don't get above your business ; you have a noble, and mighty work to do, but you must go where it is, and take hold“without mittens,” to do it. Sacrifice, if need be, your own comfort and convenience, that you may gain the full acquaintance, the true friendship, the abiding confidence, the hearty co-operation and lasting good will, of parents and pupils. In this way only, can you render the greatest assistance to parents, in their noble work of developing the physical, intellectual, and moral nature of their children. With regard to the pecuniary view of this custom, I do not see the force of the gentleman's remarks. The question has not been raised whether the Teacher should be boarded upon the Grand List or by the Scholar, since, if boarded by the scholar, both might be boarded in one place. Again, where does this public money come from that is drawn by scholars ? Although this does not touch the question, I will concur with D. M. C. that it is good policy, as well as good principle, for any State to maintain a liberal system of education at the expense of the property of the State. At the same time, I must say in couclusion, that I still believe the custom of“ boarding round" is the best for the common school, and that the Teacher who objects, is either a penny selfish or a pound lazy. H. C. 0. POLITENESS.—Boys, be courteous. In tbis fast age, school boys, and girls too, are in danger of failing in this duty,-a duty that radiates and reflects pleasure. Be polite in little things. Show your respect for your teachers, parents, or superiors in age, by respectful attention to their words, by deference of manner, by offering for their acceptance a desirable seat, by gentle and quiet behavior, by a cheerful yielding of your wishes, and, sometimes, your comfort, to that of others. Such babits formed in youth will secure you many friends; and, if prompted by right motives, the approbation of Him "who pleased not himself." G. MATHEMATICAL DEPARTMENT. ERNEST C. F. KRAUSS, A. M. EDITOR. PROF. KRAUSS : Will you explain the following examples and much oblige A Teacher. 1. From 10000 subtract 999. 3 2 6 2. Multiply by Why? 2 12 Why? 5 3 10 Ad. 1. ,We begin as usually by subtracting the units of the Subtrahend from those of the Minuend. But, the latter having no units, we propose to take one of the tens of the Minuend and dissolve it into ten units; but the Minuend has no tens either, nor any hundreds, nor units of thousands, which can be dissolved into the lower orders. But we have one ten thousand, and to obtain hundrede, tens and units, we first divide this into two parts, 9000 and 1000, then this 1000 into 900 and 100, and finally this 100 into 90 and 10. If then we dissolve the Subtrahend in the same manner into 900, 90 and 9, we have 9000- 0=9000 9 1 10000— 999=9001 This once understood, we can shorten the operation. We write as usually 10000 999 First, we subtract 9 units of the Subtrahend from ten units, which we regard as having been taken from the tens of the Minuend, leaving 1 unit in the Remainder. We further suppose one of the Minuend's hundreds to have been dissolved into 10 tens, one of which had to be transferred to the units, wherefore only 9 tens are left in the Minuend, and subtracting 9 tens from them, no tens are left in the Remainder. In the same manner we suppose one of the thousands to be divided into ten hundreds, one of which had been transferred to the tens, leaving 9 hundreds in the Minuend, and subtracting 9 hundreds from them, no hundreds are left in the Remainder. Finally, having taken one of the 10 thousands, 9 thousands are left in the Remainder, and so we have 10000 999 9001 Ad. 2. The Germans say, “Er ist in die Bruechc gekommen", (he has got into the fractions,) of one who finds himself in a predicament from which he does not know how to extricate himself. Experience proves the propriety of the proverb, for a great many who can work examples with only integral numbers well enough, are at a loss when they have to deal with fractions. And yet there is no necessity for this. The common arithmetical operations with fractions are no more difficult than with whole numbers, if once the nature of fractions is properly understood, which will soon be done, if we accurately analyze it. Taking it for granted, that the definitions of Numerator, Denominator, Integral and Fractional Unit,are understood, we proceed to show that we can take two different views of a fraction. m no - B E -F. If A B (fig. 1) represents the integral unit, and we divide it into four equal parts, our fractional unit is 1-4, for instance Am, and taking 3 of them we have \, i. e. 3 fourths or A 0. The Numerator, which designates the fig. 1 $ 1. number of fractional units given, or taken into consideration, is the number proper with which we have to deal ; the Denominator is merely the name of the fractional units, and 3 fifths (3-5) differ from 3 eighths (3-8), in a similar manner as 3 yards differ from 3 feet. $ 2. If CF, which is three times as great as A B, represents 3 integral units and we divide it by 4, we obtain Cr, which is equal to A o or 3-4. Or we may say: dividing each of the 3 integral units into fourths we get 12 fourths, and dividing this number by 4, we obtain 3 fourths (3-4). Hence a fraction may always be regarded as a Quotient, whose Dividend is the Numerator, while the Denominator is the Divisor ; 3-4 is the same as 3 • 4 and 2-5 = 2: 5. To multiply and divide fractions by whole numbers. A. Given a. 2-5 X 3. b. 6-7 • 3. $ 3. The Denominator indicating only the name of the fractional units given, whose number is expressed by the Numerator, the latter only is in case a. to be multiplied, in case b. to be divided, while the same Denominator is retained. As 3 times 2 pounds are 6 pounds, thus 3 times 2 fifths are 6 fifths ; hence, 2 2 X 3 6 5 5 As 6 feet divided by 3 gives 2 feet, thus 6 sevenths divided by 3 gives 2 sevenths ; i. e. 6 6 • 3 2 7 7 B. Given a. 3-4 X 2. b. 3-4 ; 2. $ 4. Instead of multiplying or dividing the Numerator, i. e. increasing or decreasing the number of fractional units given, we may retain this number, but increase or decrease their size in a manner corresponding to the given Multiplier or Divisor. To multiply a fraction by 2, 3, 4, с m fig. 2. { n we make the fractional units 2, 3, 4 times as large, to di. vide it by 2, 3, 4, we make them one half, one third, one fourth as large as those given. d A 1 B B' $ 5. A B (fig. 2) represents the integral unit, Ac is 1.4 (a fractional unit), A d is 3-4. To multiply this 3-4 by 2, we make fractional units of the size A'f, which is twice ås large as A c, take three of them, and have then A'n. Now there being only two such parts in A' B', each of them is and is twice as much as 1-4, and therefore A'n, which contains 3-2, must be twice as much as Ad, which contains 3-4; that is, 3 3 3 4 • 2 2 By comparison, we see that Am, which is twice 3-4 or 6-4, is as much as A'n, which is 3-2. 8 q $ 6. If A B (fig. 3) again represents the integral unit, A c is 1-4, A d is 3-4. To divide this 3-4 by 2, we make fractional units of the size A'g, which is one half as large as A c, take 3 of them and have then A'q. Now there being twice 4, i. e. 8 such parts in A' B', each of them is 1-8 and is only half as much as 1.4, and therefore A'q, which contains 3-8, must be one half as much as A d, which contains 3-4; that is, 3 3 3 4 x 2 8 By comparison we see that A p wbich is one half of A d, i. o. of 3-4, is as large as A'q or 3.8. § 7. From this it is plain that the size of the parts, |