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 extravagantly 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 (?) publishers of the Encyclopædia some time within the next twenty years.†

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, &c., Lond., 1816), for instance, speaks of number as "possessing an essence separate from sensibles, and a transcendency fabricative and at the same time paradigmatic" !!!

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

Thibet.

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 expressed by

3579,

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. g., thirty thousand three hundred and three would be written

Millions. Hundreds of thousands. Tens of thousands. Thousands. Hundreds. Tens. Units. 0 3 0 3 0

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

13 THE NOTION OF A PLURALITY OF INHABITED WORLDS, ETC.

249

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.

Philosophy.

B. S.

IS THE NOTION OF A PLURALITY OF INHABITED WORLDS CONSONANT WITH SCIENCE AND REVELATION?

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 question. We knew well that few would have the temerity to oppose the theory which we are a second time called upon to advocate, without being either compelled to object in toto to the strict limits which we felt obliged to make, or prevented from ignoring the real point and the true philosophical view of the subject under a cloak of sophistical generalities. The natural consequence is, that we have been assailed most vehemently on this, and we are told that there can be no pretext whatever for thus confining the argument to man; and even if there were any, that it would be monstrously absurd and incomprehensible. But, surely, it lies with the objectors, and not with the defenders of any opinion, to frame their objections in whatever way they deem necessary; and inasmuch as the present discussion has mainly arisen from the publication of the "Plurality of Worlds," which only maintains that men do not inhabit any other world than our own,

the foundation on which our opponents, as well as ourselves, ought to base the discussion. Not only is it just, but it is the only way in which an argument can be conducted at all. For, knowing by experience that the bodies of irrational creatures may be so formed as to be capable of a thousand various conditions of existence, and suited to almost every state, we may in answer to every scientific objection against the plurality of worlds state, that God is able to create, and actually has created, animals of so many different qualities, that there is no reason to suppose he has not created them to suit other systems; while every moral objection is necessarily removed by the fact that the brutes are not moral beings. And, again, if we include moral and intellectual beings of a different nature from man's, we, being ignorant of the conditions of their existence, are evidently incapacitated from reasoning in any way about them. Thus this question 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 a 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 66 Iman can do without the orbs of heaven," and therefore that they are not created for him! If ever there was a premise which required substantiation, surely this would be the one. 66 Threlkeld," however, does not demean himself so much as to satisfy the rational doubts of his opponents; but, like an indifferent physician, who gives large quantities of medicine without regard to the condition of his patients, "Threlkeld" compounds a most extravagant dose of nonsense and error, without thinking that it is extremely likely to send us into such a violent fit of hysterics as to cause a speedy transmission to Hades.

"L'Ouvrier" has, indeed, attempted to show that geology, so far from being our support, is entirely antagonistic to us. This attempt he illustrates by the instance of a discovery of a fish's tooth, which "seemed " to have belonged to a true fish, and "indicated" a higher organization than a mere fish. It will, however, be observed that not only is the language in which this discovery is intimated extremely hesitating and undecided, as though there was great uncertainty about the matter, but, even if it were true, that it would be beside the question; for we neither affirm nor deny the existence of organic life in other systems, but we merely deny the existence of human life. So that when he asks, "Why may not Mars and Saturn also be possessed of these conditions of animated existence, if they are now turbid masses of lava and mud?" our readers must not be confounded at the solemn paraphernalia in which he dresses his sentiments, but simply be ready with this reply, "It is just as probable as not, and has no reference whatever to the point at issue."

"Philalethes," also, had a similar design of nullifying the argument which we derive from geology. But, though we have read

So much for geology; what of astronomy? "Threlkeld" has given us a little small-talk, for the sake of appearance, about Jupiter. He tells us, on the authority of Dr. Lardner, that its attraction "does not exceed terrestrial gravity in a proportion which regains the admission of any difference of organization of the inhabitants, exceeding what may be imagined, without removing Jupiter from the general analogy of the earth." The gravity at Jupiter's surface is nearly 2} times that on the earth; and consequently, suppose a man to be in the former, everything he carried about him, even his own limbs, would appear to him to be 2 times heavier than before! And yet we are told that no change in a man's constitution would be requisite! We advise "Threlkeld," "L'Ouvrier," "Philalethes," and other Pluralists, to read Edgar Poe's story of Hans P., in which they will find full particulars for making a journey up there. Let them only starve themselves before they go, that their consequent leanness and lightness may counterbalance Jupiter's gravity!

Passing by the moon, to gratify "L'Ouvrier,"-who, as it appears to us, wrongly objects to it having any analogical force in the argument-we must make a few additional remarks on the planetoids. In our previous article we quoted a passage as a conclusion to an argument derived from their existence, which stated that "the fact of a planet being inhabited is rather the exception than the rule, and therefore must be proved, in each case, by special evidence." Have our opponents done this? Have they given us distinct and well-authenticated facts, showing that men inhabit even those few planets which they have alluded to in the course of their articles. The onus probandi lies upon them, but with ill-dissembled ingenuity they have endeavoured to transfer

it to us. They demand from us the evidence of our view, while in reality it is incumbent on themselves alone to produce theirs. They ring in our ears the hackneyed cry of pugilists, "Prove it, prove it," while they forget that their own opinion is a vapoury theory, which requires substance to make it conceivable. The à priori probability is on our side, but they endeavour to wrest it from our grasp and appropriate it to themselves. But we shall not be duped, we shall not be inveigled. We require every case to be established, every instance to be substantiated, and, until this be done, we bid defiance to all their sneers and sophistry, and take our stand on the broad basis of science and revelation.

entirely forgotten his credentials, which ought to be especially convincing in such a case, where a deduction so false is made from a premise so true. Moreover, no one can deny that God possesses an omniscient mind, but many true Christians will reject with disdain the five questions which are appended to the declaration of that fact. To the first we return a decided affirmative, to the last four a decided negative; and we hope that this answer will be satisfactory.

As to the interrogations following the proposition that "God is all-wise," we can scarcely read them without a shudder, in spite of "Threlkeld's" attempts to drown such feelings by his ridicule. They imply, and, indeed, he says so shortly after, that each star not inhabited is a useless thing, and created with an unwise purpose! We would refer "Threlkeld," with all seriousness, to the statement of one of his own side on this point:-" Neither could it be supposed, without offence to the Divine majesty, that these things were created in vain because they have no inhabitants" (p. 145), just to show that we were not far wrong when we made some severe strictures on those who thus reasoned.

With respect to the first moral argument* which we adduced, viz., that if men inhabited other worlds, Christ must have died over and over again for their respective transgressions, we can only say that, considering their being inhabited by men to be a fact, no one can deny the necessity of a frequent repetition of a Saviour's death. For man, prior to his fall, was made a natural creature (1 Cor. iv. 46), i. e., one who stood in his own 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 answered before (p. 23).

Again, we are told that God is almighty, and therefore must create other races of being. Though "Threlkeld," to judge from the confidence with which he draws his conclusion, might appear to be a prophet divinely appointed to communicate this fact to us, we must beg to remind him that he has

We must acknowledge that our division into scientific and moral arguments in our former article was not very accurately followed; but we were so hurried that this was overlooked.

make? Surely that, inasmuch as analogy teaches that all matter with which we are experimentally acquainted is created for man alone, all other matter (stars, &c.) is likewise created for him alone.

How delightful such a belief is we need scarcely pause to inquire. What a confirmation it is to the truth of revelation! That the Father should have thought us worthy of the creation of the myriads of crystalline palaces which gem the firmament, and that the Son should have thought that the same persons were sufficiently worthy for him to