tract one Root out of any poffible Equation in Numbers, either exactly or very nearly true, taken from Mr. Kerfey's Elements of Algebra; a Remark on an Error in the Reafoning of Clairaut, in that Part of his Elements of Algebra, in which he endeavours to prove the Rules of Multiplication laid down by Writers on Algebra concerning negative Quantities, by the Baron; a general Method of inveftigating the two, or three, firft Figures of the leaft Root of an Equation, that has more than one real and affirmative Root, by the fame, and reprinted from the third Volume of the "Scriptores Logarithmici" a Specimen of Vieta's Method of refolving Algebräic Equations of any Order, or Degree, by Approximation, by the fame; and Remarks on the Number of negative and impoffible Roots in Algebraic Equations, by Mr. Frend. From tris fummary of the contents of the Tracts before us, and the well known character of the editor, the mathematical reader will be fufficiently apprifed of their value and importance. In Baron Maferes's Remark on the Error of Clairaut, he furnishes new fupport to Mr. Frend's doctrine refpecting negative numbers, in addition to what he advanced in his "Appendix to the Principles of Algebra," announced in our Regifter for the year 1798. In our Regifter for the year 1796, we introduced to our readers the first part of "the Principles of Algebra, by William Frend." We have now to announce a fecond part of that work, entitled, "the Prin, ciples of Algebra, or the true Theory of Equations established by Mathematical Demonftration." In this continuation of his plan, the author perfeveres in fimplifying the cience by rejecting negative and impoffible quantities, and carefully refraining from the use of terms that appear abfurd or unintelligible in the folution of his problems. With respect to the propriety of that measure, the mathematical world is much divided in opinion; but to our mind the arguments in favour of it appear more fatisfactory than thofe adduced to fupport the ancient fyftem. This part of the author's work is confined to the Doctrine of Equations; which are divided into four claffes, according to the number of their unknown terms, and are all fubjected to one general rule, viz., that none in any clafs can have more roots than it has unknown terms. In the forms of each clafs, however, the number of roots depends, partly upon the co efficients, which he denominates the co-parts, and partly upon the changes of the marks of addition or fubtraction. Mr. Frend's folutions of his problems are direct, accurate, and perfpicuous; and are followed by fome general obfervations, which convey to the learner much useful practical information. On the whole, we think that his work deferves to be recommended as a valuable elementary treatife to those inftructors who wish to teach, or to thofe young perfons who with to acquire a knowledge of, the principles of algebra, in the most easy and expeditious manner. The "Analyfis Fluxionum, Auctore Guil. Hales, D. D. Rectore de Killeflandra, et nuper Trin. Col. Dublin. Socio, ac Linguarum Orientalium Profeffore," is divided into two parts. The first part treats of the Hiftory of Fluxions; in which the author neither awards the merit of their prior difcoyery to Newton, nor Leibnitz, but fuppofes that both thofe great men were led to the invention of the the fame principle, although differ- By Alexander Ingram, Philomath." ing in their conception, and ufing Precluded as we are by the nature different characters in their defcrip- of our work from entering into a tion of it, about the fame time. detail of the particulars of Mr. InThe bafis of the whole he makes gram's corrections and amendto be the law of prime and ultimate ments of the labours of preceding ratios. The fecond part is em- editors of Euclid, we can only obployed in illuftrating the Method ferve in general, that they show of Fluxions. Subjoined to the work him to be poffeffed of that degree are two appendixes. In the first of fcientific learning, acutenefs, and appendix we are prefented with an accuracy, which abundantly quainvestigation of the analysis of the lified him for engaging in fuch a ancient geometricians; and, in the work, and that they cannot fail of fecond, with a vindication of fir proving useful to ftudents in geIfaac Newton from the infinuations ometry. The Treatife which the of profeffor Robifon, refpecting author has added on Plane and his doctrine of a vibrating æther, Spherical Trigonometry, is judiand with other matter, critical, cioufly executed; as is, likewife, theological, and political. The the tract which follows it, on Pracwhole difplays much profound tical Geometry, illuftrated with fcience, folid learning, and a com- plates, containing diagrams, &c. mendable zeal in the interests of genuine philofophy, religion, and morals. But the author has written only for the learned; and their patience will often be put to the trial, by the want of judgment in the arrangement of his materials, and the ftrange mixture of languages, ancient and modern, in which they are exhibited. From the ample title of the next work which we have to notice, our readers will receive fufficient information respecting the nature of the task which the author has undertaken. It is "the Elements of Euclid, viz. the firft Six Books, with the Eleventh and Twelfth. In which the Corrections of Dr. Simpton are generally adopted, but the Errors overlooked by him are corrected, and the Obfcurities of his and other Editions explained. Alfo, fome of Euclid's Demonftrations are reftored, others made fhorter and more general, and feveral useful Propofitions are added: together with Elements of Plane and Spherical Trigonometry, and a Treatife on Practical Geometry, In our Regifter for the year 1798, we noticed the appearance of the first volume of a complete Syftem of Aftronomy, by the Rev. S. Vince, M. A. F.R.S. Piumian Profeffor of Aftronomy and Experimental Philofophy in the University of Cambridge." That truly va luable and useful work is now completed by the publication of a fecond volume, and entitles the author to the warm acknowledgements and commendation of every English lover of the science which it teaches. For it contains the fulleft, moft perfect and fatisfactory information, not only refpecting plane, but alfo phyfical aftronomy, that has yet appeared in our language. And with respect to the manner in which it is executed, the teftimony which we formerly gave to the author's merits as difplayed in the first volume, is frictly applicable to the proofs of them in the volume before us. The body of the work is divided into eleven chapters, in which the author fucceffively treats of the general doctrine of centripetal forces; the theory theory of the moon; the figure of the earth; the preceffion of the equinoxes, and the nutation of the earth's axis; the denfities, quantities of matter, light and heat of the planets; the motions of the planes of the orbits of the planets, from their mutual attractions; the effects produced on the motions of the planets in the planes of their orbits, from their mutual attractions; the tides; the principles of projection, and the conftruction of geographical maps; the ufe of interpolations in aftronomy; and the hiftory of aftronomy. Thefe chapters are followed by a forcible, perfpicuous, and fatisfactory argument in favour of the existence of an intelligent, omnipotent, and infinitely good First Caufe, deduced from the fimplicity and regularity obfervable in the construction and procedure of the universe. To the whole the author has added numerous luable tables, for facilitating aftronomical calculations, with precepts and examples to each; Dr. Bradley's Catalogue of 389 fixed Stars; M. de la Caille's Catalogue of 515 Zodiacal Stars, and his Catalogue of 307 principal Stars; Zach's Catalogue of 301 principal Stars, and his Catalogue of the Declination of 162 principal Stars; and Mayer's Catalogue of 992 principal Stars. va To military men the following work will prove acceptable and ufeful, from the fullness and excellence of the inftructions contained in it, and the judicious intelligible manner in which they are conveyed. It is entitled "the firft Principles of Field Fortification: containing concife and familiar Precepts for the Conftruction, Attack, and Defence of Field Works; with a preliminary Introduction to the Science of Fortification in general. By Charles Auguftus Struenfee. Translated from the German by William Nicolay, Captain-Lieutenant in the Corps of Royal Engineers." It is illuftrated with thirteen correct and well executed plates. The fecond and third parts of "a Collection of Papers on Naval Architecture, &c.," complete the fecond volume of a very useful and interefting work, which increafes in value and importance as it fwells in magnitude. The publication of the preceding divifions of this Collection was announced in our Regifters for the years 1791 and 1795. The Effays on Gothic Architecture, by the Rev. T. Warton, Rev. J. Bentham, Captain Grofe, and the Rev. J. Milner, &c." conftitute a judicious and pleafing compilation, which furnishes the reader, at an easy expence, with fome of the beft treatifes on Gothic architecture, extracted from voluminous or coftly works, and forming together an ufeful manual on that interefting fubject. They confist of an Effay on Saxon and Norman Architecture, from Bentham's elaborate Hiftory of Ely Cathedral; Captain Grofe's Preface to his Antiquities of England; Profeflor Warton's concife Hiftory, in his Notes on Spenfer's Fairy Queen; and the Hiftory of the Origin and Progrefs of the pointed Arch, fromMr. Milner's Hiftory and Antiqui ties of Winchefter. They are preceded by a preface from the pen of Mr. Taylor, the editor, which displays a confiderable acquaintance with the fubjects difcuffed; and "Obfervations on the Means neceffary for further illuftrating the ecclefiaftical Architecture of the middle Ages," by the Rev. Mr. Milner. And they are illuftrated with ten plates of ornaments, &c. felected from ancient buildings, calculated to exhibit the various ftyles of different periods. The "Anecdotes of the Arts in England, or comparative Remarks on Architecture, Sculpture, and Painting, chiefly illuftrated by Specimens at Oxford, by the Rev. James Dallaway, M. B. F. S. A. &c." are divided into three parts, each of which is fubdivided into fections. The first part treats of architecture, commencing with fome remarks on the origin of Gothic architecture, and afterwards enlarging on the feveral varieties of ecclefiaftic Gothic architecture; military and domeftic Gothic architecture; mixed architecture; the revived Grecian; and modern architecture. The fecond part, which is devoted to fculpture, exclufive of an abridged hiftory of the origin and progrefs of fculpture among the ancients, and of the practice of amaffing the treafures of antiquity in Italy before other nations of Europe acquired a fimilar tafte for the arts, confifts of diftinct catalogues of the feveral collections in England, with remarks, and fome curfory notice of the most celebrated of our living artists. The third part gives a fummary view of the hiftory of painting in England; of public and private collections; with notices of the merit of particular pictures, and of the excellences of individual painters of the modern English fchool. These anecdotes, which the author acknowledges to be only "ketches, from many caufes neceffarily lefs complete than he could with them," while they exhibit evidence that he is not devoid of knowledge or tafte, on the fubjects which he has undertaken to difcufs, discover occafional marks of negligence and inaccu racy which are highly blameable. In fome inftances, his chronological blunders, and confusion of characters, are grofs and ludicrous. When thefe errors are carefully corrected, his work, although not calculated to extort fupereminent commendation from the profound connoiffeur, may afford inftruction and amufement to young ftudents in the arts. During the prefent year Mr. Jofeph Halfpenny, by the publication of the twentieth number of "Gothic Ornaments of the Cathedral Church of York," has completed his elegant and acceptable prefent to the admirers of Gothic architecture, of which fome of the earlier numbers were announced in our Regifter for the year 1796. It confiits, in the whole, of 105 plates, and has been defervedly patronifed by a refpectable lift of fubfcribers. With refpect to the two following works, which relate to the theory and practice of a science of which we have very imperfe& knowledge, we must content ourfelves with barely announcing their titles; premifing only, that the ableft profeffional judges have concurred in beftowing very high praife on their respective merits and utility, and in recommending them to the careful attention of mufical ftudents. The first is "a general Treatife on Mufic, partiticularly on Harmony or Thoroughbafs, and its Application in Compofition: containing alfo many effential and original Subjects, tending to explain and illuftrate the whole; by M. P. King." The fecond is entitled "an Introduction to Harmony, by William Shield," well known to the public by his numerous, excellent, and favourite compositions. Ia [In the Philofophical Tranfactions of the prefent year, the first publication we ufually notice in the department of Natural Philofophy, we find several very important papers by Mr. Herschell, which materially extend the bounds of fcience. That "on the Power of penetrating into Space by Telefcopes, with a comparative Determination of the Extent of that Power in Natural Vifion, and in Telescopes of various Sizes and Constructions," is very interefting. The immenfity of penetrating power is almoft incredible. Even with his reflecting telefcope, ftars of 1340th magnitude, fhould fuch exift, may be feen; and, when affifted by the united luftre of fiderial fyftems, it will penetrate near twelve hundred millions of millions of millions of miles, more than 300,000 times the distance of the nearest fixed ftar. Other avocations feem to have drawn for a time his attention from aftronomy, and we meet with nothing on the fubje&t in our annual lift of publications, but a fecond appendix" to the improved Solution of a Problem in phyfical Aftronomy, inferted in the Tranfactions for 1788, by Mr. Hellins." But if Mr. Herfchell has left the field in which he has acquired fo much glory, for a time, he has gathered laurels in another. In this volume of the Tranfactions, we find a feries of important articles on the fubjects of light and heat. The firit is an Inveftigation of the Powers of the prifmatic Colours to heat and illuminate Objects, with Remarks to prove the different Refractions of radiant Heat; to which is added, an Enquiry into the Method of viewing the Sun advantageoufly with Telefqopes of large Apertures" and high magnifying Powers. The fecond is entitled ་་ "Experiments on the Refrangibility of the invifible Rays of the Sun;" and the third, divided into two parts, "Experiments on the folar and the terreftrial Rays that occafion Heat, with a comparative View of the Laws to which Light and Heat, or rather the Rays that occafion them, are fubject, in order to determine whether they are the fame or different." The defign of thefe papers was originally fubfervient to our author's favourite purfuit; for, as glaffes of fome colours heated without brightly illuminating the object, others, which heated little, were not fufficiently illuminating. The maximum of illumination appeared to refide in the middle of the fcale, in a light of the brightest yellow or the paleft green. Our author was obliged, however, to facrifice a little light, for the purpose of avoiding heat, and he adopted, in his telefcopes, a blueish green on a deep blue glafs. In the fecond paper, Mr. Herfchell fhows that, in the prifmatic fpectrum, the greatest heat is in the neighbourhood of the red rays, but in uncoloured light. In the third, it appears that thofe rays which are without colour, and of courfe invisible, are still refracted; and the propofition is extended to heats raised, on the furface of the carth, by our own powers. They are capable of repeated refractions, without diminishing the heat, and are not rendered vifible by condenfation. The prifm is found to difperfe the heating rays, in a ratio different front that in which the illuminating ones are affected; and the focus of heat is of courfe not the fame with that of light. From the whole, our author thinks heat and light really different. Dr. Holmes's paper, containing "Obfervations and Experiments on the Light which is fpontaneously emitted with fome |