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possessed it to the full. The dialect which grows Carlyle. The style of this great man has to reup among boys at their games and quarrels is ex-ceive the malignant attacks of those who cannot cellent. Their compositions are painfully con- reach the level of his thought. But Carlyle is strained. I find, when I overhear boys bantering above criticism. Grant that the style is the man, each other with sarcasms, that they need not my that, consequently, Carlyle is an odd man. exposition of principles and enforcement of rules then? An odd man! in order to put language to its proper use. In The rhetoric lesson should reveal the real rhetor, their exercises and in their conversation with me the utterer of thought; and unveil the machinery their speech is emasculated, and evidently needs with which the false and affected one makes his grammatical treatment. loud noise. The virtue of a style will not be tested This natural language of children should be ac- by comparison with a production of the past. Let knowledged by the teacher as more proper than us be ready to welcome a manner of speech more that to which he is trying to train them. They dreadful than even that of Carlyle, whenever such have their own jus, if not their norma, loquendi. becomes the appointed embodiment of a new charBut the normal discipline is not to be esteemed the acter. Peculiarities come not within the province less, because it is seen not to reach the root of the of criticism; nor is precedent its law. The best matter. The attitude of the teacher towards the language is not that which comes nearest to Addipupil will, however, be modified by a recognition son's, but that which best expresses the soul. Let of the higher element. Submission to grammati- us ever expect new thoughts made known to us cal law within its jurisdiction, is indispensable. with new tongues. At the same time, the thoughtful teacher will observe a certain generosity with regard to the application of rules. We would not esteem a friend the less for occasional neglect of the principles of grammar, nor would we deem it our duty to chide him for such faults. Let such a relation of friend-drassed to A. J. MANCHESTER, Providence. ship prevail to some degree with the pupil. Train him as well to forgive your inaccuracies as to strive against them himself. The developing youth will surely see these things one day in their true light, and probably remember his pedagogue as a pedantic, one-sided man. One need never condescend to talk with boys on their own level. One will do well to try to come up to it sometimes.

QUESTIONS FOR

Written Examinations.

COMMUNICATIONS for this Department should be ad

ARITHMETIC.

1. What is the quotient of 1-2 of 3-8 of 89 1-5 divided by 15-16?

2. Add three thousand fifty and one one-hundredth, and fourteen and fifty-five thousand and one hundred thousandths, subtract from this sum, one and one-ten-thousandth, and divide by ten and ten hundredths.

3. If 1 1-2 bushels of wheat are worth 21-4 bushels of rye, and 1 bushel of rye is worth 2 7-8 bushels of oats, how many bushels of wheat will pay for 75 bushels of oats?

4. What is the interest of $245.73 for 1 year 5 months 23 days at 6 per cent. ?

The student who has progressed to the study of rhetoric and the criticism of authors may at the outset be taught a higher rule than that of elegance. Buffon's epigrammatic saying, -"The style is the man," -needs no modification. Cultivation of style should be self-culture. Only to the mere critic does the style possess interest of itself. The developed man comes at last to under- 5. A merchant bought 1002 bbls. of flour on 3 stand this, and has his favorite authors. That months credit, at $5 per bbl. After keeping them would be a rare devotee to the neat and pretty, one month, he sold one-third of them at $5.50 per who should spend the literary leisure of his life in barrel on a credit of 4 months, and the remainder seeking and enjoying the elegant writers simply at $5.75 per bbl. on a credit of 6 months. Did he for their elegance. Dr. Johnson recommends that gain or lose? and how much, supposing that he one "devote his days and nights to the study of gets his notes discounted at the bank at 6 per ct. ? Addison," if he wishes to become master of the 6. How long will it take $500 to earn $37.50 at English language. Goethe advises a clear and 6 per cent. interest? sure comprehension of the thought, as the condi

7. 16-19 of 315 5-17 is 2-3 of what number?

8.

tion of the like success. The difference is that be- In a certain orchard 1-6 of the trees are aptween a pedant and a seer. The recommendation ple trees, 1-8 are pear trees, 2-5 are peach trees, in the rhetorics of authors suitable to the young, 1-10 are quince trees and the remainder plum trees. seems paltry, even to disgust, as soon as an en There are 25 plum trees. How many trees in the thusiasm has taken possession of the mind. The orchard? teacher's zeal, and not his enforcement of rhetorical canons, is effectual and contagious. No man was ever yet zealous for a negative excellence.

The arch-heretic and most notable despiser of models at the present day is, of course, Thomas

9. If twelve men can dig a ditch 118 feet long 6 feet wide and 4 feet deep in 4 days, how many days will it take 9 men to dig one 135 feet long 5 feet wide and 6 feet deep?

10. Reduce 3-5 + 4-7 — (1-6 x 2-3 7-8.)

GRAMMAR.

1. Name the principal classes of adverbs.

2. Give examples of the several classes of ad

jectives.

Mathematics.

COMMUNICATIONS for this Department, if relating to the higher branches, should be addressed to J. M. Ross,

3. Compare the adjectives bad, happy, infinite, Lonsdale; otherwise to N. W. DEMUNN, Providence. far.

4. Write a sentence containing a common noun, an active intransitive verb, and an objective case. 5. Write a sentence containing an active transitive verb, and a preposition.

6. What is a predicate?

For the Schoolmaster.

Solutions of the Problem in Equations in the
November Number.

MR. EDITOR: -Your correspondent, P. G., has

7. Write a sentence containing a modified sub-given us a singular solution (?) of the equations, ject, and a modified predicate.

x2+y=a, and y2+x=b. x2+ y2 + x + y −8 = 0,

8. Write a sentence containing a verb, an ad-(1), is a single equation, containing two unknown jective in the superlative degree, and an adverb in the comparative degree.

9. Give the plural of knife, goose, index, army,

Vortex.

10. Give the passive voice, indicative mode, future perfect tense of the verb to teach.

GEOGRAPHY AND HISTORY.

1. On what hemisphere and continent, in what grand division, country, state, county and zone do you reside?

quantities; yet he professes to solve it completely,
without combining it with any other equation, ex-
cepting such as may be obtained by transformations
of the original. But suppose the values which he
has found for x and y will satisfy equation (1), we
have no reason to infer that they will also satisfy
the given equations. For let the problem be stat-
ed thus.

(x + y = a}
xy, or thus, { x2+x= a 2

{ y2 + y = b $ ;

in either case we shall get, by adding,

2. What are the principal islands in and around 2++x+y=a+b=s•••x2+y2+x+y—s=9, (1) ; Europe? thence the solution is effected in the same manner 3. What is the torrid zone and what countries as before, and we find the same values for x and y. But, in these cases, the results are, evidently, in

does it include?

4. How many degrees of latitude does New correct; indeed, why should they be otherwise? England include ?

5. Name the Barbary States.

6. Mention the principal causes that led to the American Revolution.

7. How long was it from the discovery of Amer

ica to the settlement of Plymouth?

8. State what you know of the Stamp Act.

9. What was the act called the Boston Port

Bill?

Let us examine, for a moment, the solution of equation (1). The author finds the equation,

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the second member of this equation may be any 10. Why were the early settlements of this assignable quantity. Combining (3) with (1), we country called colonies?

SCIENCE OF COMMON THINGS.

What is snow?

What is the cause of snow?

Why does snow fall in winter time?

What is the cause of sleet?

What is the use of snow?

Does snow keep the earth warm and why? Why are some mountains always covered with snow?

Why is snow white?

What is hail?

What are essential to cause hail?

Why does rain fall in drops?

Are rain-drops always of the same size?
What causes the difference?

What are the uses of rain?

Where do all rivers really have their sources?
What are the uses of rivers?

Persons often say "it is too cold to rain." Why?
What are the swift and mighty rain-carriers ?

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[The criticisms of our correspondent on the solution presented by "P. G." in the last number, are worthy of attention.

It will be remembered that we suggested that neither of the solutions was complete, and might have added that, we suspect the complete solution by quadratics to be an impossibility. We were unwillingly forced to this conclusion years ago. We do not wish, however, to modify our remarks of commendation in the last number. Even though he has not generalized the solution, nor effected it by the ordinary processes of elimination, the author is worthy of much credit for having given us a partial solution without resorting to higher equations. Mr. L. used the particular numbers 13 and 19, for a and b, and brought out the answers x=3,

y=4, for the first values, the second values not verifying; but there was no guess work, nor resort to the methods of higher equations; but he finally produces two quadratics, y2+ y = 20, x2 + x = 12. It is due to "P. G." to state, that in a subsequent communication he says: "I have discovered that I have not after all generalized the solution of the problem, but only Mr. L.'s particular solution, and the old problem, therefore, remains unsolved, at least as to generality."-ED.]

NOTE ON THE SOLUTION OF BIQUADRATICS.

The general equation of the fourth degree, when freed of its second term, has the form,

x2+ ax2+ bx + c = 0, or x1 + ax2 -- ―bx-c; if to both members of this equation we add ha, h being arbitrary, there results,

x++ (a + h) x2=hx2-bx-c; or, completing the square, and evolving, a + h (a + h)2 - 4c)}

2

hx2 bx +

4

;

now, since may be disposed of in such a way as to suit our convenience, let us assume the equa

tion,

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(a+h)2—4c)
4

This equation reduces to,

h3 +2ah2 + (a2 — 4c) h — b2 = 0,

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....

(1)
(2)
(3)

2, 9, 36, 100, 225, &c., (4) The first is the natural series; the second, the series of cube numbers, and the one whose sum is required; the third is the triangular series, derived from the first by adding to each term in the first all the terms preceding; the fourth series is derived from the third by squaring each term, or from the second by adding to each term in the second all the terms preceding; hence any term in the fourth series is the answer to the question of that number of terms, that is, the sum of any number of terms of the series of cube numbers 1, 8, 27, 64, &c., is equal to the square of the sum of the natural series, or roots of these numbers; hence (18 + 28 +33 + 43 + ........) = (1 + 2 +3 + 4 +..........)?

S.or, algebraically expressed,

13+23+33 +48 + n3 =
....

After solving this cubic, we have only to reduce a hence the following simple quadratic, and the solution is complete.

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Sn (n + 1)) 2

2

RULE: Multiply the number of terms by the uumber of terms plus 1, and the square of half the product will be the sum required.

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when a is. By factoring, the expression may take so as to find the sum of n terms by means of the usual formula for summing the squares of the natural numbers."

the form:

x(x+1)(x-1)
2.3

If x is any whole number divisible by 3, the first factor will be divisible by 3; and then, if x is an odd number (as the three factors represent consecutive numbers in the order, [x-1] x [x + 1],) either of the other factors, (x + 1), or (x-1), will be even, and hence divisible by 2; but if x is an even number divisible by 3, the other factors will be odd, and hence not divisible by 2; but in that case, since x is an even number divisible by an odd number, the quotient of x divided by 3 is an even num ber, and hence divisible by 2. But if x is not a multiple of 3, one of the other factors is, being 1 more, or 1 less than ; and then, if a is even, the first factor is divisible by 2, and if odd, then the one of the other factors, not a multiple of 3, since both are even if x is odd; or the same even factor divisible by 3 is also divisible by 2, as above.

J. M. R.

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18+ 19 +34 + 53 + 76 + &c., 1 + 4 + 9 + 16 + 25 + 36 + &c., 0+ 4 + 10 + 18 + 28 +49 + &c., 0 +1 + 4 + 9 + 16 + 25+ &c., The third series is obtained by subtracting the se0+3+ 6+ 9 + 12 + 15 + &c., cond series of square numbers from the first, or given series, in the manner indicated by the operation. The fourth is the series of square numbers beginning with 0, for convenience, and is subtracted from the third, giving the fifth, which is an arithmetical series; hence

18+ 19+ 34 + 53 + 76 + &c. = 2(1+49 + 16 + 25 +36 + &c.) — n3 +(0 + 3 + 6 + 9 + 12 + 15 + &c.).

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By examining the operation above we see that we have separated the terms respectively thus: 1+ (4+1+3) + (9+4+6) + (16+9+9) + &c. = 12 + (22+12+3) + (32+22+6) + (42+3+9) + &c.,

from which we discover the law of the series, which is that, any term

= n2 + (n − 1) 2+ 3 (n−1) = n (2n + 1) — 2, n being the number of the term.

Natural Science.

J. M, R.

COMMUNICATIONS for this Department should be addressed to I. F. CADY, Warren.

For the Schoolmaster.

The Microscope.

from time to time, or to obtain objectives and eyepieces of varying powers, they can always be secured of such size and structure as precisely to adapt them to all requisite portions of the apparatus already in one's possession.

An instrument intended for anything beyond quite common observation can scarcely be procured for less than from thirty to sixty dollars. Ten or fifteen dollars will procure a French or German microscope of sufficient excellence to afford entertainment and instruction enough to make it a capital investment; but if it is desired to advance from the examination of the more common microscopic objects to those of minuteness and delicacy of structure, the purchaser will find it difficult to stop short of an expenditure of less than from fifty to one hundred dollars. The Messrs. Grunow, of New Haven, manufacture a very satisfactory instrumert, which they supply with two objectives and two eye-pieces, with magnifying powers, by combination, of forty, seventy, one hundred and eighty, and three hundred and fifty diameters, for sixty dollars. In this the adjustments are sufficiently delicate to admit of the use of objectives of any desirable power.

The cheapest instrument made by the Messrs. Grunow is their "educational" microscope. This is furnished with the same objectives and eyepieces as the one just described, but the adjustWe have already noticed the fact that the com- ments of the stand are less delicate. Its price is pound, achromatic microscope is indebted to Eng- forty dollars. One of the sixty dollar microscopes lish ingenuity and invention for some of its most (No. 3) made by the Messrs. Grunow for the writer, valuable qualities. Since the commencement of about one year ago, together with accessory appathe improvements based upon the discoveries of ratus, which increased the price to a little upwards Mr. Lister, -whom we were made, by an error of of eighty dollars, has proved to be very satisfactothe types in our last article, to misname Leister,-ry in its use, paying, in our humble opinion, as the English instruments have steadily maintained high a per centage upon the amount invested as their superiority over those manufactured in con- the ordinary courses of commercial pursuit are tinental Europe. The German and French micro- wont to yield. This is not, however, the appropriscopes of the same power are much cheaper than ate place to discuss the preponderance of intellecthe best English and American; but by the most tual and material wealth. In respect to this we accomplished observers they are uniformly pro- can hardly expect harmony of opinion. The posnounced inferior for the purposes of nice and ac- session of neither is to be despised. curate observation. Of the English instruments It is a very common impression, but a very inthose manufactured by Messrs. Smith and Beck correct one, that the practical value of a microare regarded as among the most perfect. Their scope corresponds directly with the degree of its "students'" and "educational" microscopes are magnifying power. A magnifying power of from those most commonly offered for sale. They can five hundred to twelve hundred diameters is supbe obtained in this country from any of the exten-posed capable of giving a satisfactory view of any sive dealers in optical instruments, among others, object that we may desire to examine; that is, one we are glad to know, for the sake of the lovers of more satisfactory than can be obtained by the use science in our own State, from Messrs. Gorham, of a lower power. This is a great mistake. The Co. & Brown's, of Providence. The glasses man- number of objects which require a higher magniufactured by Messrs. Spencer & Eaton, and espe- fying power than three hundred and fifty diamecially by Mr. Robert B. Tolles, of Canastota, N. ters for their most satisfactory examination is Y., and also those of Messrs. J. & W. Grunow & comparatively small, while multitudes are more Co. of New Haven, Connecticut, are said to be successfully observed under a power not exceeding fully equal in excellence to those of the best Eng- seventy diameters than by any higher power. For lish manufacturers. There is one advantage in ob- instance, suppose we wish to examine the structaining instruments of American makers, viz: in ture of the foot of a fly. A good objective with a case of a desire to procure accessory apparatus magnifying power of forty diameters will give a

simultaneous view of the foot together with a por- dition of objects for observation. Wherever it is tion of the leg of about one eighth of an inch in possible it is necessary that they shall be rendered length. Every portion will be distinctly brought quite transparent. Opaque objects can be viewed to view. The foot will be found to be terminated only by light reflected from their surface, and hence by two claws having much the appearance of the they require illumination by means of condensing horns of an ox, and having an apparent length of lenses and reflectors. When an object can be renfrom one-fourth to one-half of an inch. The an- dered transparent so as to permit examination by kle will be found to consist of several sections transmitted light, the result is rendered much more nicely united, and in their connection exhibiting a satisfactory. A very large proportion of the objects structure very much resembling a portion of the which it is desirable to examine with the microvertebral column of a fish. The whole will be scope are susceptible of being rendered sufficiently seen to be covered with stout, sharp spines, thick- transparent to admit of very perfect examination ly set, and directed downward, which together with when mounted in Canada balsam or some other the terminal claws, will sufficiently explain the transparent substance, upon slips of glass. reason of the tickling sensation produced by flies The preparation of objects for the microscope is in their pedestrian excursions across the nose and a delicate task, requiring considerable patience face of some day-light dreamer. Hence, too, and ingenuity. Skill and success in this work can more seriously-the excessive annoyance which be acquired only by persevering effort. The exthey occasion at the pillow of the sick. At the perimenter, however, is constantly rewarded by base of the terminal claws we shall discover two new and surprising discoveries, and he is constantcushion-like pads. These constantly secrete the ly allured onward by the mines of wealth everyglutinous substance by whose adhesive power the where displayed in the world of novelty and beaufly is enabled to traverse with such enviable facility opened to his view by the wonderful instrument ty the vertical window pane, or to sustain himself that so extends his range of observation. It would with ease upon the horizontal ceiling overhead. not, perhaps, be easy to prescribe a more rational All these parts appear distinct, and are seen at a employment than that afforded by the use of the single view. Now if we substitute a power of sevmicroscope, for a portion of our leisure hours, or enty diameters for that of forty we make no new one that is better calculated to make us acquainted discovery. The only effect is to increase the ap- with some of the most interesting objects which parent size of the parts. It is of about the same the Creator "has made beautiful in their time," advantage that it would be to increase the size of or to inspire us with emotions of wonder, love and the horse to that of an elephant for the sake of ob- veneration in contemplating the works of Him taining a more satisfactory view of his structure. whose alone are the treasures of knowledge, power Not a single new organ presents itself for observa- and wisdom. tion; not even an additional hair makes its appearance, or is seen wth greater distinctness. If we now employ an objective having a magnifying SUCCESS prompts to exertion, and habit facilipower of three hundred and fifty diameters the tates success. Habit also gives promptness, and foot and claws will extend through the entire field the soul of dispatch is decision. One may write a of view, while, owing to its thickness, the upper book or paint a picture, while another is deliberatand lower portions of a single claw cannot, at the ing about a plan or a title-page. The more we do same time, be seen with equal distinctness, but the more we CAN do. If we go forward with spirit will require a variation of the focal distance in or- and confidence, we shall succeed. der to bring the different portions distinctly into idle half the time, and he who does nothing, renview. Nor can we very mnch wonder at this fact ders himself incapable of doing anything, while when we consider that the objects under examina- our capacity is invigorated by occasions of necestion are presented to us with an apparent magni- sity. Our expenditure of intellectual wealth makes tude of more than one hundred and twenty thousand us rich, and we acquire ideas by imparting them. times their actual size. If we wish to examine the -HAZLITT'S TABLE TALK. corpuscles contained in the blood of animals, or the delicate flutings of a single scale from the wing A FELLOW once handed a shilling he had found of a butterfly, or the minute animalculæ contain- in a room which he was cleaning, to the owner, ed in a drop of some vegetable infusion, or the ex- and he was told to "keep it for his honesty." A quisitely fashioned silicious shells found in peat few days afterwards, he was asked if he had found formations or in the soft deposits at the bottom of a valuable ring, and he said he had, but had kept tranquil pools in a mountain rivulet, we shall find it for his honesty! appropriate use for a magnifying power of three hundred and fifty diameters. and shall derive great satisfaction from its employment.

I. F. C.

The best are

THE COST OF KNOWLEDGE.-In 1825 the king of Sardinia decreed that no one should be allowed

Another natural mistake respecting the use of to read and write who was not in possession of the microscope, exists in regard to the proper con- 1500 livres - about $200.

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