Mathematicians From Scotland

"Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines."

"Symbolical algebra is … the science which treats of the combination of operations defined not by their nature, … but by the laws of combination to which they are subject....[W]e suppose the existence of classes of unknown operations subject to the same laws."

"In 1841 Gregory published his Examples of the Processes of the Differential and Integral Calculus, a work which produced a great change for the better in the Cambridge mathematical books. It is the first in which constant use is made of the method known by the name of the separation of the symbols of operation, and the author has enlivened its pages by occasionally introducing historical notices of the problems discussed... His other mathematical work was A Treatise on the Application of Analysis to Solid Geometry, which was left unfinished at his death, and was completed and published by Walton in 1845. This is the first treatise in which the system of solid geometry is developed by means of symmetrical equations, and is a great advance on those of Leroy and Hymers."

"Since the beginning of the century, the general aspect of mathematics has greatly changed. A different class of problems from that which chiefly engaged the attention of the great writers of the last age has arisen, and the new requirements of natural philosophy have greatly influenced the progress of pure analysis. The mathematical theories of heat, light, electricity, and magnetism, may be fairly regarded as the achievement of the last fifty years. And in this class of researches an idea is prominent, which comparatively occurs but seldom in purely dynamical enquiries. This is the idea of discontinuity. Thus, for instance, in the theory of heat, the conditions relating to the surface of the body whose variations of temperature we are considering, form an essential and peculiar element of the problem; their peculiarity arises from the discontinuity of the transition from the temperature of the body to that of the space in which it is placed. Similarly, in the undulatory theory of light, there is much difficulty in determining the conditions which belong to the bounding surfaces of any portion of ether; and although this difficulty has, in the ordinary applications of the theory, been avoided by the introduction of proximate principles, it cannot be said to have been got ‘rid of. The power, therefore, of symbolizing discontinuity, if such an expression may be permitted, is essential to the progress of the more recent applications of mathematics to natural philosophy, and it is well known that this power is intimately connected with the theory of definite integrals. Hence the principal importance of this theory, which was altogether passed over in the earlier collection of examples. Mr Gregory devoted to it a chapter of his work, and noticed particularly some of the more remarkable applications of definite integrals to the expression of the solutions of partial differential equations. It is not improbable that in another edition he would have developed this subject at somewhat greater length. He had long been an admirer of Fourier’s great work on heat, to which this part of mathematics owes so much; and once, while turning over its pages, remarked to the writer,—“ All these things seem to me to be a kind of mathematical paradise.""

"Mr. Gregory: Late Fellow of Trinity College, Cambridge, and author of the -well-known Examples. Few in so short a life have done so much for science. The high sense which I entertain of his merits as a mathematician, is mingled with feelings of gratitude for much valuable assistance rendered to me in my earlier essays."

"There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are"

"In this chapter I shall collect those Theorems in the Differential Calculus which, depending only on the laws of combination of the symbols of differentiation, and not on the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols : but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first state? briefly the theory on which it is founded."

"It has always appeared to me that we sacrifice many of the advantages and more of the pleasures of studying any science by omitting all reference to the history of its progress: I have therefore occasionally introduced historical notices of those problems which are interesting either from the nature of the questions involved, or from their bearing on the history of the Calculus. ...[T]hese digressions may serve to relieve the dryness of a mere collection of Examples."

"The chief object of the present work is, as its title indicates, to furnish to the student examples by which to illustrate the processes of the Differential and Integral Calculus. In this respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation. But as Professor Peacock informed me that he had not leisure to superintend the publication of a second edition of his "Examples" which had been long out of print, I thought that I should do a service to students by preparing a work on a similar plan, but with such modifications as seemed called for by the increased cultivation of Analysis in this University."

"[[History of logarithms|[L]ogarithms]] are one of the most disorderly battlegrounds in mathematical history. ... Disputes like this and the other over the calculus have made more than one man of science envy his successors of ten thousand years hence, to whom Newton and Leibniz, Napier and Bürgi, and scores of lesser contestants for individual fame will be semimythical figures as indistinct as Pythagoras."

"By the time I was a student in high school I was reading the classic Men of Mathematics by E. T. Bell and I remember succeeding in proving the classic Fermat theorem about an integer multiplied by itself p times where p is a prime."

"He was admired for his science fiction and his Men of Mathematics. I was shocked when, just a few years later, Walter Pitts told me the latter was nothing but a string of Hollywood scenarios; my own subsequent study of the sources has shown me that Pitts was right, and I now find the contents of that still popular book to be little more than rehashes enlivened by nasty gossip and banal or indecent fancy."

"The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death."

"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics."

"Science makes no pretension to eternal truth or absolute truth; some of its rivals do. That science is in some respects inhuman may be the secret of its success in alleviating human misery and mitigating human stupidity."

"Some, of my unmathematical friends have incautiously urged me to include a note about the origin of modern calculating machines. This is the proper place to do so, as the Queen of queens has enslaved a few of these infernal things to do some of her more repulsive drudgery. What I shall say about these marvelous aids to the feeble human intelligence will be little indeed, for two reasons: I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly."

"Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos."

"Some of his deepest discoveries were reasoned out verbally with very few if any symbols, and those for the most part mere abbreviations of words. Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week."

"Fashion as king is sometimes a very stupid ruler. As was observed a little way back, the kernel of Plücker's theory of geometric dimensionality is that the dimensionality of a given space is not an absolute constant, but depends upon the elements, accepted as irreducible, in terms of which the space is described."

"The so-called obvious was repeatedly scrutinized from every angle and was frequently found to be not obvious but false. "Obvious" is the most dangerous word in mathematics."

"Out of fifty mathematical papers presented in brief at such a meeting, it is a rare mathematician indeed who really understands what more than half a dozen are about."

"The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so also can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular."

"Objections... inspired Kronecker and others to attack Weierstrass' "sequential" definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth — whatever that may be, if anything — precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals."

"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness."

"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. Then, in the alleged proof, be alert for inexplicit assumptions. Euclid's notorious oversights drove this lesson home."

"The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry."

"I feel my disease, and I feel that my want of alarm and lively affecting conviction forms its most obstinate ingredient; I try to stir up the emotion, and feel myself harassed and distressed at the impotency of my own meditations. But why linger without the threshold in the face of a warm and urgent invitation? "Come unto me." Do not think it is your office to heal one part of the disease, and Christ's to heal the remainder."

"It has been said that there is nothing more uncommon than common sense."

"Every man is a missionary, now and forever, for good or for evil, whether he intends or designs it or not. He may be a blot, radiating his dark influence outward to the very circumference of society; or he may be a blessing, spreading benediction over the length and breadth of the world: but a blank he cannot be. There are no moral blanks; there are no neutral characters. We are either the sower that sows and corrupts, or the light that splendidly illuminates, and the salt that silently operates; but being dead or alive, every man speaks."

"Live for something! Do good and leave behind you a monument of virtue that the storm of time can never destroy. Write your name in kindness, love, and mercy on the hearts of the thousands you come in contact with, year by year, and you will never be forgotten. Your name, your deeds, will be as legible on the hearts you leave behind, as the stars on the brow of evening. Good deeds will shine as the stars of heaven."

"To be benevolent in speculation, is often to be selfish in action and in reality. The vanity and the indolence of man delude him into a thousand inconsistencies. He professes to love the name and the semblance of virtue, but the labour of exertion and of self-denial terrifies him from attempting it. The emotions of kindness are delightful to his bosom, but then they are little better than a selfish indulgence—they terminate in his own enjoyment—they are a mere refinement of luxury. His eye melts over the picture of fictitious distress, while not a tear is left for the actual starvation and misery with which he is surrounded. It is easy to indulge the imaginations of a visionary heart in going over a scene of fancied affliction, because here there is no sloth to overcome—no avaricious propensity to control—no offensive or disgusting circumstance to allay the unmingled impression of sympathy which a soft and elegant picture is calculated to awaken. It is not so easy to be benevolent in action and in reality, because here there is fatigue to undergo—there is time and money to give — there is the mortifying spectacle of vice, and folly, and ingratitude, to encounter."

"The benevolence of the Gospel lies in actions"

"Not till we come to a simple reliance on the blood and mediation of the Saviour, shall we know what it is either to have trust in God, or know what it is to walk before Him without fear, in righteousness and true holiness."

"With the magnificence of eternity before us, let time, with all its fluctuations, dwindle into its own littleness."

"Be assured, my dear Anne, that it is only by taking our lesson from God and doing the will of God, that we can either please Him in time, or be happy with Him in eternity."

"The sum and substance of the preparation needed for a coming eternity is that you believe what the Bible tells you, and do what the Bible bids you."

"O Heavenly Father, convert my religion from a name to a principle! Bring all my thoughts and movements into an habitual reference to Thee!"

"The grand essentials of life are something to do, something to love, and something to hope for"

"O God, impress upon me the value of time, and give regulation to all my thoughts and to all my movements."

"I want to feel my own nothingness, I want to give myself up in absolute resignation to God, to lie prostrate and passive at His feet, with no other disposition in my heart than that of merging my will into His will, and no other language in my mouth than that of prayer for the perfecting of His strength in my weakness. I desire from the abyss of my own nothingness and vileness to cry unto God that He might cause me to do as I ought, and to be as I ought."

"The Bible is like a wide and beautiful landscape seen afar off, dim and confused; but a good telescope will bring it near, and spread out all its rocks and trees and flowers and v__ulant fields and winding rivers at one's very feet. That telescope is the Spirit's teaching."

"Christ came to give us a justifying righteousness, and He also came to make us holy — not chiefly for the purpose of evidencing here our possession of a justifying righteousness — but for the purpose of forming and fitting us for a blessed eternity."

"I take one decisive and immediate step, and resign my all to the sufficiency of my Saviour."

"This character wherewith we sink into the grave at death is the very character wherewith we shall reappear at the resurrection."

"What may appear as a towering peak to one may seem but an ordinary eminence to another."

"Scientific theory and its application to the growing needs of mankind advance hand in hand."

"We are perhaps too near the age of transition to see clearly the interplay of all that made for progress. Each of us has had his own peculiar training, his own personal contact with the mighty ones of the immediate past; and this forms as it were a telescopic tube determining limits to our field of vision. No doubt we may range the whole horizon; but after all we look from our own point of vantage."

"The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions."

"Oh, that's nothing – I could coach a coal scuttle to be Senior Wrangler."

"[Examiners] spend their lives in discovering which pages of a text-book a man ought to read and which will not be likely to 'pay'."