An Account of Books

An Account of BOOKS. I. Tractatus Mathematicus de Figurarum Curvilinarum Quadraturis & Locis Geometricis. Autore Johanne Craig. Londini apud Sam. Smith & Benj. Walford, Soc. Regiae Typographos. This Treatise consists of two Heads. In the First the Author undertakes to give a General Method for determining the Quadratures of Curvilinear Spaces, and this he shews may always be done by finding another Curve-line from the known Property of its Tangent; which Curve-line he therefore calls the Quadratrix of the proposed Space. Now this Quadratrix (for all Spaces comprehended by Geometrical Curves) is either a Geometrical or a Mechanical Curve-line: And accordingly he divides the whole into two Parts. In the First of these he treats of such Spaces, whose Quadratrix's are Geometrical Curves, where he illustrates his Method by a great variety of Examples; discovering in several of them a new Method of finding Infinite Series's (without the trouble of Division or Extraction of Roots) by assuming an Arbitrary Series with unknown Coefficients, which are easily determined in the Progress of the Problem. And since the Publication of this Treatise we have some Instances of the like Method of finding Series's by that excellent Mathematician, Mr. Leibnitz, Printed in the Acta Eruditorum of April, 1693. Afterwards he gives a Method of comparing the Area's of Figures with the simplest Figures of the same kind; from which he deduces many Theorems, each of which do exhibit the Quadratures of an Infinity of Figures without any trouble. ible of Calculation: And withal shewing to what Theorem any proposed Figure is to be referred, and how the like Theorems may be continued by the help of his general Method. He concludes this First Part with a Discourse concerning the Analytical Expression of Quadratures, wherein is shewn, that though the Area answering to the Abscissa be that which is commonly sought, yet the general Quadrature found by his Method, is for the most part either greater or lesser: Therefore he gives both a Geometrical and Analytical Rule for knowing whether the said general Quadrature be deficient or exceeding, and what that deficient or exceeding Quantity is. The Second Part treats of the Quadratures of those Spaces, whose Quadratrix's are Mechanical or Transcendent Curves, as he (after Mr. Leibnitz) chooses rather to call them; where he gives a general Method for finding their Tangents. And having given a Rule for defining the Transcendent Quadratrix of any Algebraical Curve (that is not capable of a Geometrical one) he Illustrates the same by finding the Transcendent Quadratures of the Circle, Hyperbola, and two other Figures. There is added at the end, the Author's Answer to a Letter of Mr. D.T.'s relating to a Controversie between them concerning a Method of Quadratures, published by Mr. D.T. in the Acta Eruditorum. The Second Head of this Treatise is concerning the Geometrical Places, wherein is shewn how to determine any Solid Place, by comparing the Equation with a general Theorem comprehending all Places of that kind: Whereby are avoided all those many Rules about the Reduction of the given Equation, and the variety of the Signs + and −, which have hitherto rendered this Piece of Geometry so troublesome. II. The History of the Church of Malabar, from the time of its being Discovered by the Portuguezes in the Year 1501. Giving an Account of the Persecutions and violent Methods of the Roman Prelates to reduce them to the Subjection of the Church of Rome, together with the Synod of Diamper, celebrated Anno 1599. With some Remarks upon the Faith and Doctrine of the Christians of St. Thomas in the Indies, agreeing with the Church of England, in Opposition to that of Rome: Done out of Portuguez into English by Michael Geddes, Chancellor of the Cathedral Church of Sarum. Lond. Printed for S. Smith and B. Walford. In Octavo. 1694. The Author of this History begins with a short Account of the Country of Malabar, as to its Situation, its Kingdoms, and Ports, mostly inhabited by Heathens, only the South end of the Serra or Gate, a Ridge of Mountains running 200 Leagues North and South, is Inhabited by Christians, as they call themselves of St. Thomas, which Apostle, they say, first converted them. They have been for 1300 Years under the Patriarch of Babylon. Next, he shews us how the first News of them came to Portugal, Anno 1501. and upon what Account they begged the Portuguez Assistance against the Infidels. Anno 1505. two Christians sent by the Venetians, taught the Indians the use of Great Guns, thereby to oppose the Portuguez and their Trade there. Till about the Year 1544. little more was heard of them, them, at which time the Crofs, and other pretended Reliques of St. Thomas were found at Meliapor, of which the Author gives the Legend, which leads him to give an Account of the Violences used by the Roman Catholicks to bring them to the Obedience of their Church, for that purpose erecting a Colledge at Cranganor, Anno 1546, and in the Year 1587, another to instruct them in the Chaldee and Syriack Tongues. The Extremes they there made me on, as the Author Observes, had like to have cost them all that they had in the Indies; and was the greatest cause of the War between the Portuguez and the Indians. There are several Remarkable Passages of the Jesuites Contrivances, and the Indians Resentment thereof, for which I refer to the Book, as less proper to our Design. The Author shews the Doctrines wherein the Church of Malabar agrees with that of England, and differs from the Roman. And concludes with the Acts and Decrees of the Synod of Diamper, with the occasion of its being called; which, with a Preface to a Missal imposed upon that Church by the Arch-Bishop Menezes, ends the whole Book. ERRATA. Numb. 208. Page 48. line 23. read Hyppon Dhini, or Phynnon Dhini. pag. 49. l. 17. r. Harlech. pag. 70. l. 17. r. 58 ½. LONDON: Printed for Sam. Smith, and Benj. Walford, Printers to the Royal Society, at the Prince's Arms in St. Paul's Church-yard. 1694.