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This large square and the three strips may be ar
Fig. 4. ranged as in Fig. 4; viz., a strip and a half on each of two adjacent sides of the large square. This * Strip will be an exact square if we fill out the corner C,
C. which contains a square, each side of which is one
Strip. and a half times the width of a strip (24 in. X 11 1 in.) Adding this emall
square ( (in.) 24 sq. in.), we have 293 sq. in. + 2 sq. in. Large Square. 1142 sq. in. = area of the square, Fig. 4, completed. Then, „1849 sq. in.
length of a side of the square, or the live D F.
F Comparing this with Fig. 3, we see it wants half the thickness of the plank, (24 in. = ? 14 in.), of being the external height of the box. Then, 4 in. + 1} in. =
12 in. = external height of the box; 12 in. x 2 24 in. external width of the box, and 24 in. X 2 = 48 in. external length of the box. Subtract from each of these dimensions twice the thickness of the plank, and we have 12 in. — 5 in. = 7 in. internal height of box; 24 in. — 5 in. = 19 in. = internal width of the box; 48 in. — 5 in. — 43 in.. internal length of the box, and the product of the internal dimensions 5719 cubic inches the contents of the box.
Messrs. EDITORS:— The following is the method I have been accustomed to employ, in explaining the first of the examples for which you requested solutions :
Having procured six square pieces of board of equal size, I show my scholars the only manner in which they can be put together to form a cubical box. This being done, I show them that there are two small corners that will need filling, in order that the box may be perfect; these corners are as large square as the board of which the box is made is thick. I also show them that each side of the box will be as much longer than the pieces of which it is made, as the thickness of the board from which they are taken. Having done this, I am ready for the solution of the question. 495 sq. ft. 7146
sq. in.; and (1.5 in.) ? X 2=4.5 sq. in., required for the 2 small corners. Then, 7146 - 4.5=7141.5, the area remaining for the six square pieces; and 7141.5 ; 6 1190.25, area of one of these pieces. Then, V1190.25 = 34.5 in., side of one piece; and 34:5 + 1.5 36 in., length of one side of the box.
J. M. L.
If a teacher wishes to solve these questions by arithmetical processes, let him not attempt to analyze them, but simply apply the Golden Rule of Double Position. It is manifest, in the first question, that 3 feet on the outside of a cube will give 54 feet of surface. From this subtract the edges of the top and bottom boards, 24 feet multiplied by 1 of a foot, and it is reduced to 51 feet. Next subtract the ends of two sides, 11 feet multiplied by f, and it is reduced to 495 feet, which is the amount of board given; and my first “ position,” or guess, of three feet, is right. Can anything be more natural and simple ?
Take the other box, whose dimensions are to be in the proportion of 1, 2, and 4. Suppose the smallest dimension to be one foot. Then the surface will be 28 feet. But from this subtract the edges of the larger sides, 24 feet multiplied by of a foot, and it is reduced to 23 feet. Subtract the four ends of the other two sides, each being seven inches by 21; that is, subtract 70 square inches, and it leaves, as the size of the boards, 2247 feet; which is the amount given ; and so the first position does not need correction. **
No. 1.- Divide the whole external surface of the box into 6 large squares, 24 strips, and 24 small squares, as in Fig. 1, above. Then, of the two strips that meet at each edge, one must be made by the thickness of the board, and should therefore be left out of account, as taking no part of the given surface of the board; and of the three small squares that meet at each corner, two must be made by the thickness of the board, and should be left out of account. There remain the 6 large squares, half the strips (=12), and one third of the small squares (=8). But 6 large squares, 12 strips, and 6 small squares, can be arranged as in Fig. 2, so as to compose 6 squares. These take up the whole given surface of the board, except so much as is required for the 2 small squares remaining, of which each side is equal to the thickness of the board, 14 in. = } ft. But (ft.) ? X 2
64 sq. ft.; and 495 - A=4984 Then, 4911 = 6=811=*; and = ** ft., length of A B in Figure 2. But A B is shorter by the width of one strip, or j ft., than a side of the box; and, therefore, 2+}= 3 ft., Ans.
No. 2. Divide the whole surface of the box into 28 large squares, 112 strips, and 112 small squares, as shown in Fig. 3, above, or explained in connection with it. Upon examination, it will be readily seen that 28 strips and 48 small squares must be left out of account, as made by the thickness of the plank. There remain the 28 large squares, 84 strips, and 64 small squares. But 1 large square, 3 strips, and 21 small squares, compose a square like that in Fig. 4, (since the corner C is evidently equal to 24 small squares); so that 28 of these composite squares take up the 28 large squares, the 84 strips, and 63 of the small squares; that is, the whole given surface of the plank, except what is required for 1 small square, (24 in.,)
= 64 sq. in. But, 2247 sq. ft. 3242 sq. in.; and 3242 — 61 = 3235 sq. in., the contents of the 28 composite squares. Then, 32353 ;-28=14143 132 sq. in., the contents of one composite square; and 1949
4 in., length of D F, which is shorter by the width of half a strip, or 14 in., than the external height of the box. Therefore, 4+ 14 = 4 = 12 in., external height of the box ; 24 in.= external width; and 48 in. = external length. The internal dimensions are less by 5 inches, twice the thickness of the plank, than the corresponding external dimensions; that is, internal height = 7 in., width='19 in., and length = 43 in. The product of these dimensions gives 5719 cubic inches, the contents of the box. X. Y.
In giving the solutions of X. Y., which do not differ essentially from those of S., we have adapted them to the Figures of S., to avoid a needless multiplication of wood-cuts. We find ourselves obliged, from want of room, to defer the various answers we have received to the remaining question proposed in January, till our next number.
QUESTIONS FOR SOLUTION. Messrs. Editors:- One of my school-boys, who has a taste for mechanical pursuits, and who is now engaged in constructing a turning.lathe, brings me the following problems :
[Question 1.] Given two fixed pulleys whose diameters are respectively 4 inches and 20 inches, and whose centres are 48 inches apart. Required the length of a band which shall pass tightly around the pulleys.
[QUESTION 2.) If the diameter of the smaller pulley be increased to 6 inches, and the centres of the two pulleys remain at the same distance apart as before, what must be the diameter of the larger pulley, in order that the same band may be used as in the first case ?
D. B. H.
[QUESTION 3.] I tried a freezing experiment the other day after the following manner. Taking a glass bulb with a long stem, I filled the bulb with water and placed it in a mixture of snow and salt. Gradually adding the mixture above, I covered the bulb; and after some minutes took it out, and found the water solid and the bulb unbroken. The water bad risen in the stem several inches.
I afterwards tried the experiment, immersing the bulb wholly in the freezing mixture. The water rose to a less height than before; and on taking out the bulb, found the upper half filled with solid ice, while the lower balf was shattered. What was the cause of the unfortunate result of the second experiment?
A READER. [QUESTION 4.] Why cannot the arithmetical questions answered in the present number be performed by the Rule of Position, unless we happen to be so fortunate as guess the right answer?
[QUESTION 5.] Let me now take my turn at playing Sphynx, and ask, Is it of material importance to have the school walls nearly directed to the cardical points of the compass ? I have a fancy that it is; that it materially assists a child if he can sit facing the north while studying a map; although for a country south of him, I should rather he should face south and bold the map upside down. At all events, let the house walls be, when practicable, nearly N. E. S. W.
[QUESTION 6.] Given the diameter of a drum wheel, the distance between the centre of the drum and that of its pulley, and also the length of the band ; to find the diameter of the pulley.
INTELLIGENCE AND MISCELLANY, READING OTHER PEOPLE'S LETTERS. - We know that our readers will thank us for allowing them to peep into the following letter from the accomplished authoress of the work on Physical Geography noticed in our last number. How many associations, from historic incident, ivy-clad ruins, and the tearful fiction of the Wizard of the North, cluster around the spot from which it is dated. - KENILWORTH, Warwickshire, 15th January, 1856. SIR:- Permit me to express my thanks to you for obligingly sending me a copy of your edition of my small work on Physical Geography, It gives me unfeigned satisfaction that this little volume should have been ushered into its new sphere in the United States, in so pleasing a form, and under the erlitorship of one who appears so fully to enter into and coöperate with my desire of rendering the subject conducive to the promotion of all that is beneficial to the rising generation ; and allow me to assure you, that in speaking of the “rising generation,” the term is by no means limited to this land, but also extends with a deep feeling of interest to that great people, our sister-land, so closely united to us by a common language, and in great measure, by a common descent. - With every good wish for the success of this little volume, I remain, sir, yours truly, ROSINA M. ZORNLIN.
• W. L. Gage, Esq.”
THERE is a voice within me
And 't is so sweet a voice
Till tears start to my eyes.
Like hidden melody,
This song of songs to me:
As other Worlds above,
It might be full of love." - London Times.
WESTFIELD NORMAL SCHOOL EXAMINATION. — The large crowd of persons who attended the exercises of this examination last week, testified by their presence to the deep interest felt by the community in this school. Under the superintendence and sound instruction of its present highly popular and experienced Principal, William H. Wells, and his able assistants, the school has attained to a degree of usefulness and prosperity hitherto unexampled in its history. When the Normal School House was built, ten years ago, the school was very small, and the house was considered by many larger than the school would ever need; butthe Normal Hall is now too small to accommodate the scholars, and the Legislature has been petitioned for aid to enlarge it. The members of the Legislative Committee, in their remarks, at the close of the examinations, promised their influence to procure the necessary State aid.
The examination was unusually interesting, and interspersed with extempore teaching exercises by the pupils, which added much to the interest of the occasion. The recitations were all excellent, manifesting great readiness and comprehension on the part of the pupils ; faithful and systematic instruction on the part of the teachers. The occasion was one of rare interest to all concerned, and clearly demonstrated the usefulness and great excellence of Normal Schools, especially of the one located in our village. — Westfield Newsletter.
There are tones that will haunt us, though lonely
Our path be o'er mountain or sea ;
When memory ceases to be ;
Though toilsome and steep be the way;
With a light that is clearer than day. — Praed. GRAMMATICAL LOVE-MAKING. — [The follo has irly haunted us. It has thrown itself again and again in our way; and again and again we have closed our eyes, determined not to see it. But now, on opening a very properlooking buff envelope, what should come forth but this, enclosed to us by one of our associate Editors as his contribution to our olla podrida ? We yield; but throw on him the responsibility of its publication. If any harm comes from it, either to "gramer" or to “morils,” he must answer for it.]
The Perils of Teaching Grammar to Young Damsels. — Mr. Editor, - I have been sendin' my darter Nancy to school to a schoolmaster in this naborhood. Last Friday I went over to the school just to see how Nancy was gittin' along, and I see's things I did ’nt like by no means. The schoolmaster was larnin' her things entirely out of the line of eddycation, and, as I think, improper.
I set a while in the school-house and heered class say their lesson. They was a spellen, and I thot spelled quite exceedingly. Then cum Nancy's turn to say her lesson. She said it very spry. I was shockt! and determined she should leave that school. I have heered that gramer was an oncommon fine study, but I do'nt want any more gramer about my house. The lesson that Nancy sed was nothin' but the foolishest kind uv talk, the ridiclest luv talk you ever did see. She got up, and the first word she said was, • I love!' I looked rite at her hard for doing so improper, but she went rite on and sed, • Thou lovest, he loves,' and I reckon you never heered such a riggermyrole in your life — love, love, love, and nothing but love. She said one time, I did love.'
Sez I, • Who did you love?' Then the scholars laffed, but I wa’nt to be put off, and I sed, "Who did you love, Nancy? I want to know, who did you love?' The schoolmaster, Mr. McQuillister, put in, and said he would explain when Nancy finished the lesson. This sort of pacyfied me, and Nancy went on with awful love talk. It got wus and wus every word. She sed, I might, could, or would love.'
I stopped her again, and sed I reckon I would see about that, and told her to walk out of that house. The schoolmaster tried to interfere, but I would'nt let him say a word. He sed I was a fool, and I nockt him down and made him holler in short order. I taukt the strate thing to him. I told him ide show him how hede learn my darter gramer.
I got the Nabers together, and we went to Mr. M'Quillister and sent him off in a hurry, and I reckon tharl be no more gramer teechin' in these parts soon. If you know of any rather oldish man in your reegen that doant teech gramer, we wood be glad if you wood send him up. But in the footure we will be keerful how we employ men. Yong schoolmasters wont do, specially if they teeches gramer. It is a bad thing for morils. Missouri Democrat.]
Yours till deth, Thomas JeFFERSON SOLE. [A friend at our elbow suggests, that Miss Nancy ought to have commenced with the passive voice, “I am loved”; and that then the active, “I love", might have followed properly enough. Mr. Sole might have been then content; and the schoolmaster, paying this proper regard to order, that essential in every good school, might have kept his standing and place.
But seriously, it is time that the word "love" should be discarded from the place which it has so long unworthily held, in our grammars, as the paradigm or perfect model of regular verbs. It is not strictly regular in its form; it does not readily show to the young student the distinction between the root and the affix in grammatical forms; and, to say nothing of the mirth and blushes which it sometimes excites in the midst of a grave exercise, it owes its place not at all to its own merits, but simply to the fact that it is a translation of the Latin verb, amo, the study of the English grammar having been derived from that of the Latin, But though amo, the immemorial paradigm in the Latin grammar, is an admirable example of a verb perfectly regular in its structure, it has no power of transferring this grammatical excellence to words of like signification in other languages. And besides, according to old usage, it was so whipped into the learner's memory, that there was no danger of its becoming associated in his mind with any sentiments of love. It was a far more potent stimulant of the opposite passion.)
NORMAL SCHOOLS. — At the recent examinations of candidates for these schools, 16 were admitted at Framingham, 35 at Salem, and 32,— 7 gentlemen and 25 ladies,— at Bridgewater. The number of graduates who have returned to form advanced classes, is 15 at Salem, and 8 at Bridgewater, making the whole number in the Salem School 116, and in the Bridgewater School (in which there are this term only two undergraduate classes, the extension of the course from a year to a year and a half taking effect at this time) 75. The number in the Framingham School we have not learned. The term at Westfield begins too late in the month for us to receive intelligence from this school before going to press.
Miss Tefft, now Mrs. Munger, has resigned her situation as teacher in the Salem School; and is succeeded by Miss Phæbe A. Breed, of Lynn, and Miss Sarah R. Smith, of Marblehead, recent graduates of the School, - an increased number of teachers being desirable on account of the fourth class.
PLEASING INCIDENT. - One of our friends, a retired school teacher, having recently applied to one of his former pupils, to take some stock in a new Insurance Company, in which the former had an interest, was received with truly filial kindness by the other — now a successful and honorable young merchant -- but who was unable to take any of the stock - circumstances haying rendered it impracticable at that time; “but,” said the merchant, putting a check for two hundred and fifty dollars into the veteran's hand, "take this, not as a gift, but in part payment of the debt I contracted to you as my teacher in days gone by. – Transcript.
Dangers of Precocity. - Willie Herbert, youn est child of Mr. J. B. Berry of this city, died on Thursday last, aged thirteen months and nineteen days, being the fourth of the prize children at the exhibition in the Music Hall last fall that has since deceased. — Boston Journal.