(3) The accusative is used with such terms as wiegen, to weigh; Foften, to cost; gelten, to pass for; werth, worth; schwer, heavy; reid), rich; lang, long; weit, wide; to mark definitely the measure or distance indicated by these words; as, tiefer Stock ist ein Fuß lang, this stick is a foot long; er ist vier Monate alt, he is four months old. In the earlier German, these words of measure or distance were put in the genitive: as, einer Spanne weit, a span wide. (4) As words expressing time indefinitely are put in the genitive (§ 128. 1.), so those denoting a particular point, or du. ration of time, are put in the accusative; as, ich erwartete den zweiten Tag, I waited two days. (5) A substantive construed with a participle, is sometimes put absolutely in the accusative; as, diesen Umstand ausgenommen, finde ich Alles recht, this circumstance excepted, I find all right. § 133. RULE. A noun or pronoun used merely to explain or specify that which is signified by a preceding noun or pronoun, must be in the same case: as, Cicero, ein großer Redner, Cicero, a great orator. 161. An equation may be cleared of fractions by multiplying both members by all the denominators. 162. In clearing an equation of fractions, it often happens that a numerator becomes a multiple of its denominator (i. e. can be divided by it without a remainder), or that some of the frac tions can be reduced to lower terms. When this occurs, the operation may be shortened by performing the division indi Der Rath meines Bruders, des Rechtsgelehrten, the advice of my cated and by reducing the fractions to their lowest terms. brother, the lawyer. OBSERVATIONS. (1) The explanatory noun is said to be in apposition with that which it explains: the latter being called the principal term. Between these two, that is, between the principal and the ex. planatory term, there often intervenes some connective particle. Thus er hat sich als Gesezgeber verdient gemacht, he, as a lawgiver, has rendered himself meritorious; mein Nachbar, nämlich der Bauer, my neighbour, namely, the farmer. This latter mode of specifying (that is, with the word nämlich), is far more common in German than in English. (2) The proper names of months, countries, towns, and the like appellatives, are put in apposition with their common names; where, in English, the two words stand connected, for the most part, by the preposition of; as, ter Monat August, the month (of) August; die Stadt London, the city (of) London; die Universität Oxfort, the university (of) Oxford. 18. Reduce the equation x b в h m Ans. x= LESSONS IN ALGEBRA.-No. XI. REDUCTION BY MULTIPLICATION. 159. When the unknown quantity is connected with a known quantity by the sign of division, the reduction is effected by multiplying both members of the equation by the latter, if it be the divisor; and by the former, if it be the divisor. In this case, it will be particularly useful to remember a rule formerly given: viz., that a fraction is multiplied by its denominator, by removing the denominator; o, in other words, putting down the numerator as the product. Also, that after this process has been performed, transposition is still to be employed as in the preceding examples. 21. Reduce 3 REDUCTION BY DIVISION. 164. When the unknown quantity contains any known quantity, as a factor, the equation is reduced by dividing every term on both members by this known quantity. 25. Reduce the equation ax + b — 3h = d. d+ 3h 2-4 Ans. a 166. If any quantity, either known or unknown, is found as a factor in every term, both members of the equation may be divided by it. On the other hand, if any quantity is a divisor in every term, both members of the equation may be multiplied by it. In this way, the factor or divisor will be removed, and the reduction may be effected as before. 30. Reduce the equation ax + 3ab = 6ad+a. Here, dividing by a, we have x+366d+1; and by transposition, x 6d + 1—3b. Ans. 3x 4 42. Reduce +6=84. Ans. x = 104. 4500 43. Reduce + 350 7000 10. Ans. x 3275. a (be-a) (m+n) + с (abc — d) (l — m —n) Ans. = Here multiplying by x, we have x+1-bh-d; and, by transposition, ah-d+b-1. Ans. 32. Reduce the equation ≈ × (a+b) — a − b = d × (a+b). Ans. x=d+ 1. 167. A proportion is converted into an equation by making the product of the extremes, one member of the equation; and the product of the means, the other member 33. Reduce to an equation ax: b:: ch: d. Here the product of the extremes is adx, and the product of the means bch; the equation is, therefore, adxbch. Whence 4 2. Reduce + = 7 - 2 + d. с 34. Reduce to an equation a +6: : h e (h- m) a+b 3 6x 16 120 14. 5. Reduce 3 =20 + 20 — x Ans. x 23. Ans. 139. 1 -a 6. Reduce add 70, and from this sum subtract 50, the remainder will be equal to 220 pounds. In order to solve this question, we must first translate the conditions of the problem into such an algebraic expression as will form an equation. Let z be the price of the watch. This price is to be multiplied by 4, which makes 42; to the product 70 is to be added, making 4x+70; from this, 50 is to be subtracted, making 4x + 70—50. Here we have a number of the conditions, expressed in algebraic terms; but we have as yet no equation. We must observe, then, that by the last condition of the problem, the preceding terms are said to be equal to 220. We have, therefore, this equation 4x+70- 50=220; which reduced, gives x 50. Ans. = Here the value of x is found to be 50 pounds, which is the price of the watch. PROOF.-The original equation is 4x + 70-50220; substituting 50 for x, it becomes 4 x 50 +70-50 220.; that is, 220 220. Prob. 2.-What number is that to which, if its half be added, and from the sum 20 be subtracted, the remainder will be a fourth of the number itself? In stating questions of this kind, where fractions are concerned, it should be recollected, that is the same as -- 19. Reduce + = 3 4 reducing the equation, we have x=16. Ans. Ans. x 23. ard 4x 3 5 7 2 4 6 Prob. 3.-A father divides his estate among his three sons in such a manner, that the first has £1,000 less than the whole; the second has £800 less than one-third of the whole; the the third has £600 less than one-fourth of the whole? what is the value of the estate? Ans. £28,800. Prob. 4.-Divide 48 into two such parts, that if the less be divided by 4, and the greater by 6, the sum of the quotients and, by the conditions of the problem, we have, + 4 +81. Ans.x=420. 9. Whence x = 12; therefore, 12 is the less part, and 36 the greater part. SOLUTION OF PROBLEMS. RULE. 1st. Translate the statement of the question from the ordinary language into algebraic language, in such a manner as to form an equation; that is, put the question into the form of an equation. 2nd. Clear the equation of fractions by multiplying every term in both members by all the denominators successively, or by their least common multiple. 3rd. Transpose all the terms containing the unknown quantity to the one side of the equation, and all the known quantities to the other, taking care to change the signs of the terms transposed, and incorporate the terms that are alike. 4th. Remove the co-efficient of the unknown quantity, by dividing all the terms in the equation by it; the result will be the solution required. PROOF.-Substitute the value of the unknown quantity for the letter which stands for it in the equation; and if the number satisfies the conditions of the question, it is the answer sought. Problem 1. A man being asked how much he gave for his watch, replied: If you multiply the price by 4, to the product 171. Letters may be employed to express the known quantities in an equation, as well as the unknown. A particular value is assigned to the letters, when they are introduced into the calculation; and at its close, the numbers are restored. Prob. 5.-If to a certain number, 720 be added, and the sum be divided by 125, the quotient will be equal to 7392 divided by 462. What is the number? d Let a be the number required; and let a 720, b = 125, 7392, and h = 462. = Prob. 6. Divide 11 into two parts, such that the sum of twice the first and half the second may be 16. Ans. 7 and 4. Prob. 7. Divide 39 into four parts, such, that if the first be increased by 1, the second diminished by 2, the third multiplied by 3, and the fourth divided by 4, the results may be all equal. Ans. 5, 8, 2, 21. Prob. 1. If a certain number is divided by 12, the quotient, Prob. 9. An estate is divided among four children, in such a manner that the first has £200 more than of the whole, the second has £340 more than of the whole, the third has £300 more than of the whole, the fourth has £400 more than of the whole. What is the value of the estate? Ans. £4800. Prob. 10. What is that number which is as much less than 500, as a fifth part of it is greater than 40? Ans. 450. Prob. 11. There are two numbers whose difference is 40, and which are to each other as 6 to 5. What are the numbers? Ans. 240 and 200. Prob. 12. Suppose two coaches to start at the same hour, one from London for Glasgow, and the other from Glasgow for London, the former travelling 10 and the latter 9 miles per hour: where will they meet, the distance between the two cities being 400 miles? Ans. 210 miles from London. Prob. 13. Suppose every thing to be as in the last question, except that the coach from Glasgow starts two hours earlier than the other; where will they meet? Ans. 200 miles from London. Prob. 14. A dealer purchases 60 yards of cloth for 30%.; and by selling one part of it at 128., another, twice as great, at 148., and the rest at 10s. per yard, he gains 81. How many yards were in the several lots? Ans. 16, 32, and 12. Prob. 15. Suppose two dealers each annually to double his capital, except an expenditure of 1007.; and, that at the end of three years, the capital of one is found to be doubled, while the other has only half what he had at first; how much had each to commence with? Ans. 1167. 13s. 4d. and 931. 6s. 8d. Prob. 16. If a person each year double his capital except an expenditure of 300l. the first year, 4007. the next year, and 500. the third, and at the end of three years be found to be worth 55007., what was his original capital? Ans. 1000. Prob. 17. A father's age is now treble of his son's, while five years ago it was quadruple: what are their present ages? Ans. 45 and 15 years. Prob. 18. Divide 1000%. between A, B, and C, giving A 1007. nore, and B 507, less, than C. Ans. A's share 4167. 13s. 4d. ; B's 266, 13s. 4d.; and C's 3167. 13s. 4d. Prob. 19. A spirit merchant finds that if he add 10 gallons to a ask of brandy, the mixture will be worth 21s. per gallon; but that if he add ten gallons more, the value will be reduced to 188. How many gallons were in the cask? Ans. 50, Prob. 20. Find a number, such that if it be divided successively by 2, 3, 4, 5, 6, 7, 8, 9, and 10, half the sum of the first four quotients increased by 20 shall be equal to the sum of the remaining five. Ans. 5040. Prob. 21. Find two numbers differing by 6, and such that three times the less may exceed twice the greater by 7. Ans. 25 and 19. Prob. 22. Find a number such, that if it be increased successively by 1, 2, and 3, the sum of one-half of the first result and one third of the second shall exceed one-fourth of the third by 8. Ans. 13. your voice suspended as if some one had stopped you before you had read all that you intended to read. 26. In the following examples keep your breath suspended when you come to the comma: but let the short pause or stop which you make, be a total cessation of the voice. Diligence, industry, and proper improvement of time, are material duties of the young. He is religious, generous, just, charitable and humane. By wisdom, by art, by the united strength of a civil community, men have been enabled to subdue the whole race of lions, bears and serpents. The genuine glory, the proper distinction of the rational species, arises from the perfection of the mental powers. Courage is apt to be fierce, and strength is often exerted in acts of oppression. laws, to pursue right measures, to correct power, to protect weakWisdom is the associate of justice. It assists her to form equal ness, and to unite individuals in a common interest and general welfare. Heroes may kill tyrants, but it is wisdom and laws that prevent tyranny and oppression. 27. When a note of interrogation occurs at the end of a sentence, the parts, and even the words, of the sentence separated by commas, should each be read like a question. Examples. Did you read as correctly, speak as properly, or beha'? › as well as James? Art thou the Thracian robber, of whose exploits I have heard so much? Who shall separate us from the love of Christ? shall tribulation, or distress, or persecution, or famine, or peril or surd? How are the dead raised up, and with what body do they come? For what is our hope, our joy, or crown of rejoicing? Have you not misemployed your time, wasted your talents, and passed your life in idleness and vice? Have you been taught any thing of the nature, structure and laws of the body which you inhabit? Were you ever made to understand the operation of ciet, air, exercise, and modes of dress, upon the human frame? that preceding a period, with the falling inflection of twice. 28. Sometimes the word preceding a comma, is to be real like Examples. It is said by unbelievers that religion is dull, unsocial, un charitable, enthusiastic, a damper of human joy, a morose intruder upon human pleasure. Nothing is more erroneous, unjust, or untrue, than the statement in the preceding sentence. Perhaps you have mistaken sobriety for dulness, equanimity for moroseness, disinclination to bad company for aversion t society, abhorrence of vice for uncharitableness, and piety for enthusiasm. Henry was careless, thoughtless, heedless, and inattentive. This is partial, unjust, uncharitable, and iniquitous. The history of religion is ransacked by its enemies, for instances of persecution, of austerities, and of enthusiastic irregularities. Religion is often supposed to be something which must be practised apart from every thing else, a distinct profession, a LESSONS IN READING AND ELOCUTION.-No. III. peculiar occupation. 29. Sometimes the word preceding a comma, is to be read like that preceding an exclamation. Examples. procured for you! that beatiful top, those polished marbles, How can you destroy those beautiful things which your father that excellent ball, and that beautifully painted kite, oh how can you destroy them, and expect that he will buy you new ones! How canst thou renounce the boundless store of charms that ing shore, the pump of groves, the garniture of fields, all that the Nature to her votary yields! the warbling woodland, the resoundgenial rav of morning glids, and all that echoes to even, all that the mountain's shitening bosom shirids, and an the dread magnificence of heaven, how canst thou renounce them and 25, In reading, when you come to a comma, you must keep hope to be forgiven! Oh winter! ruler of the inverted year! thy scattered hair with sleetlike ashes filled, thy breath congealed upon thy lips, thy cheeks fringed with a beard made white with other snows than those of age, thy forehead wrapped in clouds, a leafless branch thy sceptre, and thy throne a sliding car, indebted to no wheels, but urged by storms along its slippery way, I love thee, all unlovely as thou seemest, and dreaded as thou art! Lovely art thou, O Peace! and lovely are thy children, and lovely are the prints of thy footsteps in the green valleys. 30. Sometimes the word preceding a comma and other marks, is to be read without any pause or inflection of the voice. ashamed to read aloud. I consider it my duty, at this time, to tell you, that you have done something of which you ought to be ashamed. The Spaniards, while thus employed, were surrounded by many of the natives, who gazed, in silent admiration, upon actions which they could not comprehend, and of which they did not foresee the consequences. The dress of the Spaniards, the whiteness of their skins, their beards, their arms, appeared strange and surprising. Yet, fair as thou art, thou shunnest to glide, beautiful stream! by the village side, but windest away from the haunts of men, to silent valley and shaded glen. But it is not for man, either solely or principally, that night is made. We imagine, that, in a world of our own creation, there would always be a blessing in the air, and flowers and fruits on the earth. Share with you! said his father-so the industrious must lose his labour to feed the idle. 31. Sometimes the pause of a comma must be made where there is no pause in the book. Spaces are left in the following sentences where the pause is proper to be made. Examples. The Europeans were hardly less amazed at the scene now set before them. Their black hair long and curled floated upon their shoulders or was bound in tresses around their head. Persons of reflection and sensibility contemplate with interest the scenes of nature. The succession and contrast of the seasons give scope to care and foresight diligence and industry which are essential to the dignity and enjoyment of human beings. The eye is sweetly rested on every object to which it turns. It is grateful to perceive how widely yet chastely, nature hath mixed her colours and painted her robe. Winter compensates for the want of attractions abroad by fireside delights and homefelt joys. In all this interchange and variety we find reason to acknowledge the wise and benevolent care of the God of seasons. 32. The pupil may read the following sentences; but before reading them, he should tell after what word the pause should be made. The pause is not printed in the sentences, but it must be made when reading them. And here it may be observed, that the comma is more frequently used to point out the grammatical divisions of a sentence, than to indicate a rest or cessation of the voice. Good reading depends much upon skill and judgment in making those pauses which the meaning of the sentence dictates, Lut which are not noted in the book; and the sooner the pupil is taught to make them, with proper discrimination, the surer and the more rapid will be his progress in the art of reading. Examples. The golden head that was wont to rise at that part of the table was now wanting. For even though absent from school I shall get the lesson. It is now two hundred years since attempts have been made to civilize the North American savage. Doing well has something more in it than the fulfilling of a duty. You will expect me to say something of the lonely records of the former races that inhabited this country. There is no virtue without a characteristic beauty to make it particularly loved by the good, and to make the bad ashamed of their neglect of it. made up to us by self-approval, and the consideration of what A sacrifice was never yet offered to a principle, that was not our degradation would have been had we done otherwise. The succession and contrast of the seasons give scope to that care and foresight, vigilance and industry, which are essential to the dignity and enjoyment of human beings, whose happiness is connected with the exertion of their faculties. A lion of the largest size measures from eight to nine feet from the muzzle to the origin of the tail which last is of itself about four feet long. The height of the larger specimens is four or five Whose towers are feet. A benison upon thee gentle huntsman. these that overlook the wood? The incidents of the last few days have been such as will probably never again be witnessed by the people of America and such as were never before witnessed by any nation under heaven. To the memory of André his country has erected the most magnificent monument, and bestowed on his family the highest honours and most liberal rewards. To the memory of Hale not a stone has been erected and the traveller asks in vain for the place of his long sleep. CORRESPONDENCE. MUTUAL INSTRUCTION CLASSES. SIR,-This is an old mode of instruction, but one too much neglected. Its importance will be disputed by few, affording as it does such facilities for acquiring and exercising knowledge, quickening the perceptive faculties by wholesome and stimulating competition, and enlarging and correcting our views by association. It has struck me that the success and prosperity of the "POPULAR EDUCATOR" Would be even more permanently established, and the students themselves more interested and benefited by its lessons, if classes of this description could be generally established in connexion with this publication. The students of the "POPULAR EDUCATOR" do not sufficiently know their brother students. Many of them in the same town may be individually studying the same subject, unknown to each other, who would be very glad to meet for the purpose of mutual instruction. Difficulties frequently occur to one person, which are easily removed by another; and, where necessary, an instructor for the whole class might be engaged at a trifling expense to each, instead of paying exorbitantly for private lessons. Now, Sir, if a plan could be devised for collecting together all the students belonging to a particular locality, little difficulty would be experienced by them in subdividing themselves into classes, according to their different studies. The machinery required for this would be very simple. You would have to exercise your indulgence by sparing a corner of the P. E. for it; and any person wishing to collect together the students of the locality in which he resides, would merely have to send you his name and office. address for insertion, thus:-NORWICH, W. B. "News Any students in the place referred to, seeing this announcement, could at once communicate with him, and he might immediately bring them together by calling a meeting, when arrangements could be made for the formation of classes. I should like that this experiment were made in Norwich, believing that many students here would at once embrace the opportunity. If such persons, therefore, will send to the address I have just given as an example (viz. W. B. "News" office, Norwich), I shall feel most happy in calling them together, and assisting in the necessary arrangements. Should this effort prove effectual for Norwich, of course many other places could do the same. A system of National Adult Education of no mean character would thus be established throughout the kingdom; and the anxious desire of many great and good men of the present day, would be in a great measure realised, through the simple instrumentality of the POPULAR EDUCATOR and the exertions of a few spirited individuals in each locality. Trusting that you will lend your influence in aid of this move. |