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Scaliger (besides his correction of i Sam. xiii. 1. 2 Kings viii. 16. and xxiv. 8.) accounts for some errors in the numbers, from their having been formerly expressed by numeral letters. Buxtorff the father shews the force of prejudice, in asserting the absolute agreement of all the ancient manuscripts. Even after that corruptions, in the Hebrew text, had been proved by Cappellus, by every argument excepting that of manuscripts, Buxtorff the son, in his reply, affirmed, that no Hebrew manuscript in the world contained any various reading which agreed with either of the ancient versions. Had he been now living, he would have seen thousands of instances to the contrary: and yet this critic owned the existence of various readings, and recommended a collection of them. Glafius, though he allows many variations in the ancient manuscripts, afferts, that all these have been corrected ; and that our present copies are perfect, notwithstanding one or two corruptions. Mede contends, very justly, that the Hebrew copies used by the Apostles may at least as fafely be followed, as the copies handed down by the Masorets: and he gives reasons why some chapters, now at the end of Zachariah, probably belong to Jeremiah. His criticisms are excellent on Ifa. xxix. 13. Zach. xi. 13. 2 Kings xxv. 3. i Chron. vi. 13. and Gen. xi. 32. His Correspondent de Dieu, on the other hand, makes a very different figure, in maintaining, that the Keri were not various readings-that the Hebrew copies all agreed--and particularly, that the very fame differences, between 2 Sam. xxii. and Psalm xviii. existed universally. User, though a favourer of the Hebrew integrity, allowed Hebræum Veteris Testamenti codicem fcribarum erroribus non minus effe obnoxium, quam novi codicem et libros omnes alios. After him are celebrated Morinus, Beveridge, and Walton; whose sentiments are well known. Hammond considers a comparison of parallel places as a safe method of correction ; particularly in 2 Sam. xxii. and Psal. xviii. which he supposes to have been originally the same. Bochart strongly confirms the Samaritan text, as to the number 145 in Gen. xi. 32. ; and accounts for the corruption in the Hebrew, from the mistake of a numeral letter. Hottinger's opinion, in favour of the Hebrew text, is contrasted with that of Huetius; who allows, that marginal glofses may have crept into the Hebrew text; and he shews, how the same mistakes

may

exist in the Samaritan text, and in the Hebrew. Pocock, though he did not hold the absolute integrity of our Hebrew text, was so strongly prejudiced in favour of it, that in his edition of Maimonides, where he prints the old and true reading of Jeremiah (xxviii. 8.) as preferved by that Rabbi, he has put opposite to it the Latin word of the prefent falle reading -printing malum, instead of fames. And because this Profeflor ,

did not know what to make of the Keri, and yet considered them as 'marks of profound erudition, the futility of this opinion is proved by fix decisive examples. Jablonski was the first editor of an Hebrew Bible, who spoke of any Hebrew manuscripts ; fpecifying their contents, with the places where preserved; and be names four, by the help of which he made a few corrections. But, though he was convinced that the Masoretic text is sometimes wrong, he omitted the two' neceffary verses in Joshua : and as he recommends an examination of Hebrew manuscripts in the most diftant countries, Dr. Kennicott observes, that, in consequence of this examination, these two verses are established by 149 copies. After the mention of Le Clerc's fentiments, on improper separations made by the points, and wrong combinations of the letters into words, Opitius is taken notice of, who declares, that in his edition he obeyed the Masora, in defiance of all the manuscripts and editions of the world united. Vitringa satisfactorily accounts, how'a mistake, made in one manuscript, may obtain afterwards in many, from the practice of correcting many manuscripts by one as their standard. His conjectural emendation of 2 Chron. xxvi. 5. is confirmed by fifty copies: and his reading of Ifa. xix. 18. is established by the Talmud and fixteen Hebrew copies.

In the Bible printed at Hall, in 1720, John Hen. Michaelis published various readings, extracted from five manuscripts at Erfurth : but it has lately been discovered, that he omitted many variations of great moment; perhaps out of tenderness to the advocates of the Hebrew integrity. One of thefe advocates was Wolfius ; who maintained, that mistakes might exist in some copies, but not in all; because some one manuscript, or some one edition, always (he thinks) had the true reading. Carpzovius 'contends, that the Hebrew text is come down to us in the same purity with which it was first penned: not, indeed, so pure now in all the copies, but in those of the better fort: not, indeed, in these separately, but in these altogether. Nor yet does he think it necessary that all these be collected from every quarter of the globe ; because those which are near at hand will do the business. By this concession his former doctrine is abolished.

In the year 1729, were published Notes on the Holy Scriptures by Mr. Hallet, whose opinion is here given ; which is--thac the reason, why the New Testament frequently differs from the Old, as to the quotations, is, because the Hebrew copies have been altered fince the days of the Apostles. Our learned Editor's catalogue of the Christian writers ends with the testimony of Bishop Hare, who contends earnestly for admitting the Hebrew, text to be much corrupted ; rejects the titles of many of the Psalms, as not from the authors of these Plalms; condemns the practice of varnishing over, instead of correcling, the

corrupte!

Y 4

corrupted readings; and laments, that the chief support of cria ticism was here wanting, namely, Hebrew Manuscripts.

Such have been the sentiments of the most learned men, both Jews and CHRISTIANS, concerning the state of the Hebrew text. And though the weight of evidence, against the integrity of that text, so much preponderates; yet certain it is, that the perfection of the printed Hebrew text was generally believed till the middle of the present century.

[To be concluded in another Article.]

rona.

Art. II. A Differtation on the Summation of infinite converging Series

with algebraic Divisors. Exhibiting a Mehod not only entirely new, but much more general than any other which hath hitherto appeared on the Subject. Translated from the Latin of A. M. Lorgna, Professor of Mathematics in the Military College, at Ve

With illustravive Notes and Observations. To which is added an Appendix, &c. by H. Clarke. 4to. Ios. 6 d. Boards.

Murray. ART. III. Obfervations on converging Series, occafioned by Mr.

Clarke's Translation of Mr. Lorgna's Treatise on the fame Subject, by J. Landen, F.R.S.

4to. I s. 6 d. Nourse. HE latter of these two publications is rather a severe re

view of the former, and the design of it is to thew, that the late Mr. T. Simpron, in his Mathematical Differtations, published in 1743, has pointed out a very ready method of computing the sums of a great number of series, comprehending, at leaft, all that can be done by the method exhibited in Mr. Lorgna's book.

Mr. Lorgna begins his Dissertation with the summation of the series of fractions, that have unity for the common numerator, and whose denominators are p +9p+29, p + 39, &c.

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These fums he fnews to be equal to the fluent of

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9 which, placing m in the denominator instead of unity, he shews at his sth proposition, will be the sum of the series, when each respective denominator above, is multiplied by its corresponding term of the geometrical progresion m, mạ, m3, &c.

In his second section he proposes to find the sum of a series, when the terms have either unity, or any other common numerator, and when the denominators consist of any number of fimple factors, or have the form p+qz.m-l-nz.rts.ituz. &c. Z being the index of the terms, the sum of such a series he Thews to arise, from the fluent exhibiting the sum as in the first

mn : 11:9I section, when it is first multiplied by

and

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and the fluent taken, which fluent will be the sum of the series when the denominators consist of two factors; and this fluent in

qismin-I like manner multiplied by

ä and its fluent taken, will be the sum when the denominators conGft of three factors; and so we may proceed on without limit. He then in a Lemma confiders these compound fluxions, as members of the Auxion of a compound product of unknown variable quantities, on this foundation, that xj ty ze being the fluxion of xy, if the fluent of xj be known, or can be exprefled, that of yä may be exprefled also, being equal to xy minus that fluent; and thus he resolves compound fluentials, meaning those fluents so found one from another, into simple ones. And this is the foundation of his whole method of procedure, not only in this section, the remainder of which is taken up in illustrating the method when applied to series having two factors in the denominators, but through the whole book.

The third is employed in the application of this method, to series with three simple factors in the denominators. The fourth section treats on those with four. The fifth, on those whose numerators constitute an arithmetical progresion, the denominators being as before. Now, as these numerators would be the indices of a series of powers in geometrical progression, it is manifest, that such a series will be produced, by taking the fluxions of the several terms of one that has its numerators in geometrical progression, and dividing each of them by the fluxion of the common ratio. And thus fluential expresions for the summation of these ferics are produced. And the exemplification of this, in series whose denominators consist of two factors, either drawn into the terms of a geometrical progreffion or not, is the subject of this fifth section.

The fixth is an exemplification of the same, when the denaminators consist of three fimple factors.

The seventh proposes to find the sum of a series, whereof the numerators consist of two simple factors, which would form two arithmetical progrellions: the denominators being as before. -And here it is manifestly necessary to double the operations in the fifth section, twice taking the fuxions, and dividing by 3 that of he common ratio. The application of this to ferics with three factors in the denominators takes up the remainder of this section ; and the subject of the eighth is those with four factors.

The ninth section proposes to investigate the sums of series being the reciprocals of the powers of the natural numbers, by nieans of the areas of transcendent curves, found by the method of Mr, Cotes in his Tract on the Newtonian differential Method, published at the end of his Harmonia Mensurarum.

The

The method followed by Mr. Lorgna in his eight first fec. tions is certainly curious, regular, and extensive ; but whether it has any peculiar advantages, so as to warrant Mr. Clarke in extolling it to the skies, can only be seen by comparing it a little with others.

The moft fimple and perfpicuous method that has hitherto been given for the summation of these series, is doubtless that of Mr. James Bernoulli; who, in his Tract on Infinite Series, published about the beginning of this century, explains an artifice, by which we may find as many series of this kind, all fummable, as we please ; and, moreover, thews how, in most cases, a regress may be made from a given series to its sum, in finite terms when thus determinate, and without quadratures or fluxions; and when not determinable in finite terms, he shews how to do it with quadratures or Auxions. This artifice consists in assuming some certain series beginning with unity (whether it be accurately fummable or not, it does not signify, provided its terms continually converge to nothing), from which he subtracts the same series when its first term unity, or its two first terms, or its three first, &c. are wanting; from whence it follows, that the remainder, or series produced by such subtraction, shall either be equal to the first term of the assumed series, or to the two first, or to the three first, &c. And the operation may be repeated with the series produced by such subtraction, from whence new series at pleasure will arise, and all of them fummable. M. De Moivre has given a way to contract the work in very complicated cases, but it is neither so simple nor perspicuous; nor is it more extensive, as the author seems to intiinate, since whatever can be done by it, may likewise be done by Mr. Bernoulli's method, a little improved by some additional similar artifices and contractions, which would doubtless have been given by the author himself if he had lived. And it will be found, that the foundation of Mr. Lorgna's method is near of kin to it, Mr. Bernoulli subtracting the series themselves, and Mr. Lorgra the fluxions whose fluents, when x=1, express the same series. All this will be clearly understood, from what it is quite necessary for us to add, in order to shew the comparative merit of the Dissertation before us.

Mr. Lorgna at Art. 26, Sect. 2, says, 'We might produce a great many more examples, from several eminent mathematicians; but there we have already given, it is presumed, are abundantly sufficient to evince the superiority of our method to those who can judge of the subject, in respect of its elegancy, fimplicity, and generality; for it is obfervable, that it is applied with the same facility to series of which the signs change alternately from positive to negative, as to those affected with con

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