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sage (Pythagoras) who based most of his philosophy on the doctrine that “ number is the ruler of forms and ideas, and the cause of gods and demons!” *

The simplicity we have been noticing, though undoubtedly of natural origin,---being evidently derived from the use of the ten fingers (in the manual or palpable arithmetic already described) as the first arithmetic of uncultivated tribes,—was the work of time and labour. Classification by pairs, or the binary system of notation, is at once the simplest and the most ancient arithmetic. We find traces of this in the dual number, familiar to every schoolboy who has commenced the Greek Grammar, and which, we are told, is to be found in the languages of all barbarous tribes. In such a system we should describe seven as "a double pair, pair, and one;" the clumsiness of which, when compared with the decimal system, is strikingly evident. The system, nevertheless, has some advantages, and was most extravagantiy extolled by the great mathematical rival and cotemporary of Newton-Leibnitz. The next step probably would be to assume the double-pair, or four, as the root of the scale or classification; men naturally would, for expedition's sake, in counting, take a pair in each hand. We find traces of this quaternary system in our own language, in the words, throw, warp, &c. The ancient Mexicans used this system, and the famous tetractys of Pythagoras probably owed its mystical properties, in his eyes, to the existence of some such system of numeration. In other cases we see traces of various systems of numeration in their application to particular subjects, and in the special names for particular numbers which occur in different languages, as dozen, score, &c. But our space forbids enlarging upon this interesting part of the theory and history of arithmetic. We hope that enough has been said to show that this subject opens up a field for philological and historical research well worthy of cultivation and study, and we rejoice to be able to point to at least one treatise on the subject of standard worth and masterly skill. The historical portion of Dr. Peacock's " Treatise on Arithmetic," in the “Encyclopædia Metropolitana," is the work to which we refer; we live in faint hope of seeing it republished by the enterprising (3) publishers of the Encyclopædia some time within the next twenty years.f

Arithmetic, as has been already said, is almost necessary to man as an art, and it seems not improbable that the first faint attempt at written language consisted of a symbolical representation of numbers. Men would naturally feel the want of an inventory of their valuables long before the idea of correspondence suggested itself to their minds. But though early attempted, the written language of arithmetic was very late in arriving at anything like simplicity. The Grecian arithmetic, though perhaps the easiest and best, was a terrible matter of letters and accents, &c., while the Roman numerals were so cumbrous, that the conquerors of the world never arrived at any respectable “proficiency in figures.” To multiply cloxcix by CXLVIII is an operation of a complex and unpleasing look, to say nothing of the time which the mere writing of such awkward figures would occupy.

* We cannot, however, feel the same sympathy with the incomprehensible vagaries of some of the modern revivalists (or caricaturists ?) of ancient philosophy. A Mr. Taylor (Theoretic Arithmetic, fc., Lond., 1816), for instance, speaks of number as “possessing an essence separate froin sensibles, and a transcendency fabricative and at the same time paradigmatic"!!!

+ We may just mention, en passant, that Dr. Peucock attributes our present system of notation to


Nor were our own ancestors much better off. The following quaint old verse, of the date of 1570 (which, slightly varied, is well known in our own day), expresses nothing more than the natural result of attempting figures in those days:

“Multiplication is mie vexation,
And Division is quite as bad,
The Golden Rule is my stumbling stule,

And Practice drives me mad." But for all details on these matters we must again refer to the work of Dr. Peacock, where the reader will find a vivid picture of the miseries of clerks and schoolboys in olden time. We turn, then, to point out the nature of the modern notation. The words uttered in speaking would soon become tedious in written calculation, and yet to have a separate symbol for each number would introduce unbearable confusion; the great object, therefore, would be, if possible, to adopt something analogous to the classification by which oral numeration is rendered so simple and clear. This has been done most successfully by the use of ten symbols, which take their meaning from their position. Standing alone, the figures 1, 2, 3, &c., mean one, two, three, &c., but when two or more figures come together, each one has tenfold the value it would have if it were moved one place to the right. Thus, if the figures be 21 or 214, the 2 in each case signifies tenfold what it would do if it were occupying the place of the 1. Beginning, therefore, with the figure on the extreme right, and recollecting that it is of the lowest denomination,-i.e., under ten,and that each figure becomes of tenfold value as it moves towards the left, we are at once able to express any number by means of ten written symbols. Thus, if the number is three thousand five hundred and seventy-nine, it is espressed by—

3 5 7 9, i. e., 9 units, ten-times 7 units, ten-times-ten-times 5, or one-hundred-times 5 units, &c. In fact, our written notation is merely an elliptical mode of writing the oral notations; e.g.:-

1 (thousand) 8 (hundred and) 9 (ty, i.e., tens) 3. If, then, we can keep the positions distinctly marked, our notation will be as perfect when written as when spoken. This last point is accomplished by the use of the cypher (0), which may be regarded just in the same way as the blank space between each word in printing or writing, i. e., as having no value or use in itself, but merely distinguishing, and thereby determining, the value of the other figures or letters. In writing down any sum, therefore, we in fact write down that it consists of so many millions, hundreds of thousands, tens of thousands, thousands, hundreds, tens, and units, or individuals, omitting these denominations for brevity's sake; e. 9., thirty thousand three hundred and three would be writtenMillions. Hundreds of thousands. Tens of thousands. Thousands. Hundreds. Tens. Units,



3 0 3 or 0030303. The first two cyphers, however, being useless, are omitted, and we write 30303.

We have now slightly sketched out the preliminary fields of investigation, and illustrated the manner in which the principles and nature of every rule should be sought out

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and studied. What can be more interesting than the ingenious machinery of numeration, when considered in its reasons and essence; and yet what is more uninteresting than the account given of it in most books of arithmetic?

The limits prescribed to us compel us here to close our remarks; if we have succeeded in explaining the views we hold, it is now in the student's own power to work out the plan by the aid of his own persevering thought. In closing our remarks, however, we cannot but commend the attention of all to “ Elements of Arithmetic," by Augustus De Morgan, &c., fifth edition. Our reasons for this preference are explained fully in the following quotation from one of the earlier editions :

“Since the publication of the first edition, though its sale has sufficiently convinced me that there exists a disposition to introduce the principles of arithmetic into schools as well as the practice, I have often heard it remarked that it was a hard book. I never dared to suppose it would be other. wise. All who have been engaged in the education of youth are aware that it is a hard thing to make them think."

Other works which we commend especially are Colenso's “ Arithmetic;” Tate's “ Commercial Arithmetic;" and (as a collection of examples for practice) Walkinghame's Tutor's Assistant,” the terror of our grandfathers, which has been lately re-issued, with considerable improvements, by Routledge and Co.

B. S.




NEGATIVE REPLY. It was with some reason that we gave, in we think it but just that this also should be our opening article, a definition of this ques- the foundation on which our opponents, as tion. We knew well that few would have well as ourselves, ought to base the discusthe temerity to oppose the theory which we sion. Not only is it just, but it is the only are a second time called upon to advocate, way in which an argument can be conducted without being either compelled to object in toto at all. For, knowing by experience that the to the strict limits which we felt obliged to bodies of irrational creatures may be so make, or prevented from ignoring the real formed as to be capable of a thousand various point and the true philosophical view of the conditions of existence, and suited to almost subject under a cloak of sophistical generali- every state, we may in answer to every ties. The natural consequence is, that we scientific objection against the plurality of have been assailed most vehemently on this, worlds state, that God is able to create, and and we are told that there can be no pretext actually has created, animals of so many Whatever for thus confining the argument to different qualities, that there is no reason to man; and even if there were any, that it suppose he has not created them to suit Would be monstrously absurd and incompre- other systems; while every moral objection hensible. But, surely, it lies with the ob- is necessarily removed by the fact that the Jectors, and not with the defenders of any brutes are not moral beings. And, again, if opinion, to frame their objections in whatever we include moral and intellectual beings of way they deem necessary; and inasmuch as a different nature from man's, we, being. the present discussion has mainly arisen ignorant of the conditions of their existence, from the publication of the “Plurality of are evidently incapacitated from reasoning

orlds," which only maintains that men do in any way about them. Thus this question bt inhabit any other world than our own, I must be regarded simply as pertaining to man.

We accordingly pass over all that “ Threl | what he says very carefully, we cannot perkeld” has said about the “Almighty putting ceive any common sense in the passage; an intellect or conscience under the feathers and he must have mistaken the reason [why not under the skin also?] of a dove," for which we here adduced the analogy of &c., and turn to the geological view of our | geology. Let us, therefore, state it again: subject. “Threlkeld” acknowledges that - Geology shows us that the world at one geology proves that God has bestowed pecu- time was unfit for habitation, and after å liar care on man in time, but shrinks from similar manner, we may suppose, that many the analogical supposition that he has done of the stars are now also unfit for habitation. so equally in space. His reason for this is We do not intend to prove here so much as original; but, we should imagine, more conclu- “Philalethes" appears to think, and we cansive to himself than to any one else, viz., not see why our argument is denied. that“ man can do without the orbs of hea- So much for geology; what of astronomy? ven," and therefore that they are not created “ Tbrelkeld" has given us a little small-talk, for him! If ever there was a premise which for the sake of appearance, about Jupiter. required substantiation, surely this would be He tells us, on the authority of Dr. Lardner, the one. “ Threlkeld,” however, does not that its attraction “ does not exceed terresdemean himself so much as to satisfy the trial gravity in a proportion which regains rational doubts of his opponents; but, like an the admission of any difference of organizaindifferent physician, who gives large quanti- tion of the inhabitants, exceeding what may ties of medicine without regard to the condi. be imagined, without removing Jupiter from tion of his patients, “ Threlkeld” compounds the general analogy of the earth.” The a most extravagant dose of nonsense and gravity at Jupiter's surface is nearly 2} error, without thinking that it is extremely times that on the earth; and consequently, likely to send us into such a violent fit of suppose a man to be in the former, everyhysterics as to cause a speedy transmission thing he carried about him, even his own to Hades.

| limbs, would appear to him to be 24 times "L'Ouvrier” has, indeed, attempted to heavier than before! And yet we are told show that geology, so far from being our that no change in a man's constitution would support, is entirely antagonistic to us. This be requisite! We advise“ Threlkeld," attempt he illustrates by the instance of a "L'Ouvrier,” “Philalethes," and other Pluraldiscovery of a fish's tooth, which “ seemed ” lists, to read Edgar Poe's story of Hans P., in to have belonged to a true fish, and “indi- which they will find full particulars for cated” a higher organization than a mere making a journey up there. Let them only fish. It will, however, be observed that not starve themselves before they go, that their only is the language in which this discovery consequent leanness and lightness may is intimated extremely hesitating and unde- counterbalance Jupiter's gravity! cided, as though there was great uncertainty Passing by the moon, to gratify “ L'Ouvabout the matter, but, even if it were true, rier,"—who, as it appears to us, wrongly that it would be beside the question; for we objects to it having any analogical force in neither affirm nor deny the existence of the argument-we must make a few addiorganic life in other systems, but we merely tional remarks on the planetoids. In our deny the existence of human life. So that previous article we quoted a passage as & when he asks, “Why may not Mars and conclusion to an argument derived from Saturn also be possessed of these conditions their existence, which stated that “the fact of animated existence, if they are now turbid of a planet being inhabited is rather the exmasses of lava and mud?” our readers must ception than the rule, and therefore must be not be confounded at the solemn parapher | proved, in each case, by special evidence." nalia in which he dresses his sentiments, | Have our opponents done this? Have they but simply be ready with this reply, “ It is given us distinct and well-authenticated just as probable as not, and has no reference facts, showing that men inhabit even those whatever to the point at issue."

few planets which they have alluded to in “Philaletbes," also, had a similar design the course of their articles. The onus proof nullifying the argument which we derive bandi lies upon them, but with ill-dissembled from geology. But, though we have read ingenuity they have endeavoured to transfer it to us. They demand from us the evidence entirely forgotten bis credentials, which ought of our view, while in reality it is incumbent to be especially convincing in such a case, on themselves alone to produce theirs. They where a deduction so false is made from a ring in our ears the hackneyed cry of pugil- premise so true. Moreover, no one can deny ists, “ Prove it, prove it," while they forget that God possesses an omniscient mind, but that their own opinion is a vapoury theory, many true Christians will reject with disdain which requires substance to make it con- the five questions which are appended to the ceivable. The à priori probability is on our declaration of that fact. To the first we side, but they endeavour to wrest it from return a decided affirmative, to the last four our grasp and appropriate it to themselves. a decided negative; and we hope that this But we shall not be duped, we shall not be answer will be satisfactory. inveigled. We require every case to be As to the interrogations following the established, every instance to be substantia- proposition that “God is all-wise,” we can ted, and, until this be done, we bid defiance scarcely read them without a shudder, in spite to all their sneers and sophistry, and take of “ Threlkeld's” attempts to drown such feeloạr stand on the broad basis of science and ings by his ridicule. They imply, and, indeed, revelation.

he says so shortly after, that each star not With respect to the first moral argument* | inhabited is a useless thing, and created with which we adduced, viz., that if men inbabited an unwise purpose! We would refer “ Threlother worlds, Christ must have died over and keld,” with all seriousness, to the statement over again for their respective transgressions, of one of his own side on this point:-“Nei. we can only say that, considering their being ther could it be supposed, without offence to inhabited by men to be a fact, no one can the Divine majesty, that these things were deny the necessity of a frequent repetition of created in vain because they have no inhabia Saviour's death. For man, prior to his tants” (p. 145), just to show that we were fall, was made a natural creature (1 Cor. not far wrong when we made some severe iv. 46), i. e., one who stood in his own strictures on those who thus reasoned. strength and by his own wisdom. The very! There is only one more argument at which first temptation by which he was assailed we would glance; for, being expressly reproved the small amount of that strength quested to be brief, we feel obliged to omit and wisdom, and exhibited him as a weak several of the others. "Threlkeld” says: and frail being, totally incapable of self-sup- “ All the works of God upon this globe are, port. Now, if he be created in another in some way or other, conducive to the welplace, it makes no difference, so that where- fare of intelligent beings. Everything upon ever in the universe he may be stationed, this earth is created for the benefit of manthe first temptation would, according to kind” (p. 53), and therefore, forsooth, the analogy, make him a sinner, and, conse- stars, being useless to man, are inhabited by quently, in need of a Redeemer. Accordingly, other intelligences! But, suppose that the Christ must either have died in each world stars are useful to man, (and who in his where man existed, or the efficacy of his senses will deny that they are?) what is the atonement must have spread to them. The deduction which “ Threlkeld” himself, as first our opponents concede to be unreason- well as every other rational being, would able; the second has been sufficiently an- make? Surely that, inasınuch as analogy swered before (p. 23).

teaches that all matter with which we are Again, we are told that God is almighty, experimentally acquainted is created for man and therefore must create other races of alone, all other matter (stars, &c.) is likebeing. Though “ Threlkeld,” to judge from wise created for him alone. the confidence with which he draws his con- How delightful such a belief is we need clusion, might appear to be a propbet divinely scarcely pause to inquire. What a confirmappointed to communicate this fact to us, ation it is to the truth of revelation! That we must beg to remind him that he has the Father should have thought us worthy

of the creation of the myriads of crystalline * We must acknowledge that our division into palaces which gem the firmament, and that scientific and moral arguments in our former

the Son should have thought that the same article was not very accurately followed; but we were so hurried that this was overlooked.

persons were sufficiently worthy for him to

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