is written again on that point, as our Lord did with Satan at the time of his temptation.

There is an explaining away system which has greatly led to the mutilation of divine truth.

If we would but follow the Church's teaching we should do well, and not often be at a loss. In her articles and formularies we find positive statements on each side, as it were, of a subject; that is, a doctrine is set forth, and the guard against its misapprehension appears in the form of another positive statement of another truth. Now were we to compare one part with another, in order to discover harmony, I think we should see more and more striking features of agreement with the word of God.

We should find the principle recognised of stating positively all that is said in the word of God relative to a particular doctrine or duty, and then as decidedly setting forth what is "written again," to prevent our erring; thus directing each individual to an exercise of his faculties and graces, in order to ascertain the bearing of each equally true statement upon his faith and practice.

I believe this rule may be observed in every arrangement in the Book of Common Prayer, for bringing the great and leading truths of the Bible before our minds. There is no undue prominence given to one doctrine to the suppression of another, but all are fully stated and carefully adjusted with admirable wisdom. I look in vain for such closeness to Scripture, such bringing out of the divine mind (be it spoken with reverence), out of the church.

Upper Clapton.

Your obedient servant,

WILLIAM SPENCE.

HOW TO TEACH NUMERATION.

THE first thing to be done in teaching arithmetic, seems to be to give children some idea of number; perhaps this may best be effected by comparing quotity, or the idea of how many? with quantity, or the idea of how much? How much butter? how many eggs? will answer the purpose of illustration well enough; and, after one or two other instances have been given by the teacher, the children may be called upon to furnish a few;-they will soon see, that to answer the former question, the sizes of what we speak of is to be looked to; to answer the other, how often it presents itself must be considered. By taking instances, where the things we inquire about are of different kinds, and the "how many?" is applied to the total, they will see that each marble, top, slate, book, apple, &c., is no longer thought of as a marble, top, slate, &c., but merely as something-no matter what— that serves with others, whether of the same kind or not, to make up the collection. Sometimes, a less general term than “ things" may be used, and yet be applicable to all the items of the total; while, from its being less general, it will, of course, be more descriptive; and, when

once the children are aware of this, they may be required to find such terms; their doing so will be good exercise-not in arithmetic, certainly, but in language; the practice thus slightly begun, may be extended and continued in the grammar class, and be made the occasion of much useful information, respecting the extent and the comprehensiveness of nouns, and the way in which the latter is increased, and the former lessened, when we put an adjective to the noun; large box, for instance, being an expression, more comprehensive in its description than box alone, and, therefore applying to a smaller number of things than bor would apply to. To answer the question, how much? we measure, weigh, or guage the amount, as well as we can under the circumstances; to answer the question, how many? we count, using a different name for the total at every fresh item we find. But, such a supply of fresh names, soon leads us into difficulties. Suppose it be required to count the men in a very large army; the number of names that would enable us to go on as we began, giving a fresh name to the total for every man we counted, would be greater than the number of words of all kinds in the English language; yet, even this immense number, if we had them, would go but a very little way towards enabling us to count the people in London; and the number of names with which we could do this, though considerably larger than that of all the words of all the languages of Europe put together, would leave us completely in the lurch, even if we got them, and could remember them, when it came to be question, as it is every 10 years, of taking an account of the number of people in the kingdom.

The fresh name plan, then, being quite unmanageable, we should soon be obliged to give it up, and we consult our convenience by giving it up after the first few numbers. These we make into a collection, to which we give a name, and which we follow up by others, containing the same number of items as the first, and called by the same name, not using a fresh one, till we have made up as many similar collections as each of them contains items; so that if we began with 6 items, we should go on making collections of 6 till we got 6 of them, then we should make those 6 sixes into a fresh collection, to which we should give a name, and with this new kind of collection we should do just what we did with the first, go on till we got 6 of them, then make these 6 into a new collection, with a new name, and so on. We do something of this sort, when we call 12 quills a dozen, and 12 dozen a gross, only that we do not keep it up as we do in counting. The first collection we do actually use in counting, has just as many single items in it, as we have fingers on both hands; this would seem to show that counting on the fingers is by no means a new invention. This first collection we call a ten, and to show that it is one collection, and not so many single items, we express it in figures by the same mark as one of the single items, only we put the marks one remove further on; just as in writing money, the same figure serves to express one shilling and one penny, but we put the one that means a shilling in a different column, away from the one that means a penny. If we are speaking of a ten and one or two single things more, we put the 1 or 2 more down by the side of the one that means a ten, thus-11, one ten

can

and one, 12, one ten and two, just as we say a shilling and a penny, a shilling and two pence. With this first sort of collection, we count up to nine tens; and 9, if we add one, the nine will become another ten, and so we have ten tens, that is to say as many of the first sort of collection, as that collection contains of single items; this, therefore, gives us a fresh sort of collection; we call it a hundred, and, as before for one ten, so now for one hundred, we use just the same figure as for a single item, only, we put it not one remove onwards, as for the ten, but two removes. Thus, if we have occasion to speak of one hundred, two tens and four, (meaning four single things, which is what we always mean when no collection is named), we put the 2 for the tens, one remove from the beginning, and the 4 at the beginning. In writing figures the beginning is on the right hand side of the slate or paper, though in writing words it would be on the left hand side; in the part of the world from which our figures came to us, words are written from right to left, and not as we unite them in Europe, from left to right. With hundreds we proceed just as we did with the tens; when we have ten of them we make them into a fresh collection, which we call a thousand; still using the same mark as for one single item; we put it 3 removes from the beginning, and if we have with it any hundreds of tens, we put them at their proper distance from the beginning, as before explained; 1235 would be one thousand, two hundreds, three tens and five. It may happen, that we have thousands and tens, but no hundreds; in such a case we put a mark in the thousand column, or place as we generally call it, not exactly to show that there are no hundreds, for that would be needless, but to prevent the thousands from seeming to be in the hundreds' column, as they would if the place were not otherwise filled up. The mark we use is 0, we call it a cypher, or a naught; aught without the "n" means any thing; therefore naught means not any thing or nothing; just as we say none for not one, neither for not either, never for not ever. Were you ever in Scotland? Never.

With thousands, as before with hundreds, we go on till we have ten of them; for this fresh collection we move the one that marks it 4 removes from the beginning, but here we begin to be more sparing of our fresh words; we use two of the old names instead of a new one, and we call this collection a ten-thousand, just as we say a six-pence; in like manner, we call ten of these ten-thousands a hundred thousand, because we had before called ten tens a hundred; but when we get to the third sort of fresh collections, since we used a fresh name, and which, if we did not use one now, we should call a thousand-thousand, we call it a million. With these millions we do not just what we did with the tens, using a new word, the first time we made a collection, but what we did with the thousands, keeping the fresh word till the third time we have occasion for a name, and using two old ones the other times; the fresh one is billion. The following numbers 5,676,245,689, therefore, would be five billions, six hundred millions, seven ten millions, six millions, two hundred thousands, four ten-thousands, five thousand, six hundreds, eight tens and nine. For two tens, we say twenty; for three tens, thirty; for four, forty; the "ty" holding the

place of the word tens, and the rest of each word showing how many tens there are; so for three single items and one ten, we say thirteen ; for four and one ten, fourteen; and so on.

The two names that come next after ten seem to be a departure from this way of making up new names out of old materials, and to be altogether fresh names; they are not so however, eleven means one left,* after the collection of ten has been made up, and twelve means two left. Throughout the whole of the row of figures before given, the principle, that every collection is to consist of ten of the collection, before it has been observed, and would continue to be observed how far soever we might stretch our row of figures. In like manner, 5, 6, 7, continue to mean 5, 6, 7, etc. sometimes what these somethings are, the column in which the figure stands must determine, but a 5 for instance in the 12th column, would be just as much a 5, and would just as much mean 5, as it would if it were at the beginning; whether the 5 somethings were 5 tens of thousands, or five hundreds of billions, can no more prevent this from being the case, than does the circumstance of whether the 5 somethings at the beginning would have been 5 men, 5 ants, or 5 drops of water. A billion, or a thousand millions, is more than the highest probable number of human beings on the earth; we have therefore carried our counting or numeration quite far enough, to show how easily amazingly large numbers may be expressed with a very small supply of really fresh names, though we are still able to give a somewhat different name to the total; for every single item we add no two of the names of totals from one to a billion, and beyond it being exactly the same, though the different parts of which almost all of them are made up, are used again and again, forming altogether a most useful and ingenious piece of patchwork." Counting, then, is adding one by one, giving the proper name each time to the new total, and if we want to put two numbers together, 26 and 43 for instance, we may do this too by counting; we have only to begin with one of them as our starting point, and count away until we have counted, and therefore added one by one, as many as there are in the other number; the total we get to, will clearly be the total made by both these numbers put together. This way of adding two numbers is by no means imaginary; it is often practised, not merely by children, but by grown persons, and to make the reckoning still safer people often do it on the fingers; this is a very common way, indeed, of finding out what day of the month some day of the week after next will be. But if we were always to add in this way, it would be nearly as difficult and tiresome as having to use fresh names for all our totals in counting.

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Instead, then, of taking only one at a time we take several, and to do this requires nothing more than a little experience in counting; thus if we know that the next number but one to 6 is 8, and we have to add 3

* By a reference to any dictionary in which the etymologies are given, it will be seen that the two words eleven and twelve are made up in Anglo-Saxon, (to which language, like most of our household words, they belong), of the first two numerals and the participle of the verb " to leave."

to 5, instead of adding one which would give 6, or two, which of course would give the next number to 6, we add the three at once, which must give the next number but one to 6; we say then, that 5 and 3 are 8, instead of saying 5 and 1 are 6, and 1 7, and 1 8; if afterwards we want to add 5 and 4 we see it must be 1 more than 5 and 3, and therefore the next number to 8, or 9; if we have 4 and 3, that must be 1 less than 8, or 7; so that with a little management and a very slight knowledge of counting, we need never be at a loss to add together any two small numbers. Suppose we have 9 and 7; if the 9 were 1 more it would be a ten, and then the total would be clearly a ten, and seven or 17, as it is, the total must be one less than this; a ten and six, or 16; again if it had been 8 and 7, 8 being two less than the first collection, the total must be two less than 17, or 15; 5, as we should readily remember, being the next number but one before 7, which little matter serves to remind us, that a knowledge of counting enables us, as we see here, to take away one number from another, as well as to add numbers together.

And this taking may we often have occasion to practice while we are adding numbers together-7 and 5 are how many? taking 3 of the 5 and putting them to the 7 will give us 10, and 5 being the next but one to 3, there will be only 2 left; these put on to the 10 will of course give a ten and 2, or 12; here then we have split the 5 into two parts to add them separately,* and while we are gaining readiness and experience in counting, this way of adding will be found useful, especially in helping us to keep clear of blunders, which generally give a great deal more trouble than any little contrivances we might have used to prevent them would have done. Suppose now that we have two really large numbers to add together for instance, 29,425, and 62,543, beginning by adding to the larger number the 5 single items of the other we make it

62, 5 4 8, next adding the 2 tens we have

62

5 6 8, then the hundreds, which will give

6 2 9 68, after these the thousands, giving

6(11,)9 6 8.

Now here we have in the thousands' column 11, and therefore one more than is wanted to make a collection of the kind next higher than thousands; in such a case we do just as in counting, we make up a collection of the next kind, add it to the 6 ten-thousands, making them 7; and put down in the thousands' column only what is left of the thousands, after the new collection has been made up; here there will be only one thousand left, and the total will be 91,968. When

* Persons unacquainted with the difficulties which children commonly find in their early attempts at figures, may be inclined to consider all these minutia a mere caricature of explanatory teaching; those who know something of the difficulties will, I think, be less likely to do so. The teacher should endeavour to draw from the children's previous knowledge and mere common sense a portion of the necessary information, and a great deal of what cannot be thus elicited may be suggested, instead of told. Simultaneous questioning, however, especially with a large class, will seldom, I believe, be found to operate beneficially upon more than a small section of the children subjected to it, and that section will consist precisely of those who are least in need of the awakening effect meant to be produced. By simultaneous questioning I mean, of course, questions to which the answers are simultaneous.