David Gregory (mathematician)

"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."

"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."

"[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."

"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."

"[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."

"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."

"After Kepler’s bold and fruitful efforts to advance natural philosophy by the help of geometry, there should have appeared any philosopher and particularly a geometer, namely Descartes, who should leave this one narrow path and try to investigate the causes of things logically, or rather, sophistically. What is to be said of him who while certainly learned in geometry would build his cosmic system (which he valued so highly and of which he boasted so grandiloquently) from vortices, without previously examining whether bodies carried around by a vortex at different distances from the centre would have periodic times whose squares were as the cubes of the distances from the centre? But he was intoxicated by easier and less composite laws, and, not applying his geometric ability in the slightest, fell into errors from which we were at length liberated by the aid of geometers."

"[W]e have come into the age where questions that were once cosmographical are being transformed into geometrical problems. For it has now been shown not only that the areas which bodies driven in a circuit describe by the radii drawn to the centre of forces are in immobile planes and proportional to the times, but also conversely that every body moved in this way is impelled by a centripetal force that tends towards the aforesaid point. By this proposition alone the Ptolemaic system is destroyed, for the primary planets by radii drawn to the Earth describe areas in no way proportional to the times, while with the radii drawn to the sun it is established that they run over areas sufficiently proportional to the times."

"Since two or more mutually gravitating bodies describe orbits around a common immobile centre of gravity, and since by common consent there is an immense difference between the quantity of matter in the sun and that in the Earth, it is clear that neither the sun nor, much less, the sun in the company of five planets can revolve around an immobile Earth. Thus is shown not only the falsity but the impossibility of the ."

"Mr Issac Newton in addition to the geometric figure in any orbit of a projectile sought also to find the measure of the (tending to a given centre) of the body borne in that orbit, from whatever cause that force may arise, be it from a deeper mechanical one or from a law imposed by the supreme creator of all things. He inquires geometrically into the law of centripetal force of a body moved in the circumference of a circle with the force tending to a given point either on the circumference or anywhere outside it or inside it, or even infinitely removed. By the same method he seeks the law of centripetal force tending to the centre of a plane nautical spiral (that is one that the radii cut in a given angle) which will drive a body in that spiral. Also the law of centripetal force that would make a body rotate in an ellipse when the centre of the ellipse coincides with the centre of forces. If the ellipse is changed into a hyperbola and the centripetal force into a centrifugal one the same things apply to the hyperbola. Also the resolution of the same problem when the centre of forces coincides with either focus of the ellipse shows that the law of centripetal force is reciprocally in the duplicate ratio of the distance [as the inverse square of the distance]; others had long before shown that this was the one and only law that would satisfy the other phenomenon observed by Kepler in the motion of the planets. These results also apply to the hyperbola and the parabola when the centre of forces is situated in a focus of the conic section."

"[W]ith this one problem... three years ago, after the above was published, Gottfried Leibniz... produced a construction... in terms of his system—not without the blemish of paralogism."

"But... Kepler’s problem was to be resolved, to find the position of a body moved in an elliptical orbit at a given time. As concerns an algebraic resolution... adapted to the construction of tables, we... also have produced a work not... to be ashamed of."

"But, since the law of centripetal force employed by nature is to be discovered from its symptoms, the indisputably elliptical orbit and the sesquialteral ratio of the periodic times and the distances from the centre of forces, the same great Newton solved not only the universal problem of determining the trajectory and the motion in it for any given centripetal force, but also its converse. After this universal problem had been solved the sequel was to find other [quantities] in the geometric figure that are measures of physical qualities; for example, that the periodic times in ellipses are in the sesquiplicate ratio of the transverse axes [the squares of the times are as the cubes of the axes], and as many other things similar to these as possible. Also, for instance, to compare this force, which we experience in the planets, with another given force near to us, namely gravity. But also the new philosophy was to concern itself with movable elliptical orbits, in which the line of apsides either advances or retires. Also, for instance, a more exact [theory] of rectilinear descent and of the motion of pendulous bodies than the Huygenian one, since that supposes the centre to be infinitely removed. Therefore also, other s different from the common one and variously devised according as the pendulum oscillates inside or outside the surface of the Earth. And let that suffice for this problem. But also on account of the mutual actions of bodies moving around a centre the orbits usually turn out to be deformed, and also an investigation of these actions and of the deformity arising from them, whence arise many minor inequalities of the planets, such as the motion of the nodes, the variation of maximum latitude, and other things in the moon."

"Although the celestial spaces in which the planets move around are... unresisting, yet media are considered in which the moving body is resisted, and this resistance is considered in conjunction with gravitation or centripetal force. Among others, this problem now presents itself for solution: Given the direction, the law of centripetal force, and the law of resistance, to construct the path of the projectile. In particular, if the law of centripetal force is posited as reciprocally duplicate to the distances and the resistance is in the duplicate ratio of the speed, then indeed the problem of Galileo will be solved, as is fitting."

"Descartes's cosmic system, which he jokingly called his fable of the world, is shown to be a fable indeed."

"[T]he motion of comets is found to be not at all dissimilar to the motion of the planets, depending on the same principles and undergoing continual revolutions. But, since these planets move in ellipses in which the distance of the foci has a great ratio to the transverse axis, and that major axis is immense, their periodic times greatly exceed the periodic times of the common planets, for they are in the sesquialteral ratio [3:2] of the transverse axes [their squares are as the cubes of the axes]. Thus Descartes’s rectilinear cometary trajectory, which he filched from Kepler, collapses, and many things about comets in his fable of the world may be added, which almost two thousand years ago had been shown to be impossible by Lucretius. Such is the free motion in full spaces, to say nothing of the fact that in his system the motion is from time to time against the motion of the vortex. But, in place of so eccentric ellipses whose more remote parts cannot be observed because the comet is not visible there, parabolae may be assumed in calculation, and many things useful for improving astronomy and physics and advancing them further may thus be deduced with the help of the more intricate geometry."

"The phenomena prove that the tail of a comet is vapour from its head which, when the comet is greatly heated in the neighbourhood of the sun in perihelion, continually rises up and moves away into the regions opposite the sun."

"[T]he increases of optics, geography and other sciences... are also due to the application of the more intricate geometry to philosophical matters. Hence has been made clear the curvature of the rays of light in the same medium; hence the causes of extraordinary s have been laid bare; hence, given one surface of a lens, another may be determined by means of which a ray entering the lens with given position will have a given position in emerging from it; hence in geography the excess of the normal diameters of the axis over the axis is found, and also the al figure of any planet; hence the varying gravity of the same body in different parts of the Earth, and the varying length of an isochronous pendulum according to the latitude of its place, and then indeed, after the due correction, the construction of a universal measure and of a perfect ."

"[F]or the further improvement of natural philosophy a more advanced geometry must be found. ...[T]he reason why physical science has here been brought to a level that is the envy of foreigners is the knowledge... of some more universal geometry. Of what part of this the learned owe to this renowned university and in it to the prince of geometers I shall not speak lest I appear to be fawning, which in a mathematician would be unseemly."

"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."

"[T]he Physics, it is all taken out of the above mention'd Authors; but is here intermix'd with Astronomy, in such places as seem'd proper and convenient; the Geometry to be met with in it, I have either borrowed elsewhere, and quoted... or delivered it Lemmatrically."

"[T]hose who are less vers'd in the more abstruse parts of Geometry, or less concerned about the Physical parts, may pass over, and only read the Astronomy separately and distinct..."

"The Celestial Physics, or Physical Astronomy, is not only the first in dignity of all inquiries into Nature... but the first in order, because it is the easiest."

"For the Sun and Planets are separated from one another by so immense a distance, as renders them incapable of exerting most of those forces whereby all Bodies act upon one another; so that they have no other force left them whereby they can affect one another, but the single force of universal Gravity: Whereas in the production of several Phænomena, that are observ'd upon our Earth, innumerable other forces are exerted, such as are very hard to be distinguish'd from one another; which notwithstanding, if not accurately done, in vain do we attempt Nature, and make any inquiry into it."

"Upon this account it is, that every Problem in the Terrestrial Physics is very operose and perplex'd, on the contrary, in the Celestial Physics, much more easy and simple; tho' even the latter has its difficulties, arising from the different distances and magnitudes of the Celestial Bodies, For the Fix'd Stars are so vastly distant asunder, that they have no mutual action upon each other, observable by us..."

"[T]hose persons seem to apply their thoughts but to a very indifferent purpose in the study of Nature, that overlook this part of Astronomy, from whence the principal and most simple Laws of Nature are to be learn'd."

"[T]he Physics delivered in the following Work... was both known and diligently cultivated by the most ancient Philosophers. ...[T]he true System of the World, approv'd of Pythagoras, and others among the Ancients..."

"[W]e do still tread in the steps of the Ancients in this Physical Astronomy; inasmuch as they knew that the Celestial Bodies gravitated towards each other, and were retain'd in their Orbits by the force of Gravity; and were also apprized of the Law of this Gravity."

"[I]f we look back to the first Rise of Astronomy... we shall find nothing better approv'd of, nothing more universally entertained among the several Sects of Philosophers, than this notion of the Gravity of the Celestial Bodies."

"That saying is well known, so often used by Anaxagoras, and his Scholars, Achelaus and Euripides, Namely, "That the Sun and Stars were fiery or red-hot Stones and Golden Clods." Of the same mind also were Democritus, Metrodorus, and Diogenes..."

"[A]s we are told by Democritus these notions about the Sun and Moon are not to be ascrib'd to Anaxagoras as their original... He had them from his Master Anaximenes whose Opinion... was, that the Stars were of a fiery nature and substance, that there were also mingled with them certain Earthly Bodies... [H]e plainly means, Planets of a terrestrial nature, performing their revolutions in the System of every Fix'd Star."

"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."

"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."

"'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."

"[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."

"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."

"[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."

"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.'"

"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."

"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."