To find the Remainder.-Take the sum of the figures in the divisor and subtract from the divisor, this will give you the remainder. 7402) 104875858755/14168583, inwhich 1+8=9,&c. Any master may with the assistance of one of his monitors soon find the dividends of 200 sums, which can be entered and numbered on a board, and then given out promiscuously. I have tried the method ten years, and can strongly recommend it. The master should arrange the divisor, quotient, and remainder himself, leaving the labour of multiplying to the monitor. The following are examples from my own board. No. 1. Divide 350804382012 by 5986 30. A MAN laid out 5 shillings in apples and 4 shillings in pears,—how much money did he spend altogether? In order to find the right answer to this question, we will, by means of numeration, add all the units in the number 4 to those in the number 5. That is, we show first, that 5+1=6 Or in other words, the number 9 contains as many units as the two numbers 4 and 5 taken together; man has spent in all 9 shillings. and hence we say, that the 31. We have treated the numbers 5 shillings and 4 shillings as if they were the abstract numbers 4 and 5; but it will be evident that our doing so does not affect the result: for, whatever these numbers be, whether abstract or concrete, they equally represent the same number of units. Let the next question be, how many are 4 + 2 + 3 + 6? By the preceding method we have Therefore 4+2= 6 6 + 3 = 9 4 + 2 + 3 + 6 = 15. 32. The above process is called Addition; therefore addition is that process by which we find a number which shall contain as many units as two or more numbers taken together contain. The result of this process is called the sum of these numbers; that is, 9 is the sum of 5 and of 4, and 15 is the sum of 4, 2, 3, and 6. Addition, we have therefore seen, depends on numeration; but in order to save time, the pupil should learn by heart the sums of all the simple numbers taken together 2 at a time. 33. When there are two or more numbers to be added together, it is of no consequence in what order the addition is effected. Thus 4 + 5 = 5 + 4; 6 + 2 + 7 = 6 + 7 + 2 2 + 6 + 7 = 2 +7 +6= 7 + 6 + 2 = 7 + 2 + 6. That is, the sum total remains the same, whatever may be the order in which the numbers are placed.* 34. A man who has 9 shillings in his purse, owes 5 shillings to a friend; how much will he still have, when he has paid his debt? He will have 5 shillings less than he had at first; that is, he will have 9 shillings less 5 shillings, which is expressed by figures or signs, thus: 95, for the sign or stroke, and which is usually called "minus,” signifies less; and if we call that which remains x, we shall have 9 5 = x. It is evident, that to find what number is represented by x in this case, we must take away from the 9 units all the * It is often found useful to represent numbers by letters, in which case they are termed algebraic quantities. Ex. gr., let a, b, c, be 3 numbers, of which the sum is 16, and we shall then have a + b + c = 16, to illustrate what we mean by addition. units in the number 5, (which is the converse of the process in numeration,) and we shall thus have successively Therefore 9 8 7 6 5 = - 1 8 1=7 Or, in other words, when the man has paid away 5 shillings, he will still have 4 shillings. I could also have ascertained this by finding what number, added to 5, will make 9; for if I add 4 to 5 (the number of shillings paid away,) I shall have 9, the number of shillings which were in the purse in the first place. 35. This process or operation is called Subtraction, and subtraction therefore is that operation of arithmetic by which one number is taken away from another. The result of subtraction is called the remainder. 36. John has 9 apples, whilst Thomas has 5; how many more has John than Thomas? We here inquire what is the difference between the two numbers or quantities of apples, that is, how many Thomas has less than John : which, in other words, will be as many as John has less those which Thomas has, or 9 - 5; so that by subtraction we also ascertain what is the difference of two numbers. And 95 = 4, as was shown above.* Therefore subtraction is likewise an operation by which we learn the difference of two numbers; for which reason the result is called the diffe rence. In the case before us, 4 is the difference of 9 and of 5. 37. Hence we see that addition and subtraction are converse operations, the one to the other; for addition is but a more simple form of numeration, and subtraction being the converse of it, may therefore be viewed as denumeration. Or again; if, for instance, after having added 4 to the number 6, I wish to take away 4 from the sum (or 10), I ought to find the original number (or 6), for I shall have in fact that which I had at first, or 64 4 : 6. = In such a case + 4 and 4 are said to destroy each other. In the same manner, if, after having taken away one number from another, we add the same number to the difference, we have again the first or original number. Thus 6 = 6; where 4 and 4 destroy each other. 38. We give the name of equals, or more properly of an equation, to the expression of two quantities which are equal the one * From (Lat.) to the other; thus (954) is an equation.* æquus, equal. *It is evident that if the present treatise on arithmetic were published for the use of schools, a greater number of examples for practice in all the rules would be desirable, and we might in this case, take advantage of a suggestion contained in pages 326 and 327, Vol. II. of this Journal, (November 1844), "to give lessons in justice, economy, brotherly kindness and charity." Now it is evident that, if to each of two equal quantities we add the same quantity, the equality or equation will still subsist; or, ex. gr., if to 95, and again, to 4, we add 5, we shall then have 9 5+5 = 4+ 5; or, as 4 + 5. 5 and 5, destroy each other, we shall have 9 Hence, if, as we have already seen, we add to the difference of two numbers the lesser of those numbers, we shall have the greater. Again, because 4 + 5 === 5+4, we can say that the lesser number added to the difference is equal to the greater. 39. Addition and subtraction may assist each other. Thus, if we would add 8 to 25, as we have 8 instead of adding 8 I may add 10; but as I shall then have added 2 more than 8, I must subtract 2 to have the sum required, or 25 +8 25 + 10 2 = 35 2 = 33. = Again, in order to take away 8 from 25, if I take away 10, I shall have taken away 2 too much; it will therefore be necessary to add 2, that the difference sought may be ascertained, or 258 25— 10 + 2 = 15217.* Extracts from Charges. THE LORD BISHOP OF SALISBURY ON THE PROSPECTS OF NATIONAL EDUCATION. FROM the baptism of our children it is natural to pass on to that which engages the anxious attention and continuous care of the parish priest who truly feels his responsibilities, and rightly understands how these are to be in any measure effectually discharged-I mean the endeavour to train up in christian faith and holiness those children who in baptism are made members of Christ and children of God, and as such are committed to the oversight and charge of his ministers. The more I consider the whole subject of the education of the poorer classes in this country, the more I see the difficulties which encompass it, inasmuch as we have not only to contend against the obstacles arising from the original corruption and indwelling sinfulness of our fallen nature, but also against those occasioned by existing evils in our social state, which are beyond our controlthe deep poverty of the labouring class in our agricultural villages, which incapacitates them from providing for the education of their own children,-and the unhappy religious differences which, by separating large bodies of people from the communion of the Church, make it impossible for the state to establish, at the public expense, any satisfactory system of combined education. I say "impossible," because I presume that the result of the attempt which was made two years ago to effect this object, as regards the manufacturing districts, has convinced most persons that no such endeavour is likely to be successful. The scheme in question was honestly intended; it was carefully and considerately framed; and it was as little open to objection as any scheme can be, * Let a and b be two numbers, and d their difference, and we shall have a―b=d; in adding to the two terms, or sides of the equation, we shall have a−b+b=d+b, but-b+b destroy each other; so that a=d+b, which result indeed we have already proved in the case of numbers. By an algebraic expression, however, such as the note contains, the general principle itself will be better understood by the pupil, inasmuch as a and b may represent any numbers whatever. which attempts to unite in the same system of religious education two things, which are in their own nature irreconcileable, namely, free scope for the exposition of the truth as the Church must require it for her members, and satisfactory security for the religious scruples of those who are unhappily separated from her communion, and reject her teaching. This, however, of good resulted from the attempt, that the real bearing of the question became better known; a prejudice which existed in many quarters, that it was the Church which alone hindered, by needless objections, a combined system of education, was extensively removed; the necessity was more clearly shown for increased efforts of individual benevolence; and the hearts of the members of the Church were stirred to strengthen the hands of the National Society for the Education of the Poor by an amount of pecuniary aid, which, though inconsiderable if compared either with the means of those from whom it was derived, or the necessity which it was designed to relieve, was nevertheless large in comparison with the ordinary measure of similar contributions, and has been productive of the greatest good in those districts for which it was specially designed. Our agricultural parishes have, however, derived from this source only an indirect and comparatively slight advantage. But I hope that, independent of this, some improvement has been made, and still is in progress, and that the foundations of its more extensive development are being laid.—From a charge by the Bishop of Salisbury, 1845. TRAINING SCHOOLS; THEIR IMPORTANCE AND MEANS OF SUPPORT. Especially, I regard with the greatest satisfaction, as a means of this, the permanent establishment of our training-school for mistresses by the purchase of the lease of very commodious premises in the Close at Salisbury. And having now had a four years' experience of the working of this institution, I may say, that there is no single thing, since my connexion with this diocese, which I look back upon with more satisfaction, than the having been in any degree instrumental in its establishment. I confidently hope that we are training up in it, under the judicious care of the excellent mistress, who has from the first conducted it, and the indefatigable superintendence of the secretary of our board of education, a class of mistresses in some measure at least qualified morally and intellectually for the difficult task of educating the young; and herein especially, that they are taught that the work of instruction is not a mere hireling's task, but a christian calling—a ministration in the Church of Christ, which, though a subordinate one, yet lays the foundation upon which the superstructure of faith and godly obedience is to be built. And I beseech you, my brethren, to look in this light upon the office of those who under you have the care of the little ones of your flocks. Encourage them to feel that they are associated with you in a common work. Uphold their rightful influence and authority, and they will the more readily strengthen yours. Teach them to place confidence in you, that so you may assist their judgments, and supply their deficiences. Of those who have been regularly trained as pupils in our training-school, twenty-six have been sent out to situations, and our accounts of nearly the whole of these are extremely satisfactory. Two more are appointed to schools, and are on the point of taking charge of them; and there have been on an average during the last six months, twenty-eight pupils in the school. The above are in addition to those who have been received for shorter periods, as being either already mistresses of existing schools, or appointed to schools about to be established. Of this class of pupils we have had twenty, who, we hope, have profited by the instruction they have received, limited as this has necessarily been. Let me, however, take this opportunity of pointing out how the successful working both of our training-school, and of the diocesan board of education generally, must depend in great measure upon the interest which is felt in |