Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[ocr errors]

ot.

[ocr errors]

ad l ah; again, multi

(3) The accusative is used with such terms as wiegen, to weigh; ( 160. Though it is not always necessary, yet it is often convekoften, to cost; gelten, to pass for ; werth, worth; Tehwer, heavy; nient, to remove the denominators from fractions consisting of reich, rich ; lang, long; weit, wide; to mark definitely the measure known quantities only. This is done in the same manner as or distance indicated by these words; as, dieser Stod ift ein Fuß in the preceding rule. lang, this stick is a foot long; er ift viet Monate alt, he is four

h. months old. In the earlier German, these words of measure or 17, Reduce the equation distance were put in the genitive: as, einer Spanne weit, a span wide.

(4) As words expressing time indefinitely are put in the ge- Here, multiplying by a, we have x= őt nitive (§ 128. 1.), so those denoting a particular point, or duration of time,' are put in the accusative; as, id erwartete den

abh

; zweiten Tag, I waited two days.

plying by b, we have bx=adt lastly, multiplying by c, (5) A substantive construed with a participle, is sometimes put absolutely in the accusative; as, tiefen Umstand ausgenommen,

acd tabh finde ich Alles recht, this circumstance excepted, I find all right. we have bcxacd tabh. Whence x=

Answer.

bc $ 133. RULE.

161. An equation may be cleared of fractions by multiplying both A noun or pronoun used merely to explain or specify that members by all the denominators. which is signified by a preceding noun or pronoun, must be in 162. In clearing an equation of fractions, it often happens that the same case : as,

a numerator becomes a multiple of its denominator (i. e. can be

divided by it without a remainder), or that some of the fracCicero, ein großer Kebner, Cicero, a great orator.

tions can be reduced to lower terms. When this occurs, the Ihm, meinem Wohlthäter, to him, ray benefactor. Der Rath meines Bruders, des Rechtsgelebrten, the advice of my cated and by reducing the fractions to their lowest terms.

operation may be shortened by performing the division indibrother, the lawyer.

5

h, OBSERVATIONS.

18. Reduce the equation

+

Ans. = (1) The explanatory noun is said to be in apposition with that which it explains : the latter being called the principal term. abgm+ademadgh. Between these two, that is, between the principal and the ex. dgm planatory term, there often intervenes some connective particle.

2 4 Thus er hat sich als Gefeßgeber verdient gemacht, he, as a lawgiver, has 19. Reduce the equation

+

Ans. rendered himself meritorious; mein Nachbar, nämlich der Bauer,

2 my neighbour, namely, the farmer. This latter mode of speci-9-. fying (that is, with the word nämlich), is far more common in

163. In clearing an equation of fractions, it will be necessary German than in English: (2) The proper names of months, countries, towns, and the that the whole value is to be subtracted, which is done by

to observe, that the sign - prefixed to any fraction, denotes like appellatives, are put in apposition with their common names; where, in English, the two words stand connected, for changing the signs of all the terms in the numerator. the most part, by the preposition of; as, der Monat Auguft, the

EXAMPLES month (of) August; die Stadt London, the city (of) London; die Universitat Oxfors, the university (0) Oxford.

36-2hr-on, 20. Reduce

Ans. =

e

[ocr errors]
[ocr errors]

6

2

[ocr errors]
[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]
[blocks in formation]

159. When the unknown quantity is connected with a known quantity by the sign of dirision, 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 dirisor.

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; or, 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.

8

23. Reduce 2x

[ocr errors]

25+

Ans. 2 10.

5

5

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

1476+-dək. Ans. x=(a+)x(hd).

+(6-7)+22

6

16. Reduce the equation

[ocr errors]

10.

}

1.

b2 = a

[ocr errors]
[ocr errors]
[ocr errors]

EXAMPLES.
?67. If ihe unknown quantitity has co-efficients in several
terms, che tyuation must be divided by the sum of all these co.
efficients.

3

41. Reduce + 27. Reduce the equation 3x

d.

750 375 Here, 3x - bix (3 - 5)x; and (3 — 5) x x = a d.

Here, by substituting a for 750; 6 for 3; and o for 375; the ad

8

Ans.
Whence, dividing by 3-6, we have x =
3

+ =1. Now clearing of fractions,
28. Reduce the equation ax + x=1- 4.
Ans. *

ab h

we have cx tab=ac: and 2 sa On restoring the
a-4-1
sono

3 x 750
atd
numbers, we have <= 750

- 744. Ans.
29. Reduce the equation X

Ans. =

375
h
4

32
(a +d) - 13

42. Reduce

+6=84. Ans. = 104.

4 4 (1-1)

4500 - 166. If any quantity, either known or unknown, is found as a 43. Reduce

10. Ans. x = 3275.

350 7000
factor in very term, both members of the equation may be
divided by it. On the other hand, if any quantity is a divisor

(bc-a) (mton) in every term, both members of the equation may be multiplied

44. Reduce

to
:8. Ans. aC

=
by it. In this way, the facter or divisor will be removed, and

mtn
the reduction may be effected as before.
30. Reduce the equation ax + 3ab = bad + a.

45. Reduce

+

1-m-Here, dividing by a, we have x + 36 = 60+l; and by

外 transposition, x = 6d+1-36. Ans.

(abc --- ) (

M") 2+1 To -d

Ans. at 31. Reduce the equation

+

[ocr errors]

a

C

= ab.

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

46. Reduce

[ocr errors]
[ocr errors]

m т

btcta

[blocks in formation]

Eure multiplying by x, we have & +1-b=hd; and, by transposition, x =k d+b-1. Ans.

32. Reduce the equation 2 x (a + b)-2-b=dx (a+). Ans. =dt 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. Rurage to an equation ax : 6 ::ch:d.

Here the product of the extremes is ada, and the product of the means bch; the equation is, therefore, ada = bch. Whence bok

Ars.

[merged small][merged small][ocr errors][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][subsumed][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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 ac 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 a =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 + 70-50 =220; substituting 50 for , 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 it is the same as

23 that x =

&c.

õ'
Let æ be the number required.

3

; 3

[merged small][ocr errors][merged small]
[ocr errors]

+7.

[ocr errors]
[ocr errors]

6

[ocr errors]
[ocr errors]
[ocr errors]

+

4

+

3

:21. Rduce 23 – 9=72 +
Ans, x = 45.

20

and 2

4 5

reducing the equation, we have x=16. Ans. 艺十2 22. Reduwe sll= Ans. = 23.

16

16 อ้

PROOF. Thus 16 + 20

2

4 23. Reduce

Prob. 3.-A father divides his estate among his three sons +1.

Ans. = 12.
2
3

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 24. Reduce 11

Ans. * = 40.
5
4

the value of the estate: Ans. £28,800.

Prob. 4.-Divide 48 into two such parts, that if the less be 2+1

-1 23. Reduce 구

Ans. = 19. divided by 4, and the greater by 6, the sum of the quotients: 4

will be 9.

3 X +9
32 +7

Let æ be the smaller part; then 48 * is the greater part; 26. Reduce +

+3. Ans. =51.
8
12
20

48 and, by the conditions of the problem, we have, +

4

6 23 42

32 27. Reduce

+ +81. Ans.x=420. = 9. Whence x = 12; therefore, 12 is the less part, anů 36 3 5 7 2

6

the greater part.
2
3

171. Letters may be employed to express the known quantities 28. Reduce

= 6. Ans. x = ll.

in an equation, as well as the unknown. A particular value is 2 4

assigned to the letters, when they are introduced into the cal

culation; and at its close, the numbers are restored. SOLUTION OF PROBLEMS.

Prob. 5.-If to a certain number, 720 be added, and the arm 170. For the solution of problenis in Simple Equations, we be divided by 125, the quotient will be equal to 7392 divided derive from the preceding principles the following general by 462. What is the number? rule:

Let z be the number required ; and let a : 720,b= 125, RULE. Ist. Translate the statement of the question from the ordi- a 7392, and h

462. nary language into algebraic language, in such a manner as to form an equation; that is, put the question into the form of an equation. Then, by the conditions of the problem, we have

2nd. Clear the equation of fractions by multiplying every terin in both members by all the denominators successively, or by their least ed

ba ah common multiple.

m; and reducing, we have x
h

hr
3rd. Transpose all the terms containing the unknown quantity to
the one side of the equation, and all the known quantities to the

(125X7392)-(720X 462) other, taking care to change the signs of the terms transposed, and

Restoring the numbers; we have x =

462 incorporate the terms that are alike.

=1280. 4th. Remove the co-efficient of the unknown quantity, by dividing all the terms in the equation by it; the result will be the

Prob. 6. Divide 11 into two parts, such that the sum of twice the first and half the second may be 16.

Ans. " and 4. solution required.

Proof.-Substitute the value of the unknown quantity for the Prob. 7. Divide 39 into four parts, such, that if áhe first be inletter which stands for it in the equation ; and if the number satsar creased by 1, the second diminished by 2, the third: naultiplied by fies the conditions of the question, it is the answer sought.

3, and the fourth divided by 4, the results may be all equal.

Ans. 5, 8, 2, 24. Problem 1. A man being asked how much he gave for his Watch, replied : If you multiply the price by 4, to the prouot . Ptobi, 1'.. Ifalsertain number is divided. by 12, the quotient,

[ocr errors]
[ocr errors]

dividend, and divisor, added together, will amount to 64. pour voice suspended as if some one had stopped you before you What is the number?

Ans. 48. had read all that you intended to read. Prob. 9. An estate is divided among four children, in such

26. In the following examples keep your breath suspended a manner that the first kas £200 mcre than of the whole, when you come to the comma: but let the short pause or stop the second has £340 more than of the whole, the third has which you make, be a total cessation of the voice. £300 more than of the whole, the fourth has £400 more than

Diligence, industry, and proper improvement of time, are of the whole. What is the value of the estate? Ans. £4800. material duties of the young.

Prob. 10. What is that number which is as much less than He is religious, generous, just, charitable and humane. 500, as a fifth part of it is greater than 40 ?

Ans. 450. By wisdom, by art, by the united strength of a civil community, Prob. 11. There are two numbers whose difference is 40, and men have been enabled to subdue the whole race of lions, bears which are to each other as 6 to 5. What are the numbers and serpents.

Ans. 240 and 200. The genuine glory, the proper distinction of the rational species, Prob. 12. Suppose two coaches to start at the same hour, one arises from the perfection of the mental powers. from London for Glasgow, and the other from Glasgow for Lon

Courage is apt to be fierce, and strength is often exerted in acts don, the former travelling 10and the latter 93 miles per hour:

of oppression. Where will they meet, the distance between the two cities being laws, to pursue right measures, to correct power, to protect

weak,

Wisdom is the associate of justice. It assists her to form equal 400 miles Ans. 210 miles from London.

ness, and to unite individuals in a common interest and general Prob. 13. Suppose every thing to be as in the last question, welfare. except that the coach from Glasgow starts two hours earlier than

Heroes may kill tyrants, but it is wisdom and laws that prevent the other; where will they meet? Ans. 2000 miles from London. tyranny and oppression. Prob. 14. A dealer purchases 60 yards of cloth for 301.; and

27. When a note of interrogation occurs at the end of a senby selling one part of it at 128., another, twice

as great, at 148., tence, the parts, and even the words, of the sentence separated by: and the rest at 10s. per yard, he gains 81. How many yards commas, should each be read like a question. were in the several lots ?

Ans. 16, 32, and 12. Prob. 15. Suppose two dealers each annually to double his

Examples capital, except an expenditure of 1001.; and, that at the end of three years, the capital of one is found to be doubled, while the Did you read as correctly, speak as properly, or beha'y ? as well other ħas only half what he had at first; how much had each to as James ? commence with ? Ans. 1161. 13s. 4d. and 931. 6s. 8d. Art thou the Thracian robber, of whose exploits I have heard

80 much Prob. 16. If a person each year double his capital except an expenditure of 3001. the first year, 4001. the next year, and 5001.

Who shall separate us from the love of Christ? shall. tribulathe third, and at the end of three years be found to be worth tion, or distress, or persecution, or famine, or peril or swird ? 55001., what was his original capital?

How are the dead raised up, and with what body do they come ? Ans. 10007,

For what is our hope, our joy, or crown of rejoicing ? Prob. 17, A father's age is now treble of his son's, while five Have you not misemployed your time, wasted your talents, years ago it was quadruple : what are their present ages? and passed your life in idleness and vice?

Ans. 45 and 15 years. Have you been taught any thing of the nature, strukture and Prob. 18. Divide 10001. between A, B, and C, giving A 1002. laws of the body which you inhabit? more, and B 501. less, than C. Ans. A's share 4161. 13s. 4d.;

Were you ever made to understand the operation of diet, air, B's 2661. 13s. 4d.; and C's 3161. 135. 4d.

exercise, and modes of dress, upon the human frame? Prob. 19. A spirit merchant finds that if he add 10 gallons to a that preceding a period, with the falling inflection of tevi vsise.

28. Sometimes the word preceding a comma, is to be re a i like Eask of brandy, the mixture will be worth 21s. per gallon; but that if he add ten gallons more, the value will be reduced to 18s. How many gallons were in the cask ?

Ans, 50,

Examples Prob. 20. Find a number, such that if it be divided successively

It is said by unbelievers that religion is dull, unsocial, urs by 2, 3, 4, 5, 6, 7, 8, 9, and 10, half the sum of the first four charitable, enthusiastic, a damper of human joy, a morose intruder quətients increased by 20 shall be equal to the sum of the remain- upon human pleasure. ing five.

Ans. 5040. Nothing is more erroneous, unjust, or untrue, than the state

ment in the preceding sentence. Prob. 21. Find two numbers differing by 6, and such that

Perhaps you have mistaken sobriety for dulness, equanimity three times the less may exceed twice the greater by 7.

for moroseness, disinclination to bad company for aversion to Ans, 25 and 19.

society, abhorrence of vice for uncharitableness, and piety for Prob. 22. Find a number such, that if it be increased succes- enthusiasm. sively by 1, 2, and 3, the sum of one-half of the first result and Henry was careless, thoughtless, heedless, and inattentive. one third of the second shall exceed one-fourth of the third by 8. This is partial, unjust, uncharitable, and iniquitous.

Ans. 13. The history of religion is ransacked by its enemies, for instan

ces 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. PUNCTUATION.

29. Sometimes the word preceding a comma, is to be read liks that preceding an exclamation.

THE COMMA.

[ocr errors]

Examples.

How can you destroy those beautiful things which your father 22. The mark used for the comma is a round dot with a small procured for you! that besytiful top, those polished marbles, curve appended to it, turning from right to left.

that excellent ball, and that beautifully painted kite, oh how can 23. When you come to a comma in reading, you must, in you destroy them, and expect that he will buy you new ones! general, make a short pause or stop, so long as would enable you

How canst thou renounce the boundless store of charms that

Nature to her votary yields ! the warbling woodland, the resound24. The last word before a comma is most frequently read with ing shore, the pomp of groves, the garniture of fields, all that the the falling inflection of the voice.

genial ray of morning gilds, and all that echoes to the song of

even, all that the mountain's sheltering bosom shields, and all the Examples.

dread magnificence of heaven, how canst thou renounce them and 25. In reading, when you come to a comma, you must keep i hope to be forgiven !

to count one.

Oh winter! ruler of the inverted year! thy scattered hair with It is now two hundred years since attempts have been made to sleetlike ashes filled, thy breath congealed upon thy lips, thy civilize the North American savage. cheeks fringed with a beard made white with other snows than Doing well has something more in it than the fulfilling of a those of age, thy forehead wrapped in clouds, a leafless branch duty. thy sceptre, and thy throne a sliding car, indebted to no wheels, You will expect me to say something of the lonely records of but urged by storms along its slippery way, I love thee, all un- the former races that inhabited this country. lovely as thou seemest, and dreaded as thou art !

There is no virtue without a characteristic beauty to make it Lovely art thou, 0 Peace ! and lovely are thy children, and particularly loved by the good, and to make the bad ashamed of lovely are the prints of thy footsteps in the green valleys. their neglect of it. 30. Semetimes the word preceding a comma and other marks, made up to us by self-approval, and the consideration of what

A sacrifice was never get offered to a principle, that was not is to be read without any pause or inflection of the voice.

our degradation would have been bad we done otherwise.

The succession and contrast of the seasons give scope to that Examples.

care and foresight, vigilance and industry, which are essential to You see, my son, this wide and large firmament over our the dignity and enjoyment of human beings, whose happiness is heads, where the sun and moon, and all the stars appear in their connected with the exertion of their faculties.

A lion of the largest size measures from eight to nine feet from turns.

the muzzle to the origin of the tail which last is of itself about Therefore, my child, fear and worship, and love God. He that can read as well as you can, James, need not be four feet long. The height of the larger specimens is four or five

feet. ashamed to read aloud. I consider it my duty, at this time, to tell

A benison upon thee gentle huntsman. Whose towers are that you have you,

these that overlook the wood ? done something of which you ought to be ashamed. The Spaniards, while thus employed, were surrounded by

The incidents of the last few days have been such as will many of the natives, who gazed, in silent admiration, upon probably never again be witnessed by the people of America and actions which they could not comprehend, and of which they did such as were never before witnessed by any nation under braven. not foresee the consequences. The dress of the Spaniards, the

To the memory of André his country has erected the most magwhiteness of their skins, their beards, their arms, appeared strange nificent monument, and bestowed on his family the highest

honours and most liberal rewards. and surprising

To the memory of Hale not Yet, fair as thou art, thou shunnest to glide, beautiful stream! a stone has been erected and the traveller asks in vain for the by the village side, but windest away from the haunts of men, to place of his long sleep. 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

CORRESPONDENCE. always be a blessing in the air, and flowers and fruits on the earth,

MUTUAL INSTRUCTION CLASSES. Share with you! said his father-so the industrious must lose his labour to feed the idle.

SIR,-This is an old mode of instruction, but one too much 31. Sometimes the pause of a comma must be made where neglected. Its importance will be disputed by few, affording as it there is no pause in the book. Spaces are left in the following ening the perceptive faculties by wholesome and stimulating com

does such facilities for acquiring and exercising knowledge, quick. sentences where the pause is proper to be made.

petition, and enlarging and correcting our views by association. It

has struck me that the success and prosperity of the “ POPULAR Ecamples.

EDUCATOR" would be even more permanently established, and

the students themselves more interested and benefited by its lessons, The Europeans were hardly less amazed at the scene now set if classes of this description could be generally established in conbefore them.

nexion with this publication. The students of the “ POPULAR Their black hair long and curled floated upon their shoulders EDUCATOR" do not sufficiently know their brother students. or was bound in tresses around their head.

Many of them in the same town may be individually studying the Persons of reflection and sensibility contemplate with interest sanie subject, unknown to each other, who would be very glad to

meet for the purpose of mutual instruction. Difficulties frequently the scenes of nature,

occur to one person, which are easily removed by another; and, The succession and contrast of the seasons give scope to where necessary, an instructor for the whole class might be engaged care and foresight diligence and industry which are essential to at a trifling expense to each, instead of paying exorbitantly for the dignity and enjoyment of human beings.

private lessons. The eye is sweetly rested on every object to which it turns.

Now, Sir, if a plan could be devised for collecting together all It is grateful to perceive how widely yet chastely nature the students belonging to a particular locality, little difficulty hath mixed her colours and painted her robe.

would be experienced by them in subdividing themselves into classes, Winter compensates for the want of attractions abroad by according to their different studies. The machinery required for fireside delights and homefelt joys. In all this interchange this would be rery simple. You would have to exercise your and variety

we find reason to acknowledge the wise and indulgence by sparing a corner of the P. E. for it; and any benevolent care of the God of seasons.

person wishing to collect together the students of the locality in

wbich he resides, would merely have to send you his name and 32. The pupil may read the following sentences ; but before address for insertion, thus :-NORWICH, W.B. "News" office. reading them, he should tell after what word the pause should be Any students in the place referred to, seeing this announcement, made. The pause is not printed in the sentences, but it must be could at once communicate with him, and he might immediately made when reading them. And here it may be observed, that bring them together by calling a meeting, when arrangements the comma is more frequently used to point out the grammatical could be made for the formation of classes. divisions of a sentence, than to indicate a rest or cessation of the believing that many students here would at once embrace the

I should like that this experiment were made in Norwich, voice. Good reading depends much upon skill and judgment in opportunity. If such persons, therefore, will send to the address. I making those pauses which the meaning of the sentence dictates, have just given as an example (viz. W. B."News" office, Norwich), but which are not noted in the book; and the sooner the pupil is I shall feel most happy in calling them together, and assisting in taught to make them, with proper discrimination, the surer and the necessary arrangements. the more rapid will be his progress in the art of reading.

Should this effort prove effectual for Norwich, of course many

other places could do the same. A system of National Adult Esamples.

Education of no mean character would thus be established throughThe golden head that was wont to rise at that part of the table men of the present day, would be in a great measure realised,

out the kingdom; and the anxious desire of many great and good was now wanting.

through the simple instrumentality of the POPULAR EDUCATOR and For even though absent from school I shall get the lesson. the exertions of a few spirited individuals in each locality, For even though dead I will control the trophies of the capitol, Trusting that you will lend your influence in aid of this move

« ΠροηγούμενηΣυνέχεια »