Fellows Of The Royal Society

"Here then we have all the essentials for the calculus; but only for explicit integral algebraic functions, needing the binomial expansion of Newton, or a general method of rationalization which did not impose too great algebraic difficulties, for their further development; also, on the authority of Poisson, Fermat is placed out of court, in that he also only applied his method to certain special cases. Following the lead of Roberval, Newton subsequently used the third definition of a tangent, and the idea of time as the independent variable, although this was only to insure that one at least of his working variables should increase uniformly. This uniform increase of the independent variable would seem to have been usual for mathematicians of the period and to have persisted for some time; for later we find with Leibniz and the Bernoullis that d(dy/dx) = (d2y/dx2)dx. Barrow also used time as the independent variable in order that, like Newton, he might insure that one of his variables, a moving point or line or superficies, should proceed uniformly; ...Barrow... chose his own definition of a tangent, the second of those given above; and to this choice is due in great measure his advance over his predecessors. For his areas and volumes he followed the idea of Cavalieri and Roberval."

"Fermat... adopted Kepler's notion of the increment of the variable becoming evanescent near a maximum or minimum value, and upon it based his method of drawing tangents. Fermat's method of finding the maximum or minimum... involved the differentiation of any explicit algebraic function, in the form that appears in any beginner's text book of today (though Fermat does not seem to have the "function" idea); that is, the maximum or minimum values of f(x) are the roots of f'(x) = 0, where f'(x) is the limiting value of [f(x+h) - f(x)]/h; only Fermat uses the letter e or E instead of h."

"What Cavalieri and Roberval did for the integral calculus, Descartes... accomplished for the differential branch by his... application of algebra to geometry. Cartesian coordinates made possible the extension of... drawing... tangents to... curves of any kind. ...[H]e habitually used the index notation ...this had a very great deal to do with ...Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make... great progress... in... drawing of s, owing to... an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three:—"

"The next step was... more analytical... [B]y the method of indivisibles, Wallis... reduced... many areas and volumes to... the series (0^m + 1^m + 2^m +... n^m) / (n + 1)n^m, i.e. the ratio of the mean of all the terms to the last term, for integral values of n; and later he extended his method, by a theory of , to fractional values of n. Thus the idea of the Integral Calculus was in a fairly advanced stage in the days immediately antecedent to Barrow."

"In 1635 Cavalieri published a theory of "indivisibles," in which he considered a line as made up of an infinite number of points, a superficies as composed of a succession of lines, and a solid as a succession of superficies, thus laying the foundation for the "aggregations" of Barrow. Roberval seems... first, or... an independent, inventor of the method; but he lost credit... because he did not publish it, preferring to keep the method... for his own use... a usual thing... of that time, due perhaps to... professional jealousy. The method was severely criticized... especially by Guldin, but Pascal... showed that the method of indivisibles was as rigorous as... exhaustions... they were practically identical. ...[T]he progress... is much indebted to this defence by Pascal. Since this method is... analogous to... integration, Cavalieri and Roberval have... claim... as... inventors of... one branch of the calculus; if it were not for the fact that they only applied it to special cases, and seem... unable to generalize... owing to cumbrous algebraical notation, or to have failed to perceive the inner meaning... concealed under a geometrical form. Pascal... applied the method with great success, but also to special cases only; such as his work on the ."

"Galileo... would appear to have led the way, by the introduction of the theory of composition of motions into mechanics; he also was one of the first to use infinitesimals in geometry, and from... what is equivalent to "virtual velocities" it is... inferred that the idea of time as the independent variable is due to him. Kepler... was the first to introduce... infinity into geometry and to note that the increment of a variable was evanescent for values of the variable in the immediate neighbourhood of a maximum or minimum; in 1613, an abundant vintage drew his attention to the defective methods in use for estimating the cubical contents of vessels, and his essay on the subject (Nova Stereometria Doliorum) entitles him to rank amongst those who made the discovery of the infinitesimal calculus possible."

"The beginnings of the Infinitesimal Calculus... arose from determinations of areas and volumes, and the finding of tangents to plane curves. The ancients attacked the problems in a strictly geometrical manner, making use of the "s." ... This was the method by means of which Archimedes proved most of his discoveries. But there seems to have been some distrust of the method, for we find... many... discoveries... proved by a '..."

"I have used three distinct kinds of type: the most widely spaced type has been used for Barrow’s own words; only very occasionally have I inserted anything of my own in this, and then it will be found enclosed in heavy square brackets, that the reader will have no chance of confusing my explanations with the text... Barrow makes use of parentheses very frequently, so that the reader must understand that only remarks in heavy square brackets are mine... The small type is used for footnotes only. In the notes I... use the Leibniz notation, because it will... convey my meaning better; but there was really no absolute necessity for this, Barrow’s a and e, or its modern equivalent, h and k, would have done quite as well."

"I have to all intents rewritten Barrow’s book; although throughout I... adhered... to Barrows own words. I have only retained those parts which seemed... essential for the purpose in hand. This was necessary... that room might be found for... critical notes on the theorems.., proofs omitted by Barrow, which when given in Barrow’s style, and afterwards translated into analysis, had an important bearing on the point as to how he found out the more difficult of his constructions; and lastly for deductions therefrom that point steadily, one after the other, to the fact that Barrow was writing a calculus and knew that he was inventing a great thing."

"[T]he conclusion is the effect of a gradual accumulation of evidence... I have given a wholly inadequate account of the work of Barrow’s immediate predecessors; but... to a sufficiency for... showing... the time was... ripe for the work of Barrow, Newton, and Leibniz."

"Only on completing my annotation of the last chapter of this volume, Lect. XII, App. III, did I come to the conclusion that is given as the opening sentence of this Preface; for I then found that a batch of theorems.., on careful revision, turned out to be the few missing standard forms, necessary for completing the set for integration; and that one of his problems was a practical rule for finding the area under any curve, such as would not yield to the theoretical rules he had given, under the guise of an "inverse-tangent" problem."

"My attention was arrested by a theorem in which Barrow... rectified the cycloid, which... has usually been ascribed to Sir C. Wren. ...What I found induced me to treat a number of the theorems ...I came to the conclusion that Barrow had got the calculus; but I queried even then whether Barrow himself recognized the fact."

"Leibniz bought a copy of Barrow’s work in 1673, and was able "to communicate a candid account of his calculus to Newton" in 1677. In... the face of Leibniz’ persistent denial that he received any assistance whatever from Barrow’s book... bear... in mind Leibniz’ twofold idea of the "calculus":— (i) the freeing of the matter from geometry, (ii) the adoption of a convenient notation. ...[O]n these two points ...he derived not the slightest assistance from Barrow’s work; for the first ...would be dead against Barrow’s practice and instinct, and of the second Barrow had no knowledge whatever. ...[F]or [these points] ...the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. ...the differentiation and integration of any function whatever (such as were known in Barrow’s time), must be ascribed to Barrow."

"During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per Æquationes. This, though composed in 1666, was not published until 1711."

"He further developed... [infinitesimal calculus] in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he communicated to Newton the great secret of his geometrical constructions... and it was probably this that set Newton to... attempt to express everything as a sum of powers of the variable."

"Barrow was familiar with the paraboliforms, and s and areas connected with them, in from 1655 to 1660 at the very latest; hence he could... differentiate and integrate by his own method any rational positive power of a variable, and thus also a sum of such powers."

"By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a product, a quotient, or a power of a function; and also a recognition and demonstration of the fact that differentiation and integration are inverse operations.""

"Isaac Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow’s work, obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow’s book that he purchased in 1673."

", "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."

"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."

"Mathematics is the fruitful Parent of, I had almost said all, Arts, the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to Human Affairs. In which last Respect, we may be said to receive from the Mathematics, the principal Delights of Life, Securities of Health, Increase of Fortune, and Conveniences of Labour: That we dwell elegantly and commodiously, build decent Houses for ourselves, erect stately Temples to God, and leave wonderful Monuments to Posterity: That we are protected by those Rampires from the Incursions of the Enemy; rightly use Arms, skillfully range an Army, and manage War by Art, and not by the Madness of wild Beasts: That we have safe Traffick through the deceitful Billows, pass in a direct Road through the tractless Ways of the Sea, and come to the designed Ports by the uncertain Impulse of the Winds: That we rightly cast up our Accounts, do Business expeditiously, dispose, tabulate, and calculate scattered 248 Ranks of Numbers, and easily compute them, though expressive of huge Heaps of Sand, nay immense Hills of Atoms: That we make pacifick Separations of the Bounds of Lands, examine the Moments of Weights in an equal Balance, and distribute every one his own by a just Measure: That with a light Touch we thrust forward vast Bodies which way we will, and stop a huge Resistance with a very small Force: That we accurately delineate the Face of this Earthly Orb, and subject the Oeconomy of the Universe to our Sight: That we aptly digest the flowing Series of Time, distinguish what is acted by due Intervals, rightly account and discern the various Returns of the Seasons, the stated Periods of Years and Months, the alternate Increments of Days and Nights, the doubtful Limits of Light and Shadow, and the exact Differences of Hours and Minutes: That we derive the subtle Virtue of the Solar Rays to our Uses, infinitely extend the Sphere of Sight, enlarge the near Appearances of Things, bring to Hand Things remote, discover Things hidden, search Nature out of her Concealments, and unfold her dark Mysteries: That we delight our Eyes with beautiful Images, cunningly imitate the Devices and portray the Works of Nature; imitate did I say? nay excel, while we form to ourselves Things not in being, exhibit Things absent, and represent Things past: That we recreate our Minds and delight our Ears with melodious Sounds, attemperate the inconstant Undulations of the Air to musical Tunes, add a pleasant Voice to a sapless Log and draw a sweet Eloquence from a rigid Metal; celebrate our Maker with an harmonious Praise, and not unaptly imitate the blessed Choirs of Heaven: That we approach and examine the inaccessible Seats of the Clouds, the distant Tracts of Land, unfrequented Paths of the Sea; lofty Tops of the Mountains, low Bottoms of the Valleys, and deep Gulphs of the Ocean: That in Heart we advance to the Saints themselves above, yea draw them to us, scale the etherial Towers, freely range through the celestial Fields, measure the Magnitudes, and determine the Interstices of the Stars, prescribe inviolable Laws to the Heavens themselves, and confine the wandering Circuits of the Stars within fixed Bounds: Lastly, that we comprehend the vast Fabrick of the Universe, admire and contemplate the wonderful Beauty of the Divine 249 Workmanship, and to learn the incredible Force and Sagacity of our own Minds, by certain Experiments, and to acknowledge the Blessings of Heaven with pious Affection."

"For to pass by those Ancients, the wonderful Pythagoras, the sagacious Democritus, the divine Plato, the most subtle and very learned Aristotle, Men whom every Age has hitherto acknowledged as deservedly honored, as the greatest Philosophers, the Ring-leaders of Arts; in whose Judgments how much these Studies [mathematics] were esteemed, is abundantly proclaimed in History and confirmed by their famous Monuments, which are everywhere interspersed and bespangled with Mathematical Reasonings and Examples, as with so many Stars; and consequently anyone not in some Degree conversant in these Studies will in vain expect to understand, or unlock their hidden Meanings, without the Help of a Mathematical Key: For who can play well on Aristotle’s Instrument but with a Mathematical Quill; or not be altogether deaf to the Lessons of natural Philosophy, while ignorant of Geometry? Who void of (Geometry shall I say, or) Arithmetic can comprehend Plato’s 218 Socrates lisping with Children concerning Square Numbers; or can conceive Plato himself treating not only of the Universe, but the Polity of Commonwealths regulated by the Laws of Geometry, and formed according to a Mathematical Plan?"

"Smiling always with a never fading serenity of countenance, and flourishing in an immortal youth."

"Virtue is not a mushroom, that springeth up of itself in one night when we are asleep, or regard it not; but a delicate plant, that groweth slowly and tenderly, needing much pains to cultivate it, much care to guard it, much time to mature it, in our untoward soil, in this world's unkindly weather."

"The Mathematics which effectually exercises, not vainly deludes or vexatiously torments studious Minds with obscure Subtilties, perplexed Difficulties, or contentious Disquisitions; which overcomes without Opposition, triumphs without Pomp, compels without Force, and rules absolutely without Loss of Liberty; which does not privately overreach a weak Faith, but openly assaults an armed Reason, obtains a total Victory, and puts on inevitable Chains; whose Words are so many Oracles, and Works as many Miracles; which blabs out nothing rashly, nor designs anything from the Purpose, but plainly demonstrates and readily performs all Things within its Verge; which obtrudes no false Shadow of Science, but the very Science itself, the Mind firmly adheres to it, as soon as possessed of it, and can never after desert it of its own Accord, or be deprived of it by any Force of others: Lastly the Mathematics, which depend upon Principles clear to the Mind, and agreeable to Experience; which draws certain Conclusions, instructs by profitable Rules, unfolds pleasant Questions; and produces wonderful Effects; which is the fruitful Parent of, I had almost said all, Arts, the 47 unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human Affairs."

"The second part of the book... contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity."

"The quality of the human mind, considered in its collective aspect, which most strikes us, in surveying this record, is its colossal patience."

"We are able to appreciate the difficulties which in each age restricted the progress which could be made within limits which could not be surpassed by the means then available; we see how, when new weapons became available, a new race of thinkers turned to the further consideration of the problem with a new outlook."

"In the third period, which lasted from the middle of the eighteenth century until late in the nineteenth century, attention was turned to critical investigations of the true nature of the number π itself, considered independently of mere analytical representations. The number was first studied in respect of its rationality or irrationality, and it was shown to be really irrational. When the discovery was made of the fundamental distinction between algebraic and transcendental numbers, i.e. between those numbers which can be, and those numbers which cannot be, roots of an algebraical equation with rational coefficients, the question arose to which of these categories the number π belongs. It was finally established by a method which involved the use of some of the most modern of analytical investigation that the number π was transcendental. When this result was combined with the results of a critical investigation of the possibilities of a Euclidean determination, the inferences could be made that the number π, being transcendental, does not admit of a construction either by a Euclidean determination, or even by a determination in which the use of other algebraic curves besides the straight line and the circle are permitted. The answer to the original question thus obtained is of a conclusive negative character; but it is one in which a clear account is given of the fundamental reasons upon which that negative answer rests."

"The second period, which commenced in the middle of the seventeenth century, and lasted for about a century, was characterized by the application of the powerful analytical methods provided by the new Analysis to the determination of analytical expressions for the number π in the form of convergent series, products, and continued fractions. The older geometrical forms of investigation gave way to analytical processes in which the functional relationship as applied to the trigonometrical functions became prominent. The new methods of systematic representation gave rise to a race of calculators of π, who, in their consciousness of the vastly enhance means of calculation placed in their hands by the new Analysis, proceeded to apply the formulae to obtain numerical approximations to π to ever larger numbers of places of decimals, although their efforts were quite useless for the purpose of throwing light upon the true nature of that number. At the end of this period no knowledge had been obtained as regards the number π of the kind likely to throw light upon the possibility or impossibility of the old historical problem of the ideal construction; it was not even definitely known whether the number is rational or irrational. However, one great discovery, destined to furnish the clue to the solution of the problem, was made at this time; that of the relation between the two numbers π and e, as a particular case of those exponential expressions for the trigonometrical functions which form one of the most fundamentally important of the analytical weapons forged during this period."

"The first period embraces the time between the first records of empirical determinations of the ratio of the circumference to the diameter of a circle until the invention of the Differential and Integral Calculus, in the middle of the seventeenth century. This period, in which the ideal of an exact construction was never entirely lost sight of, and was occasionally supposed to have been attained, was the geometrical period, in which the main activity consisted in the approximate determination of π by the calculation of the sides or the areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork of the method was the Greek method of Exhaustions. In the earlier part of the period the work of approximation was much hampered by the backward condition of arithmetic due to the fact that our present system of numerical notation had not yet been invented; but the closeness of the approximations obtained in spite of this great obstacle are truly surprising. In the later part of this first period methods were devised by which the approximations to the value of π were obtained which required only a fraction of the labour involved in the earlier calculations. At the end of the period the method was developed to so high a degree of perfection that no further advance could be hoped for on the lines laid down by the Greek Mathematicians; for further progress more powerful methods were required."

"The history of our problem falls into three periods marked out by fundamentally distinct differences in respect of method, of immediate aims, and in equipment in possession of intellectual tools."

"As in Mathematics in general, the really great advances, embodying new ideas of far-reaching fruitfullness, have been due to an exceedingly small number of great men... there are periods when for a long series of centuries no advance was made; when the results obtained in a more enlightened age have been forgotten. We observe the times of revival, when the older learning has been rediscovered, and when the results of the progress made in distinct countries have been made available as the starting points of new efforts and a fresh period of activity."

"A new point is determined in Euclidean Geometry exclusively in one of the three following ways: Having given four points A, B, C, D, not all incident on the same straight line, then (1) Whenever a point P exists which is incident both on (A,B) and on (C,D), that point is regarded as determinate. (2) Whenever a point P exists which is incident both on the straight line (A,B) and on the circle C(D), that point is regarded as determinate. (3) Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate. The cardinal points of any figure determined by a Euclidean construction are always found by means of a finite number of successive applications of some or all of these rules (1), (2) and (3). Whenever one of these rules is applied it must be shown that it does not fail to determine the point. Euclid's own treatment is sometimes defective as regards this requisite. In order to make the practical constructions which correspond to these three Euclidean modes of determination, correponding to (1) the ruler is required, corresponding to (2) both ruler and compass, and corresponding to (3) the compass only. ...it is possible to develop Euclidean Geometry with a more restricted set of postulations. For example it can be shewn that all Euclidean constructions can be carried out by means of (3) alone..."

"The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry."