University Of Aberdeen Faculty

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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 ..."

"In 172S, Lord Polwarth... engaged Maclaurin to go as a tutor and companion to his eldest son... on his travels. After... Paris, and... other towns in France, they fixed in Lorrain; where he wrote his piece, On the Percussion of Bodies, which gained... the prize of the Royal Academy of Sciences... 1724. But his pupil dying soon after at Montpelier, he returned... to his profession at Aberdeen."

"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."

"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."

"[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..."

"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 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."

"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."

"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..."

"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."

"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."

"[H]e was always ready to lend his assistance in contriving and promoting any scheme which might contribute to the public service."

"When the Earl of Morton went... 1789, to... his estates in Orkney and Shetland, he requested... Maclaurin to assist him in settling the geography... very erroneous in all our maps; to examine their natural history, to survey the coasts, and to take the measure of a degree of the meridian. ...[F]amily affairs would not permit him to comply... [so] he drew up a memorial of what he thought necessary to be observed, and furnished proper instruments... recommending Mr. Short, the noted optician, as... operator..."

"Mr. Maclaurin had... another scheme for the improvement of geography and navigation... the opening of a passage from Greenland to the South Sea by the North Pole. That such a passage might be found, he was so fully persuaded, that he used to say, if his situation could admit... he would undertake the voyage even at his own charge."

"But when schemes... were laid before the Parliament in 1744, and... before he could finish the memorials he proposed to send, the premium was limited to the... North West passage: and he used to regret that the word West was inserted, because he thought that passage, if at all to be found, must lie not far from the Pole."

"In 1745, having been... active in fortifying the city of Edinburgh against the rebel army, he was obliged to fly from thence into England, where he was invited by Dr. Herring, Archbishop of York, to reside with him... however, being exposed to cold and hardships, and... of a weak and tender constitution... much more enfeebled by close application to study, he laid the foundation of an ilness which put an end to his life in June 1746, at 48 years of age, leaving his widow with two sons and three daughters."

"Mr. Maclaorin was a very good as well as a very great man, and worthy of love as well as admiration."

"His... merit as a philosopher was, that all his studies were accommodated to general utility; and we find, in many places of his works, an application even of the most absruse theories to the perfecting of mechanical arts. For the same purpose, he had resolved to compose a course of Practical Mathematics, and to rescue several useful branches of the science from the ill treatment... often met with in less skilful hands. These intentions... were prevented fay his death; unless we... reckon, as a part of his intended work, the translation of... David Gregory's Practical Geometry, which he revised, and published with additions, in 1745."

"Alexander Bain was probably the first modern thinker whose primary concern was with psychology itself He has been credited with writing the first 'comprehensive treatise having psychology as its sole purpose'. His two-volume systematic work, The Senses and the Intellect (1855) and The Emotions and the Will (1859), was the standard British text for almost half a century, until Stout's replaced it. He also founded Mind (1876-), the first psychological journal in any country. His work requires close attention, because it is the meeting-point of experimental sensory-motor physiology and the association psychology. His influence on the conceptions of later workers was direct and extremely important. Ferrier studied classics and philosophy under Bain at Aberdeen (first class honours, 1863). When he and Jackson acknowledge their intellectual debts or make references to psychology, the names most often mentioned are Bain and Spencer-the figures whose work was the culmination of the association psychology in its traditional form."

"The arguments for the two substances - mind and body - have, we believe, entirely lost their validity; they are no longer compatible with ascertained science and clear thinking. One substance with two sets of attributes, two sides (a physical and a mental), a double-faced unity, would appear to comply with all the exigencies of the case."

"Disinterestedness is as great a puzzle and paradox as ever. Indeed, strictly speaking, it is a species of irrationality, or insanity, as regards the individual’s self; a contradiction of the most essential nature of a sentient being, which is to move to pleasure and from pain"

"He that could teach mathematics well, would not be a bad teacher in any of the rest [physics, chemistry, biology, psychology] unless by the accident of total inaptitude for experimental illustration; while the mere experimentalist is likely to fall into the error of missing the essential condition of science as reasoned truth; not to speak of the danger of making the instruction an affair of sensation, glitter, or pyrotechnic show."

"Instinct is untaught ability."

"The method of arithmetical teaching]] is perhaps the best understood of any of the methods concerned with elementary studies."

"Those that can readily master the difficulties of Mathematics find a considerable charm in the study, sometimes amounting to fascination. This is far from universal; but the subject contains elements of strong interest of a kind that constitutes the pleasures of knowledge. The marvellous devices for solving problems elate the mind with the feeling of intellectual power; and the innumerable constructions of the science leave us lost in wonder."

"What renders a problem definite, and what leaves it indefinite, may best be understood from mathematics. The very important idea of solving a problem within limits of error is an element of rational culture, coming from the same source. The art of totalizing fluctuations by curves is capable of being carried, in conception, far beyond the mathematical domain, 65 where it is first learned. The distinction between laws and coefficients applies in every department of causation. The theory of Probable Evidence is the mathematical contribution to Logic, and is of paramount importance."