respect. By an earnest request, Mr. Chase repeated his Essay. Wednesday morning a large and attentive audience listened to the discussion of the “ Jurisdiction of Teachers over their Pupils out of the School House." The morning exercises were closed by an excellent address "To Teachers,” by Joseph E. Woodbury. A portion of the afternoon was occupied in reflections upon the apathy of the public mind to the interests of our schools. Also in listening to reports on the condition of schools and school houses in the country. After recess, the President, Rev. William Sewall, delivered an address, of which the following notes are given: Home Culture in its Bearings on the Common School. 1st. It is important, because the authority of the parent over the child is higher than that of the teacher, necessarily so. And because the home culture is first in order. The parent has made his mark upon the child before he enters the District School, 2d. The nature of this home culture is two-fold. There is, first, the home instruction as indirectly touching the school, though just as really and unavoidably. This embraces the general principles of deportment and propriety which apply to children everywhere, and, secondly, that which directly reaches the child's conduct at school, viz: that which relates to his treatment of the teacher, of his fellow pupils, of the passers-by, or of the visitors of the school. The indirect culture embraces these points: 1st. Place a good example before your child ; an example of gentleness, patience, manliness, purity, truthfulness, sincerity, frankness; and these in action, as well as word. 2d. Let home discipline, which is sometimes unavoidable, be never administered in anger, but always in love. 3d, Take care to use every means within your power to beget in the minds of your children a taste for good company and good reading. 4th. Keep the sweet confidence of your children, which is yours so entirely in their earlier years, and which is one of the strongest bonds of your influence over them. 5th. Make home attractive. The home instruction more directly relating to the school, may be considered under the following heads : 1st. Teach your child to respect the teacher. 2d. Tell him always to give implicit obedience to the teacher. 3d. Check, peremptorily, any inclination on the part of your child, to find fault with the teacher or the management of the school. To this end, be careful never to find fault with the teacher, in the presence of your children. 4th. As to their studies, give your children to understand that the teacher is to be the judge as to their text books, their particular classes, and their degree of advancement. A vote of thanks was tendered to the speakers and singers, for their able and timely service. The people had become so “waked up" that it was voted to have an evening session, which was mostly spent in a merry discussion upon the barbarism of board ing around." A limited one upon the extent of moral and mental qualifications superintendents should require in the examination of candidates for teaching, was also had. Adjourned to meet in Lunenburg, November next. This meeting was truly a successful one,-more so than the most ardent dared to hope for. It was noted throughout for social discussion, much of which was by parents. And the citizens truly reflected credit upon themselves, by the kind care and courtesy extended to those who were not of them." CHAS. W. KING, Recording Secretury. 66 MATHEMATICAL DEPARTMENT. ERNEST C. F. KRAUSS, A. M., EDITOR. $ 5. In Arithmetic we have not to deal with Opposite Quantities. It is immaterial whether we say, the quantities used there are all absolute or all positive. Certain it is, that we meet no Negative Quantities in it. In consequence of this, the definitions of Addition and Subtraction will in Algebra differ from those in Arithmetic. While in Arithmetic Addition is the operation of finding a number which contains as many units as two or more given numbers together, in Algebra we must define it as the operation of reducing two or more given quantities to their simplest form. While Subtraction in Arithmetic consists merely of the operation of taking one quantity from another, to find the remainder; in Algebra we must define it as the operation of COMPARING two quantities with each other, in order to obtain their difference. The difference between two quantities which are given without any special condition, is an absolute number (§ 1 April number) and remains the same, whichever of the given quantities be named first. Thus, the difference between 8 and 5, as well as between 5 and 8, is 3; the dif. ference between + 4 and 3, as well as between 3 and + 4, is 7. But, when it is stated, which of the two . given quantities is the Minuend, (the other being the Subtrahend,) the question is understood to be this: How many units are to be added to the Subtrahend, to produce the Minuend? Is the Minuend above the Subtrahend, when viewed on the scale in $ 4 (April number,) the dif. ference is found by ascending from the Subtrahend to the Minuend and therefore positive; in the opposite case, by descending, and hence negative. $ 6. The rules according to which Opposite Quantities are to be added and subtracted, are found in every class- , book on Algebra, and if they are not understood by the pupils, we work some examples on the scale of $ 4, or we let the given numbers express miles, positive to indicate the direction North, hence negative the direction South, some geographical point being taken as the starting point, whose representation is 0, as was demonstrated in $ 3. If this is donte properly and some pupil does not comprehend it, he may safely be given up as a hopeless case. His knowledge in Mathematics will always remain a negtive quantity. To comprehend these principles of Mathematics, neither genius nor special talent is required, but only common sense. Yet, what close observer has not made the experience, that “common sense” is rather an uncommon thing? $ 7. There is, however, a case, and that in Multiplication, which is not so obviously clear. Even intelligent scholars find it difficult, if not impossible, to reconcile with their reason the rule, that the product of two negative factors should be positive. How, if all that is operated with and operated upon is negative, can the result be positive? We will endeavor to make it clear. The signs + and — are signs of operations (Addition and Subtraction,) as well as inherent signs of quantities, indicating whether these are positive or negative. We therefore ask: Can they always be taken for the one, as well as the other ? Certainly they can; and a rational consideration will, in every special case, readily decide which view is the more appropriate. But if a quantity is given with the inherent sign + or -, and we separate the same from the quantity, to regard it as a sign of operation, what will then the quantity be, positive or negative ? It will evidently be neither, but will become absolute ($ 1.) Any positive or negative quantity can be multiplied. But to say, we take a quantity a positive or negative number of times, would be as absurd as to say, we multiply three dollars by four dollars. We can only take a quantity a certain number of times, and hence a Multiplier must always be an absolute quantity. If, therefore, we meet, as we frequently do, two factors, which both 1 . appear as relative quantities, we must, for the sake of a rational understanding of their multiplication, separate the one, which we take as Multiplier, from its sign, whereby the quantity becomes absolute and the sign, a sign of operation; and, whatever be the product of this absolute Multiplier into the relative Multiplicand, it has, conscquently, to be added or subtracted, according to the sign preceding the Multiplier being plus or minus. 1. To multiply + 7 by + 4, wo have + (4 x(+7)) = + ( + 28)= + 28. = = 2. To multiply — 7 by + 4, we have +(4x.(-7)) =+(-28)= - 28. +/- = . 3. To multiply + 7 by — 4, we have — ( 4 x (+7)) =-(+ 28 )=- 28. 4. To multiply — 7 by 7 by — 4, we have — ( 4 X (-7)) =-1— 28 )= + 28. The first three cases are easy. To explain the fourth, let us make a practical application. Suppose a business man makes four transactions, losing 7 dollars in each of them. This is case 2. The minus inside the parenthesis shows the $28 to be a loss. The plus outside of it, that this is an additional item to be entered on his debit side, being loss, after which bis book shows an actual loss of $28. Now suppose this entry has been made and afterwards turns out to have been a mistake, and we have case 4. The minus inside the parenthesis shows this item to be a loss, for as such it was entered; the minus outside of it, that this item is to be subtracted, i. e., erased, after which his book shows a gain of $28, compared with its amount before this subtracting or erasing. $ 8. If Multiplication is well understood, Division can offer no difficulty, if we keep in mind that Division is an inverted Multiplication, and that the Quotient is in all cases that quantity which, multiplied into the Divisor, pro duces the Dividend. There are only two fundamental rules in Arithmetic. |