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11 x number of inhabitants This number will therefore be

33 X number of sponsors number of inhabitants

And we come to the conclusion, that the 3 x number of sponsors. greatest number of children under the charge of each sponsor would never be likely to exceed the number discovered, upon dividing the whole number of the inhabitants of the district by three times the number of persons who would be willing to act as sponsors, which fact we may represent by the following algebraical formula:

number of inhabitants Number of children =

3 x number of sponsors ; and therefore conversely,

number of inhabitants Number of sponsors =

3 x number of children. "This conclusion is of great value, since, if we have a certain number of persons willing to enter upon this work in any parish or district, it enables us to ascertain at once the utmost limit to which the charge of each of them is ever likely to amount. And vice versá, if we determine the limit to which we think each sponsor might safely charge himself, it enables us to pronounce the number of sponsors that we ought to have available in order to carry on the work with comfort and completeness.

To apply these deductions to a particular case. Take a parish where the number of inhabitants is 4,238. Now, supposing that 30 communicants of each sex (i. e. 60) could be engaged to act as godfathers and godmothers to the childre of such a parish, we should have the greatest number of children ever likely to be under the charge

4,238 4,238 of any one sponsor =

= 23.5 nearly, which means

3 X 60 180 23 or 24. Should this number appear too large by five or six, and should it be thought desirable to begin to act with such a body of sponsors, that the number under the charge of each should never exceed 18, we shall easily, by the inverse process, find out how many sponsors must be engaged for this purpose, for we shall have necessary

4,238 4,238 number of sponsors =

= 78 nearly, or 39 of each sex.

3 x 18 54 “ These numbers have been chosen without design, and merely to

or

number of inhabitants

will give the whole number of attendances that must be 11 given by sponsors at the public baptisms of the district or parish during the same

number of inhabitants period, and consequently 11 x number of sponsors,

will give us the number of times that each sponsor must attend, or the number of children to whom he must stand sponsor every year. But as we proceed on the supposition, that only one of the three sponsors takes the baptized child under his or her more especial superintendence, one

number of inhabitants third of the last found number, or

will be the number

33 x number of sponsors, of children brought every successive year under the special superintendence of each sponsor.”—(p. 38.)

show how the two formulæ may be made use of; and nothing is easier than to apply the calculations we have made with reference to the above parish to any other in the kingdom. The great value of the formulæ is, that they enable any clergyman to form a judgment as to the means he has in his power for undertaking this labour of love.(pp. 40–44.)

“ The following table gives the number of sponsors that would be necessary in nine parishes of different amounts of population, in order that the number of children under the care of each might never be likely to exceed 30, 25, 20, &c., as might be determined.

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“ There is,” observes the author, “perhaps, no spot in England where the difficulty of applying the described principle would be greater than in the last parish but one of the above list; and yet what are three or four hundred out of eighteen thousand ?”—(pp. 86–88.) For, as the above table shows, if in a population of 18,000 there were only 400 communicants willing to become sponsors, the number of children under the care of each would never exceed 15. “ Let these considerations," he adds, " stir us up at least to an active inquiry into the resources we possess for carrying out this object. If they were to lead us to nothing further than a diligent examination into the number, the zeal, the spiritual attainments, and the practical piety of our communicants, they would confer a most important benefit on the established church." -(p. 88.)

The author makes some remarks upon the number of godchildren that a sponsor might, without extravagance, hold himself adequate to have at any one time under his care; and also suggests various expedients that might be adopted for setting the machinery of the proposed plan in motion. Another paper, giving the substance of these sugges. tions and remarks, will conclude our notice of his work.

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Rev. Sir,—In my former paper on Arithmetic, which you did me the honour to insert in your Journal, I confined myself strictly to an explanation of the Separate Sum System, carefully avoiding any digression

which might draw your readers' attention from the subjects I desired to explain, viz., the nature of the system, and the method of applying it.

I will now, with your permission, lay a few observations before them on the same subject; but not so exclusively explanatory. Among these remarks, the query from yourself claims precedence.

Allow me to answer, that when a child has acquired a habit of counting by the aid of his fingers, it is long before he leaves it off; some never do. One motive, then, for preventing them from using the assistance of objects, in learning the first rules of arithmetic, is to hinder them from acquiring a habit which would materially impede their future progress. But there is another motive; the principal object in the first cases of each rule, is to fix the different tables firmly on the memory of the child, as may be inferred from the statement in the former paper, that although they were allowed the free use of tables when working the sums, yet they were not to be advanced to a higher case till they could work the lower with facility without their aid. Now if they were permitted to use their fingers, the tables would not be learned, and the object in making the lower cases defeated.

As it would be a very vain and foolish thing to imagine that instructors will discard their methods of teaching, and adopt others, without substantial reasons for believing, that the superseding method is so much superior to their own as to justify and demand the change; it is necessary for me, before I can expect any to adopt my plans, to explain the points in which I conceive the superiority of the separate sum system consists.

The pleasure it affords the children, particularly the younger ones, is an important feature in this system; for surely no one will deny that, the methods being equal in other respects, the children will make quicker progress when the method is one they like, than they will when the system fails to excite their interest. Does any one object, that the systems at present in use do excite such feelings in the children? my answer is, that in the upper classes, with a competent teacher, they may; but look at the children in the lower classes, when ciphering under a boy teacher, and observe whether the lesson affords them pleasure and absorlys their attention. Perhaps some of your readers may reply, “The pleasure they take in your method is owing to its novelty, and would soon wear off.” This is a mistake. I have tried the system sufficiently to know, that the children do not get tired of it, nor exhibit any of that listlessness which we so often see among children who are ciphering, all the class in the same sum. But the system must be properly administered, or this, as well as all other advantages, will vanish. Thus, if a child be advanced into a case or rule for which he is not fully prepared by properly going through the preceding cases; or if the assistance he needs is not promptly rendered, he will get careless and indifferent, and will not exert himself for the remainder of the lesson.

One cause of the listlessness of which I have spoken is, that some boys in the class are further advanced than the others; and they imagine that the sum is given out, not for their instruction, but for those whom they consider so far behind themselves, and this opinion must, to some extent, be correct. I think it will be generally found, that in

large classes one-fourth will be in advance of the medium, and another fourth as far behind it; and this is especially the case where the children are arranged according to their attainments in reading and general information. Teachers of British schools, as is well known, endeavour to meet this difficulty by making the ciphering class distinct from the reading class; and by making the classes smaller, and consequently increasing their number. With the unsoundness of this expedient, particularly the latter part, your readers must be well acquainted. The separate sum system, however, effectually cures the evil; for if each boy in a class, however large, belonged to a case or rule different from all the others, there would be no difficulty in suiting them, each with his proper case; because the hearer can hear three boys in three different cases, almost as easily as one.

Another great advantage consists in the rapidity and correctness with which children perform the various operations of arithmetic; and the short time, comparatively speaking, in which they acquire this facility. Several causes combine to produce this effect : first, the gradual development of the rules; next, the readiness with which the children are obliged to work one case, before they are advanced to another, thus teaching them one thing at a time, and that well;” next, the emulation which the system creates. Lest I should be misunderstood in the application of the word emulation, I beg to observe, that by it I mean an anxious desire to work the greatest number of sums, but by fair and legitimate exertion, without attempting to obtain any unjust advantage. Sometimes it means, that boys who have been too short a time in the case to hope to be at the head of the list, are proud and happy if they work three or four sums more in the present lesson than they did in the preceding. If there be a lurking principle of evil in such emulation, I must confess my inability to discover it. Another cause is the eager attention which the children pay to their instructors, when the latter are teaching them how to work a new case or rule, and consequent quickness with which they acquire the necessary information. If any doubt this eagerness to learn, let them reflect that the child who is put in a new case has already mastered the preceding ones, and consequently has the remembrance of these conquests to encourage him to attack new difficulties with vigour and perseverance; with this aim he will be glad to avail himself of all the assistance he can obtain. Besides, it is probable that other children will be advanced into the new case at the same time with himself, and the hope of being first in the race about to be run, or the fear of being left behind will be another stimulus to exertion.

The habit of steady and persevering attention to one object, which this system demands, will naturally have the effect of fixing their minds more steadily on their other lessons when engaged on them.

I will mention as another advantage, the pleasure it affords the parents to hear their little ones* (as I know by experience they will), talking of

* I care not how often I use these words, for I can scarcely too strongly impress on the minds of your readers, that it is for little ones, for children from five to nine years of age, that this method is chiefly intended.

the number of sums they have done, as they will call it ; not that they pay much attention to the children's statements, but they will rejoice to observe that their children are fond of school. This may not be an improper place to mention, that the master who has this system in full operation in his school, need not be afraid of having his scholars removed to other schools ; the children who have been taught on this system will have a decided disrelish for other methods.

Another important feature in this system is, that the teacher can give each child information suited to his peculiar case, without interrupting the studies of the other children. In a class of from twenty to thirty pupils, there will rarely be more than one requiring assistance at the same time.

Some teachers to whom I have explained the plan, have expressed their belief of its being an easy method for the master. In this opinion I do not concur: it may indeed leave his hands more at liberty, and thus allow his eyes, those most necessary organs in a school, larger opportunities for exerting their influence; but, like all other systems, it requires the most vigilant superintendence; without this, it would quickly degenerate into the fruitful source of confusion, noise, petty animosities, dishonesty, and deceit. But under the watchful eye of a teacher who has that complete and undisputed control of his school, which every competent master who tempers firmness with discretion in his management can most certainly acquire, unless improperly interfered with by the managers, this system will produce the advantages I have described, without causing the master greater anxiety or more labour than other systems.

As masters who have at heart, the welfare of the children committed to their care are naturally desirous to seize every

favourable opportunity to impress upon them some useful lesson, they would of course be ready to avail themselves of any fit circumstance which might occur during ciphering time. For instance, a master might observe a hearer, by accident or design, give two sums instead of one to a boy ; should he return one, the master would commend him for his just and praiseworthy conduct, by telling him that it caused him much pleasure to find that he was so honest, and had such proper spirit as to reject an unfair advantage. He might then, for the instruction of the rest of the class, inquire his reasons for refusing to profit by the opportunity. These ought to be, a fear and dislike of doing wrong, and because the accepting it might be a real disadvantage ; for if he could have sums given him without working them, he would probably become idle and careless; and then, should he be advanced to another case on the faith of the number marked against his name on the list, it would quickly be found that he was totally unfit for it, and he would have to suffer the disgrace of being put back to the lower case again. When reasons such as these are laid before young children, in language they can clearly comprehend, they are not, in general, slow to perceive, and act on the conviction, that it is their duty and interest “ to be true and just” in their ciphering. And might not such a salutary conviction have some beneficial effect“ in all their dealings ?" On the other hand, should the child attempt to secure the advantage which the accepting two sums

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