ed by the furgeons who attend the collieries at Newcastle, which has prevailed, he fays, more than a century, and which he at-first adopted. This confifted in anointing the burned or fcalded parts with linfeed oil, and then covering them with a foft cerate fpread on lawn paper. The veficles were fnipped to let out the effufed ferum, and plenty of oil left, with directions to the attendants to raife the plaifter, and anoint the parts from time to time as they become dry, or on the pain increafing. So attached are the people to this mode, the author fays, that he has known a gallon of oil to be used in the space of twentyfour hours, where the burn has been very extenfive. This procefs was continued until the fire was fuppofed to be subdued; that is, until the pain ceafed, which generally happened on the fourth day. When it was protracted beyond this time, the patient was exhaufted by the fymptomatic fever, the fwelling of the parts fubfided, the fkin became pale, dark brown or black fpots appeared, and the patient died on the eighth day. Internally oily emulfions, with nitre and other cooling drugs, with purgatives and opiates, were adminiftered, until fuppuration took place, when a more liberal diet, with beer or wine, were allowed, and bark and other tonics given. The author faw five cafes treated after this manner, in all of which the patients died. He was thence led to confider the subject more maturely, and to attempt correcting what he conceived to be wrong in the practice. The principal error he thought confifted in pursuing the antiphiogific or debilitating plan. By bleeding, purging, and a low dict, the ftrength of the patient was reduced, and the digeftion of the wound, and confequent feparation of the efcars prevented. He therefore determined to follow the oppofite, or ftimulating plan; that is, to fupport the powers of nature, by having recourfe, immediately after the accident, to wine, bark, and opium, and by applying oil of turpentine, alcohol, &c. to the burned parts, inftead of oil. In the first cafe treated in this manner, life was protracted to the twelfth day; that is, four days after the time on which patients treated in the ordinary mode ufually died. The phænomena which occurred in the progrefs of this cafe affifted in confirming, as well as in fome degree in correcting, the author's ideas on the fubject. In the fecond cafe, which is detailed at length, he obtained complete fuccefs. The author now confidered various modifications of his new method of practice, to be adopted according to the place hurt, or the extent of the injury. These appear to be well conceived, and the fuccefs has been fuch as will be likely to attract the attention of practitioners to this hitherto too much neglected branch of furgery. In an Appendix the author examines the method of curing burns 299 burns by the application of vinegar, communicated by Mr. David Cleghorn, brewer of Edinburgh, to the late Mr. John Hunter, and published in the fecond volume of Medical Facts and Obfervations. Mr. Kentish attributes the effects of the vinegar to the alcohol it contains. He obferves alfo, that Mr. Cleghorn avoided purging and debilitating medicines, and recommends a generous diet. From the analy fis we have given, our readers will fee, that this is a work, though fall in fize, of confiderable importance; and as fuch we recommend it to their notice. ART. XV. The Doctrine of Permutations and Combinations, being an effential and fundamental Part of the Doctrine of Chances, as it is delivered by Mr. James Bernoulli, in his excellent Tract on the Doctrine of Chances, intitled Ars Conjectandi, and by the celebrated Dr. John Wallis, of Oxford, in a Tract intitled from the Subject, and published at the End of his Treatife on Algebra: in the former of which Tracts is contained, a Demonftration of Sir Ifaac Newton's famous Binomial Theorem, in the Cafe of Integral Powers, and of the Reciprocals of Integral Powers. Together with fome other ufeful Mathematical Tracts. Publifhed by Francis Maferes, Efq. Curfitor Baron of the Court of Exchequer. Large 8vo. 606 pp. 125. Whites. 1795. THE irk fome toil of reading the trash that fo often iffues from the prefs, to which a Reviewer is condemned, is fometimes relieved by the pleasure of peruting folid and useful books, with which the volume now before us may justly be claffed. Of the valuable matter contained in it, a confiderable part is printed in the Third Vol. of the Scriptores Logarithmici, a work which, although begun before the commencement of our Review, is not yet finished, and which, on account of the valuable new materials, as well as old ones, of which it is compofed, muft hereafter have its proper share of attention from us; and, therefore, on our first reading this octavo volume, we thought that one account might serve for both the quarto and octavo; but, upon the fecond reading of * This work was mentioned in our Review for January, 1794: fee vol. iii, p. 3. it, we were perfuaded that its contents are fo highly valuable to the ftudents of the Mathematics, that we determined to review it separately. The doubt on this point has occafioned the lateness of our critique. The number of tracts of which this volume consists is nine; each of which deferves particular notice. The first of these tracts contains, in the original Latin, the three first chapters of the fecond part of James Bernoulli's Treatife on the Doctrine of Chances, together with an English tranflation of them. These three chapters, as the learned tranflator obferves, "contain a most accurate and diftinct explanation of the fundamental parts of the doctrine of Permutations and Combinations, and of the most remarkable properties of the Figurate Numbers, which, it is well known, are of the most extenfive ufe in various branches of the Mathematics." Pref. p. iii. Amongst the ufes to which this doctrine was applied by Mr. Bernoulli, is a very neat demonftration of Sir Ifaac New ton's Binomial Theorem, in the cafe when the index is an affirmative whole number, which indeed is the cafieft cafe of it and it was a defire of making this demonftration more generally known, that induced the tranflator to publish this volume. He fays, "As there are many perfons in England that are fond of the mathematical fciences, without having much acquaintance with the Latin language, I have, in order to render the contents of these three valuable chapters acceffible to fuch perfons, tranflated these chapters into English, and fubjoined the tranflation to the original text in Latin; fo that the reader may choofe in which of the two languages he will perufe them. And in this tranflation I have expreffed myself in a fuller manner than Mr. Bernoulli had adopted in the original, because I had obferved that the great degree of brevity with which Mr. Bernoulli had expreffed himself, had rendered fome parts of the original rather obfcure. And I have likewise added a few notes, both to the original and the tranflation, where the text seemed to me to require them." Pref. p. iii. This is a juft and modeft account; for, befides the notes, the learned writer has illuftrated it with many examples which are not in the original, and has produced a demonstration of the Binomial Theorem, when the index is any negative whole number, no lefs neat and elegant than Bernoulli's demonftration of the easier cafe. The fecond tract in this volume, is the tenth effay of the late Mr. Thomas Simpson, On finding the Sum of a Series of Numbers, of which the Roots are in Arithmetical Progreffion. This being nearly connected with the fubject of the preceding tract, and of confiderable utility, is here reprinted, and cannot fail of being acceptable to those who have not Simpson's Essays, which is a book that begins to be scarce. The third tract contains an Investigation and Demonftration of the Binomial Theorem, in the cafe of integral and affirmative powers. Here the investigation of the co-efficients of the terms, as the writer very fairly acknowledges," was fuggefted by Profeffor Saunderfon, in the fecond volume of his Algebra (p. 607) and the demonftration is nearly the fame with that which was given by Mr. John Stewart, of Aberdeen, in the fixth fection of his Commentary on Sir Ifaac Newton's tract, intitled Analyfis per Æquationes numero terminorum infinitas." But the fuli and clear manner in which both the Inveftigation and Demonftration are here explained, will render this a very valuable paper to those who are entering on these speculations. Before we difmifs this tract, we have to obferve, that the method of Demonftration which was used by Stewart, is, in effect, the fame that was used by Ronayne in his Algebra, p. 215 and 216 of the fecond edition", which was printed eighteen years before Stewart's book. The fourth tract, is a Difcourfe of Combinations, Alternations, and Aliquot Parts, by Dr. John Wallis, Profeffor of Geometry, at Oxford. This valuable tract was published with his algebra, in 1685, and it is too well known to need any commendation at this time. The fifth tract, is the Appendix to the English Tranflation of Rhonius's Algebra, made by Thomas Brancker, A. M. and publifhed London in the year 1668; containing a table of odd numbers, and of all primes, lefs than 100,000; by means of which table (which will be very ufeful to thofe who have frequent occafion to make calculations in the higher parts of the mathematics) any odd number lefs than 100,000, if not a prime, may quickly be refolved into its component parts; and if it be a prime, that will be difcovered. The fixth tract, is Of rational Numbers that express the Sides of Right-angled Triangles. We here find very elegant and masterly folutions of thefe two problems; 1. To find as many right-angled triangles as we please, of which the three fides fhall be expreffible in rational numbers. "2. To divide a given fquare number into two other fquare numbers, either whole numbers, or fractions, or mixt numbers." After thefe, many fets of rational numbers (difcovered by thefe folutions) which exprefs the lengths of the sides of right * Whether this demonftration was in the first edition of the book, printed in 1717, we cannot fay, not having it by us. angled angled triangles, are fet down; and then a table of the fquares of all the whole numbers from 1 to 100, together with their first and fecond differences, to facilitate the finding of fuch rational numbers; which table will be found useful on many other occafions. The feventh tract, is chiefly On the Extraction of the CubeRoot, by M. de Lagny's method, the investigation of which is given, together with feveral examples of its ufe. It con tains alfo a large extract from a letter of the celebrated M. Leibnitz to M. Oldenburgh, dated February 3, 1672-3, respecting the feveral orders of the differences of cube numbers, and the fums of certain feries. It contains likewise a table of the cubes of the numbers 1, 2, 3, &c. to 100, together with their first, fecond, and third differences, which may be useful on many occafions. This paper ought not to pass through our hands without a remark, that, if either of M. de Lagny's rational expreffions of the fecond near value of the cube-root, 2c4a3 be reduced to the form of a fraction, it will be +245X9, where c denotes any number of which the cube-root is wanted, and a the firft near value of that root; and that M. de Lagny's theorems were first published in the year 1691. The eighth tract contains a statement of M. de Lagny's Method of extracting any higher Roots whatsoever of Numbers by Approximation, together with the Investigation of his Theorems, and an Illuftration of them, by a proper Number of wellchofen Examples. Here again we find ourselves obliged to remark, that, if either of M. de Lagny's rational expreflions of "VN be reduced to a fraction*, it will be +1. N+m—1.aTM m 1. N+m+1.ana; and that the learned writer of this tract informs us, the original was published in French, in the year 1697. The gentleman, therefore, who has lately published thefe theorems, as his own invention, is no more than the fecond inventor of them. The last tract in this volume is intitled, Obfervations on Mr. Raphfon's Method of refolving affected Equations of all Degrees by Approximation. Here, after fome very judicious remarks on the perplexity and obfcurity which the introduction of negative quantities into algebra has occafioned, the Baron proceeds to the folution of an high equation, which (to use his own words) is performed at great length, in order to fet forth, in as clear a manner as poffible, the feveral reasonings upon * See our Review for February, p. 159. |