For the Monthly Magazine. A SKETCH of the HISTORY of PURE MATHEMATICS, tranflated from "Traité Elementaire de Mathematiques Pures, par LEMOINE, Profeffeur de Mathema. tiques et de Phyfique, &c. ALGEBRA. [Continued from page 527, Vol. xiii.] HE rectification of a curve line, 79. T firft peformed by Neil, was only an extenfion of the new views which Wallis had opened in his Arithmetic of Infi. nites, a work pregnant with genius, the appearance of which may be regarded as an epocha in the progrefs of modern geometry. Guided by the thread of analogy, of which he always knew how to avail himself, the English analyst observed, that the denominators of fractions may be confidered as powers with negative exponents. This remark enabled Wallis to measure all the spaces, whofe elements are inversely as fome powers of the abfciffe. 80. We alfo owe to Wallis the method known by the name of interpolation, which confifts in inferting, in a progreffion of quantities, proceeding by a certain law, one or more intermediate terms, conformable to that law. In endeavouring to interpolate, in a certain progreffion, a term which he expected would give him the area of the circle, Wallis only found an infinite feries of terms, which converged more and more to the true value. Not being fatisfied with this refult, he invited Lord Brounker to fecond his efforts; and his Lordship, by means of continued fractions, of which he was the inventor, gave a more just approximation. 81. To Brounker, geometry is also indebted for having first expressed the area of the hyperbola in an infinite feries. Mercator, who had discovered a fimilar feries, published that fine difcovery in his • Lord Brounker, Viscount of Caftle Lions in Ireland, was born about the year 1620, and died in 1684. When the Royal Society was first established, he was elected Prefident, and was annually continued in that office for a period of about fifteen years. + Nicholas Mercator was a native of the Dutchy of Holftein, but went to England about the year 1660, and refided in that country till his death. He was one of the first Members of the Royal Society. That celebrated mathematician was faid to have been fo weak, as to believe in judicial aftrology. Logarithmotechnia, which was printed in 82. Barrow, the cotemporary of Wallis, published his Geometrical Lectures in 1669. Among the excellent inventions, which he has explained in that work, filled as it is with profound inquiries, we ought particularly to observe his method of drawing tangents to curves. That English Geometrician confidered the Ifaac Barrow was born in London about the year 1630. His youth juftified no great expectations; but having arrived at manhood, he made a rapid progrefs in all kinds of knowledge. His merit procured him the Greek profefforfhip in Cambridge, which he exchanged for that of Geometry. Barrow died in 1678. Note by the Tranflator.-Inftead of any biographical addition to our author's hints refpecting Dr. Barrow, the reader may perhaps be pleased to fee a tranflation of a curious paffage in the preface which that confummate geometrician, that preceptor and precurfer of Newton, has prefixed to his edition This of Appollonius, published in 1675. paffage appears to me to poffefs all the fimplicity and fublimity, with very little of the quaintnefs, obfervable in the fingular Dedidication of Edwards's History of Birds: 40 ☺tòç yeœμestges-GOD a&is geometrically. "Tu autem, Domine, quantus es Geometra," &c. "But how great a geometrician art thou, O Lord!" For this fcience knows no limits, and even human fagacity can difcover numberless new truths: but Thou perceiveft them all at one view, without any chain of deductions, or tire fome length of demonftrations. In other fubjects, our intellect poffeffes but little power: like the imagination of brutes, it feems only to dream of fome uncertain objects, concerning which there are almost as many opinions as there are men. But in mathematical truths, there is an universal agreement; in them the human mind feems capable of fomething great and wonderful,&c.-Thee, therefore, I rejoice to love; to 1 hee I look up, ardently longing for that day, when thy immenfe and most holy benignity shall enable me to understand, not only thefe, but far more numerous and important truths, with a mind purged from error and prejudice, and without this fucceffive and laborious effort of thought."-Barrow in his Oratio præfatoria, on being placed in the chair of geometry, difplays the vaft utility of the mathematical fciences more elegantly and forcibly than any author I know. But that piece, though brief for the fubject, would be too long for this place. The original may be seen in his Leiones Mathem. Cantab. babitæ, A. D. 1664, &c. and a tolerable tranflation in Stone's Mathematical Dictionary, article Mathematics. little little triangle, whofe fides are the difference of two ordinates, infinitely near to each other, their perpendicular diftance, and the element of the curve, as fimilar to the triangle formed by the ordinate, the fub-tangent, and the tangent. He then fought, by means of the equation of the curve, the relation of the two fides of the little triangle, of which one expreffes the difference and the other the diftance of the two ordinates, by forming this proportion. As the difference of the two ordinates is to their distance, fo is the ordinate to the fubtangent. This rule, which is that of Fermat fimplified, differs from the method of the differential calculus in nothing but the notation. 83. Barrow had the honour of numLering Newton among his pupils, and Ifaac Newton was born on the 25th of December, 1642, at Wolftrop, in the county of Lincoln. In his first attempts he feemed rather to invent than to ftudy. Having but glanced over the Elements of Euclid, he paffed on to the Geometry of Defcartes, in which he found ideas proportioned to the force of his mind. Newton advanced in his fcientific career with the most firm and rapid pace. In him we trace neither errors nor failures, and to him was properly applied the idea of Lucan on the river which waters Egypt, the fource of which was unknown to the ancients:-Men bave not been permitted to trace the Nile to its weak commencement. Barrow, on refigning his academical chair at Cambridge, procured it for Newton, who was then but twenty-two years of age; but in his twenty-fourth year he was in poffeffion of two of his fineft difcoveries, namely, the theory of light, and the method of fluxions. In 1687, he gave to the learned world his Mathematical Principles of Natural Philofophy, an immortal work, in which the most profound geometry is laid down as the basis of true phyfics, and which will always be confidered as one of the moft fub. lime productions of the human mind. Newton having been appointed Mafter of the Mint in 1696, filled that station with equal genius and difinterestedness. Till his eightiech year, he poffeffed that uninterrupted health which he owed to his temperance. But he then began to decline, and in the beginning of the year 1727, he was attacked with the ftone. In this fatal conjuncture he fhewed as much firmnefs as he had difplayed fagacity during his life. The excruciating pangs which terminated his life, Extorted from him neither complaints nor murmurs. At laft, in his 85th year, he flept in that tranquillity which he had always purfued. His body was conveyed to Westminster Abbey, and laid on a bed of fate, whence it was carried in great pomp of appreciating the merit of a man who difplayed the highest powers of the hum.n intellect. 84. Scarcely had Newton commenced his mathematical ftudies, than he perfected the ancient methods, and invented to the place of interment. The family of Newton have fince erected a monument to his memo y, on which is infcribed a moft honourable epitaph, which ends thus:-Sibi gratulentur mortales tale tantumque extitiffe bu mani generis decus. Note by the Tranflator.-Here the author has added a French poetical paraphrafe of Pope's well-known couplet : Nature and Nature's laws lay hid in night, God faid, Let Newton be-and all was light. But as that paraphrafe does not rise above mediocrity, I have taken the liberty to omit them, and to fubftitute the following lines, which I flatter myself will be the more acceptable to the reader, as they do not appear to have ever been printed, except in 1741, in Caribbeana, a collection of papers which were interefting to few, and therefore known to few but the West Indians of that period. Their author was the celebrated Dr. Pitcairn, who having been a good mathematician (Cheyne addrefied his Fluxions to him in the epiftolary form) was the better able to judge of the vast extent and value of Sir Ifaac's difcoveries and improvements. DE NEWTONO. Pythagora jactat Samius fe fundus alumno, Newtono geftit terra Britanna fuo. Dum vaga Phabeis terra vebetur equis; ON SIR ISAAC NEWTON. Samos much boasts Pythagoras's birth, Nor Britain lefs th' illuftrious Newton's worth. Both ifles, from each, like glory will derive, Whiift Sol attracts the earth, their names fhall live; But varying ftill in this, that Britain's fon Much farther hath the Samian sage out-done, Than does th' extent of fam'd Britannia's inle The narrow confines of the Samian foil. In fome accounts of Newton's life, the binomial theorem is faid to be infcribed on his monument in Westminster Abbey. The fame thing is affirmed in the Mathematician, p. 273, and in Stone's Mathematical Dictionary, article, Binomial Root. But I have more than once inspected that mɔnument very attentively, without being able to difcover the leaft trace of that admirable formula, which affuredly would have been the moft lafting part of the monument. When Newton invented it, he might have juftly exclaimed-Exegi monumentum ære ferennius! new Godfrey William, Baron of Leibnitz, was born at Leipzick in 1646. He began his ftudies in his native country, which afforded him but indifferent afiiftance. He formed himfelf, fo to fpeak, by his active and ardent genius, and at the age of fifteen he had poffeffed himself, with incredible zeal, of all kinds of human knowledge. Poetry, hiftory, antiquities, jurifprudence, philofophy, mathematics, phyfic, &c. in a few years came under the dominion of his genius. In the beginning of the year 1673, Leib. nitz, having been in London, became ac quainted with Mr. Oldenburg, the Secretary of the Royal Society, with whom he opened an epiftolary correfpondence. After a stay of fome months in the metropolis of England, he returned to Paris, where he had already been in 1672. Then it was that he began to apply to the higher geometry, a tafie for which he had conceived from the converfation of Huygens. In 1676 Leibnitz returned to Germany, by the way of London, where he ftaid but a few days, and repaired to the Court of the Elector of Hanover, who recailed him, and attached him to his interefts. The bufinefs with which he was charged by that Prince did not hinder him from inferting in the Leipzick A&ts a number of memoirs, both phyfical and mathematical, which all bear the ftamp of genius, and which leave us to regret that their author, distracted between his laborious employments, and by his taste for metaphyfics, † had not leifure to cultivate, exclufively, the accurate fciences, and to give to the learned world a work, of which, according to his plan, the differential and integral calculi were to have formed the moft confiderable part. It was at the inftance of Leibnitz, that Frederick 1. King of Pruilia and Elector of Brandenburg, founded the Academy of Berlin in 1701. Leibnitz was appointed Prefident of that Inflitution, and continued in that place till his death, which happened on the 14th of November, 1716, when a fit of the gout feizing on his nobler organs, almost immediately deprived him of life. † Leibnitz was a metaphyfician as well as a mathematician.-See his correfpondence with Dr. Samuel Clarke, in which, how ever, that celebrated German makes, compaTranflator. ratively, but a poor figure. enlarged the bounds of algebra. He dif- 85. It appears that Newton first difcovered the method of fluxions; but it is not lefs certain that Leibnitz invented the differential calculus, without borrowing any thing from Newton. On this fubject I believe we might refer to the teftimony of Newton himself, who, in his Principia, has thus fpoken of the German geometrician: 86. "About ten years ago (fays he) having exchanged fome letters with M. Leibnitz, and having fignified to him that I was in poffeffion of a method of determining the Maxima and Minima, and of drawing tangents, and which irrational quantities did not embarrass, and having concealed my method, by tranfpofing the letters; he returned me for anfwer, that he had fallen upon a fimilar method, which he communicated to me, and which differed from mine in nothing but the enunciation and notation, and the idea of the generation of quantities." James Gregory was born in Scotland in 1636, and lived feveral years in Italy. On returning to his native country, in 1670, he was appointed Profeflor of the Mathematics at St. Andrews. He was advancing rapidly in the fteps of Newton, and was inspiring the greatest expectations, when a premature death carried him off in 1675. Dr. James Gregory was the brother of Dr. David Gregory, the great aftronomer. Their family has been remarkable for genius, efpecially in the mathematics. See the Encyclo pædia Britannica, article Gregory. Tranflater. What we call a differential, Newton calls a Fluxion; and what we call an integral, he calls a fluent. The method of fluxions anfwers to the calculus differentialis; and the inverfe method of fluxions, to the calculus integralis. 87. This 87. This paragraph we read in the editions of 1713 and 1714; and though it has been fuppreffed in the edition of 1726, the defenders of Leibnitz will always have it in their power to appeal to the teftimony and the confcience of Newton. 88. It appears that Leibnitz might have remained in quiet poffeffion of part of the honour redounding from the difcovery of the new calculus, if he had done more justice to Newton. In fome letters which he had written to perfons in England, he claimed the invention exclusively, which drew forth fome very difagreeable remarks upon the prior rights of Newton. In 1708, Keill* published, in the Philofophical Tranfactions, a paper in which he exprefsly affirmed, that Newten was the first inventor of the method of fluxions, and that Leibnitz, when he publifhed it in the Acta Eruditorum of Leipfick, had only changed the name and the notation. 89. Leibnitz, infulted by this charge of plagiarium, demanded, in a letter to the Secretary of the Royal Society of London, that Keill fhould retract what he had advanced. Keill, instead of this, returned for answer a long letter, in which he ftated all the proofs which he had, to fhow, not only that Newton had preceded Leibnitz, but also that he had given the German geometrician fo many fpecimens of his calculus, that it could not escape a man even of ordinary underflanding. The Royal Society of London appointed a Committee to confult the original papers. They gave no opinion on the merits of the cafe; but they refolved that Keill had not injured Leibnitz, by affirming that Newton was the first inventor of the method of fluxions. 90. The controverfy, however, was continued. A common friend of Newton and Leibnitz tried to bring them to a mutual explanation. But this attempt anly ferved to increase their ill-humour; Leibnitz perfiiting in his denial of Newton's right of priority; and Newton refofing to Leibnitz what he had formerly conceded to him. At laft, the death of Leibnitz put an end to the dispute. John Keill, M.D. Profeffor of Aftronomy in Oxford, and Member of the Royal Society of London, was a native of Scotland, and died in 1721, in the 50th year of his ge. The works of that able man are in very great eftimation among the learned. His brother, James Keill, M. D. was alfo a good mathematician. Tranflater. 91. It is now generally agreed, every where but in England, that Newton and Leibnitz attained the fame object by the force of their genius, but by purfuing different methods; Newton by regarding Aluxions as the fimple proportions of nalcent and evanefcent quantities; and Leibnitz, by confidering that, in a feries of quantities increafing or decreafing, the difference between the two confecutive terms may become infinitely small, that is, lefs than any finite affignable magnitude. 92. If the method of fluxions be the moft luminous, if it has the merit of anticipating the objections which may be made against different orders of infinitely fmall quantities; the differential calculus poffeffes the advantage of conducting us to the fame results by a lefs difficult path. (Algebra to be continued.) For the Monthly Magazine. ACCOUNT of a MODE of KILLING SEALS. N On the fouth fide of the ifle of Zante, is a village named Agala. Its inhabitants, in addition to the agricultural labors which are common to them with the other islanders, avail themselves of their fituation to carry on the feal-fishery. They live at the diftance of only two miles from the fea, which, nevertheless, they cannot reach without defcending precipices, of which the bare afpect is fufficient to infpire terror. Thofe moun taineers, however, having acquired intrepidity from habit, defcend to the fea fide with aftonifhing agility, only fupporting themfelves by a thick rope fastened to a tree or the point of a rock. At the water-edge, the rocks that border the fea are full of caverns, into which the feals retire to fleep, and to bring forth their young. To penetrate into thefe caverns, it is neceffary for the adventurer to wade in the water almoft chin-deep, taking care to hold aloft the pistol with which he intends to fhoot his game. If the feal happen to be afleep at the time, fuccefs is certain: but, if awake, at the approach of his enemy he violently darts into the water; in which cafe the greatest dexterity is requisite to hit him in the head, the only place where the hot is mer be but flight, and infufficient to prevent tal in any other part, the wound would his escape. : When the mountaineer has killed the feal, he flays him in the cavern, and takes away only the fkin and fat, leaving all the reft of the animal to be devoured by the the birds or carried off by the waves. The skin, being proper¡y dresset, ferves to make fhoes, which are found more durable than those of neat's leather. The fat is melted and reduced to oil, which thofe peafanis burn in their lamps, and which gives a clearer light, and lafts longer, than olive-oil: but the fmell which it emits while burning is intolerable to any perfon whofe olfactory nerves are not habituated to it, as thofe of the inhabitants of Agala. The fpring is the most convenient feafon for this ipecies of fishery. I For the Monthly Magazine. DEFENCE OF FORESTALLING. (Continued from page 424 of Vol. xiii.) CASE V. DIP into my file of papers, and meet with the cafe of Rufby. The proceedings in this cafe have been fo well expofed by Sir Thomas Turton, that I hall pafs on, with too much certainty of foon tumbling upon another. I fhall only obferve, that Sir Thomas has made fome conceffions which he would not have made if he had had time to follow, as far as his principles would have led him, and which cannot fail to give a handle to the foes of foreftalling, if the foes of foreftalling fhould ever defcend to reafening. But how feldom has this been done by any men who could employ force! CASE VI. "The Sheep-market to-day would have been very reasonable if it had not been for those horrible vermin the engroffers. They were very abundant, and must have fold at a low price, if they had not been bought up by thefe pefts of fociety." This is the manner in which the argument is carried on by the advocates of perfecution. The real meaning of this rhetoric is this:-So great a number of theep happened to be brought on a particular day, that they must have been fold a great deal under the average price of the markets, if thofe farmers, who had not capital enough to afford to drive them home again, and keep them till another market-day, had not been affifted by the capital of the middle-men. Let us fuppofe thefe middle-men thus addreffing the owners of the fheep:-"You are obliged to fell your fheep for lefs than they will bring at any future market, on mount of the accidental glut of to-day. We will lend you money on the fecurity of your sheep. Come again at a future time: repay the money, with intereft, and the keep of the fheep, and you shall band. MISORHETOR. To the Editor of the Monthly Magazine. SIR, HAT discoveries, on their first introducTT the most useful and important tion, meet with oppofition from prejudice, intereft or envy, is a fact too well known to require much illuftration. With what violence and acrimony has not even the vaccine inoculation, one of the most important difcoveries of ancient or modern times, been opposed by some men under the influence of one or more of thefe motives? Prejudices founded upon long habits and common example, will not readily yield even to the evidence of facts. Your Magazine for June contains a new inftance of this kind. You have inferted in that number of your interefting Mifcellany, a letter from a Conftant Reader, on the fubject of the New Improvements made in Tanning. 2 The au thor |