied French it may be well to reline between two dots +, which shows that the quantity pre mark that the German û has the same sound as the ording it is to be divided by that which follows. Thus a = 0, French u. is a divided by c. ou is always to be sounded as in out, our. 2.) Division is more commonly expressed in the form of a fraction, putting the dividend in the place of the numerator, c has different sounds, according to its position in the word. gh before e and i must be pronounced like g in get, gimlet. and the divisor in that of the denominator. Thus Thus, is a In the interlinear pronunciation we shall denote it by oh, when it is pronounced like an aspirated k, or like the As well as y, divided by b. gern n. ch in the Scotch word loch. It has this sound after the | Wohnung, f. dwell- Klein, small, Umber, around, vowels a, o, u, au ; by ing (wohnen, to Gast, m. guest. about; schauen, to “y,” when it is pronounced like an aspirated y. It has dwell, reside, live). Entfkiegen, to fly & look. this sound in all other German words; by Nun, now, well. way (ent-, prefix, Uns, us. * k," when it is pronounced like k. It has this sound in Picen, to pick. away, up, forward). Ansehen, to look at most words derived from the Greek and before s, unless Die, pl. the. Den, acc. m. the. (an, prep. and prethis s belongs to the next syllable, or is an inflection ; Brosamen, pl. scraps. Nahe, near. fix, at, to, by). by Krume, f. crum. Walb, m. forest. Etwas, something, “sh," when it is pronounced like sh in English. It has Auf, up, upon, on. Bauen, to build. anything. this sound in words derived from the French. Die, relative pronoun Singen, to sing. Wollen, to wish, want, We shall transcribe g in the same manner, whenever it is pl. which. Fröhlich, merry (froh, be willing pronounced in one of these ways. Von, of, from. glad, cheerful). Antworten, to answer Tisch, m. table. Lieb, n. song, hymn, (Antwort, f. an1.-Das Rothfehlchen. Fallen, to fall. air. swer). Dass rote-keyl-den. Auch, also. Sehen, to see, to Vater, m. father, Ein Rothfehlchen · fam in der Strenge des Winters an das Kind, n, child. look. Wenn, if, when. Ine rote'-keyl-yen kahm in dair shtreng'-ai dess vin'-ters an dass Vogel , m. bird (-lein, Kehren, to turn. Sie, they, them, she, Fenster eines frommen Landmanns, als ob es hinein sign of diminutives). Abermals, again. her. fen'-ster i'-ness from’inen lant-manss, alss op ess gairn hin-ine' Lieb halten, to love. Weibchen, n. Weib, Reden, to speak (Reve, Werth halten, to es n. mate, female, f. speech). möchte. öffnete der Landmann sein Fenster und nahm teem, cherish wife. möy'-tai. Dah öf-nai-tai dair lant-man zine fen'-ster öðnt nahm Können, to be able, (halten, to hold). Mit, with; bringen, can. bas zutrauliche Thierchen freundlich in seine Wohnung. Nun Aber, but. to bring ; haben, to Würden, woald, dass tsoo'-trou-l-ye teer'-yen froint'-lij in zi-nai voʻ-noðnk. Noon Der, nom. m. the. have. should, sign of pidte 18 die Brosamen und Strümchen auf, die von seinem Frühling, m. spring. Sammt, together the conditional pick'-tai ess dee broʻ-zah-men dönt krü'm'-yen ouf, dee fön zi-nem Wieder, again, prefiere. with. mood. Tische fielen. Auch hielten die Kinder des Landmanns das Bebūsd), n. collective Freuen, to enjoy, re Sagen, to say. tish-shai fee'-len. Ouch hee®-ten dee kin'-der dess lant-manss, dass noun, bushes, joice. Zutrauen, n. confi. copse (ge-, a prefiæ, Sehr, very, much. dence. Vöglein lieb und werth. Aber als nun der Frühling wieder in showing a mass of Beide, both. Grweden, to awake. fö'g'-line leep oont veyrt. ah'-ber alss noon dair frü"-link vee-der in things; Busch, m. Wie, how, as, like. Liebe, f. love (Lieben, das Land fam und die Gebüsche sich belaubten, öffnete bush). Aus, out, out of. to love). dass lant kahm dónt dee gai-büsh-shai ziy bai-laup'-ten, dah öf-nai-tai Sich, himself, herself, Klar, clear. Erzeugen, to engender, der Landmann sein Fenster, und der kleine Gaft entflog in das itself, themselves. Auge, n. eye (Aeuglein, beget. dair lant-man zine fen'-ster, dont dair kli'-nai gast ent-flo'ch' in dass Belauben, to cover diminutive, Oegen, preposition nahe Wäldchen, und bauete fein Neft und fang sein fröhliches with foliage (Raub, small eye, beauti and prefix, coun. nah'-hai velt-yen, dont bou’-ai-tai zine nest dönt.zank zine fröʻ-11-yess foliage). ter, against Siedchen leet -yen, KEY TO EXERCISES IN LESSONS IN GERMAN. und fiehe, als der Winter wiederkehrte, da fam das Roth EXERCISE 60 (Vol. I., page 302). oðnt zee'-hai, alss dair vin'-ter vee'-der-keyr-tai, dah kahm dass rote' 1. These great beautiful houses are all to let. 2. The one house is fehlchen abermals in die Wohnung des Landmanns und to let, the other to sell. 3. It is not to be believed that he has forkeyl-yen ah'-ber-mahlss in dee vo'-noðnk dess lant-manss ont saken us. 4. This book is to be had of Mr. Westermann in Brunshatte sein Weibchen mitgebracht. Der Landmann aber fammt wick. 5. Not one single star was to be seen in the whole heavens . hat-tai zine vipe'-yen mit-gai-bracht. Dair lant-man ah'-ber zamt 6. How is this long word to be pronounced? 7. Where are the best boots, shoes, and over-shoes to be found ? 8. The best, which I have seinen Kindern freuten fich sehr, als sie die beiden Thierchen "sahen, seen, are to be found at my old neighbour N's. 9. The fire burnt so zi-nen kin'-dern froi'-ten zij zeyr, alss zeo dee bi-den teer'-yen zah'-hen, rapidly that nothing was to be saved in the castle. 10. Nothing valuwie sie aus den klaren Peuglein zutraulich umherschauten; able is to be gained without trouble. 11. This high rock is not to be vee zee ouss dain klah'-ren oig'-line tsoo'-trou-liy dóm-heyr'-shou’-ten; this forest one cannot get. 14. He is neither to be convinced nor to be climbed. 12. This old house is to be repaired no more. 13. Through und die Kinder sagten : Die Vögelchen sehen uns an, als ob persuaded. 15. His behaviour is not at all to be pardoned. 16. What ont dee kin'-der zahch-ten: Dee fö"-ghel.yen zey'-hen dönss an, alss op is your friend's name? 17. He is called James. 18. How is this fie etwas sagen wollten. called in German ? 19. It is called Brille (spectacles). 20. The more perfect a work of art is, that is, the more parts it has, and the more zoe et'-vass zah'-ghen vol-ten. all these parts contribute to the purpose, the more beautiful it is. Da antwortete der Vater: Wenn sie reden fönnten, so Dah ant'-võr-tai-tai dair fah'-ter: Ven zee rey'-den kön'-ten, zo' EXERCISE 61 (Vol. I., page 302). tvürden sie sagen : Freundliches Butrauen 1. Die Aussprace fremder Wörter ist nur durch Uebung zu erlernen. 2. erivedet vür-den 3. Vollfommene Oludjeligfeit ist in Nichts ist ohne Mühe zu erlernen. zee zah'-ghen: Froint'--yes tsoo'-trou-en err-veck'-et dieser Welt nicht zu finden. 4. Sie sprechen so schnell, daß Sie nicht Zutrauen, und Liebe erzeuget Gegenliebe ! zu verstehen sind. 5. Gesundheit ist mit Geld nicht zu erfaufen. 6. Die tsoo-trou-en, dont lee'-bai err-tsoi-ghet ghey'-ghen-lee'-bai! Ruhe der Stadt war burch firenge Befehle nicht herzustellen. 7. Wie VOCABULARY. nennen Sie diese Blumen? 8. Sie werden Tulpen genannt. 9. Die Das, n. the. An, to. Hinein, into (hin, klugen Schüler sind zu loben. 10. Der Unterschied zwischen fausen und Rothkehlchen, n. red- Fenster, n. window. along, towards). verkaufen muß den Schülern zu dieser Zeit befannt sein. 11. Dieses Buch breast (roth, red; Eines, gen. m. and n. Da, then, there. ist bei dem Buchhändler 6. in Lonton zu haben. 12. Gin werthvolled Keble, f. throat; of a, of one. Öffnen, to open. Kunstwerk kann nicht ohne viel Mühe gemacht werden. 13. Die Rose und -chen, affix, sign of Fromm, pious. Sein, his. tas Veilchen werden wegen ihres Wohlgeruchs geschaft, die Tulpe wegen bes diminutive). Landmann, m. coun- Und, and. Glanzes ihrer Farben. 14. Jakob geht morgen nach Braunsdweig. 15. Gin, a, one. tryman. Nehmen, to take. Die Himmel verfündigen die Herrlichkeit Gottes. EXERCISE 62 (Vol. I., page 302). 1. Where are you sending your servant ? 2. He is ill, he can go noStrenge, f. severity. Gern, willingly, with Freundlich, friendly, where. :3. Do you copy this letter! 4. I have already copied it. 5. Do you believe that the bookbinder sends me back my books? 6. Has Des, gen. m. and n. of pleasure (Gern kind (Freund, m. your sister received the flowers which I have bought for ber? 7. The the. mögen, to like; fom friend ; -lich, affix, gardener comes to-morrow, and will bring them with him. 8. When Winter, m. winter. men, understood). doas John go to school? 9. He goes there to-morrow, and little like -ly). |