University Of Glasgow Alumni

"Throw all your stagey chandeliers in wheel-barrows and move them north To celebrate my mother's sewing-machine And her beneath an eighty-watt bulb, pedalling Iambs on an antique metal footplate."

"James Murray combs the dialect from his beard And files slips for his massive Dictionary."

"Ghetto-makars, tae the knackirs' Wi aw yir schemes, yir smug dour dreams O yir ain feet. Yi're beat By yon new Scoatlan loupin tae yir street..."

"... Listen — Not to dour centuries of trudging, Marching, and taking orders; Today I have heard the feet of my country Breaking into a run."

"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."

"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."

"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."

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

"Many... were... published in the Philosophical Transactions; as the following: 1. On the Construction and Measure of Curves, vol. 30.---2. A New Method of describing all Kinds of Curves, vol. 30.---3. On Equations with impossible Roots, vol. 34.---4. On the Roots of Equations, &c. vol. 34.---5. On the Description of Curve Lines^ vol. 39.---6. Continuation of the same, vol. 39.---7. Observations on a Solar Eclipse, vol. 40.---8. A Rule for finding the Meridional Parts of a Spheroid with the same Exactness as in a Sphere, vol. 41.---9. An Account of the Treatise of Fluxions, vol. 42.---10. On the Bases of the Cells where the Bees deposit their Honey, vol. 42."

"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."

"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."

"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."

"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."

"In his life-time..., he had frequent opportunities of serving his friends and his country by his great skill."

"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."

"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 mean time, he was continually obliging the public with some observation or performance of his own, several of which were published in the 5th and 6th volumes of the Medical Essays at Edinburgh."

"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."

"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."

"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."

"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."

"In 1734, Dr. Berkley, , published a piece called ... which he took occasion, from... disputes... concerning the grounds of the fluxionary method, to explode the method... and... charge mathematicians... with infidelity in religion."

"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."

"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."

"[M]athematical classes soon became very numerous... generally upwards of 100 students attending his Lectures... who being of different standings and proficiency, he was obliged to divide them into four or five classes..."

"Maclaurin thought himself included in this charge, and began an answer to Berkley's book: but [so many] other answers... discoveries... new theories and problems occurred to him, that, instead of a vindicatory pamphlet he produced a Complete System of Fluxions, with their application to the most considerable problems in Geometry and Natural Philosophy."

"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."

"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."

"He was hardly settled... when he received an invitation to Edinburgh... University... that he should supply the place of Mr. James Gregory, whose great age and infirmities had rendered him incapable of teaching."

"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."

"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."

"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 had here some difficulties to encounter, arising from competitors... and... from the want of an additional fund... which, however, at length were all surmmounted, principally by the means of Sir Isaac Newton."

"This work was published at Edinburgh in 1742, 2 vol. 4to.; and as it cost him infinite pains, so it is the most considerable of all his works, and will do him immortal honour, being indeed the most complete treatise on that science... yet..."

"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."

"In his 15th year he took the degree of Master of Arts; on which occasion he composed and publicly defended a Thesis on the Power of Gravity, with great applause."

"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."

"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."

"After this he quitted the University, and retired to a country seat of his uncle, who had the care of his education; his parents being dead some time."

"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."

"His great genius for mathematical learning discovered itself... at twelve years of age; when, having accidentally met with a copy of Euclid's Elements in a friend's chamber, he became in a few days master of the first 6 books without... assistance: and... in his 16th year he had invented many of the propositions which were afterwards published as part of his work entitled Geometria Organica."

"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."

"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."

"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."

"These, with other observations concerning this method, and its application, led me on gradually to compose a Treatise of a much greater extent than I intended, or would have engaged in, if I had been aware of it when I began this Work, because my attendance in the University could allow one to bestow but a small part of my time in carrying it on."

"And as this has been the occasion of my delay in publishing... I hope it will serve for an apology, if some mistakes have escaped me in treating of such a variety of subjects, in a manner different from that in which they have usually been explained."

"[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."

"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."

"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."