ception, we might continuously advance to the conception of a number one degree higher, so long as we were able to keep accurate count of the precise amount of repetition by which that particular step in the numerical scale was characterised. But such a limit, without some artificial aid of the memory, would very speedily be reached, and in the lowest stage of mental cultivation would probably not be placed beyond the number three or four. The requisite aid, however, is not far to seek, and is found by all the families of man in the quinary division of the hand, the fingers of which afford a ready scale on which to tell off the units of any group, up to five, of which one might wish to take count. Thus beginning with the thumb of the left hand, the first finger would mark a single repetition of the kind under enumeration, or a second member of the group; the middle finger a second repetition, or a third member of the group, and in this way primitive man would learn to associate a definite amount of repetition with each of his fingers, and might attain to a clear conception of the first five numbers antecedent to the use of any vocal designation. But sooner

or later the demands of language would give rise to the use of spoken names, one, two, three, four, five, denoting the numbers told off on each successive finger; and these, being constantly repeated in regular order, constitute a series so rooted in the memory that each name serves at once to bring before the mind the preceding portion of the series, and thus affords a standard of the extent of repetition to which it corresponds, as distinct as that supplied by the fingers passed over in telling numbers on the hand. When the fingers on one hand are exhausted, we may either go through a second series with names of the form five-one, five-two, five-three, &c., which are actually found in many rude dialects, or the higher numbers may be told off on the other hand with a fresh set of names, six, seven, eight, nine, ten, corresponding to the fingers of the second hand.

When the ten digits are exhausted, we advance, by the continued addition of one, to the conception of higher numbers under designations of the form, ten-one, ten-two, &c.; two-tens, two-ten-one, &c. ; three-tens, &c.; ten-tens, ten-ten-one, &c.; and so on, to an indefinite extent, using the convenience of compendious names for such of the powers of ten as may be convenient for resting-places in the process of numeration.

The composition of such a system of numbers is enounced in the following definitions

:

(2) Two is the aggregate of one and one, or, shortly,

Two is one and one.

(3) Three is two and one, &c.

(11) Eleven is ten and one, &c.

(20) Twenty is ten and ten, or two tens, &c.

(100) A Hundred is ten tens, &c.

By reference to these definitions the numerical value of all arithmetical expressions may be ascertained or compared with each other, because the definitions afford the means of reducing each of the systems in question, when necessary, to its constituent units, or of

building it up out of them, and thus of ticking off against each other the systems to be compared, unit by unit. To show, for example, that seven and six are thirteen, we have, by defn. (8),

Seven and one are Eight.

Adding one to each side,

Seven and one and one are Eight and one,

Or, by defns. (2) and (9),

Seven and two are Nine.

Adding one again,

Seven and two and one are Nine and one,

Or, by defns. (3) and (10),

Seven and three are Ten. And so on, till we come to

Seven and six are Thirteen.

As the number of a set of things depends exclusively upon the length of the series, one and one and one, &c., where each 'one' of the series answers to an individual of the enumerated class as it is successively brought under review in the process of counting, without reference to any difference between one individual and another, it is plain that the aggregate number of the class can in no wise be affected by the order in which the individuals of the series are counted. If I have a series of balls, black, white, green, and red, the aspect under which I regard them in counting will be, one and one and one and one, whether I take them in the order of black, white, green, red, or of red, green, white, black. And so, if I jumble together a set of (m) white balls and a set of (n) black ones, the tale of the whole will be the same, whether in counting I pick out first the white and then the black, or first the black and then the white. In other words, the sum made by the addition of (n) to (m) is the same as that made by the addition of (m) to (n), or algebraically,

m + n = n + m.

In a similar way it may be shown that the product of two factors (m) and (n) is independent of the order in which the factors are taken; that (n) times (m) is the same as (m) times (n); or algebraically that

nm = mn.

Suppose that we have five groups of seven balls each, it is obvious that the number will be seven times as great as if there were only one in each group, when the number would be only five in all; so that five times seven is the same as seven times five. Or to take the question more in detail, let the balls of each group be marked 1, 2, 3, &c., 7. Then there will in the aggregate be five ones, five twos, &c., and five sevens; making seven sets of five each. Thus it appears, from the nature of the conception, that things which are known as making five groups of seven each may be otherwise arranged in seven groups of five each, or, in other words, that five times seven is equal to seven times five.

If now we look back for a summary answer to the inquiry with which we set out, we find that our assurance in the universal truth of Arithmetic arises from seeing that the numerical equations which

form the body of the science are necessary consequences of the fundamental constitution of the numbers in question, as distinguished in thought or apprehended in actual existence.

The conception of every phase of Number consists, as we have seen, in a reference, more or less explicit, to a succession of units of definite length, wholly independent of the nature of the enumerated objects; and the demonstration of the numerical equation consists in showing, from the essential constitution of the numbers concerned, that the units contained in the combination on one side of the equation may be otherwise arranged in the groups indicated by the numbers on the other side. We show, for instance, that 7 times 8 is 56 by taking the units contained in 7 rows of 8 each, and showing, from the definitions, that they may be arranged in 5 rows of 10 each and one of 6. We find, from a gradual decomposition of the conceptions, that the mental operation, by which we enumerate the aggregate of seven groups of eight each, whether of balls or books or anything else, is identical with that by which we enumerate a group of 56, and thus we know with absolute certainty that things which are given us in the form of seven lots of eight each may be enumerated under the form of fiftysix. HENSLEIGH WEDGWOOD.

IX.-NEW BOOKS.

Darwinism tested by Language. By FREDERIC BATEMAN, M.D., &c. With a Preface by Edward Meyrick Goulbourn, D.D., Dean of Norwich. London, &c.: Rivingtons, 1877. Pp. 224.

The

The author's special argument is imbedded in a number of observations on the doctrine of Evolution generally. He seeks to establish three positions: (1) that articulate speech is a distinctive attribute of man, the ape and lower animals possessing no trace of it; (2) that it is also a universal attribute, all races having either a language or the power of acquiring it; (3) that the faculty of speech is immaterial. This last proposition is opposed by the author to all the different. attempts yet made to assign a local seat of speech in the brain the pathological and other evidence, he maintains now, as he has maintained before, is dead against them all, Broca's included. positive import of his proposition is thus disclosed :-"With these facts before me, I am tempted to ask whether speech, like the soul, may not be an attribute-an immaterial nescio quid, the comprehension of which is beyond the limits of our finite minds". He further declares for a spirit or organ of God-consciousness in man, which "differentiates him from the brute" possessing only a body and soul. Upon this it occurs to one to ask what Dr. Bateman means by "soul in the earlier sentence. If he means all that is not body in man, he degrades the "spirit," with the animal life, into a mere attribute". which looks very like materialism. If, on the other hand, he means the kind of life we share with animals,-how, by comparing language

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66

therewith, does he establish its distinctively human character? And, once more, is it language or is it spirit ("the organ of God-consciousness") that we are to take as the really differential element in man's nature? Dr. Bateman is not a very careful reasoner or writer.

The Dean of Norwich, who stands forward as sponsor for the work, argues about Evolution in Dr. Bateman's general strain, only more pointedly.

General Sketch of the History of Pantheism. 2 vols. Vol. I. From the Earliest Times to the age of Spinoza. London: Deacon & Co., 1878. Pp. 395.

The anonymous author describes his work as "merely an outline or epitome of a history," and as "chiefly a compilation, taken more frequently from translations and abridgements of the originals than from the originals themselves". After compiling in regard to Oriental and Greek Pantheism and sketching, in a fashion of his own, "the paganisation of Christianity and consequent decay of Pantheism" as far as the Rise of Scholasticism, he passes by a sudden stride to Servetus, Bruno, and Vanini, and will resume with Spinoza. It cannot be said that he compiles with such discrimination as to justify his work.

Proteus and Amadeus: A Correspondence. Edited by AUBREY DE VERE. London: Kegan Paul & Co., 1878. Pp. 184.

A veritable correspondence, under assumed names, carried on in 1876 between two friends-twenty years before pupil and master in a Catholic College-on the Existence of God and the human Soul. Proteus, the pupil, had strayed into "materialism" and Darwinism, accepting them intellectually but unhappy over them. Amadeus seeks to maintain the old orthodox positions against the modern objections. In the end the pupil is more than shaken; Darwin, as he allows, having "been hewed to pieces" for him by the master's "and Mivart's sword," and even Evolution being "emasculated and left harmless henceforth for ever". But still he cannot quite come back to the fold.

On the Nature of Things. A Science Primer. By JOHN G. MACVICAR, A.M., LL.D., D.D. With Illustrations. Edinburgh and London: Blackwood & Sons, 1878. Pp. 112. "This work is grounded on the belief of an Almighty Being possessing unity, omnipresence, and ever-blessedness, and awarding existence to a creation for the sake of manifesting Himself and extending blessedness beyond Himself, and, in a word, to be a mirror of Himself, so far as the finite can bear a likeness to the Infinite. After setting out with this cosmical law of assimilation, by its aid alone bearing on only one kind of created substance or energy (mind-stuff'), the author deduces the creation of the world of Spirits, and as their home the Universal Ether or medium of light. Then, as a beautiful cloudwork in the_azure of the Spirit World, he gives the genesis of Matter and the molecular system, culminating in this planet in the construction of the myo-cerebral organism, whose characteristic function is to construct a powerful tissue of organised ether or

the matter of light, which, being unified in its focus of vital action into an element of energy so powerful as to have recovered the primal attribute of energy-namely, mental power-is a spirit. And thus creation, after a lapse into matter, becomes the mother and nurse of spirits again, destined, if the design of the Creator is fulfilled, to find a home in heaven, the realm of light, and there to experience the final fulfilment of the cosmical law of assimilation and be blessed for ever.

"The author, anticipating the criticism that all this is merely the fond imagination of one who disregards the now prevailing views of men of science, and who still clings to his theological education, has devoted more than half the volume to the verification of his theory by a detailed appeal to natural phenomena and experiments in physics and chemistry, which his theory enables him to deduce and account for, but which the most recent speculations in the science of the day leave still in the dark."

Comparative Psychology; or, The Growth and Grades of Intelligence. By JOHN BASCOM. New York: Putnam's Sons, 1878. Pp. 297.

The author in his preface says:—

"Without tracing the history of intelligence, we are not prepared to decide what is primitive and what is acquired, what is original material and what is the deposit of growth. The empiricist cannot be fully and fairly met without travelling with him these spaces of evolution, and determining at least their general character and laws. This I have undertaken in the present volume. It is my purpose to test the nature and extent of the modifications put upon human psychology by its relations in growth to the life below it, and in doing this to reach a general statement of each stage of development. I have derived great benefit from many forms of the Empirical Philosophy: these I cheerfully acknowledge, while I must remain its unflinching adversary. The Intuitional Philosophy can and should appropriate these excellent fruits, and this volume is the result of such an effort."

The Balance of Emotion and Intellect: An Essay introductory to the Study of Philosophy. By CHARLES WALDSTEIN, Ph. D.

London Kegan Paul & Co.

"The title of this forthcoming Essay indicates that it is meant to form an introduction to the study of philosophy. Its object is to contribute to the development of the philosophical attitude of mind. The author first attempts to counteract prevailing fallacies with regard to the false opposition of Emotion and Intellect, Common and Scientific Thought, the Exact Sciences and Philosophy. He then gives a short Sketch of the History of Philosophy." Moralische Briefe.

Von A. HORWICZ.
Pp. 126.

Magdeburg: Faber, 1878.

The author of Psychologische Analysen here appears in the character of a censor, exposing the sores of the German body politic, and only not despairing of his country's future. The Germans, he declares, are suffering from "blue-devils," manifested especially in the socialistic madness. The follies and affectations of fashion have laid hold on men and women alike. Trade and industry are vitiated by deception and sham. And while a gross materialism is the only creed of the masses, true culture in the higher grades is becoming ever more