Στε μαν | ΑΣΤΥΑ---νακτας | μακαρας... (Suppl. 1025. (Prom. Vinct. 137. • Nor were the dactyl and ipondee the only feet fubftituted for the anapæft. The pyrrichius, the trochee, the tribrachys, were likewife admitted, as in the following examples from Afchylus and Seneca. In the former part of this quotation, method and propriety required an exemplification of a Spendaic anapæft by the production of fuch a verfe; and, in the latter part, the verfe from fchylus is the 127th in number in no edition, and the 297th in Pauw's republication of Stanley. This typographical error, however, is of trivial importance in comparifon with the author's own miftake, who does not feem to apprehend the Giftinétion between an anapaftic bafis and the common anapaftic verfe. The bafis uniformly concludes a fyftem of anapæfts, and frequently a period in the fyftem; and is univerfally two times, one long fyllable or two thort fyllables, of lefs length than the other, but fo modified as to form, we believe, univerfally the conclufion of an hexameter. Our author then refers us to p. 139 for fome further remarks on the anapæft; in which alfo he is inaccurate. No final fyllable in an anapæftie verfe of the Greek tragedians is common, in the grammatical fenfe of that term, except the last fyllable of the bafis ; which, like the fame fyllable in hexame ters, or pentameters, may be either long or thort by nature: but in the anapæstic series, if a short fyllable terminate an anapæftic verse, this either happens in the cafe of a word which admits the paragogic v, or that fyllable is lengthened by the introductory confonants of the following verte; or again, no fyllable of an anapaftic verfe can be short at the end, though naturally fhort, if fucceeded in the next verie by a word whose initial letters would lengthen that fyllable in a lineal arrangement. This peculiarity was pointed out by Dr. Bentley in his difputations on Phalaris, and is technically denominated the Synaphia of the anapaftic feries. Some other inaccuracies may be found in this volume, which will not escape the learned author on a revifal of his work; and, as another edition, we fhould think, will foon be demand. ed by the public, we will fuggeft a few hints, by which, in our judgement, that edition may be confiderably improved. As all poffible brevity, confiftent with perfpicuity, is of high moment in such a publication, both on account of the expence, which is always prejudicial to the reception of a school-book, and the impolicy of burdening the fcholar's memory or fatiguing his attention by the accumulation of a fingle line not abfolutely neceffary; we fhould recommend the shortening of fome occafional digreffions, and the entire fuppreffion of others. In our opinion, Alvarez's rules fhould be totally banished not only in their collective ftate, as exhibited in the introduction, but in their detailed form, as conftituting the texts and divifions of our author's reinarks and illustrations throughout the volume. Such fyftems, however ingenious. in themselves, contain of neceffity fo much nonfenfical and crabbed verse as to become very irksome to the learner, at a time of life when their utility is but imperfectly apprehended; and, whatever may be the practice in a neighbouring ifland, no judicious mafter, we think, will choose to adopt them here. We should prefer Mr. Carey's own ftatement of his rules, in the fame order if he please, but in his own English. The fynopfis alfo, which occupies the thirteen laft pages of the book, we should gladly find omitted. We will venture likewife to pronounce decidedly against fuch a multitude of examples in plain cafes; by which the book is, on the whole amount, very unneceffarily fwollen. Take the following specimen, as an illustration of our remarks. The plural increments I and U are fhort, as Quibus, Tribus, Montibus, Lacubus, Verubus:except Bubus, which has the penultima long, for the reafon alleged in page 34. Vivite felices, quibus eft fortuna peracta... (Virgil. Necte tribus nodis ternos, Amarylli, colores. (Virgil. Montibus in noftris folus tibi certet Amyntas. (Virgil. Præmia de lacubus proxima mufla tuis. (Ovid. Pars in frufta fecant, verůbusque trementia figunt. (Virgil. (Ovid.' P. 45. Half of thefe inftances, in our opinion, would have been fufficient and very numerous curtailments of a fimilar kind may be allowed without impairing the intrinfic value of this truly refpectable and edifying performance. The Principles of Algebra. By William Frend. 8vo. 5s. Boards. Robinsons. The Principles of Algebra: or the true Theory of Equations" eftablished on Mathematical Demonftration. Part the Second. By William Frend. 8vo. 35. Boards. Robinsons. THE first of these two volumes we have already briefly notired in our XXIId Vol. N. A. p. 345. The publication of the second seems to have completed the author's plan, and we shall therefore be more full in our investigation of it. The fcience of algebra is as pleasant to the fpeculative reclufe as it is to the practical geometrician, and we have often been furprised at its not being more widely cultivated in the prefent ftudious and enlightened age; for, excepting the univertities themfelves, there are few public fchools, even among, those of very confiderable reputation, where it is ever introduced as a branch of learning at all; and even where it is fo admitted, from want of fufficient knowledge or method in the teacher, the pupil feldom advances beyond the portico of this truly elegant and magnificent ftructure, and to generally retreats difgufted with the labour it has already coft him, and incapable of difcerning the connection of part with part, or the general ufe and benefit of the entire fyftem. Much of this evil, we are ready to believe, refults from a want of a proper elementary treatife or introduction to this important fcience, which has often been denominated abstrufe, but which is no otherwife fo than in confequence of fuch. a deficiency. In reality, from the quaint dialogue introduction of Fenning to the voluminous and operofe quartos of Saunderfon, we are acquainted with no one book of rudiments, in our own language, which we could readily recommend to ftudents as comprising the very neceffary qualifications of concifenefs and perfpicuity, or conveying to them the principles. of algebraic algorithm. We are happy, therefore, to meet with the publication before us, which, notwithstanding a variety of innovations of which we cannot altogether approve, is compofed with a far more lucid order and fimplicity than any elementary book we are at prefent acquainted with; and as fuch we feel no hetitation in recommending it to our private tutors engaged in domeftic education, as well as to the public academies in which algebra forms a part of the learning diffeminated. If upon this fuggeftion,' fays the author, any master of a fchool fhould adopt the mode propofed, I fhould be much obliged to him to acquaint me with the refult of his experience; and indeed if the mafters, ufhers, or tutors of fchools, academies, or colleges, should, on examination, find this work adapted to their ufe, I should esteem it a favour to have any faults in it pointed out to me by them, and to receive their hints for future improvement. To prepare boy for the reading of this book, we cannot begin too early; and the preparation is fimple. As foon as he begins to write figures, the algebraical marks should be introduced into his copies. Thus his first copy might be in addition, after a time in fubtraction, soon after in multiplication, then in divifion. 3+4=7 8 5 = 3 9 x9= 81 2483 'On fhowing the copy-book, the boy fhould always read his copy to the master. Thus three and four equal feven; from eight take five, the remainder equals three; nine into 9 equals eighty-one; twentyfour divided by eight equals three. By degrees a letter may be placed in his copies. Thus a 6. :. 5a = 30; and thus with very little trouble a boy will by mere reading become not only as well acquainted with the marks +−x÷<=>.. as with the figures 0 1 2 3 4 5 6 7 8 9, but understand the principles laid down for the folution of fimple equations. During this time, the boy, it is prefumed, is learning the first rules in arithmetic; and as foon as he can add, fubtract, multiply, and divide whole numbers, and can just do a fum in the rule of three, I recommend that he should enter upon the principles of algebra. Allowing this mode to be good, fome one may perhaps afk me, why I fhould think of adding to the number of books already written upon this fubject, and not content myself with referring to the authors in common ufe? I am prepared to answer the question. Half a dozen years experience, as tutor of a college in the university of Cambridge, taught me the difficulties under which young men labour in endeavouring to learn algebra by the common mode. Some throw away their books before they can do a fimple equation; others, with more courage, get through equations of the se cond order, but are afraid to venture on the fecond part of Mac, laurin's Algebra; others wade through a few chapters, but are frightened, and with good reafon, at Cardan's Rule; the bold ones ruth forward through thick and thin, till after having made, as they think, fome curious difcoveries on the limits of negative roots, they finish their courfe in defpair, in endeavouring to find the number of impoffible roots in an equation of a dimenfions.' Part i. P. vii. We have already obferved that there are a variety of innovations in the work before us, for which we fee little or no reafon, and confequently of which we cannot approve. We here particularly allude to the violent expulfion, without a due and legal trial by jury (our author is a politician-he alludes to the politics of the prefent day even in this isolated treatise upon algebra-and he will forgive us, therefore, if we compare his conduct in the inftance before us to that which expatriated many of the legiflators of France to Cayenne during a Jate revolution)-we fay without a fufficient trial by jury of thofe old-established and very valuable members of the inflitu-. tion before us, the terms negative quantities, fquare, cube, biquadrate, furfolid, &c. "The firft error in teaching the principles of algebra is obvious on perufing a few pages only in the firft of Maclaurin's Algebra. Numbers are there divided into two forts, pofitive and negative; and an attempt is made to explain the nature of negative numbers, by allufions to book-debts and other arts. Now, when a perfon cannot explain the principles of a fcience without reference to metaphor, the proba bility is, that he has never thought accurately upon the fubject. A number may be greater or lefs than another number; it may be added to, taken from, multiplied into, and divided by another num-. ber; but in other refpects it is very untractable: though the whole world should be destroyed, one will be one, and three will be three ; and no art whatever can change their nature. You may put a mark before one, which it will obey: it submits to be taken away from another number greater than itself, but to attempt to take it away from a number lefs than itfelf is ridiculous. Yet this is attempted by algebraifts, who talk of a number lefs than nothing, of multiplying a negative number into a negative number and thus produ-, cing a pofitive number, of a number being imaginary. Hence they talk of two roots to every equation of the fecond order, and the learner is to try which will fucceed in a given equation: they talk of folving an equation, which requires two impoffible roots to make it folvible: they can find out fome impoffible numbers, which, being multiplied together, produce unity. This is all jargon, at which common fenfe recoils; but, from its having been once adopted, like many other figments, it finds the moft ftrenuous fupporters among those who love to take things upon truft, and hate the labour of a ferious thought.' Part i. P. x. Some care has been taken in endeavouring to adapt the language for the perfons to whofe ufe this book is dedicated, that is, to English boys and girls. Hence the terms quadratic, cubic, biquadratic, and the like, as applied to equations, are exploded; and the words fquare, cube, folid, furfolid, as applied to numbers, are for the fame reafon rejected. But habit will fometimes prevail over our beft defigns. Thus, from the long ufe of the word fquare, it escaped my correcting hand in page 105, line 12, where, for fquare, the words "fecond power" fhould be inferted. Square and cube are modes of continued quantity, and cannot be applied to numBers: the abfurdity is fecn in the ufe of the word furfolid; for, if there could be fuch a thing as a folid number, there might be a furfolid number, and a thing might be more than folid, which is atfurd. People err much in fuppofing that a word is of little confquence, if it is explained. If that word has a very different mean. ing in other refpects, the learner will confound frequently the dif ferent meanings, and pass through life without having a clear idea |