Scaliger (befides his correction of 1 Sam. xiii. 1. 2 Kings viii. 16. and xxiv. 8.) accounts for fome errors in the numbers, from their having been formerly expreffed by numeral letters. Buxtorff the father fhews the force of prejudice, in afferting the abfolute agreement of all the ancient manufcripts. Even after that corruptions, in the Hebrew text, had been proved by Cappellus, by every argument excepting that of manufcripts, Buxtorff the fon, in his reply, affirmed, that no Hebrew manufcript in the world contained any various reading which agreed with either of the ancient verfions. Had he been now living, he would have feen thousands of instances to the contrary: and yet this critic owned the existence of various readings, and recommended a collection of them. Glaffius, though he allows many variations in the ancient manuscripts, afferts, that all these have been corrected; and that our prefent copies are perfect, notwithstanding one or two corruptions. Mede contends, very juftly, that the Hebrew copies ufed by the Apoftles may at least as fafely be followed, as the copies handed down by the Maforets and he gives reasons why fome 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. 1 Chron. vi. 13. and Gen. xi. 32. His Correfpondent 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 Pfalm xviii. exifted univerfally. Usher, though a favourer of the Hebrew integrity, allowed – Hebræum Veteris Teftamenti codicem fcribarum erroribus non minus effe obnoxium, quam novi codicem et libros omnes alios. After him are celebrated Morinus, Beveridge, and Walton; whofe fentiments are well known. Hammond confiders a comparison of parallel places as a fafe method of correction; particularly in 2 Sam. xxii. and Pfal. xviii. which he fuppofes to have been originally the fame. Bochart ftrongly 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 contrafted with that of Huetius; who allows, that marginal gloffes may have crept into the Hebrew text; and he fhews, how the fame mistakes may exift in the Samaritan text, and in the Hebrew. Pocock, though he did not hold the abfolute integrity of our Hebrew text, was fo ftrongly 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 oppofite to it the Latin word of the prefent falle reading -printing malum, inftead of fames. And becaufe this Profeflor did not know what to make of the Keri, and yet confidered them as marks of profound erudition, the futility of this opinion is proved by fix decifive examples. Jablonski was the first editor of an Hebrew Bible, who spoke of any Hebrew manufcripts; fpecifying their contents, with the places where preferved; and he names four, by the help of which he made a few corrections. But, though he was convinced that the Maforetic text is fometimes wrong, he omitted the two neceffary verses in Joshua and as he recommends an examination of Hebrew manufcripts in the moft diftant countries, Dr. Kennicott obferves, that, in confequence of this examination, these two verses are eftablished by 149 copies. After the mention of Le Clerc's fentiments, on improper feparations 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 Mafora, in defiance of all the manuscripts and editions of the world united. Vitringa fatisfactorily accounts, how a mistake, made in one manufcript, may obtain afterwards in many, from the practice of correcting many manufcripts by one as their ftandard. 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 fome one manufcript, or fome one edition, always (he thinks) had the true reading. Carpzovius contends, that the Hebrew text is come down to us in the fame purity with which it was firft penned: not, indeed, fo pure now in all the copies, but in those of the better fort: not, indeed, in these feparately, but in thefe altogether. Nor yet does he think it neceffary that all these be collected from every quarter of the globe; because those which are near at hand will do the bufinefs. By this conceffion his former doctrine is abolished. In the year 1729, were publifhed Notes on the Holy Scriptures by Mr. Hallet, whofe opinion is here given; which is that the reason, why the New Teftament frequently differs from the Old, as to the quotations, is, because the Hebrew copies have been altered fince the days of the Apoftles. Our learned Editor's catalogue of the Chriftian writers ends with the teftimony of Bishop Hare, who contends earneftly for admitting the Hebrew text to be much corrupted; rejects the titles of many of the Pfalms, as not from the authors of these Pfalms; con-demns the practice of varnishing over, instead of correcting, the Y 4 corrupte corrupted readings; and laments, that the chief fupport of cri ticifm was here wanting, namely, Hebrew Manufcripts. Such have been the fentiments of the moft learned men, both JEWS and CHRISTIANS, concerning the ftate 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 prefent century. [To be concluded in another Article.} ART. II. A Differtation on the Summation of infinite converging Series with algebraic Divifors. Exhibiting a Method not only entirely new, but much more general than any other which hath hitherto appeared on the Subject. Tranflated from the Latin of A. M. Lorgna, Profeffor of Mathematics in the Military College at Verona. With illuftrative Notes and Obfervations. To which is added an Appendix, &c. by H. Clarke. 4to. 10 s. 6 d. Boards. Murray. ART. III. Obfervations on converging Series, occafioned by Mr. Clarke's Tranflation of Mr. Lorgna's Treatife on the fame Subject, by J. Landen, F. R. S. 4to. 1 s. 6d. Nourse. THE latter of thefe two publications is rather a fevere re THE view of the former, and the defign of it is to fhew, that the late Mr. T. Simpfon, in his Mathematical Differtations, published in 1743, has pointed out a very ready method of computing the fums of a great number of feries, comprehending, at leaft, all that can be done by the method exhibited in Mr. Lorgna's book. Mr. Lorgna begins his Differtation with the fummation of the feries of fractions, that have unity for the common numerator, and whofe denominators are p + q, p + 2 q, p + 34, &c. Thefe fums he fhews to be equal to the fluent of and which, placing m in the denominator instead of unity, he fhews at his 5th propofition, will be the fum of the feries, when each refpective denominator above, is multiplied by its correfponding term of the geometrical progreffion m, m2, m3, &c. In his fecond fection he propofes to find the fum of a feries, when the terms have either unity, or any other common numerator, and when the denominators confift of any number of fimple factors, or have the form p+qz.m+nx.r+sz.t+uz. &c. z being the index of the terms, the fum of fuch a feries he fhews to arife, from the fluent exhibiting the fum as in the first m:n-pq-I fection, when it is firft multiplied by I -· X and and the fluent taken, which fluent will be the fum of the feries when the denominators confift of two factors; and this fluent in I like manner multiplied by . * r:sm: n-I * and its fluent taken, will be the fum when the denominators confift of three factors; and fo we may proceed on without limit. He then in a Lemma confiders thefe compound fluxiors, as members of the fluxion of a compound product of unknown variable quantities, on this foundation, that xyy being the fluxion of xy, if the fluent of xy be known, or can be expreffed, that of yx may be expreffed alfo, being equal to xy minus that fluent; and thus he refolves.compound fluentials, meaning those fluents fo found one from another, into fimple ones. And this is the foundation of his whole method of procedure, not only in this fection, the remainder of which is taken up in illuftrating the method when applied to feries having two factors in the denominators, but through the whole book. The third is employed in the application of this method, to feries with three fimpie factors in the denominators. The fourth fection treats on those with four. The fifth, on those whofe numerators conftitute an arithmetical progreffion, the denominators being as before, Now, as thefe numerators would be the indices of a series of powers in geometrical progreffion, it is. manifeft, that fuch a feries will be produced, by taking the fluxions of the feveral terms of one that has its numerators in geometrical progreffion, and dividing each of them by the fluxion of the common ratio. And thus fluential expreffions for the fummation of thefe feries are produced. And the exemplification of this, in feries whofe denominators confift of two factors, either drawn into the terms of a geometrical progreffion or not, is the fubject of this fifth fection. The fixth is an exemplification of the fame, when the denominators confift of three fimple factors. The seventh proposes to find the fum of a feries, whereof the numerators confift of two fimple factors, which would form two arithmetical progreffions: the denominators being as before.And here it is manifeftly neceffary to double the operations in the fifth fection, twice taking the fluxions, and dividing by that of the common ratio. The application of this to feries with three factors in the denominators takes up the remainder of this fection; and the fubject of the eighth is thofe with four factors. The ninth fection propofes to inveftigate the fums of feries being the reciprocals of the powers of the natural numbers, by means of the areas of tranfcendent curves, found by the method of Mr. Cotes in his Tract on the Newtonian differential Method, published at the end of his Harmonia Menfurarum. The method followed by Mr. Lorgna in his eight firft fections is certainly curious, regular, and extenfive; but whether it has any peculiar advantages, fo as to warrant Mr. Clarke in extolling it to the fkies, can only be feen by comparing it a little with others. The moft fimple and perfpicuous method that has hitherto been given for the fummation of thefe feries, is doubtlefs 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 feries of this kind, all fummable, as we pleafe; and, moreover, fhews how, in moft cafes, a regrefs may be made from a given feries to its fum, in finite terms when thus determinate, and without quadratures or fluxions; and when not determinable in finite terms, he fhews how to do it with quadratures or fluxions. This artifice confifts in affuming fome certain feries beginning with unity (whether it be accurately fummable or not, it does not fignify, provided its terms continually converge to nothing), from which he fubtracts the fame feries when its first term unity, or its two firft terms, or its three firft, &c. are wanting; from whence it follows, that the remainder, or feries produced by fuch fubtraction, fhall either be equal to the first term of the affumed feries, or to the two firft, or to the three firft, &c. And the operation may be repeated with the feries produced by fuch fubtraction, from whence new feries at pleafure will arife, and all of them fummable. M. De Moivre has given a way to contract the work in very complicated cafes, but it is neither fo fimple nor perfpicuous; nor is it more extenfive, as the author feems to intimate, fince whatever can be done by it, may likewise be done by Mr. Bernoulli's method, a little improved by fome additional fimilar artifices and contractions, which would doubtlefs 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 fubtracting the feries themfelves, and Mr. Lorgra the fluxions whofe fluents, when x1, exprefs the fame feries. All this will be clearly understood, from what it is quite neceffary for us to add, in order to fhew the comparative merit of the Differtation before us. 6 Mr. Lorgna at Art. 26, Sect. 2, fays, We might produce a great many more examples, from feveral eminent mathematicians; but thefe we have already given, it is prefumed, are abundantly fufficient to evince the fuperiority of our method to those who can judge of the subject, in refpect of its elegancy, fimplicity, and generality; for it is obfervable, that it is applied with the fame facility to feries of which the figns change alternately from pofitive to negative, as to thofe affected with con ftant |