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cannot find room. Such articles as seem to us less important than some others, are laid aside; and problems very complex or extended in their solutions, or involving principles but little known, even by fair mathematicians, must give way to those that will prove of more interest to the generality of our correspondents. Next month we shall probably have place for “ ORWELL’s” article on the “zero power.”
Some of our correspondents think No. 3, published in April, should be solved without the aid of Fluxions. Have already had two communications on the subject, and should be glad to hear from others.
QUESTIONS FOR SOLUTION. No. 15. There are three rectangular blocks of marble, all of the same shape, which is such that they may be placed together, so as to make a similar joint block. The largest is eight inches long. How long is the joint block ?
A. A. K. No. 16. Find three series of perfect squares, any term of the first of which shall be the sum or difference of the corresponding terms of the other two.
J. S. BURNHAM. No. 17. Suppose the diameter of the upper base of the frustrum of à cone to be 20 in., that of the lower base 28 in., and the altitude 40 in., what will be the perpendicular distance between the lower base and a parallel plane, dividing the solid into two equivalet frustra?
CONTRACTIONS.—The contractions spoken of by “BOND,” in the July number of the Journal, may be well applied to mental arithmetic.
If a = any number, we have (a + })? = a? +a+Ex. (7) = 49+7+1= 56. Also (a + 1)2 = aa+a+to, and (a +:)
al + atoa: Ex. (91)2 = 81 +45 + b = 8516, and (161)2 = 256+4+=26074. Or, if we know the value of a?, we may find, by a simple mental operation, that of (a +)?, (a ++)?, or (a+3)2. For products, we have (a + 1)(6+1) =
2 formula may be extended to the other cases.) Ex. 141 X 151= 210
+143 +=2244. This also illustrates another truth, which may be proved general, viz: the product of any two consecutive numbers +} added to each = the square of the greater - 4.
If we wish to square 45, we may regard the 5 as a decimal, and reduce to }; then equare, gives 204; reducing the to a decimal, and removing the separatrix, gives for (45%), 2025. By the same process, we have (75)2 = 5625; (185)2 = 34225; (225)2 = 50625.
: ab +
So, if any number ends in 25, if we know the square of the number preceding the 25, we may regard the latter as 4, and after squaring, as above, consider as whole numbers. Thus, (625)2 = 64 hundreds square= 391'6 square hundreds, or 390625 units. (1325)= 169+ 61+ ' square hundreds = 1755625. All the difficulty is that of reducing the vulgar fraction to a decimal, and then reading as whole numbers; and this can be done without much effort. If a number ends in 125, we may regard this as ļ, and square as above. Take 18125 : (183)2 = 32837; but the decimal for a't = .015625, and that for 33 =.5; and for (18125)?, we have 328515625.
As it is easy to remember the squares of the natural numbers to 25, we can mentally do any thing that is here suggested ; and it requires but little ingenuity to apply these principles to the extraction of roots, in written as well as mental arithmetic.
In determining the powers of 5, the following may be of use : }= .5, 1 = .25, š= .125, iš = .0625, z'ı = .03125, 7'4 = .015625; and, generally, tủe significant figures of a decimal corresponding to unity, divided by any power of 2, are equal to 5 raised to the same index.
At first, some of these rules may be thought too complex for mental operations; but, on experience, they will be found just sufficiently difficult to afford a good stimulus.
J. B. Dunn.
- "Stop your crying,” said an enraged father to his son, who had kept up an intolerable yell for the last five minutes; "stop, I say, do you hear?” again repeated the father, after a few minutes, the boy still crying. “You do n't suppose I can choke off in a minute, do you?” chimed in the hopeful urchin.
-“A rolling stone gathers no moss.” A restless, unsatisfied Teacher, always grumbling and always moving, is a bad investment-won't pay.
- The knowledge of man's wants, and the means of supplying them, makes the true learned man.
- In whatsoever manner or degree learning may be acquired, and minds formed, still it is true, that they become useful to mankind, only in proportion to their observations and experience.
– A mere enthusiam for doing good, if excited by vanity, and not accompanied by common sense, will seldom be very serviceable to ourselves or to others.
-"It is important to distinguish between the reward of intellectual superiority, and the approbation of intellectual effort.” Rewards should be for moral character, a recompense for something good performed.
From the Knickerbocker. DON'T SAY "YOU CAN'T."
Don't say "you can't !” there's joy in store
And there is wo
For all below,
Do what you can,
To be a man,
Or if he did,
His conscience bid
And mean the sloth,
Who quits his swath
Then, brother, hoe
Your honest row,
Don't say “you can't!” let us while here
Descend the hill
With right good will,
The clock on yonder mantle-piece
The brass, in part,
Shows man his heart,
Each tiny wheel,
That turns with zeal,
Then, brother, heed the simple text,
Don't say “you can't!”
But, like the ant,
· Welcome, Teachers and Pupils, again, to the communion and fellowship of kindred minds, in the mutual labors and pleasures of teaching and being taught! Schools have again commenced, and here and now, we should all keep in view their design. Education is but a means for achieving an end; and that is, the moral and intellectual perfection of man.
Parents should commence the education of children at home-before they go to school. The first lessons should be as to morals-a strict adherence to truth -an exact honesty ;-they should be accustomed to prompt and implicit obedience, and to look at their conduct as God regards it.
The lessons as to their intellectual education at this early age, should beEncourage the child to observe and study things, and foster in him a taste for reading. Primary Teachers—those engaged in elementary instruction, who receive the pupil, and make upon him or her the first, and perhaps lasting impression-must not be discouraged because their work does not show well. Quintillian, the Roman teacher, said long ago, what yet may be truthfully said, “The roofs of buildings are seen by every one, while the foundations escape notice. Things are not to be despised as little, without which great ones cannot be produced.” It is not the quantity at first, but the quality of their instruction, and the correctness with which it is impressed upon the mind, that eventually benefits the pupil.
If a Teacher be desirous that a pupil acquire an early habit of correctness in every thing he is afterwards to learn, it must take its rise and date from the hour he begins to learn the rudiments of any particular branch of study; for, the good or bad habits acquired in the first stage of his progress, will, most assuredly, be carried on with him into the other stages, and conveyed from one branch to another, with an almost unchangeable and unalterable effect. Strive, therefore, ye who have the beginners in school, to have them so trained that they not only learn their lessons correctly, but learn the important lesson of acquiring a habit of correctness-to be proficient as they go. Rivet this, then, into the hearts and minds of all—“Do every thing with attention."
We are told that it is not what we earn, but what we save, that makes us rich. It is not what we eat, but what we digest, that makes us fat. It is not what we read, but what we remember, that makes us learn. These are simple statements, yet Teachers who have but small salaries should consider them.
Teachers can make themselves what they want to be. Many of them, who had position and opportunity in schools last year, have lost caste and their pla. ces by inefficiency; while humble aspirants, in many cases, by meritorious conduct and earnest discharge of duty, have taken their places.
School Trustees and Directors are sometimes capricious—are mistaken in judgment, but, in the main, the safety of the Teacher lies in himself; worth will win. Make-believe Teachers, those of more show than substance, must give way; while earnest, clear-headed and warm-hearted men and women must take rank.
In Ohio, there is encouragement for the very best Teachers; they are appreciated, and will be rated and paid accordingly. The West has a great work to do;-her forests have to be felled-her prairies ploughed and planted-her streams navigated-her workshops filled with educated mechanics. Railroads