So far, the chief object should be to impart the ability to produce a eorrect manuscript-to thoroughly drill the scholar in the elementary principles of written language.

FIFTH STEP: FIFTH READER SCHOLARS.

Composition Proper, or the discussion of themes, in which Invention, Arrangement and Style largely enter. Didactic, persuasive and argumentative writings are included under this head. A more extended analysis of this step is unnecessary. It is fully presented in the common works on English Composition, or Rhetoric.

The above is a mere skeleton or outline of the subject: the skillful Teacher can easily fill it out. The adoption of this, or a better Course of Composition, securing a complete division of labor, is, in our judg ment, greatly needed. Between the different grades of school, early instruction in this important and useful branch of education, is now sadly neglected. E. E. WHITE.

Portsmouth, Aug. 1857.

Mathematical Department.

PROF. W.

H. YOUNG, ATHENS,

EDITOR.

All communications for this Department should be addressed to the Editor, Ohio University, Athens, O.; and to be in time, must be mailed by the first of the month preceding that in which they are expected to appear.]

SOLUTIONS OF QUESTIONS PUBLISHED IN JULY,

No. 10. What is the amount of $100 in ten years, at 6 per cent., supposing the interest to be compounded every instant?

SOLUTION BY JAS. MCCLUNG.-The interest on one dollar for an instant, at .06, is .06 X; and the amount for one instant is 1+; and the amount of one dollar for ten years, at .06, compounded every 06) 10-1, which, expanded,

10-1

instant, is (1+-06) '°

becomes ∞ 10-1+ (1000

=

(+06) 109-1

·1) ∞10-2.06+

(10-1) (10-2) ∞ 10-3.062+

1.2

.6n

The sum of

2.3.. n.10n'

which is 1.8221; and $100 will give $182.21+. Ans.

No. 11. Find under what circumstances vulgar fractions are convertible into finite decimals.

SOLUTION BY S. HARVEY.-A vulgar fraction can be converted into a finite decimal only, when all the prime factors, except 2 and 5, found in the denominator are contained in the numerator; for, since adding ciphers introduces no prime factor except 2 and 5, the complete division of the numerator by the denominator can only take place under the above condition.

No. 12. The sides of a rectangle are to each other as 2 to 1/3; and the diameter of a circle, drawn to touch the middle point of one of the larger sides, and passing through the corners of the opposite and adjacent sides, is 48 rods. Required the area of the rectangle.

=

=

SOLUTION.-Construct a rectangle, ABCD, and draw a circle touching E, the middle point, or longer side, AB, and passing through D and C. From E draw EG, a diameter to the circle; it will pass through the middle point, F, of CD. Let x3 BC EF, and = AB, or x EB = FC. EG = 48, and FG = = 48. x V3. FC2= EFX FG, or x2 ― x √3(48-x√3); whence x = 12 √3. Area of rectangle AB × BC = 2x × x √3 = 24 √3 × 36 : = 864 square rods. Ans.

2x

√3

=

ACKNOWLEDGMENTS.-All the Questions were solved by A. Schuyler, Jas. McClung, J. B. Dunn, and Joseph Turnbull; Nos. 10 and 11, by A. A. K.; No. 10, by Joel E. Hendricks; Nos. 11 and 12, by S. Harvey, J. S. Burnham and Jas. Goldrick; No. 12, by Lewis McKibben and Isaiah Thomas. Several correspondents furnished solutions for questions, published in June, which were not received in time for acknowledgment last month.

It is not expected, of course, that everything furnished for the Mathematical Department will be published. We have several interesting articles and problems on hand which we would like to publish, but

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.

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 a 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? C. S.

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+1)2 = a2+a+1. Ex. (71)2 =49+7+1=56. Also (a+1)2= a2+a+%, and (a+1)2 = a2+1α +1⁄2· Ex. (91)2=81+4+85%, and (161) =256+4+4=2604. Or, if we know the value of a2, we may find, by a simple mental operation, that of (a+1)2, (a+1)2, or · (a + 1)2.

2

For products, we have (a+‡)(6+1)=ab+a+0+. (This formula may be extended to the other cases.) Ex. 14 15 – 210 +14+=2243. 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-1.

If we wish to square 45, we may regard the 5 as a decimal, and reduce to; then square, gives 204; reducing the to a decimal, and removing the separatrix, gives for (452), 2025. By the same process, we have (75)2 = 5625; (185)2 = 34225; (225)2 = 50625.

So, if any number ends in 25, if we know the square of the number preceding the 25, we may regard the latter as, and after squaring, as above, consider as whole numbers. Thus, (625)2=6 hundreds square 39 square hundreds, or 390625 units. (1325)2 = 169+ 6+ square hundreds 1755625. 1755625. All the difficulty is that of reducing the vulgar fraction to a decimal, and then reading as whole I numbers; and this can be done without much effort. If a number ends in 125, we may regard this as §, and square as above. Take 18125: (183)2 = 32833; but the decimal for= .015625, and that for 32 .5; and for (18125)2, 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, .25, .125,.0625, 3.03125, = .015625; and, generally, the 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 don'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.