An Account of Some Books

An Account of some Books. I. GEOMETRIÆ PARS UNIVERSALIS, Quantitatum Curvarum transmutationi & mensura inserviens, Auth. Jacobo Gregorio Scofo, Patavii, 1668. in quarto. Numb. 30. We gave an account of a small Tract, entitled Quadratura Circuli & Hyperbola in propria sua Proportionis specie inventa & demonstrata, à Jac. Gregorio Scofo; and intimated, that it would be reprinted here, and accordingly the Impression was begun; but since then, the Author writing to a member of the R. Society, signifies, that it is now reprinting at Venice (and possibly not without some enlargement;) which hath caused the design to be laid aside here. Mean time, the same Author hath sent over his other Book of his own composure also: in the Preface whereof he observes, That the defect of Algebra (which is most manifest in the Mensuration of Curved quantities) may in some manner be supplyed, if out of some essential property of any Figure there to be given a method of changing it into another Figure (equal thereto) that hath known properties, and of that into another and so forward, until at last you change it into some known quantity: And he modestly saith, that his own Treatise hath so far pursued this Method, that it refuseth no particular figure yet consider’d by Geometry. After this, he answers such objections, as either have been, or such as himself could conceive might be made against his former Book of the Quadrature of the Circle and Hyperbola. For such, as would square a Circle organically, or divide an Angle in a given ratio, he supposeth, that there is no easier way of doing it, then by the common Linea Quadratrix, (the properties whereof are largely handled in Leotaudi Cyclomathia, Lugduni, 1663. in quarto.) Then he discourseth, that all things concerning Logarithms and the Composition of Ratio’s may be perform’d by help of a Curved Line, drawn through the Tops of a Rank of Lines in continual Proportion, standing as Perpendiculars on a right line and at equal distance; that the Operations perform’d thereby are not to be accounted a-geometrical, because they are not perform'd by the Sole aid of Ruler and Compass; which he suggests to be well observ'd à Subtilissimo Mathematico D. Carolo Reinaldino in Geo- metra suo promoto, dum tractat de novis illis Lineis, quas Medi- ceas appellat: Concerning which Author, he saith thus, p. 132. Qui autem desiderat plenam Analyseos & Equationum doctrinam, exspectet absolutissimum Caroli Reinaldini Opus de Resolutione & Compositione Mathematica, quod nunc sub prato est. It seems, that the Book will be three large Volumes in folio; the first whereof, being Introductory, and containing the Algebra of the Antients, is already in England. And for the Confirmation of what hath been asserted, the Au- thor thus demonstrates, that no Cubick Equation (that is irre- ducible to a Quadratic) can be resolv'd by the sole aid of Ruler and Compass. For every Cubick Equation hath either but one or three real roots, which if they could be found by the said sole aid of Rule and Compass, or by the Intersection of a Circle and a Right Line, then a Right Line should cut a Circle either in one Point or three; either of which is most absurd. And for the like reason a Cubick Equation, having three real roots, can ne- ver be reduced to a pure Equation, which hath but one onely root; for in these Equations, Reduction shall no wise profit, for- asmuch as 'tis impossible, by aid thereof to change an Imaginary root into a real one, and the Converse. As to the Argument of the Book itself, it contains these seve- ral Heads. 1. The Mensuration of sundry Solids, with General Methods to that purpose; concerning which the Author saith, p. 123. Totus namque Archimedis Tractatus de Sphera & Cylindro facile demonstratur ex hujus 3. ad modum hujus 46. & aliquot sequenti- um: Liber de Conoidibus & Spheroidibus, & tota Luca Valerii do- ctrina, ex hujus 21. Tota Guldini, Johannis de la Faille, & An- drae Tacqueti doctrina, ex hujus 35. & aliquot sequentium. And as a Corollary of Prop. 62, he Cubeth or measureth either of the Segments of a Parabolical Conoid cut with a Plain, parallel to the Axis. Hence we observe, that supposing such a Seg- ment, again cut with a Plain, erect to the former Plain, the Pro- position may be well apply'd to the Guaging of Cask part out, when the Liquor falls between the heads, which are supposed erect to the Horizon. 2. The Mensuration or Plaining of the Surfaces of divers Solids and Spiral Spaces, unknown to Antiquity, and not treated of by any modern Authors, till of very late years; from whom the Author differs in his Method: Particularly he finds a Circle equal to the Surface of A Parabolical Conoid, resembling a Cup or Bowl; viz. An Hyperbolical when the Revolution is about their Axes, Prop. 46. & 49. The Parabolical Hour-Glass or Solid, when the Revolution is about a Touch-line passing through the Vertex, Prop. 52. A Long Broad Spheroid, Prop. 47. 48. Generally Prop. 36. The Surface of every Round Solid is equal to a Rectangle, whose Base is the Circumference of the Figure, by the Rotation whereof the Solid is begot, and the Height equal to the Circumference, which the Center of Gravity of the Perimeter of the Figure describeth. 3. A Method for straightening of Curved lines in the first 6 Propositions; and in particular he finds a Right line equal to a Parabolical Curve, Prop. 51. 4. Divers Optick Propositions towards the end of the Book, concerning the Imperfection of the Eye, and the Confusion of the Sight; the apparent Magnitude of the Sun, low and high; the Tails of Comets; what Proportion the Earth's illumination by the Sun at the Full of the Moon bears to the illumination of the Earth by the Moon; and the like comparison between the Sun and Sirius; That vision by aid of a Telescope or Microscope, is not deceitful: And an Observation of the likeness between the Earth and the Moon. This same Author in his Letter to Mr. J. C. suggests, that Cassini hath observed the Motion of Jupiter about his Axis in 10 hours; of Mars in 23 hours; that Venus hath the like Rotations: but the precise period not yet known; that Cassini hath hath publish'd Tables of the Motion of the Satellites of Jupiter, with an Ephemeris of the same for this present year: All which are there much applauded. The like Tables have been formerly publish'd by the Learnd John Baptist Hodierna at Rome about 1656. which we intimate, because that and other Works of that knowing Author are here scarcely known nomine tenus. In another Letter of this Author to the same J. C. (which is an answer to a Quere, whether Antimo Farby, by some suppos'd to be Hon. Fabry, the Author of a Tract entituled Opusculum Geometricum de linea Sinuum & Cycloide, printed at Rome An. 1659. had publish't the Treatises promised in the Preface thereof, viz. a Century de Maximis & Minimis; and some other Geometrical Tracts, as precurory to his intended General Body of Geometry) answers, that none of these Treatises are extant; that Mich. Angelo Ricci only, (since Viviani) hath written de Maximis & Minimis in two sheets, but to extraordinary good purpose. The Argument, doubtless, concerns either the Limits of Geometrical Problems, or of Equation. Concerning the latter, we shall here intimate, that Erasmus Bartholinus hath well handled the same in his Treatise, entitul'd, Dioristice sive Methodus Equationum prima & secunda, Hafniæ, 1663; which are different from those of de Beaune formerly publish't: At the end of which Treatise the said Bartholinus promiseth a General Body of Algebra, wherein the Precepts shall be explain'd by Examples. The same Author hath publish't other Treatises, which we do not find to have been brought over; as one, De Arte Analytica inveniendi omnia Problemata Proportionalium maximé Harmonicorum, Hafniæ, 1657. in 4°. Another, de Problematis Mathematicis, ibid. A. 1665. in 4° II. AN INTRODUCTION TO ALGEBRA, Translated out of High-Dutch into English by THO. BRANDER, M.A. much alter'd and augmented Dr. J.P. Also a Table of such odd Numbers, as are less than one hundred thousand, shewing those that are Incomposit, and resolving the rest into their Factors or Coefficients. Printed at London in 4°. First, as to the Method of this Book, it is New, such as con- cains much in a little, each distinct step of Ratiocination or Operation hath a distinct Line: the Author putting small Letters for unknown Quantities, and great Letters for known ones; and the Method is such, that most of the Book, if not all, may be understood by those not vers'd in the English Tongue, that are vers'd in Specious Algebra; most of the Questions being propounded in Symbols, and the progress of the work so described by the Marginal quotations, that for those exercised in Algebra, that would transcribe a Problem in this Method, it were sufficient, only to take the Margin, omitting the work itself, till farther leisure is afforded to perform it. Next, as to the Matter, the Book consists of many excellent Problems; some whereof are such, as Bachet (that famous Commentator on Diophantus) either confesseth he did not attain, or at least left obscure: and others of them are such, as the celebrated Descartes and Van Schooten have left doubtful, as not being by them thoroughly understood. And some of them are such, as being unlimited, have for their Answers certain ranks or series of all possible whole or rational Numbers, whereby the Student may be accomplished for the resolution of other Questions of the like Nature. Thirdly, as to the Table of Incompositis, no Book but this extends it to above Ten thousands; some of the uses whereof are declared in the Title, others in the Book; and even in Common Arithmetic, it is of excellent Use for the Abbreviation of Fractions, and for giving of all the aliquot parts of a Number proposed, useful for the Depression and Resolution of Equations, as is taught by Albert Gerard, and Van Schooten. Besides, it is observable in this Treatise, that the Author declineth the Exegesis numerosa of Vieta, which following Writers use for the finding of the Roots of Equations. As to the Remaining part of the Book, as it was published by John Henry Robin in High Dutch, reasons may be given why it was omitted in this English Edition. The First Part of it handles the Taction of Circles; about which Argument some Epistles of Descartes are published in the Third Volume of his Posthumous Letters. The Second Part of it treates of the Geometrical Composition or Delineation of Equations by and of a Circle and Parabola, wherein the Author seemes to have followed Descartes. About this Subject see an Excellent Tract Intituled Mesolabum, sive Duæ medie inter extremas datas infinitis modis exhibite, Auctore Renato Francisco Slusio Canonico Leodiensi (cujus nomen subti- cetur) Leodii Eburonum 1659 4', which Book the Lear- ned Author thereof promiseth to reprint and enlarge this Sum- mer. The Third Part of it contains 105 Theoremes about Sines, Tangents, Secants, &c. the Doctrine whereof, together with what else is omitted in this Edition, and other considerable matters about Equations, may be hoped for from the Pen of that ex- cellent Person, that is mentioned in the Epistle to the Rea- der. III. AN ESSAY towards a REAL CHARAC- TER and a PHILOSOPHICAL LANGU- AGE, by JOHN WILKINS D.D. Dean of Ripon, and Fellow of the R. Society. The Reverend and Learned Author of this well-con- sider'd Work hath digested the things, which to him seem'd most proper and material to be said of this Subject, into four parts. In the First, he premises some things as Pracognita, concern- ing such Tongues and Letters as are already in being, particularly concerning those various defects and imperfections in them, which ought to be supply'd and provided against, in any such Lan- guage or Character, as is to be invented according to the Rules of Art. The Second contains that which is the great Foundation of the thing here designed, viz. a regular Enumeration and Description of all those Things and Notions, to which Markes or Names ought to be assigned according to their respective natures; which may be stiled the Scientifical Part, comprehending Universal Philo- sophy: It being the proper End and Design of the several branc- hes of Philosophy, to reduce all things and notions unto such a frame, as may express their natural order, dependence, and relati- ons. ons. All these things or notions he represents in a Scheme, and reduces them to forty Genus. The Third part treats of such helps and Instruments, as are requisite for the framing of these more simple Notions into continued Speech or Discourse; which may therefore be stiled the Organical or Instrumental Part, and doth comprehend the Art of Natural or Philosophical Grammar. In the Fourth, he shews, How these more general Rules may be applied to particular kinds of Characters, and Languages, giving an Instance of each. To which he adjoyns, by way of Appendix, a Discourse shewing the advantage of such a kind of Philosophical Character and Language, above any of those which are now known; more particularly above that, which is of most general use in these parts of the World, namely, the Latin. Lastly, There is added a Dictionary of the English Tongue, in which is shewn, How all the words of this Language, according to the various equivocal senses of them, may be sufficiently expressed by the Philosophical Tables here proposed. This is the Method, in which the Author hath treated of this considerable subject; concerning which he addresses his desires to the R. Society, to whom he dedicateth this Book, that they would appoint some of their Number, thoroughly to examine and consider the whole, and to suggest, what they judge fit to be amended in it. Which desire of his hath already been so far entertain'd, that several of the Fellows of that Society have been nominated, and desired to peruse the Book with attention, and thereupon to make a Report accordingly, for the furthering and facilitating the Practice of what is therein aimed at. IV. STANISLAI De LUBIENIETZ THEATRUM COMETICUM, duabus partibus constans; quarum Altera, Cometas A. 1664. & 1665. variis Virorum per Europam Clarissimorum; cum quibus Author de hoc Argumento contulit, Observationibus, dissertationibus, animadversionibus, descriptos, &c. 59. Figuris anis illustratos, exhibet: Altera, continet Historiam 415. Cometarum, a tempore Diluvii ad A. 1665, cum 25. Figuris, & accurato indiculo (692) sulo non tantum tristium, sed & letorum Eventuum, eos secutorum: in qua simul Synopsis quaedam Historiae Universalis propositur; & Theatri Cometici Exitus sive de significatione Cometarum. Opus Mathematicum, Physicum, Historicum, Politicum, Ethicum, Oeconomicum, Chronologicum. Amstelodami A. 1668. in Fol. ERRATA. Page 667. l. 5. r. Tympanum p. 684 l. 13. r. execta. p. 685. l. 17. r. thence be given. p. 688. l. 14. r. he answers. FINIS. In the SAVOT, Printed by T.N. for John Martyn, Printer to the Royal Society, and are to be sold at the Bell a little without Temple-Bar, 1668.