University Of Cambridge Alumni

", "Barrow's Mathematics: between ancients and moderns" Before Newton: the Life and Times of Isaac Barrow (1990) ed., Mordechai Feingold."

"[R]ecent work... removes him from a major role in the development of the calculus or in Newton's early mathematical thinking. ...[T]he new symbolic algebra... was essential to calculus. Barrow accepted neither the art nor the notion, and... avoided the technique wherever possible. The theorems in the Geometrical Lectures that historians juxtapose to form the belong to... different lines of inquiry, one... characterized by its freedom from... "tediousness of calculation" and divorced in Barrow's mind from... s or limits. The central lectures contain a program of research, but... not... the program that led Newton and Leibniz to the calculus."

"J. M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Paul Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"As a Line, I say, is looked upon to be the Trace of a Point moving forward, being in some sort divisible by a Point, and may be divided by Motion one Way, viz. as to Length; so Time may be conceiv'd as the Trace of a Moment continually flowing, having some Kind of Divisibility from an Instant, and from a successive Flux, inasmuch as it can be divided some how or other. And like as the Quantity of a Line consists of but one Length following the Motion; so the Quantity of Time pursues but one Succession stretched out as it were in Length, which the Length of the Space moved over shews and determines. We therefore shall always express Time by a right Line; first, indeed, taken or laid down at Pleasure, but whose Parts will exactly answer to the proportionable Parts of Time, as its Points do to the respective Instants of Time, and will aptly serve to represent them. Thus much for Time."

"It cannot be justly inferr'd... We do not perceive the Thing, therefore there is no such Thing, that is a false Illusion, a deceitful Dream, that wou'd cause us to join together two remote Instants of Time. But nevertheless this is very True... That is, for as much Motion as there was, so much Time seems to have been elapsed; nor, when we mention such a Quantity of Time, do we merely mean any Thing else, than the Performance of so much Motion, to the continued successive Extension of which we imagine the Permanency as Things is co-extended."

"As Magnitudes themselves are absolute Quantums Independent on all Kinds of Measure, tho' indeed we cannot tell what their Quantify is, unless we measure them; so Time is likewise a Quantum in itself, tho' in Order to find the Quantity of it, we are obliged to call in Motion to our Assistance as a Measure... and thus Time as measurable signifies Motion; for if all Things were to continue at Rest, it would be impossible to find out by any Method whatsoever how much Time has elaps'd; and the several Ages wou'd roll on imperceptibly and undistinguish'd. Do I say we shou'd not perceive how Time flows? No indeed, nor any Thing else, but remain like Stocks or Stones in a continual Insensibility. We perceive nothing, unless so far as we may be instigated by some Change affecting the Senses, or that our Souls are mov'd and excited by the internal Operation of the Mind. We esteem the Quantities and different Degrees of Things according to the Extension or Intension of Motions striking upon us either interiorly or exteriorly. So that the Quantity of Time so far as we can observe; depends upon the Extension of Motion."

"But you may perhaps wonder why I explain Time without Motion, and will say, does not Time imply Motion? I answer no, as to its absolute and intrinsic Nature; any more than it does Rest. The Quantity of Time, in itself, depends not on either of them; for whether Things move on, or stand still; whether we sleep or wake, Time flows perpetually with an equal Tenor."

"But perhaps you may ask, whether Time was not before the World was created? And if Time does not flow in the Extramundane Space, where nothing is: A mere Vacuum? I answer, that since there was Space before the World was created, and that there now is an Extramundane, infinite Space, (where God is present)... Time existed before the World began, and does exist together with the World in the Extramundane Space, because 'tis possible that some Thing might have existed long before the World was made; and there may now be something in the Extramundane Space, capable of such a Continuance: Some Sun might have given Light long before; and at present this, or some other like it, may diffuse Light thro' Imaginary Spaces. Time therefore does not imply an actual Existence, but only the Capacity or Possibility of the Continuance of Existence; just as Space expresses the Capacity of a Magnitude contain'd in it."

"Now pray tell me what Time is? ...Time (to speak abstractedly) is the continuance of any Thing in its own Being. But some Things continue longer in their Beings than others... Time absolutely... is Quantity, as admitting in some Manner the chief Affections of Quantity: Equality, Inequality, and Proportion..."

"What Mathematicians Chiefly consider in Motion is the Mode of Lation or Manner of bearing, and the Quantity of the motive Force. ...But because the Quantity of motive Force cannot be known without Time, we must say something concerning its Nature."

"Among these Ways, or any other whatever, of generating Magnitudes, the Primary and Chief is, that perform'd by local Motion, which all of them must in some Sort suppose, because without Motion, nothing can be generated or produced and therefore this must first be considered. The following Axiom of Aristotle concerning Motion is famous... He that is ignorant of Motion, must necessarily know nothing of Nature... in Nature every Thing created is produc'd by Motion, or certainly not without Motion."

"I... chose rather to publish... in puris Naturalibus, or as they were produced as first, than be at the Trouble of reducing them into any other Form... I could not bear the Pains of reading over again a great Part of these Things; either from my being tired with them, or not caring to undergo the Pains and Study in new modelling them. But I have done in this as weakly Mothers, who give up their Offspring to the Care of their Friends, either to Nurse and bring up, or abandon to the wide World. One of which is Mr. Isaac Newton, my Collegue, a Man of great Learning and Sagacity, who revised my Copy and noted such Things as wanted Correction, and even gave me some of his own, which you will see here and there interspersed with mine, not without their due Commendations. The other is Mr. John Collins (who may be deservedly called the Mersennas of our Nation, Born to promote this Science, both with his own Labours, and those of others. Who with much Trouble took care of the Edition."

"The Definition in the Elements, according to Clavius, is this: Magnitudes are said to be in the same Reason [ratio], a first to a second, and a third to a fourth, when the Equimultiples of the first and third according to any Multiplication whatsoever are both together either short of, equal to, or exceed the Equimultiples of the second and fourth, if those be taken, which answer one another.... Such is Euclid’s Definition of Proportions; that scare-Crow at which the over modest or slothful Dispositions of Men are generally affrighted: they are modest, who distrust their own Ability, as soon as a Difficulty appears, but they are slothful that will not give some Attention for the learning of Sciences; as if while we are involved in Obscurity we could clear ourselves without Labour. Both of 300 which Sorts of Persons are to be admonished, that the former be not discouraged, nor the latter refuse a little Care and Diligence when a Thing requires some Study."

"They [mathematicians] only take those things into consideration, of which they have clear and distinct ideas, designating them by proper, adequate, and invariable names, and premising only a few axioms which are most noted and certain to investigate their affections and draw conclusions from them, and agreeably laying down a very few hypotheses, such as are in the highest degree consonant with reason and not to be denied by anyone in his right mind. In like manner they assign generations or causes easy to be understood and readily admitted by all, they preserve a most accurate order, every proposition immediately following from what is supposed and proved before, and reject all things howsoever specious and probable which can not be inferred and deduced after the same manner.—Barrow, Isaac."

"It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel."

"Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from arithmetic to another Science, (which yet is no science) viz. algebra."

"An accomplished mathematician, i.e. a most wretched orator."

"These Disciplines [mathematics] serve to inure and corroborate the Mind to a constant Diligence in Study; to undergo the Trouble of an attentive Meditation, and cheerfully contend with such Difficulties as lie in the Way. They wholly deliver us from a credulous Simplicity, most strongly fortify us against the Vanity of Scepticism, effectually restrain from a rash Presumption, most easily incline us to a due Assent, perfectly subject us to the Government of right Reason, and inspire us with Resolution to wrestle against the unjust Tyranny of false Prejudices. If the Fancy be unstable and fluctuating, it is to be poised by this Ballast, and steadied by this Anchor, if the Wit be blunt it is sharpened upon this Whetstone; if luxuriant it is pared by this Knife; if headstrong it is restrained by this Bridle; and if dull it is roused by this Spur. The Steps are guided by no Lamp more clearly through the dark Mazes of Nature, by no Thread more surely through the intricate Labyrinths of Philosophy, nor lastly is the Bottom of Truth sounded more happily by any other Line. I will not mention how plentiful a Stock of Knowledge the Mind is furnished from these, with what wholesome Food it is nourished, and what sincere Pleasure it enjoys. But if I speak farther, I shall neither be the only Person, nor the first, who affirms it; that while the Mind is abstracted and elevated from sensible Matter, distinctly views pure Forms, conceives the Beauty of Ideas, and investigates the Harmony of Proportions; the Manners themselves are sensibly corrected and improved, the Affections composed and rectified, the Fancy calmed and settled, and the Understanding raised and excited to more divine Contemplation. All which I might defend by Authority, and confirm by the Suffrages of the greatest Philosophers."

"These laws must be construed according to the intention of them: and the circumstances of things at the time of enacting them ought to be taken into consideration."

"Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike for his range, his analytical power, and, not least, for his prolific production of new views and fertile theories. There is hardly a subject in the whole of pure mathematics at which he has not worked."

"Many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers, and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert. The interest in numbers seems implanted in the human mind, and it is a pity that it should not have freer scope in this country. The methods of the theory of numbers 271 are peculiar to itself, and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude; it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our University. From the moment that Gauss, in his wonderful treatise of 1801, laid down the true lines of the theory, it entered upon a new day, and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it."

"In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as 102 perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. “Pauca sed matura” was Gauss’s motto."

"The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not; he must take care not to be the slave of his symbols, but always to have before his mind the realities which they merely serve to express. For these and other reasons it seems to me of the highest importance that a mathematician should be trained in no narrow school; a wide course of reading in the first few years of his mathematical study cannot fail to influence for good the character of the whole of his subsequent work"

"Quite distinct from the theoretical question of the manner in which mathematics will rescue itself from the perils to which it is exposed by its own prolific nature is the practical problem of finding means of rendering available for the student the results which have been already accumulated, and making it possible for the learner to obtain some idea of the present state of the various departments of mathematics.... The great mass of mathematical literature will be always contained in Journals and Transactions, but there is no reason why it should not be rendered far more useful and accessible than at present by means of treatises or higher text-books. The whole science suffers from want of avenues of approach, and many beautiful branches of mathematics are regarded as difficult and technical merely because they are not easily accessible.... I feel very strongly that any introduction to a new subject written by a competent person confers a real benefit on the whole science. The number of excellent text-books of an elementary kind that are published in this country makes it all the more to be regretted that we have so few that are intended for the advanced student. As an example of the higher kind of text-book, the want of which is so badly felt in many subjects, I may mention the second part of Prof. Chrystal’s “Algebra” published last year, which in a small compass gives a great mass of valuable and fundamental knowledge that has hitherto been beyond the reach of an ordinary student, though in reality lying so close at hand. I may add that in any treatise or higher text-book it is always desirable that references to the original memoirs should be given, and, if possible, short historic notices also. I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."

"It would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress. Not only does the area widen but the subjects of study increase rapidly in number, and the work of the mathematician tends to become more and more specialized. It is, of course, merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician, but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted, even in a general way, with the progress of any of the branches of mathematics except those which form the field of his own labours. I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained.... I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject"

"The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio."

"The manner of the transaction was to gild over and conceal the truth; and whenever Courts of law see such attempts made to conceal such wicked deeds they will brush away the cobweb varnish and show the transactions in their true light."

"The statute law is the will of the legislature in writing; the common law is nothing else but statutes worn out by time; all our law began by consent of the legislature, and whether it is now law by usage or writing, it is the same thing."

"You shall not stipulate for iniquity. All writers upon our law agree in this, no polluted hand shall touch the pure fountains of justice."

"I have always thought that formerly there was too confined a way of thinking in the Judges of the common law Courts, and that Courts of equity have risen by the Judges not properly applying the principles of the common law, being too narrowly governed by old cases and maxims, which have too much prevented the public from having the benefit of the common law."

"I am not for stirring a single pebble of the common law."

"A course of precedents and judicial proceedings in Courts of justice make the law: it would be endless to cite cases upon it. A course of practice for a few years has been held to controul an Act of Parliament."

"Settlements are supposed in law to be indifferent to paupers; though they are often in fact desirous of one in preference to another."

"God forbid that the rights of the innocent should be lost and destroyed by the offence of individuals."

"The husband is not liable for the criminal conduct of his wife."

"Many of the old cases are strange and absurd : so also are some of the modern ones."

"You all very well know what deference I always pay, and ever will, to that part of the office of a jury which properly belongs to them. In regard to the law, I have always been as tenacious of the proper function of a Judge, as I have been of that of the jury. I never will, while I have the honour of executing the office of a Judge, attempt to controul or influence their minds in respect of damages; but only submit to them such observations as occur to me upon the evidence."

"We cannot explore any mode of sentencing a man to imprisonment, who is imprisoned already, but by tacking one imprisonment to the other."

"The sentences of foreign Courts have always some degree of regard paid to them by the Courts of justice here: and it is very right that an attention should be paid to them, as far as they ought to have weight in the case depending."

"I cannot help thinking that where a person appeals to the Law of England, he must take his remedy according to the Law of England to which he has appealed."

"It is a strange, wild jurisdiction, where the jurors are judges both of law and fact, and ignorant country fellows2 are to determine the nicest points of law."

"The time makes no difference in the reason of the thing."

"It is the principle of the common law, that an officer ought not to take money for doing his duty."

"There is no merit in a settlement: it depends upon positive law."

"Nothing could be of worse consequence, than that an officer of the Court should combine with a criminal to frustrate the sentence of the Court."

"If once we go upon niceties of construction, we shall not know where to stop. For one nicety is made a foundation for another; and that other for a third; and so on, without end."

"The landscape has been so totally changed, the ways of thinking have been so deeply affected, that it is very hard to get hold of what it was like before... It is very hard to realize how total a change in outlook Isaac Newton has produced."

"Time is that which is measured by a clock. This is a sound way of looking at things. A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense about this kind of thing."

"All science is full of statements where you put the best face on your ignorance, where you say: true enough, we know awfully little about this, but more or less irrespective of the stuff we don't know about, we can make certain useful deductions."

"The kind of lecture which I have been so kindly invited to give, and which now appears in book form, gives one a rare opportunity to allow the bees in one's bonnet to buzz even more noisily than usual."