m n is contained in it 6 times; therefore, if m n is equal to 1, A B will be equal to 6. For the most part, however, in all processes of the kind there is a remainder; but it is proposed to consider this part of the subject at some future time; and there are, besides, other measurements, of which it is also advisable we should defer speaking at present.

* From (Lat.) quantus, how much.

5. Quantity is, in general terms, every thing that is capable of being measured, or, in other words, whatever may be increased or diminished. Thus time is a quantity, because it is measured by days, or by years, or by centuries, and can be increased or diminished: or, again, the time which has elapsed since our birth has increased in a continuous and uniform manner, whilst that which remains to us in this world will in like manner deBy a similar system of explanation, the pupil will have little difficulty in comprehending that, for the same reason, all solid bodies, surfaces, lines, coins, weights, forces, are so many quantities.

crease.

* From (Lat.) con, together, and cresco, cretum, to grow.

6. All quantities, of whatever kind, may be represented by numbers; so that these numbers may at the same time present to the mind the idea of quantity, and that of the genus or kind to which they belong. In this case they are called Concrete Numbers.* But if the idea of genus or kind is excluded from our consideration, whilst we only think of or of how many units our subject matter consists, the number itself is then called "An Abstract Number;"* ex. gr., 5 shillings, 20 years, are concrete numbers, be

the quantity,

* From (Lat.) abs, from, and traho, tractus, to draw.

cause they express, at the same time, the idea of quantity and that of the genus or kind, which we have fixed upon to represent unity; on the other hand 5, and again 20, are abstract numbers, because we are in their case only thinking of the quantity, the number of units, without allowing the idea of genus or species to enter into the question.

All abstract numbers are homogeneous, although we 'may have at first become acquainted with them from the admeasurement of heterogeneous* quantities, or quantities of different kinds.

*From (Gr.) heteros, another, and genos, kind.

*From (Lat.) numero, to reckon.

2. Numeration* of Whole Numbers.

7. Besides obtaining the idea of whole numbers (whether from lines, surfaces, or solids), it is requisite that we should give to such numbers names, and be able likewise to represent them to the eye by certain signs, either printed or written; now numeration is that part of arithmetic which professes to do all this. We may then define numeration to be the art of forming all kinds of numbers, of giving names to them, and of representing them by a certain small number of words and of characters, which last are called figures.

In numeration, then, as generally taught and understood, ten of such characters, or signs, or figures are used. It is hence called decimal numeration; it would, however, have been as easy to have adopted any other system. There are therefore two kinds of numeration—that expressed in words, and that expressed by written or printed signs.

The first gives their names to numbers, and the second teaches us to represent them by figures. We now proceed to speak of both of these.

=

8. Every representation of unity, whatever it may be in itself, is named one, and is represented by the figure 1. If one is added to one, arithmeticians express it by one plus one, thus forming the collection two, which is represented by the figure 2. In words at length it would be said, one plus one equals two, which, written or printed, would be thus expressed, 1+1=2. That is, for the word "plus," is substituted the sign +, and for “equals," the sign and by the use of these signs, when teachers and pupils are familiar with them, it becomes more easy both to give and to receive instruction in arithmetic. continuing to add or join one unit to the preceding collection, and in representing this new collection by a new figure, we soon obtain all the simple numbers of these there are altogether nine. Thus two and one make up the number called three, and which is represented by the figure 3; or, on the former plan, we may express it thus: 2 + 1 3 and in the same manner we form the number four, or

In

4

+

3 1

4

1

5

5

1

+

1

7 + 1

16

679

8

9. Both the names and signs, therefore, of these nine numbers, are arbitrary,* whilst all other numbers, with very few exceptions, are formed in a regular manner from 10. If to nine there is added one, we obtain the following number, or ten, which, in order to avoid burdening the memory with many arbitrary words or signs, is termed a tenth, and is written 10, and is consequently equal in value to the ten units already spoken of; so that on the principle of analogy,* in the same manner as we have counted as far as nine units, we may count as far as nine tens. One-tenth in addition will then give the number a hundred, the ten-tenths forming again, on of analogy, a unit of a certain kind, called a hundreth. With this new unit it will be easy to count, in the same manner, as far as nine hundred.

* From (Gr.) ana, in proportion, and logos, a word or ac

count. the principle

11. These new collections of tenths, of hundreths, &c., considered as abstract numbers, that is, leaving out of the question what is the name and value of the collection which such unit represents, are expressed by the same figures which have already served as signs to distinguish the nine primary units, but in order to make it evident to what species they belong, (whether to tenths or hundredths), whilst the primary units are placed to the right, the tenths are next in succession to the left, and the hundreths the next to the tenths, or the third to the left. At the same time, to signify the particular value of each figure, by preserving the rank to which its belongs; a tenth sign, in addition to the nine already spoken of, is employed, viz.—0, called a nought or zero, which serves likewise to express the absence from any particular rank or ranks of any kind of unit whatever :

N

12. The names of the different collections of tens, are formed from the names of the figures which stand for them, a termination, expressing this modification in their signification, being added to the original words. Thus we have two-tens called twenty of which the sign is

20

30

40

50

60

70

80

90

We see that in these written, or printed characters, all the collections of tens occupy the second rank to the left, and are in fact, only to be distinguished by their being thus placed, from the collection of units, from which they derive their name and signification.

13. For the several collections of hundreds, which answer to the corresponding series of units or tens, we do not employ any particular designation, but express them in words as one-hundred, two-hundred, three-hundred, and so on as far as nine hundred, whilst in their written characters, they appear thus-100, 200, 300, and so on to 900.

14. It is evident, that if we join successively to all the collections of tenths, all the collection of units; and again to all the collection of tenths, those of the tenths and units together, we shall then have a series of numbers from one to nine-hundred and ninety-nine, or 1 to 999, of which the common difference, or the difference of each, from the number preceding and succeeding it, will only be one or unity.

15. It will be observed in the names of the numbers, composed of tens and the several simple numbers, that with the exception of eleven and twelve, there is throughout a perfect analogy, thus: Ten and one are called eleven, or may be expressed thus:

=

10 + 1 11 10 + 2

=

12

With regard to the written or printed signs for eleven and twelve, and for the following teens, the simple numbers, or those which represent the nine units, are successively substituted in the place of the nought (0) or zero, which marks their proper rank or situation in the written sign for ten, or 10.

(To be Continued.)

F. CURIE.

Extracts from Charges.

ARCHDEACON SINCLAIR ON THE RELATIVE POSITION OF THE NATIONAL SOCIETY AND COMMITTEE OF COUNCIL.

I. On the first mentioned subject, that of National Education, no small amount of misconception prevails, more especially as regards the relative position of the church and of the state in that department: I ought, perhaps, to say the relative position of the National Society and the Committee of Privy Council. Perhaps there is no person in the kingdom who possesses equal means with myself of removing any false impressions that may have arisen in the public mind, as to the transactions between the society and the government, and I gladly avail myself of this occasion (the only one that may be afforded me) to explain briefly the whole case, by relating each event as it occurred.

In the month of September, 1839, when I first entered on my duties as secretary to the National Society, I received letters every day from clergymen and other parties engaged in building church schools, who stated, that in compliance with their application to government for assistance, they had obtained the offer of a grant from the parliamentary vote, but that a new condition was annexed, which caused them much embarrassment. A state inspector, neither sanctioned nor directed in any way by the authorities of the church, was to have the right of entering their schools, and without inquiring into the religious education of the pupils, was to examine and report exclusively upon their secular attainments. The declared object of his visit was "to secure a conformity in the regulations and discipline established in the several schools, with such improvements as might from time to time be suggested by the Committee" of Council. As state inspection was in itself a novelty, and as the form it had assumed seemed liable to serious objection, my correspondents throughout the country expressed an anxious wish to be advised, whether they should submit to the required condition, or reject the offered grant. The whole of the parties were impatient for an immediate solution of their difficulties, inasmuch as all were called upon by the government to return an answer within a prescribed period, which would soon expire.

Applications of this kind poured in upon me from every side, and caused me much embarrassment. My embarrassment was not lessened by the circumstance, that the committee of the National Society had previously fixed their next meeting for a day subsequent to the period alluded to. The members were all dispersed over the kingdom, and yet some immediate measures must be adopted. In this emergency I issued a private circular, advising the several applicants to ask the privy council for further time, in order that, before returning a final answer, they might consult the National Society.

This circular produced the favourable result I had anticipated. On the 16th of October, 1839, when the committee assembled, I was able to inform them, that in case they should resolve on advising the applicants to decline public grants, I had already ascertained the general disposition to comply, however serious the pecuniary loss to be sustained. This fact had great influence. The committee adopted the decided measure of recommending, that until the obnoxious condition was either modified or withdrawn, public money should be refused. The grounds of this recommendation are fully stated in the society's papers and reports; and so deeply did they impress the public mind, that out of 204 applicants for government aid, only 49 accepted it; of that small number, 14 afterwards declined it. Others in the strongest terms expressed their wish to do the same, if their poverty would permit them; while several boards of education in the country intimated their desire that the society should have

recourse to stronger measures, and expel from union any school, the managers of which should throw it open to the state inspector.

This general adherence to the principles of the National Society was most encouraging. I looked forward with much confidence to the meeting of parliament. I anticipated a combination of various interests, Scotch and Irish, as well as English, in our favour; but when parliament met, the agreeable prospect was far from realised. The Scottish establishment, torn by intestine faction, submitted to the new system; the Presbyterians of Ireland followed the example of their brethren in Scotland: the organs of all political parties, however discordant on other subjects, were upon this nearly all harmoniously united against us. And to complete the isolation of the society, those members of the House of Commons to whom the church on most occasions looks especially for support, declined to bring the question of national education forward. When earnestly appealed to, and solicited at least to call parliamentary attention to the grievances of the parties disappointed of public grants, they replied that church education was not the first business to be attended to, but rather church extension.

This determination of our friends in parliament was unfortunate. The public mind was not at that time occupied about building churches, but was effectually excited upon the subject of building and improving schools. The excitement, however, like every other of the same kind, could not be expected to last. The opportunity to obtain redress would rapidly pass away; the attempt to divert the current of popular feeling into another channel would be a failure, and would only make it cease to flow at all. Accordingly, the resolution to apply for a parliamentary vote in aid of building churches was put aside as impracticable; while, in the meantime, the cause of education languished, and was in danger of ruin.

Providentially, the church escaped that calamity. Various circumstances had disposed the Committee of Council to adopt any arrangement by which national education might be fixed on a secure basis. Their lordships had not sufficiently anticipated the opposition of the clergy and promoters of schools throughout the kingdom, to their recent aggressive measures. They were embarrassed by the complaints and remonstrances of applicants desirous to obtain grants without submitting to the new condition. They began to see that the National Society was more favourable than they had supposed to the cultivation of the popular understanding. Thus it happened that both parties, the National Society-and the Committee of Council, were predisposed to any terms of accommodation by which intellectual development should be combined with moral and religious feeling.

On the negotiations for that purpose which were so ably and successfully conducted by our revered metropolitan, I shall only now remark, that our church owes his grace a debt of gratitude, which, if it be proportioned to the difficulty and delicacy of the questions at that time to be determined, can never be forgotten or repaid. By the arrangement which the archbishop concluded with government, and which, without any change whatever, still regulates the distribution of the parliamentary grant for education, all the objections stated by the National Society to the plan of inspection proposed by the Committee of Council were removed. In the regulations for the appointment of inspectors, full security was given for the selection of individuals in whom the church might have confidence. In the instructions, also, drawn up their direction, all interference on their part with the authority of the clergy and school managers was strictly and repeatedly prohibited, while due prominence was given to sound religious and moral training. The new arrangements gave general satisfaction to the church. The best thanks of the Committee of the National Society were unanimously voted to the archbishop. The managers of schools throughout the country readily accepted public grants. Nearly all those who