University Of Edinburgh Faculty

"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."

"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."

"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."

"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."

"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."

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

"If, upon the whole, the Evidence of this method be represented to the satisfaction of the Reader, some of the abstruse parts illustrated, or any improvements of this useful Art be proposed, I shall be under no great concern, though exceptions may be made to some modes of Expression, or to such Passages of this Treatise as are not essential to the principal design."

"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."

"The science being now vastly enlarged, and applied with success to philosophy and the arts, it is of greater importance than ever that its evidence be preserved perfect."

"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."

"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."

"After these came to be relished, an infinite scale of infinites and s (ascending and descending always by infinite steps) was imagined and proposed to be received into geometry, as of the greatest use for penetrating into its abstruse parts. Some have argued for quantities more than infinite; and others for a kind of quantities that are said to be neither finite nor infinite, but of an intermediate and indeterminate nature."

"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."

"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."

"He considered magnitudes as generated by a or motion, and showed how the velocities of the generating motions were to be compared together. There was nothing in this doctrine but what seemed to be natural and agreeable to the antient geometry. But what he has given us on this subject being very short, his conciseness maybe supposed to have given some occasion to the objections which have been raised against his method."

"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."

"We shall not consider any part of space or time as indivisible, or infinitely little; but we shall consider a point as a term or limit of a line, and a moment as a term or limit of time: nor shall we resolve curve lines, or curvilineal spaces, into rectilineal elements of any kind."

"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 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."

"They found, that similar triangles are to each other in the duplicate ratio of their homologous sides; and, by resolving similar polygons into similar triangles, the same proposition was extended to these polygons also. But when they came to compare curvilineal figures, that cannot be resolved into rectilineal parts, this method failed."

"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."

"But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles."

"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."

"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 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."

"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."

"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."

"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."

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

"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."

"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."

"I should be thoroughly acquainted with the real Practice & this is a thing very different from what can be learned in a College; thus for instance we are taught by our Professors that if a sick person breaths with great difficulty, one thing must be done; if his respiration is yet more laborious, another. But how shall we judge of the nice degrees of laborious breathing unless from a dayly & familiar acquaintance with, & study of, the appearances and looks of Patients &c. Most young Physicians neglect this essential point of their art in their education & very often acquire it when they come to Practice at the expense of their patients' safety."

"I have not had time this last winter to apply to it sufficiently; tho I had the opportunities, my attention was too much taken up with some of the Colleges, preparing my Thesis, & recalling to my mind everything I had learned, on account of the examinations."

"If I go to London to acquire this part of medicine, I may see a good deal of Practice, but I am a stranger there, & have no acquaintance whom I can venture to trust so much or be so familiar with as to trouble him with all my questions and doubts.—On the contrary here, medicine is allowed on all hands to be in a very flourishing condition. It is practised in the most rational & simple manner."

"I have the happiness to be lodged with a Gentleman who is justly esteemed by all his Brethren, who has extensive practice both as a Physician & Surgeon & tho no Doctor himself, yet the oldest of them are not ashamed to consult with him in private. Besides this he is my intimate & familiar Friend & is willing upon every occasion to teach me as far as he knows himself."

"When I am well instructed in a method of Practice here, a very short time of London will be sufficient; for then I need only observe the different manner of doing the same thing there, which I shall soon be master of."

"These, sir, are the chief of the reasons which have been urged to me for staying here some time longer & which I thought so good that I determined to acquaint you with them & in the meanwhile [I] will employ my time to the best advantage till I have your opinion of them. I am Dr Sir Your most affecte & Dutyfull Son,"

"The quantitative investigations of Black on the burning of lime and magnesia alba, in which the balance (previously characterized by the French chemist Jean Rey as "an instrument for clowns") was applied at every turn, led to the rejection of a hypothetical "principle of causticity," and replaced it by a "sensible ingredient of a sensible body," fixed air."

"The extension of Black's method by the physicist Lavoisier led to the downfall of the purely qualitative theory of phlogiston, and gave to chemistry the true methods of investigation, and its first great quantitative law—the law of conservation of matter."

"He had discovered that a cubic inch of marble consisted of about half its weight of pure lime, and as much air as would fill a vessel holding six wine gallons. ...What could be more singular than to find so subtle a substance as air existing in the form of a hard stone, and its presence accompanied by such a change in the properties of that stone? … It is surely a dull mind that will not be animated by such a prospect."

"Black discovered the difference between weak and strong s, i.e. between the alkali s and s. The work... is essentially quantitative, and... leads into... gas analysis, which was to play a very important role in providing a new chemical theory. ...Black ...first showed that and were two different substances. Although both effervesce when treated with s... [m]agnesium carbonate does not form common lime when heated strongly, and on cooling, the residue is insoluble in water. This product of ignition (oxide) however forms the same salts with acids as does the original salt (carbonate) with the difference that no effervescence occurs. Black also observed that during ignition "air" () is lost, and supposed that to be responsible for the loss of weight as well as for the effervescence... [H]e dissolved the magnesium oxide in sulphuric acid and then precipitated the magnesium with . ...[T]he composition of the precipitate was identical with that... before the ignition... He... concluded that alkali carbonates were not elemental substances, as had been originally thought, because they give "air" to the... oxide... [the] same "air"... responsible for the effervescence... Black then [examined] lime and limestone and applied similar experiments. He established that the air was not identical with atmospheric air... only a component... called by Black "fixed air"... that part... absorbed by lime and the alkali hyroxides. ...[T]he relationship... is similar to that between alkalis and acids... alkalis are "in some measure neutralized" by the fixed air. However, the relation between acids and alkalis is stronger as the acid drives out the fixed air."

"Black... began a new epoch in Chemistry and Physics, by his fundamental work on Heat, and on the nature of chemical combination; and his name must ever remain associated with those of other illustrious Scotsmen of his day as one who led the way in chemical research and its technical applications."

"Black's celebrated thesis ...gained for him not merely the degree of Doctor of Medicine, but also brought his name before every "philosopher" in Europe and America as that of a man who had made a discovery of more fundamental influence on the progress of Chemistry than any which had previously been described."

"In the olden days it was considered quite as marvellous that a gas could be made to occupy a small volume, or that "air" could be produced in quantity from a stone, as that an Arabian "djinn" of enormous size and ferocious mien could issue from a bottle..."

"[I]n the middle of the seventeenth century Robert Boyle had enunciated his famous discovery, "Touching the Spring of the Air"; in which he proved that the greater the pressure to which a gas is exposed the smaller the volume it will occupy. But however great the pressure, Boyle's air remained air."

"It was Black's discovery of the production of carbonic-acid gas, or, as he named it, "fixed air," from , which first directed notice to this possibility of the production of a gas from a solid; and, further, the peculiar property of this gas its power of being fixed was one which completely differentiated it from ordinary air."