Fellows Of The Royal Society

"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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"[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..."

"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..."

"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."

"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."

"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."

"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."

"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."

"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."

"[I]t is by the help of geometry that all the arts necessary for improving life,.. as geography,.. rules of navigation,.. determining of times... [etc.], have been carried to such an incredible pinnacle of distinction."

"[W]ithin the memory of ourselves and... our fathers, philosophers began to extend the limits of geometry in order to found the kingdom of astronomy. This they have carried out... with such success that now no one can be received into astronomical citizenship who is not a visiting citizen in the most abstruse geometry and has not arisen from the patrician, that is the geometrical, family of philosophers."

"Although in every age there have been those who cultivated astronomy, either by... observations... or by theories and systems made up according to the state of understanding of any period, or by a talent for exposition, yet the lucubrations of all these astronomers do not reveal the ways of the heaven any more than they reveal the skill and experience of their progenitors in geometrical matters."

"In the past many very base Remus’s leapt over the walls of the astronomical city, but now the geometers have so fortified it with a ditch and a rampart that the portals of the sun receive those whom impartial Appollonius has loved and whom Kepler, Wren, Wallis and Newton have borne to the aetherial regions, and accordingly the profane, that is ungeometrical men, are exiled and depart from the grove and wander away over the whole heaven."

"It is... most gratifying to my soul that, after spending a large part of my life in other universities, I can be linked in the University of Oxford with the prince of geometers Dr John Wallis as colleague and the most scholarly Dr as successor."

"[T]he easier, simpler and less composite these theories are, the more they will be consonant not only with what has already been observed... but with what is yet to be observed, and with the very machine of the world which was constructed by the supreme creator with the greatest simplicity."

"[W]hat sharpness of mind was employed by John Kepler... when, from there being just five regular solids... he inferred that the number of the planets was six, and by inscription of spheres within these solids and circumscription of spheres around them related the distances and ratios of the orbits. It can scarcely be said with what power of prophecy and by what labours he succeeded in arriving at that great theorem of the elliptical planetary orbits with a common focus at the sun... in such a way that the areas that the radius vector of the planet from the sun traverses are proportional to the times. Nevertheless... so great a man... owned himself unequal to... solving directly the problem of determining for a given time the place of the planet in the elliptical orbit. Here geometry, his goddess-mother, was of no avail... But... he brought forward a conjecture of great use, namely, that the squares of the periodic times are in the same ratio as the cubes of the distances between the planets and the sun. Finally, he discovered a marvellous property of bodies by which in the minimally resisting ether they seek each other and as it were attract. From this he also deduced the tides in a clear but brief discourse in his immortal Commentaries on the star Mars, and was as it were a prophet and a precursor of a great geometer born among the English."

"[G]lory has been reserved to our era and to the English people, who since the instauration of the sciences have made such advances... And passing over the immense labours undergone by the most fruitful astronomers of our people... [H]ow easy and how exact... how geometrical, astronomy has been left to us by that most acute geometer... or astronomer, the Right Reverend Dr Seth sometime Bishop of Salisbury, who while he was among men adorned this chair. How geometrically and acutely he determined the positions and species of the orbit and other related matters, following Kepler and substituting as mean motion the angle at the other focus (which he accordingly called that of the mean motion) in place of the areas to the sun that the radius vector describes and as it were sweeps out. Content with this artifice he did not detain himself over the solution of Kepler’s problem, in which the division of the area of an ellipse in a given ratio by a straight line through a focus is required. But, being a most perspicacious man, he was conscious of what delays arose hence in the construction of tables, and, in order to show the world that astronomy was to be advanced by the help of geometry whatever hypotheses it depended upon, he accomplished the same astronomical problems geometrically from the circular hypothesis."

"It was also during my being chaplain to bishop More, that I published my first work, intitled, A New Theory of the Earth, from its Original to the Consummation of all Things, wherein the Creation of the World in Six Days, the Universal Deluge, and the General Conflagration, as laid down in the Holy Scriptures, are Shewn to be perfectly agreeable to Reason and Phylosophy. ...In the New Theory, fifth edition... Lem. xxxii. Schol. instead of its latter part, read, as Sir Isaac Newton also did in his latter writing of this nature, I mean the Theory of the Moon, publimed by Dr. Gregory, and has suppofed the sun's parallax, 10″; and from this hypothesis I made these and the following calculations. Which therefore cannot be far from truth..."

"[U]niversally, the Weights which generate all Tones in Strings, are reciprocally as the Squares of the lengths of Strings of equal Tension, producing the same sound in any Musical Instrument."

"I will add another thing which I also had from Dr. Bentley himself. Mr. Halley was then thought of for successor, to be in a mathematick professorship at Oxford; and bishop Stillingfleet was desired to recommend him at court; but hearing that he was a sceptick, and a banterer of religion, he scrupled to be concern'd; 'till his chaplain, Mr. Bentley, should talk with him about it; which he did. But Mr. Halley was so sincere in his infidelity, that he would not so much as pretend to believe the christian religion, tho' he thereby was likely to lose a professorship; which he did accordingly; and it was then given to Dr. Gregory: Yet was Mr. Halley afterwards chosen into the like professorship there, without any pretence to the belief of christianity. Nor was there any enquiry made about my successor Mr. Sandersons christianity, even when the university of Cambridge had just banished me for believing and examining it so throughly, that I hazarded all I had in the world for it."

"Pythagoras... applied the proportion he had thus found by experiments, to the Heavens, and from thence learn'd the Harmony of the Spheres. And, by comparing these Weights with the Weights of the Planets, and the intervals of the Tones, produced by the Weights, with the interval of the Spheres; and lastly, the lengths of Strings with the Distances of the Planets from the Center of the Orbs; he understood, as it were by the Harmony of the Heavens, that the Gravity of the Planets towards the Sun (according to whose measures the Planets move) were reciprocally as the Squares of their Distances from the Sun."

"[T]he Opinion of the Ancients concerning Gravity... they were perswaded that Gravity was not an affection of Terrestrial Bodies only, but of the Celestial also, that all Bodies gravitate towards one another; and that the Planets are retained in their Orbits by the force of Gravity, and lastly, that the Gravity of the Planets towards the Sun are reciprocally as the Squares of their Distances from it. What the industry and skill of the Moderns have added to these inventions of the Ancients, the following Pages do declare at large."

"Gregory, David, nephew of preceeding James Gregory (1638-1675)], was born at Aberdeen in 1661, and died 1708, Savilian professor of Astronomy at Oxford. He published, 1. 'Exercitatio Geometrica de Dimensione Figurarum,' 4to. Edinb. 1684. 2. 'Catoptricæ et Diopticræ Sphericæ Elementa,' 8vo. Oxon. 1695. 3. 'Astronomæ Physicæ et Geometriæ Elementa,' fol. Oxon. 1702, and 4to. Genev. 1726. 4. 'Treatise of Practical Geometry,' originally written in Latin, and of which a translation by Mr. MacLaurin, was published in 8vo. 1745, and again in 1751. 5. 'A Short Treatise of the Nature of Arithmetic and Logarithms,' printed at the end of Keill's translation of Commandine's Euclid, besides several papers in the 'Philosophical Transactions.'"

"'Tis from this Doctrine of Gravity, that all Bodies gravitate mutually to one another, 'tis by this that Lucretius, taught by Epicurus and Democritus, labours to prove, that the Universe has no Center or lowest Place, but that there is an infinity of Worlds like ours in the immense Space. His Argument... If the nature of things were bounded any where, then the outmost Bodies, since they have no other beyond them, towards which they may be made to tend by the force of Gravity, wou'd not stand in an Equilibrio, but make towards the inner and lower Bodies, being necessarily inclin'd that way by their Gravity, and therefore having made towards one another, during an infinite space of time, would have long ago met, and lye in the middle of the whole, as in the lowest place."

"Lucretius and those whom he followed, believ'd that all Bodies did Gravitate towards the Matter placed around them, and that every single Body was carried by the more prevailing Gravity, towards that region where there was most Matter."

"[S]o also they were not unacquainted with the Law and Proportion which the action of Gravity observ'd according to the different Masses and Distances. For that Gravity is proportional to the Quantity of Matter in the heavy Body, Lucretius does sufficiently declare, as also that what we call light Bodies, don't ascend of their own accord, but by the action of a force underneath them, impelling them upwards, just as a piece of Wood is in Water; and further, that all Bodies, as well the heavy as the light, do descend in vacuo, with an equal celerity."

"From some things mention'd by Diogenes Laertius concerning Plato, which also are obscurely hinted at in his Timæus I am apt to believe with Galileo that the divine Philosopher suppos'd the Mundane Bodies, when they were first formed, were moved with a Rectilinear motion (by the means of Gravity,) but after that they had arrived to some determined places, they began to revolve by degrees in a Curve, the Rectilinear Motion being chang'd into a Curvilinear one."

"These notions Anaximenes received from Anaximander, Anaximander from Thales himself, who was the Head and Founder of the Ionic Philosophy; and spread this opinion of the Gravity of the Fix'd Stars among his Sect."

"[A]fterwards it diffused it self thro' the Italic Philosophy, the followers of which taught, that each Star was a World in the infinite Æthereal Space, containing Earth, Air and Æther; and that the Moon, not only was like our Earth, but inhabited by Animals of a larger size, and furnish'd with Plants of a more beautiful appearance."

"Nor were they so absurd in their conceptions about Gravity, as to think that it was done by the virtue of any point within the Earth, or of a Center, to which all heavy Bodies placed any where tended; but they thought it was done by the power of the whole Matter in the Terrestrial Globe attracting all things to it self: And as the power of the is composed of the powers of the several parts combin'd together, so they believed that the Gravity towards the whole Earth, resulted from the Gravity towards each single part of it. ...[T]hey believ'd there was a Gravity towards the Moon and Sun, acting in the same manner as it does towards the Earth; and that each Planet, like a Stone, whirl'd in a sling, was kept in its Orbit by the same principle, and for the same reason revolving always about us."

"[T]he famous Theorem about the proportion whereby Gravity decreases in receding from the Sun, was not unknown at least to Pythagoras. This indeed seems to be that which he and his followers would signify to us by the Harmony of the Spheres: That is, they feign'd Apollo playing upon an Harp of seven Strings, by which Symbol, as it is abundantly evident from Pliny, Macrobius and , they meant the Sun in Conjunction with the seven Planets, for they made him the leader of that Septenary Chorus, and Moderator of Nature; and thought that by his Attractive force he acted upon the Planets (and called it Jupiter's Prison, because it is by this Force that he retains and keeps them in their Orbits, from flying off in Right Lines) in the Harmonical ratio of their Distances. For the forces, whereby equal tensions act upon Strings of different lengths (being equal in other respects) are reciprocally as the Squares of the lengths of the Strings."

"After great and fruitful efforts both in the purer geometry and the more intricate and complex physics, the most skilful geometer Sir Christopher Wren, who among other luminaries of the University of Oxford graced this professorship, solved the following problem: To find the law of gravity or centripetal force by which several bodies moved around a common centre of forces are driven, given that the squares of the periodic times are as the cubes of the radii, as was observed by Kepler for the planets moved around the sun. The most renowned Wren found that the required law of gravity was such that the centripetal forces were reciprocally as the squares of the distances from the centre of forces, and that no other law would agree with what was observed."

"David Gregory’s manuscript ‘Isaaci Neutoni methodus fluxionum’ is the first systematic presentation of the method of fluxions written by somebody other than Newton. It was penned in 1694, when Gregory was the Savilian Professor of Astronomy at Oxford. ...[I]t sheds light upon Gregory’s views on how Newton’s mathematical innovations related to... other mathematicians, both British and Continental. This paper... proves that Newton, far from being—as often stated—wholly isolated and reluctant to publish the method of fluxions, belonged to a network of mathematicians who were made aware of his discoveries. Second, it shows that Gregory—very much as other Scottish mathematicians such as George Cheyne and John Craig—received Newton’s fluxional method within a tradition that was independent from England and that, before getting in touch with Newton, had assimilated elements of the calculi developed on the Continent."

"For Pythagoras as he was passing by a Smith's Shop, took occasion to observe, that the Sounds the Hammers made, were more accute or grave in proportion to the weights of the Hammers; afterwards stretching Sheeps Guts, and fastning various Weights to them, he learn'd that here likewise the Sounds were proportional to the Weights. Having satisfy'd himself of this, he investigated the Numbers, according to which Consonant Sounds were generated. Whether the whole of this Story be true, or but a Fable, 'tis certain Pythagoras found out the true ratio between the sound of Strings and the Weights fasten'd to them."

"The Science of Astronomy which is as much esteem'd and admir'd for its great and manifold uses for the Service of Mankind, as it is delightful and entertaining to the more curious and contemplative, has in all ages been cultivated and improv'd, by Men the most eminent for their parts and learnings; and is now brought... to the utmost degree of perfection, and that chiefly by the Superior Genius and Industry of those of our own Nation. But since nothing considerable therein, has been as yet writ in our own Language... I could not oblige my Country-Men more than in publishing an English Edition of the most valuable and finish'd piece of Astronomy now extant. It is generally reckoned to be a Book that contains not only all the Discoveries and Philosophical Sentiments of the great Kepler, and the various Hypotheses of the most noted Astronomers before and since his Time; but is chiefly valued by the best Judges, for the large and instructive Comments... on the Writings of the illustrious Sir Isaac Newton, as well as on the Several Astronomical Dissertations of the Sagacious Dr. Halley, which the Reader will find here every where interspers'd. ...I shall, in a very little time, present... another Volume, containing correct Astronomical Tables, for the ready computing of the Planets Places, Eclipses, &c. all done by a Person of known ability, from the true Theory of Gravity, deliver'd in this Book: For it was by no means judged proper that I should annex to so intire a piece as this, any imperfect Tables, drawn from a different Principle from what is here established, such it seems all those as yet published are."

"After I had taken holy orders, I returned to the college, and went on with my own studies there, particularly the mathematicks, and the Cartesian philosophy; which was alone in vogue with us at that time. But it was not long before I, with immense Pains, but no assistance, set myself, with the utmost zeal, to the Study of Sir Isaac Newtons wonderful discoveries in his ', one or two of which Lectures I had heard... read in the publick Schools, though I understood them not at all... Being indeed greatly excited thereto by a Paper of Dr. Gregory’s when he was Professor in Scotland; wherein he had given the most prodigious Commendations to that work, as not only right in all things, but in a manner the Effect of a plainly divine genius, and had already caused several of his Scholars to keep Acts, as we call them, upon several branches of the Newtonian Philosophy; while we at Cambridge, poor Wretches, were ignominiously studying the fictitious Hypotheses of the Cartesian, which Sir Isaac Newton had also himself done formerly, as I have heard him say."

"My design in publishing this Book, was, that the Celestial Physics, which the most sagacious Kepler had got the scent of, but the Prince of Geometers Sir Isaac Newton, brought to such a pitch as surprises all the World, might, by my... illustrating, become easier to such as are desirous of being acquainted with Philosophy and Astronomy."