admitted, by no means justifies the conclusion that they are finite, that is, expressible in finite terms. If one of the coefficients had happened to be the series 1+++ &c., it would have been perfectly definite, though not expressible in finite terms, when the terms are decimal places. No one will admit à priori the necessity that the coefficients in the developement of 1+x', are, with respect to r, expressible in finite terms. There is as much reason for the same admission in the case of a, in which, nevertheless, the fact is not true. The proof, therefore, that Q (r−1) (r−2) (r-3), &c. cannot be a part of the coefficient, is inconclusive; but even were it established, there is another class of expressions which would need consideration. We refer, for example, to the form Q ((1)2++(−1)2+) which also satisfies the condition of being always nothing when r is any whole number. We cannot, therefore, help hoping, for the sake of the mathematical sciences, that this principle is not of extensive application.

There is this admirable point about our author, that in stating a process which he feels to contain defective reasoning, he never for one moment endeavours to elude detection, by clothing the fallacy in many words. Whatever the error may be, and whether he falls into it himself, or merely gives it as a part of the usual Cambridge course, the same concise simplicity is preserved. In one instance at least, he has judged it necessary to overturn in a note, the demonstration in the text. Having given a proof that every expression has as many factors as dimensions, he adds in a note, that the proof just given is imperfect, and proceeds to show that it amounts to taking for granted the thing to be proved. He then tells the student, that the correct reasoning is too abstruse for him, and that he must take the thing for granted. This being the case, why is such a demonstration inserted in the text? It is reasonable enough that, to a learner, authority should sometimes be sufficient for the reception of a fact, though never when it can be avoided; but that bad reasoning should be furnished to avoid the appearance of appealing to authority, where good reasoning is too difficult, is a principle against which we must protest. The author seems to think he is bound to give either a proof, or something that looks like one. We hold, that the less that which is not a proof, is made to look like one, the better.

The second and third parts of the work, containing the theory of equations, the summation of series, and several other subjects, are judiciously selected, and well explained. We must, however, observe that more is necessary, particu

larly in the theory of equations. In treating of this lastmentioned subject, a notation is used, which though sufficiently simple and systematic for a student whose algebraical studies are to be bounded by what he finds here, is not well calculated for him, who is hereafter to read modern works on analysis. Algebra is a language to be learned, as well as a method of reasoning to be understood, and the pupil who comes to the works which are now written, without previously accustoming himself to their notation, has the disadvantage of being obliged to pursue his studies in a foreign tongue. When the science was in its infancy, no problem needed many letters, and it was therefore almost indifferent which were chosen. As more complicated questions were introduced, the practice was to use the letters of the English and Greek alphabets indiscriminately, until they were exhausted. The works of Euler will afford repeated specimens of this inelegant method; simple as the illustrious author always made his explanations, the reader is retarded by the necessity of recollecting the different parts of a very unconnected system of notation. The writings of Lagrange and others, introduced the method now universally adopted, of signifying different magnitudes of the same kind, not by different letters, but by the same letter with accents or figures written underneath. Thus in a problem of mechanics, in which a system of forces is considered, and also the angles which their directions make with a given line, the forces would be represented by P, P, P, &c., or P, P', P" &c., and the angles by 0, 0, 0, &c., or e, e', 0" &c. This method, to say nothing of its obvious advantages, is now generally used, and the student should therefore be early drilled into its use. It would, then, be convenient that a general equation which is made the subject of reasoning, should be written in the following manner:

an.

2

and that its roots should be represented by a, a, &c., up to Dr. Wood has followed the old system, which is a defect in his work to those who intend to make it the stepping-stone to higher studies. The same thing is remarkably prominent in the fourth part of the work, on the application of algebra to geometry. The notation is not so constructed, as to point out, without explanation, the correlations of its several parts. As an instance, we subjoin the formulæ for changing one set of co-ordinates to another, than which nothing can be conceived which will look more like the notation of Euler, and less like that of Lagrange.

y = sd + p z + s v x = f − m z + yo

On this part of the work we may say, that it is utterly unlike every modern book on the subject, which is a disadvantage in itself, to say nothing of the superior simplicity and generality to which we have now attained.

In concluding this notice, we have only to remark, that with all its faults, we think this an elementary work of a high character, and inferior to none published in England for the beginner in algebra. It would be very possible to make a book which should more nearly contain what is most necessary to be known; but looking at the manner in which elementary writers usually perform their task, we can also say, that it might easily happen that the amended treatise should be so deficient in clearness, that the pupil would learn more from the work of Dr. Wood, and be better pleased with the subject.

GOLDSMITH'S HISTORY OF GREECE.

Goldsmith's History of Greece. 2 vols. 8vo. Edinburgh: 1813. Abridged for the use of schools. Twelfth Edition, 1830.

To write good elementary histories is no easy task. We should hardly think it necessary to begin with so trite an observation, except that the signal failure of most of those who have undertaken it furnishes reason to believe that the class which writes, and the class which reads have come to different conclusions upon this subject. Yet the difficulties of the task are sufficiently evident. A history for the use of children or schoolboys must be short, or it would not be read at all; yet it must be more than a chronological summary, for unless we can interest readers of this class, the impression made on their minds will hardly outlast those made on the eye. Considerable fullness, therefore, must be given to some parts, while others must be proportionally curtailed and as those events which can be made most interesting to a young mind are, in a majority of cases, not really the most important, the author is likely to be often divided between the desire of being instructive, and the fear of being tedious. Take for example the Persian and Peloponnesian wars, and suppose that only a given space could be assigned to them jointly the unity of action,-the decisive and magnificent results of the former,-are much better calculated to arrest attention than the numerous indecisive actions, and complicated intrigues which mark the course of the other. The former, therefore, would probably be told in detail; the latter

would be a mere summary of events. Yet as a means of making us acquainted with the Greek character, the Peloponnesian is infinitely more valuable than the Persian war. And supposing these leading points to be properly selected, still we ought not to jump through history like a quagmire, over a series of disconnected stepping-stones, so that the reader looks back from his present to his last halting-place, and wonders how he got where he is: while at the same time it is no easy matter to connect our stepping-stones as they ought to be connected, by narrative at once clear, correct, and short, giving due prominence to those facts upon which important consequences have hinged, and discarding what is merely episodical, or exerts little influence upon the future.

Sound judgment, a competent knowledge at least of the subject, and a clear and pleasing style, are the minimum of qualification which can be expected from the writer of an elementary history. In the last of these points, Dr. Goldsmith was pre-eminently successful in the two former, as far as knowledge of Greece is concerned, he was eminently defective, yet his history is the only one which has obtained general currency for purposes of education. This appears to be much more easily accounted for by the absence of competitors, than by the intrinsic merits of the work: and as it is important that the young student's first impressions, defective as they may be, should at least not be fallacious, we propose to examine its claims to the patronage which it has so long, and so extensively enjoyed. As written by Dr. Goldsmith, it consists of two octavo volumes, containing about eight hundred pages; but this larger work is, we believe, almost entirely superseded by an abridgment, which from its size and price is better fitted for schools; while those who have money to buy, or time to read the original, mostly seek their information in books of higher reputation. We shall, therefore, chiefly direct our attention to the abridgment, comparing it at the same time with the original, that we may keep the blunders of each distinct. We may here notice, that the former is preceded by rather a pompous advertisement, vaunting the author's intimacy with Dr. Goldsmith, and knowledge of his sentiments upon historical composition, concluding with an assurance that this edition has been carefully revised, and several errors which had crept into former ones corrected. If the abridger really possessed any advantages, it is a pity that he did not make better use of them: and if this is to be considered as a corrected edition, melancholy indeed must have been the state of former ones. Orthography, at least, ought to be the peculiar care of an editor: if the prin

cipal cannot spell, his deputy ought to spell for him. Both original and abridgment, however, are deformed by the grossest carelessness in the spelling of names; and as the blunders of the one are continually repeated in the other, it is an unavoidable conclusion, that ignorance has had some share in their production. Thus in both we have Lelexa for Lelex, Leona for Leana, Calisthenes for Cleisthenes, Selinuta for Selinus, with a long train of others which it would be tedious to enumerate. In the first forty pages of the ori ginal, there occur at least twelve blunders in names, and a very cursory examination of the first volume has furnished us with a list of forty-four mistakes, without reckoning those which are repeated,

The chief faults of the history now before us are twofold. First with respect to the nature of its contents, it is overladen with irrelevant and childish matter, such as ought not to find place in any history of Greece; and much is told at considerable length, which in so short a work should have been merely sketched, to the exclusion of other and more important matter secondly, the execution is careless and inaccurate in the extreme, a fault produced, we suspect, by trusting entirely to modern historians, or at best consulting only the latest and least authoritative of the ancient writers. There are indications which lead us to suspect, that Herodotus, Thucydides, and Xenophon, were sealed books to the author: assuredly they might as well have been so for any use which he has made of them, where their authority would have militated against that of Plutarch or Justin. A necessary consequence of imperfect acquaintance with the Greek authors is, that the book is written in a thoroughly unscholarlike tone; and is evidently the composition of a person, who had no distinct notions of the things and times about which he was writing.

Of the mythic period, no account is given, except that a few pages are devoted to relating the origin of the chief cities of Greece. Sixty years ago, the chronology of Newton was much less generally admitted than it now is, and therefore while protesting against the extravagant antiquity assigned to these cities, we ought hardly to censure the author for adopting what was generally admitted by the authorities of his day. Yet there is a glaring absurdity in the precision with which we are told that Sicyon, a city of which no one remarkable event is recorded, during its assigned duration of 1000 years, was founded B. C. 2089, only 259 years after the flood, 233 years before the next event recorded, the foundation of Argos in 1856, 600 years before it is pretended