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mon herd of intriguers, he may 'get a pension, he may have a riband, or a peerage, perhaps he may be of consequence to a leader, he may even head a party, or mancuvre it to his interests, but he will never become a really great man; he will never be adorned with true glory : and his name will pass away and be forgotten, like that of thousands who have preceded him in the same ignoble course.' p. 360.

True glory! real greatness ! his name forgotten! What a fool the man might justly be deemed, who should care about such things as these, when he can have a pension and a peerage, and when he cares nothing about an all-seeing Judge, a future account, and a state of retribution. And about these grand considerations he will not learn from our author to care.-The book, however, contains a very large share of valuable instruction, though tainted with a most cor. fupt morality. Art. VI. A Treatise on Algebra ; in which the most essential Prin

ciples of the Science are clearly demonstrated, and applied in [to] the Resolution of a great variety of Problems of different kinds ; in

cluding New Improvements in the Solution of Cubic and Biquadratic · Equations ; designed for the Use of Schools and Places of Public

Education. By John Mole, Author of the Elements of Algebra.

12mo, pp. xii. 304. Price 7s. bound. Cadell and Davies. 1809. W e cannot convey to our mathematical readers a clearer

"description of this book in small compass, than by say, ing that its general external and internal appearance are extremely like those of a neat Introduction to Algebra by Mr. Bonnycastle, first published about twenty-four years ago, and which, like Mr, Mole's Treatise,. was expressly designed for the use of schools and places of public education. In pur opinion, a pretty close imitation of such a book in almost all the prominent parts, throwing the principal rules into the same order, assuming a similar shape, and doing little else than vary the examples, was not much wanted. There are, however, a few slight differences in the plans of the two books, which it may be proper to mention. Mr, Bonny castle's Introduction contains an epitome of the doctrine of logarithms; Mr. Mole's does not : Mr. Bonnycastle gives a neat collection of problems on the summation and interpolation of infinite series ; Mr. Mole does not, Mr. Mole, on the contrary, expatiates a little on the solution of exponens tial equations; while Mr. Bonnycastle does not: and Mr. Mole gives problems, at length, on the descent of heavy bodies, and on the maxima and minima of quantities; which Mr. Bonnycastle does not. But Mr. Mole seems to build his hopes of exclusive reputation from the present Treatise, on the excellence of his rules for the solution of biquadratics,

roblems oro le does notion of exponanr.

and one for approximation by converging series. On these subjects we shall permit him to speak for hiinself.'

I have expatiated very largely on the solution of affected biquadra, tics, have explained and illustrated by examples all the cases in which biquadratics can be solved in finite terms, and have given ample directions to know when, and how the roots of biquadratics may be so ob, tained; for it would discover a want of acquaintance with this science, to solve equations by the tedious method of approximation and converging series, when the roots can be readily found in finite terms. Those therefore who would be well versed in this essential branch of this subject, may have recourse to this work for instruction, where all the different cases in which biquadratic equations can be solved, independently of approximation and converging series, are demonstrated. In some of these cases the biquadratics have all their terms; and in others, sometimes the second, sometimes the third, and sometimes the fourth term of the bi. quadratic, is wanting. Those that want the second term, and many others, I have solved by a method of my own, which is much shorter and more simple than that of Des Cartes, or of any other author's that I have met with. This method commences on page 236.

In approximation by converging series, I have invented a method which at least is new to me, and by which the roots of cubic equations may be obtained with great facility. This method commences on page 246; and is performed by taking the cube of the difference between X, the root sought, and a number nearly equal to it, from the given equation ; and solving the quadratic equation that remains. I have likewise shewn how to find all the roots, when they are irrational, of any numeral bi. quadratic, having rational coefficients; and have spared no pains to make the work every where as easy as the nature of the subject will admit ; well knowing by experience, that a small matter turns a person, untutored, quite out of his way.

Now, respecting the methods given by Mr. Mole for the solution of biquadratics, we have only to say, very briefly, that they all depend upon some particular relations of the co-efficients of the several terms of the equation, and are, therefore, extremely confined in their application: and with respect to Mr. Mole's method of approximating to the roots of equations, it is, we think, greatly inferior to the elegant method exhibited in part IV of Simpson's Select Exercises, as well as to the simple process by trial and error, suggested by Dr. Hutton, in the first volume of his Course of Mathe, matics.

Mr. Mole professes to demonstrate clearly the essential prin. ciples of the science; being well aware that a small mat, ter turns a person,untutored," (which they of necessity must be who are at schools and places of public education') I quite out of the way. Yet we cannot help fancying, he is sometimes so unkind as to turn them quite out notwithstanding. Thus, in demonstrating clearly' the rules relative to the signs + and -- in multiplication, he goes to work in the exploded manner, by multiplying x-x=, by n, and by -- 1. But here he forgets that nothing is no quantity, that multiplication is an operation that concerns quantities, not non-entities, and that therefore, while he supposes he is performing the mathematical operation of multiplication, he is only accumulating a stock of nothings. Nil agit exemplum litem quod lite resolvit.

The 'untutoredat our places of public education' will also be sadly turned out of his way, if he places implicit confidence in what our author says respecting momentum, (p. 284.) He is to find the height of a tover, by being told that a ball of 10lbs. being let fall from the top, struck at the foot with a force of 2400lbs. In the solution given by Mr. Mole, he confounds momentum with mere weight, and so dedoces the velocity of the body at the foot of the tower : but the 'untutored ought to be told, that momen, tum is neither equivalent to velocity nor to weight, but to the product of the mass into the velocity.

The 'untutored' must, further, go quite out of his way', or put Mr. Mole quite out of his way, before he can understand the nature of equations in general. If he wishes to be convinced that any equation has necessarily as many roots as it has dimensions, he must turn to some other author. for the demonstration.

Mr. Mole, however, notwithstanding his own defects, is sufficiently on the alert to censure others. Speaking of an erroneous remark relative to the roots of biquadratics, made by Simpson in his Algebra, he says, "Mr. Bonnycastle, following Mr. Simpson implicitly, copied this remark into his Algebra, page 123. fourth edition.' (p. 225.) It would not have injured Mr. Mole's character for candour, if he had informed his readers that Mr. Bonnycastle had corrected this inadvertency in subsequent editions of his Algebra; or if he had remarked that the first person who detected and point

erl out Simpson's niistake, was Dr. Hutton, under the word · Biquadratics, in his valuable Mathematical and Philosophical Dictionary.

On the whole, we are sorry we cannot speak in commendatory terms of this performance. If there be a chasm in our mathematical libraries, waiting to be filled up by a luminous, comprehensive, and complete treatise on Algebra, according to its recent improvements, this is by no means the book for that purpose ; and a mere school-book of Algebra was not a desideratum before Mr. Mole published his.

Art. VII. The Critique in the Eclectic Review, on 1 John v. 7. Confuted

by Martin's Examination of Emlyn's Answer ; to which is added an Ap. pendix, containing Remarks on Mr. Porscn's Letters to Archdeacon Travis, concerning the Three Heavenly Witnesses. By J. Pharez.

(Concluded from po 71.) THE inestimable utility of the Ancient Versions, in judg

ing of the readings of the New Testament, is obvious to every one.. The most early of them approach near to the age of the sacred writers themselves ; and the most recent might, not improbably, be made from manuscripts more ancient than any that now exist. One of them has been so widely diffused, as to have given birth to a considerable diversity and mixture of readings ; but, from the same cause, the means of correction have been correspondently abundant. Others have been shut up within the limits of obscure and insulated communities, so that old copies have been long preserved, transcripts have been comparatively seldom taken, and the causes of various readings have had a more confined range of action. The latter of these cases applies to both the Syriac, and to some other oriental versions ; and the former, to the Latin Vulgate. We shall enumerate all the ancient versions, shewing how each stands affected to the disputed passage.

i. The Old Syriac, called the Peshito (i. e. right, or exact). Version, is shewn, by arguments arising to a bigh degree of probability, to possess an antiquity nearly reaching to the apostolic age; and it cannot be reasonably supposed later than the fourth century, by those who object to a more remote date. It furnishes no trace of the passage in question. Tremellius translated it into Syriac, and affixed it to the margin of his edition, Geneva, 1560, from whence it was transplanted into the text by Gutbirius and Schaaf, without any authority whatever. Dr. Claudius Buchanan, the benevolent advocate of Indian Christianity and missions, in his recent travels among the native Christians of India, who still use this version, which they have derived from the earliest period, examined many of their copies, and found the passage uniformly absent,

2. It is wanting in the Philoxenian Syriac, a version made in the beginning of the sixth century.

3. It is wanting in the Coptic Version, which is, on good grounds, attributed to the fourth or fifth century. .

4. It is wanting in the Sahidic, a Version in the dialect anciently spoken in Upper Egypt. Only extracts from the manuscripts of this version have yet been printed, but its antiquity is believed, by competent judges, to exceed that of the Coptic.

. 5. It is wanting in the Æthiopic Version, which is certainly of high antiquity, and is generally-attributed to the fourth century; but we do not possess a full knowledge of the history of this version. Such is the rarity of copies in Abyssinia, and the barbarism of the people, that Bruce assures us he could by no means obtain one. It was brought to Europe in the sixteenth century, and first published at Rome in 1548 and 1549.

6. The Armenian Version was made very early, in the fifth century. The disputed text doe's indeed appear in the first printed copy of this version, Amsterdam, 1666, and in two subsequent editions: but we are assured by the respectable authority of an eminent Armenian priest, Father Joannes Zohrab, that this passage was not found in a single Armenian manuscript, out of many which he had examined. (Alter, ap. Marsh's Letters, pref. p. ix.) It must, therefore, have been added, without authority, by Uscan, the editor of the first edition. Should any contend that the passage is, read in the Acts of the Armenian Council of Sis, A. D. 1307, and in the Epistle of Gregory to Haitho about 1270; we answer that this supposition rests on the suspicious fidelity of the Popish editor Galanus, in 1690 ; and that, were the assumption granted, it would be of no service to the advocates of the clause, for it would only prove that this was among the interpolations from the Vulgate which were notoriously introduced into the Armenian New Testament by Haitho, a superstitious devotee to the see of Rome, .',

7. It is not found in any of the known Arabic Versions, though it is most unwarrantably, added, without any pretence of authority from MSS., in the Arabic N. T. printed at London iu 1727, by the Society for promoting Christian Knowledge.

8. The passage under consideration is found in the common copies of the Vulgate, the authorized Latin version of the Papal church : and this, we verily believe, was its birthplace. .But let our readers take into their consideration a few important circumstances.

(1.) We reverence the Vulgate as an important relic of Christian antiquity, and in general a good and faithful version : but every biblical critic knows, that, in its passage from the fifth to the fifteenth century, it has received many corruptions and interpolations.

(2.) In the MSS. of the Vulgate, the passage is found exceedingly diverse and fluctuating, as to its readings and position. Many of them, and the printed text, even that of Pope Clement VIII, have the final clause of the 8th

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