Fellows Of The Royal Society

"And the best Copies of his Manuscript having been transmitted to the Publisher, it was easy, by comparing them, to establish a correct and genuine Text. There were, besides, several detached Papers, some of which were quite finished, and wanted only to be inserted in their proper Places. In a few others, the Demonstrations were so concisely expressed, and couched in Algebraical Characters, that it was necessary to write them out at more Length, to make them of a piece with the rest. And this is the only Liberty the Publisher has allowed himself to take; excepting a few inconsiderable Additions, that seemed necessary to render the Book more compleat within itself, and to save the Trouble of consulting others who have written on the same Subject."

"MR. MACLAURIN a most eminent mathematician and philosopher, was the son of a clergyman, and born at Kilmoddan, in Scotland, in the year 1698."

"Here he spent two or three years in pursuing his favourite studies; but, in 1717, at 19 years of age... he offered himself a candidate for the Professorship of Mathematics in the of , and obtained it after a ten days' trial against a very able competitor."

"He was sent to the University of Glasgow in 17Q9; where he continued five years, and applied to his studies in a very intense manner, and particulariy to the mathematics."

"1. To give the general Principles and Rules of the Science, in the shortest, and at the same time, the most clear and cemprehensive Manner that was possible. Agreeable to this, though every Rule is properly exemplified, yet he does not launch out into what we may call, a Tautology of Examples. He rejects some Applications of Algebra, that are commonly to be met with in other Writers; because the Number of such Applications is endless: And, however usefull they may be in Practice, they cannot, by the Rules of good Method, have place in an Elementary Treatise. He has likewise omitted the Algebraical Solution of particular Geometrical Problems, as requiring the Knowledge of the Elements of Geometry; from which those of Algebra ought to be kept, as they really are, entirely distinct; reserving to himself to treat of the mutual Relation of the two Sciences in his Third Part, and, more generally still, in the Appendix. He might think too, that such an Application was the less necessary, that Sir Isaac Newton's excellent Collection of Examples is in every body's Hands, and that there are few Mathematical Writers, who do not furnish numbers of the same kind."

"He seems, in composing this Treatise, to have had three three Objects in view."

"2. Sir Isaac Newton's Rules, in his ', concerning the Resolution of the higher Equations, and the Affectations of their Roots, being, for the most part, delivered without any Demonstration, Mr. MacLaurin had designed, that his Treatise should serve as a Commentary on that Work. For we here find all those difficult Passages in Sir Isaac's Book, which have so long perplexed the Students of Algebra, clearly explained and demonstrated. How much such a Commentary was wanted, we may learn from the Words of the late eminent Author.The ablest Mathematicians of the last Age (says he) did not disdain to write Notes on the Geometry of Des Cartes; and surely Sir Isaac Newton's Arithmetic no less deserves that Honour. To excite some one of the many skilful Hands that our Times afford to undertake this Work, and to shew the Necessity of it, I give this Specimen, in an Explication of two Passages of the '; which, however, are not the most difficult in that Book.What this learned Professor so earnestly wished for, we at last see executed; not separately nor in the loose disagreeable Form which such Commentaries generally take, but in a Manner equally natural and convenient; every Demonstration being aptly inserted into the Body of the Work, as a necessary and inseparable Member; an Advantage which, with some others, obvious enough to an attentive Reader, will, 'tis hoped, distinguish this Performance from every other, of the Kind, that has hitherto appeared."

"His father died six weeks after; but that loss was in a good measure supplied to the orphan family, by the affectionate care of their uncle Mr. Daniel Maclaurin, minister of Kilfinnan, and by the virtue and prudent œconomy of Mrs. Maclaurin. After some stay in Argyleshire, where her sisters and she had a small patrimonial estate, she removed to Dumbarton, for the more convenient education of her children: but dying in 1707, the care of them devolved entirely to their uncle."

"Colin Maclaurin was descended of an ancient family, which had been long in possession of the island of Tirrie, upon the coast of Argyleshire. His grandfather, Daniel, removing to Inverara, greatly contributed to restore that town, after it had teen almost entirely ruined in the time of the civil wars; and, by some memoirs which he wrote of his own times, appears to have been a person of worth and superior abilities. John, the son of Daniel, and father of our author, was minister of Glenderule; where he not only distinguished himself by all the virtues of a faithful and diligent pastor, but has left, in the register of his provincial synod, lasting monuments of his talents for business, and of his public spirit. He was likewise employed by that synod in Completing the version of the Psalms into Irish, which, is still used in those parts of the country where divine service is performed in that language. He married a gentlewoman of the family of Cameron, by whom he had three sons; John, who is still living, a learned and pious divine, one of the ministers of the city of Glasgow; Daniel, who died young, after having given proofs of a most extraordinary genius; and Colin born at Kilmoddan in the tnonth of February 1698."

"Had the celebrated Author lived to publish his own Work, his Name would, alone, have been sufficient to recommend it to the Notice of the Publick: But that Task having, by his lamented premature Death, devolved to the Gentlemen whom he left entrusted with his Papers, the Reader may reasonably expect some Account of the Materials of which it consists, and of the Care that has been taken in collecting and disposing them, so as best to answer the Author's Intension, and fill up the Plan he had designed."

"3. After having fully explained the Nature of Equations, and the Methods of finding their Roots, either in finite Expressions, when it can be done, or in infinite converging Series; it remained only to consider the Relation of Equations involving two variable Quantities, and of Geometrical Lines to each other; the Doctrine of the Loci; and the Construction of Equations. These make the Subject of the Third Part."

"In the first class he taught the first 6 books of Euclid's Elements, Plane Trigonometry, Practical Geometry, the Elements of Fortification, and an Introduction to Algebra. The second class studied Algebra, with the 11th and 12th books of Euclid, Spherical Trigonometry, Conic Sections, and the General Principles of Astronomy. The third... in Astronomy and Perspective... a part of Newton's Principia, and... experiments... illustrating them: he afterwards... demonstrated the Elements of Fluxions. Those in the fourth class read a System of Fluxions, the Doctrine of Chances, and the remainder of Newton's Principia."

"In delivering the principles of this method, we apprehend it is better to avoid such suppositions: but after these are demonstrated, short and concise ways of speaking, though less accurate, may be permitted, when there is no hazard of our introducing any uncertainty or obscurity into the science from the use of them, or of involving it in disputes."

"The difficulty in presenting a rigorous as well as clear statement of the theory of limits is inherent in the subject. ...If the reader has found some difficulty in grasping it he may be less discouraged when he is told that it eluded even Newton and Leibniz. ... Many contemporaries of Newton, among them ... taught that the calculus was a collection of ingenious fallacies. ... decided that he could found calculus properly... The book was undoubtedly profound but also unintelligible. One hundred years after the time of Newton and Leibniz, Joseph Louis Lagrange... still believed that the calculus was unsound and gave correct results only because errors were offsetting each other. He, too, formulated his own foundation... but it was incorrect. ...D'Alembert had to advise students of the calculus... faith would eventually come to them. This is not bad advice... but it is no substitute for rigor and proof. ... About a century and a half after the creation of calculus... Augustin Louis Cauchy... finally gave a definitive formulation of the limit concept that removed doubts as to the soundness of the subject."

"The method of demonstration, which was invented by the author of fluxions, is accurate and elegant; but we propose to begin with one that is somewhat different; which, being less removed from that of the antients, may make the transition to his method more easy to beginners (for whom chiefly this treatise is intended), and may obviate some objections that have been made to it."

"[C]onsider the steps by which the antients were able... from the mensuration of right-lined figures, to judge of such as were bounded by curve lines; for as they did not allow themselves to resolve curvilineal figures into rectilineal elements, it is worth examin[ing] by what art they could make a transition from the one to the other: and as they... finish their demonstrations in the most perfect manner... by following their example... in demonstrating a method so much more general than their's, we may best guard against exceptions and cavils, and vary less from the old foundations of geometry."

"The Gregory-Newton interpolation formula was used by Brook Taylor to develop the most powerful single method for expanding a function into an infinite series. In his Methodus Incrementorum Directa et Inversa Taylor derived the theorem... he praises Newton but makes no mention of Leibniz's work of 1673 on finite differences, though Taylor knew this work. Taylor's theorem was known to James Gregory in 1670 and was known... by Leibnez, however these two men did not pubish it. John Bernoulli did publish practically the same result in the Acta Eruditorium of 1694; and though Taylor knew his result he did not refer to it. ...Colin Maclaurin in his Treatise of Fluxions (1742) stated that... [Mclaurin's theorem] was but a special case of Taylor's result."

"This way of considering what is called the sublime part of geometry has so far prevailed, that it is generally known by no less a title than the Science, Arithmetic, or Geometry of infinites. These terms imply something lofty, but mysterious; the contemplation of which may be suspected to amaze and perplex, rather than satisfy or enlighten the understanding... and while it seems greatly to elevate geometry, may possibly lessen its true and real excellency, which chiefly consists in its perspicuity and perfect evidence; for we may be apt to rest in an obscure and imperfect knowledge of so abstruse a doctrine... instead of seeking for that clear and full view we ought to have of geometrical truth; and to this we may ascribe the inclination... of late for introducing mysteries into a science wherein there ought to be none."

"He [Kepler] supposes, in that treatise [epitome of astronomy], that the motion of the sun on his axis is preserved by some inherent vital principle; that a certain virtue, or immaterial image of the sun, is diffused with his rays into the ambient spaces, and, revolving with the body of the sun on his axis, takes hold of the planets and carries them along with it in the same direction; as a load-stone turned round in the neighborhood of a magnetic needle makes it turn round at the same time. The planet, according to him, by its inertia endeavors to continue in its place, and the action of the sun's image and this inertia are in a perpetual struggle. He adds, that this action of the sun, like to his light, decreases as the distance increases; and therefore moves the same planet with greater celerity when nearer the sun, than at a greater distance. To account for the planet's approaching towards the sun as it descends from the aphelium to the perihelium, and receding from the sun while it ascends to the aphelium again, he supposes that the sun attracts one part of each planet, and repels the opposite part; and that the part which is attracted is turned towards the sun in the descent, and that the other part is towards the sun in the ascent. By suppositions of this kind he endeavored to account for all the other varieties of the celestial motions."

"After I saw that so much had been written upon it to no good purpose, I was rather induced to delay the publication of this Treatise, til I could finish my design."

"I accommodated my Definition of the Variation of Curvature in Chap. xi. to Sir Isaac Newtons, to prevent mistakes, as I have observed in Article 386, but made no material alteration in any thing else."

"A LETTER published in the year 1734, under the title of ' first gave occasion to the ensuing Treatise; and several reasons concurred to induce me to write on this subject at so great a length. The Author of that Piece had represented the as founded on false Reasoning, and full of Mysteries. His Objections seemed to have been occasioned in a great measure, by the concise manner in which the Elements of this Method have been usually described; and their having been so much misunderstood by a person of his abilities, appeared to me a sufficient proof that a fuller Account of the Grounds of them was requisite."

"In explaining the Notion of & , I have followed Sir Isaac Newton in the first Book, imagining that there can be no difficulty in conceiving Velocity wherever there is Motion; nor do I think that I have departed from his Sense in the second Book; and in both I have endeavoured to avoid several expressions, which, though convenient, might be liable to exceptions, and, perhaps, occasion disputes. I have always represented Fluxions of all... Orders by finite Quantities, the Supposition of an infinitely little Magnitude being too bold a Postulatum for such a Science as Geometry."

"The greatest part of the first Book was printed in 1737: but it could not have been so useful to the Reader without the second; and I... recommend... to peruse the first Chapters of the second Book, before the five last of the first; there being a few passages... that will be better understood by... [having] some knowledge of the principal Rules of the Method of Computation... in the second Book."

"Though there can be no comparison made betwixt the extent or usefulness of the antient and modern Discoveries in Geometry, yet it seems to be generally allowed that the Antients took greater care, and were more successfull in preserving the Character of its Evidence entire."

"His main design seems to have been to explain only those parts off Newton's Philosophy which have been controverted: and this is supposed to be the reason why his grand discoveries concerning light and colours, are but transiently and generally touched upon; for it is known, that whenever the experiments, on which his doctrine of light and colours is foimded, had been repeated with due care, this doctrine had not been contested; while his accounting for the celestial motions, and the other great appearances of nature, from gravity, had been misunderstood, and even attempted to be ridiculed."

"The Algebra, though not finished by himself, is... excellent in its kind; containing, in no large volume, a complete elementary treatise of that science, as far as it has hitherto beea carried; besides some neat analytical papers on curve lines."

"His Account of Newton's Philosophy was occasioned in the following manner:---Sir Isaac dying in the beginning of 1728, his nephew, Mr. Conduitt, proposed to publish an Account of his Life, and desired Mr. Maclaurin's assistance. The latter, out of gratitude to his great benefactor, cheerfully undertook, and soon finished, the History of the Progress which Philosophy had made before Newton's time; and this was the first draught of the work in hand, which, not going forward on account of Mr. Conduitt's death, was returned to Mr. Maclaurin. To this he afterwards made great additions, and left; it in the state in which y it now appears."

"Since his death... two... volumes have appeared; his Algebra, and his Account of Sir Isaac Newton's Philosophical Discoveries."

"But, because the Method of Infinitesimals is much in use, and is valued for its conciseness, I thought it was requisite to account explicitly for the truth, and perfect accuracy of the conclusions that are derived from it; the rather, that it does not seem to be a very proper reason that is assigned by Authors, when they determine what is called the Difference (but more accurately the ) of a Quantity, and tell us, That they reject certain Parts of the Element, because they become infinitely less than the other parts; not only because a proof of this nature may leave some doubt as to the accuracy of the conclusion, but because it may be demonstrated that those parts ought to be neglected by them at any rate, or that it would be an error to retain them."

"This determined me, immediately after that Piece came to my hands, and before I knew any thing of what was intended by others in answer to it, to attempt to deduce those Elements after the manner of the Antients, from a few unexceptionable principles, by Demonstrations of the strictest form."

"But it has been objected on several occasions, that the modern improvements have been established for the most part upon new and exceptionable maxims, of too abstruse a nature to deserve a place amongst the plain principles of the ancient geometry: and some have proceeded so far as to impute false reasoning to those authors who have contributed most to the late discoveries, and have at the same time been most cautious in their manner of describing them."

"He made... calculations relating to the provision, now established by law, for the children and widows of the Scotch clergy, and of the professors in the Universities, entitling them to certain annuities and sums upon the voluntary annual payment of a certain sum by the incumbent. In contriving and adjusting this wise and useful scheme he bestowed a great deal of labour, and contributed not a little towards bringing it to perfection."

"I perceived that some Rules were defective or inaccurate; that the Resolution of several Problems which had been deduced in a mysterious manner, by second and third s, could be completed with greater evidence, and less danger of error, by first Fluxions only; and that other problems had been resolved by Approximations, when an accurate Solution could be obtained with the same or greater facility."

"GEOMETRY is valued for its extensive usefulness, but has been most admired for its evidence; mathematical demonstration being such as has been always supposed to put an end to dispute, leaving no place for doubt or cavil. It acquired this character by the great care of the old writers, who admitted no principles but a few self-evident truths, and no demonstrations but such as were accurately deduced from them."

"Whatever difficulty occurred concerning the constructing or perfecting of machines, the working of mines, the improving of manufactures, the conveying of water, or the execution of any public work, he was always ready to resolve it."

"In the method of indivisibles, lines were conceived to be made up of points, surfaces of lines,and solids of surfaces; and such suppositions have been employed by several ingenious men for proving the old theorems, and discovering new ones, in a brief and easy manner. But as this doctrine was inconsistent with the strict principles of geometry, so it soon appeared that there was some danger of its leading them into false conclusions: therefore others, in the place of indivisible, substituted infinitely small divisible elements, of which they supposed all magnitudes to be formed; and thus endeavoured to retain, and improve, the advantages that were derived from the former method for the advancement of geometry."

"In 1740, he... shared the prize of the... [Royal] Academy with ... D. Bernoulli and Euler, for resolving the problem relating to the motion of the tides from the theory of gravity... He bad only ten days to draw up this paper in, and could not... transcribe a fair copy; so ... the Paris edition... is incorrect. He afterwards revised the whole, and inserted it in his Treatise of Fluxions."

"[T]he Defense of the , and of the great Inventor, was not neglected."

"Besides an answer to '... the Author concealed his real name... a second, by the same hand, in Defense of the first, a Discourse by Mr. Robins, a Treatise of Sir Isaac Newton, with a Commentary by Mr. Colson, and several other Pieces, were published on this Subject."

"Circles are the only curvilineal plane figures considered in the elements of geometry. If they could have allowed... these as similar polygons of an infinite number of sides (as some have done who pretend to abridge their demonstrations), after proving that any similar polygons inscribed in circles are in the duplicate ratio of the diameters, they would have immediately extended this to the circles themselves and would have considered the second proposition of the twelfth book of the Elements as an easy corollary from the first. But there is ground to think that they would not have admitted a demonstration of this kind. It was a fundamental principle with them, that the difference of any two unequal quantities, by which the greater exceeds the lesser, may be added to itself till it shall exceed any proposed finite quantity of the same kind: and that they founded their propositions concerning curvilineal figures upon this principle... is evident from the demonstrations, and from the express declaration of Archimedes, who acknowledges it to be the foundation...[of] his own discoveries, and cites it as assumed by the antients in demonstrating all their propositions of this kind. But this principle seems to be inconsistent with... admitting... an infinitely little quantity or difference, which, added to itself any number of times, is never supposed to become equal to any finite quantity whatsoever."

"They proceeded therefore in another manner, less direct indeed, but perfectly evident. They found, that the inscribed similar polygons, by increasing the number of their sides, continually approached to the areas of the circles; so that the decreasing differences betwixt each circle and its inscribed polygon, by still further and further divisions of the circular arches which the sides of the polygons subtend, could become less than any quantity that can be assigned: and that all this while the similar polygons observed the same constant invariable proportion to each other, viz. that of the squares of the diameters of the circles. Upon this they founded a demonstration, that the proportion of the circles themselves could be no other than that same invariable ratio of the similar inscribed polygons; of which we shall give a brief abstract, that it may appear in what manner they were able... to form a demonstration of the proportions of curvilineal figures, from what they had already discovered of rectilineal ones. And that the general reasoning by which they demonstrated all their theorems of this kind may more easily appear, we shall represent the circles and polygons by right lines, in the same manner as all magnitudes are expressed in the fifth book of the Elements."

"When the certainty of any part of geometry is brought into question, the most effectual way to set the truth in a full light, and to prevent disputes, is to deduce it from s or first principles of unexceptionable evidence, by demonstrations of the strictest kind, after the manner of the antient geometricians. This is our design in the following treatise; wherein we do not propose to alter Sir Isaac Newtons notion of a , but to explain and demonstrate his method, by deducing it at length from a few self-evident truths, in that strict manner: and, in treating of it, to abstract from all principles and postulates that may require the imagining any other quantities but such as may be easily conceived to have a real existence."

"He was employed to terminate some disputes of consequence that had arisen at Glasgow concerning the gauging of vessels; and for that purpose, presented to the commissioners of the excise two elaborate memorials, with their demonstrations, containing rules by which the officers now act."

"If an Accountant, that pretends to a scrupulous exactness, should tell us that he had neglected certain Articles, because he found them to be of small importance, and it should appear that they ought not to have been taken into consideration by him on that occasion, but belong to a different account, we should approve his conclusions as accurate, but not his reason. This method, however, may be considered as an easy and ready way of distinguishing what Parts of an Element are to be rejected, and which are to be retained, in determining the precise Fluxion of a Quantity, or the rate according to which it increases or decreases."

"Several Treatises have appeared while this was in the press, wherein some of the same Problems have been considered, though generally in a different manner. I have had occasion to mention most of them in the last Chapter of the second Book; but had not there an opportunity to take notice, that the Problem in 480 has been considered by Mr. Euler in his Mechanics."

"Upon this Plan Mr. Mac-Laurin composed a System of Algebra, soon after his being chosen Professor of Mathematics in the University of Edinburgh; which he, thenceforth, made use of in his ordinary Course of Lectures, and was occasionally improving to the Perfection he intended it should have, before he committed it to the Press."

"In 1719... Maclaurin visited London... where he became acquainted with Dr. Hoadley... Bishop of Bangor, Dr. Clarke, Sir Isaac Newton, and other eminent men; at which time... he was admitted... [to] the ..."

"There were some, however, who disliked the... use of infinites and infinitesimals in geometry. Of this number was Sir Isaac Newton (whose caution was almost as distinguishing a part of his character as his invention), especially after he saw that this liberty was growing to so great a height. In demonstrating the grounds of the method of fluxion, he avoided them, establishing it in a way more agreeable to the strictness of geometry."

"In most of the instances wherein my conclusions did not agree with those given by other Authors, I have not mentioned their names."