University Of Edinburgh Alumni

"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."

"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."

"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."

"[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."

"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]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."

"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."

"[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."

"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."

"[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."

"'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."

"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."

"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."

"[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."

"[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."

"[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."

"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.'"

"[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."

"[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."

"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."

"[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]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."

"[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..."

"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..."

"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."

"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."

"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."

"[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."

"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."

"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..."

"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."

"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."

"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."

"[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."

"In his major work, Principles of Gestalt Psychology, Koffka (1935) devoted over 60 pages to the hypothesized role of ego in memory... Koffka distinguished between self and ego, the former being defined as a central subsystem of ego, but his analysis of memory was given in terms of the broader system, ego."

"If an environmental trace is in close connection with the Ego system it will not only be in communication with the particular time structure of that system with which it communicated at the time of its formation; but because of the coherence of the whole temporal Ego system it will be in communication with later strata also."

"In our theory the total field of excitation is divided into two major sub-systems, each containing numerous sub-systems of its own: the Ego and the environment. And the trace field which is created by the excitation field contains the same dichotomous organization."

"The second factor, equally important, is that of good continuation and good shape. The things which we see have a better shape, are bounded by better contours, than the holes which we might see but do not. Therefore, when in exceptional circumstances these conditions are reversed, we see the hole and not the things, as the shape of a gap between two projecting rocks with sharp profiles, which may look like a face, an animal, or some other object, while the shape of the rock disappears."

"A complete theory would have to answer, among others, the following questions, given two adjoining retinal areas of different stimulation, under what conditions will the corresponding parts of the behavioural (perceptual) field appear of different whiteness but equal brightness (or "illumination"), when of different brightness but equal whiteness? A complete answer to this question would probably supply the key to the complete theory of colour perception in the broadest sense."

"Why We See Things and Not the Holes Between Them. We can now attempt an answer to the question why we do so. Two of the factors of organization which we have so far discussed seem to me to be the most important causes of this effect. In the first place, the segregation and unification which occurs will separate areas of different degrees of internal articulation, and according to our law, the more highly articulated ones will become figures, the rest fusing together to form the ground. Look at any landscape photograph. You see the shape of the things, the mountains, and trees and buildings, but not of the sky."

"Conduct, of course, is possible without science. Humans carried on in their daily affairs long before the first spark of science had been struck. And today there are millions of people living whose actions are not determined by anything we call science. Science, however, could not but gain an increasing influence on human behaviour. To describe this influence roughly and briefly will throw a new light on science. Exaggerating and schematizing the differences, we can say: in the prescientific stage man behaves in a situation as the situation tells him to behave. To primitive man each thing says what it is and what he ought to do with it: a fruit says, "Eat me"; water says, "Drink me"; thunder says, "Fear me," and woman says, "Love me.""

"We have discussed quantity, order and meaning with regard to their contributions to science in general and to psychology in particular. We extracted each of our categories from a different science, but we claimed that despite their different origins, they are all universally applicable. And as a matter of fact, in our treatment of the issues involved in each of our three categories we have found the same general principle: to integrate quantity and quality, mechanism and vitalism, explanation and comprehension or understanding, we had to abandon the treatment of a number of separate facts for the consideration of a group of facts in their specific form of connection. Only thus could quantity be qualitative, and order and meaning be saved from being either introduced into the system of science as new entities, the privileges of life and mind, or discarded as mere figments."