Mathematics

"It was not until the nineteenth century, chiefly through... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...[I]t called attention... to the fact that a proof can be given of the impossibility of proving certain propositions within a given system. ...Gödel's paper is a proof of the impossibilty of formally demonstrating certain important propositions in number theory. ...[T]he resolution of that parallel axiom question forced the realization that Euclid was not the last word on the subject of geometry, since new systems of geometry can be constructed... incompatible with those adopted by Euclid. ...[I]t gradually became clear that the proper business of pure mathematicians is to derive theorems from postulated assumptions, and that it is not their concern whether the axioms are actually true."

"We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name."

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ... arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof."

"When we are engaged in investigating the foundations of a science, we must set up a system of s which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science... is held to be correct unless it can be derived from axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all, I wish to designate the following as the most important among numerous questions which can be asked with respect to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results."

"In a never-ending search for good mathematical problems and fresh mathematical talent, Erdős crisscrossed four continents at a frenzied pace, moving from one university or research center to the next. His modus operandi was to show up on the doorstep of a fellow mathematician, declare, "My brain is open," work with his host for a day or two, until he was bored or his host was run down, and then move on to another home. ...He did mathematics in more than 25 different countries, completing important proofs in remote places and sometimes publishing them in equally obscure journals."

"The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle."

"Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks, mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians."

"The difficulty in presenting a rigorous as well as clear statement of the theory of limits is inherent in the subject. ...If the reader has found some difficulty in grasping it he may be less discouraged when he is told that it eluded even Newton and Leibniz. ... Many contemporaries of Newton, among them ... taught that the calculus was a collection of ingenious fallacies. ... decided that he could found calculus properly... The book was undoubtedly profound but also unintelligible. One hundred years after the time of Newton and Leibniz, Joseph Louis Lagrange... still believed that the calculus was unsound and gave correct results only because errors were offsetting each other. He, too, formulated his own foundation... but it was incorrect. ...D'Alembert had to advise students of the calculus... faith would eventually come to them. This is not bad advice... but it is no substitute for rigor and proof. ... About a century and a half after the creation of calculus... Augustin Louis Cauchy... finally gave a definitive formulation of the limit concept that removed doubts as to the soundness of the subject."

"In 1920 logic was mostly a philosopher's garden. There were also a few mathematicians there, cultivating the logical roots of the mathematical tree. Today, Theory, Set Theory, and Proof Theory, logic's major subdisciplines, have become full-fledged branches of mathematics."

"are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof's overall strategy, delightfully, is not. ...They belong to a branch of mathematics known as formal logic or mathematical logic, a field which was viewed, prior to Gödel's achievement, as mathematically suspect."

"The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned."

"The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth: synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths."

"Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from the Book!". This viewpoint expresses... that mathematics, as the... foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God..."

"Another roof, another proof."

"It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word."

"The specific aim of the was to find an elementary proof of a special case of the density (DHJ)... The theorem was known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights. There were two reasons to want a new proof... First... [it] had just one—a long and complicated proof that relied on heavy mathematical machinery. An elementary proof—one that starts from first principles instead of relying on advanced techniques—would require many new ideas. Second, DHJ implies another famous theorem, called , novel proofs of which led to several breakthroughs over the past decade, so there was reason to expect that the same would happen..."

"More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence.""

"At the age of forty he was, for the first time, introduced to the works of Euclid, and at once 'fell in love with geometry,' being attracted, he says, more by the rigorous manner of proof employed than by the matter of the science. (Mathematics, we must remember, were then only beginning to be seriously studied in England. Hobbes tells us that in his undergraduate days geometry was still looked upon generally as a form of the 'Black Art,' and it was not until 1619 that the will of Sir Henry Savile, Warden of Merton College, established the first Professorships of Geometry and Astronomy at Oxford.)"

"The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes."

"The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why?"

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."

"The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!"

"Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an , which allows a certain amount of self-reference. And by this time everyone was pretty bored."

"Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy."

"In the summer of 1914 I attended Frege's course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. ...Unfortunately, his admonitions go unheeded even today."

"Proof is the idol before whom the pure mathematician tortures himself."

"On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically: as is particularly noticed by Bhaskara himself, towards the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities."

"Proofs have gaps and are... inherently incomplete and sometimes wrong. ...There is another reason ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification."

"One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them."

"Perhaps nobody should be particularly surprised by this, as, after all, the laws of nature physicists acknowledge also seem to be timelessly true independently of whether anyone takes them to be true. Where do they come from? Since there are somewhat mundane interpretations of what laws of nature are – including the possibility that they are accidental generalities valid in this particular universe and/or within a certain time-span – the case posed by mathematical constructs seems to be even more clear and powerful. Math, like diamonds, truly seems to be forever. If one ‘goes Platonic’ with math, one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder."

"How is it that mathematical ideas can be communicated in this way? I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts. ... When one 'sees' a mathematical truth, one's consciousness breaks through into this world of ideas, and makes direct contact with it ('accessible via the intellect'). I have described this 'seeing' in relation to Gödel's theorem, but it is the essence of mathematical understanding. When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'. (Indeed, often this act of perception is accompanied by words like 'Oh, I see'!) Since each can make contact with Plato's world directly, they can more readily communicate with each other than one might have expected. The mental images that each one has, when making this Platonic contact, might be rather different in each case, but communication is possible because each is directly in contact with the same externally existing Platonic world!"

"I don’t call myself a “Platonist,” since I don’t even know what it even means to say that mathematical truths “exist in a Platonic realm.” Indeed, that concrete way of putting things even strikes me as a step backwards—as if astronomers could one day send probes to Plato’s realm to find out what’s in it, or prove that it never existed. The concepts of “existence” and “reality” that we use when discussing physical objects are just flat-out irrelevant here. I instead call myself an “anti-anti-Platonist”: someone who’s not literally committed to a “Platonic realm” of mathematical truth, but who given the choice, would much rather that people believe in such a realm than that they spout the self-evident garbage that goes with denying math’s intellectual autonomy."

"What’s the greatest discovery in the history of thought? Of course, it’s a silly question – but it won’t stop me from suggesting an answer. It’s Plato’s discovery of abstract objects. Most scientists, and indeed most philosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum. Mathematicians, on the other hand, are at least slightly sympathetic. Working day-in and day-out with primes, polynomials and principal fibre bundles, they have come to think of these entities as having a life of their own. Could this be only a visceral reaction to an illusion? Perhaps, but I doubt it."

"Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say."

"Berkeley Bishop of Cloyne was a man of first-rate talents, distinguished as a metaphysician, a philosopher, and a divine. His geometrical knowledge, however, which, for an attack on the , was more essential than all his other accomplishments, seems to have been little more than elementary. The motive which induced him to enter on discussions so remotely connected with his usual pursuits has been variously represented; but whatever it was, it gave rise to , in which the author professes to demonstrate, that the new analysis is inaccurate in its principles, and that, if it ever lead to true conclusions, it is from an accidental compensation of errors that cannot be supposed always to take place. The argument is ingeniously and plausibly conducted, and the author sometimes attempts ridicule with better success than could be expected from the subject; thus when he calls ultimate ratios the ghosts of departed quantities, it is not easy to conceive a witty saying more happily fastened on a mere mathematical abstraction."

"[W]hile we put number into objects, on the other hand we derive our idea of number only from the presence of the world external to the mind. We see a group of people, and we begin by making an abstraction ("people"), and we say, "Here are ten people"—thus calling them all by the one abstract name, even though the individuals be very different. "A careful observation shows us, however, that there are no objects exactly alike; but by a mental operation of which we are quite unconscious, although it holds within itself the entire secret of mathematical abstraction, we take in objects which seem to be alike, rejecting for the time being their differences. Here is to be found the source of calculation." So the idea of number is generated in the mind by the sense perception of a group of things supposed to be alike."

"In solving a problem, be it one in the calculus, in algebra, or in the second year of arithmetic, we begin by substituting for the actual things certain abstractions represented by symbols; we think in terms of these abstractions, aided by symbols, and finally from our result we pass back to the concrete and say that we have solved the problem. It is all a matter of "one to one correspondence," it being easier for us to work with the abstract numbers and their corresponding figures than to work with the actual objects. Fundamentally the process is something like this: 1. By abstraction we pass to numbers. 2. Thence we pass to symbols, and we make an equation, either openly, as in algebra, or concealed, as in many forms of arithmetic. This equation we solve, the result being a symbol. 3. We find the number corresponding to this symbol, and say that the problem is solved. All this does not mean that primary number is to be merely a matter of symbols. It means that in mathematics we find it more convenient to work purely with symbols, translating back to the corresponding concrete form as may be desired. And so those teachers who fear lest the child shall drift into thinking in symbols instead of in number, are really fearing that the child shall drift into mathematics."

"Geometry can in no way be viewed... as a branch of mathematics, instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry."

"Change of state involves what is meant by the word "time," which, like space, is a necessary condition for thought, or the formation of concepts. These occur in succession, as do all phenomena involving a change, and whenever motion or change is involved, time is required, and the universal knowledge of this fact is an elementary abstraction. Unity apart from any concrete thing... is the primary abstraction in arithmetic, and number and the theory of numbers grow from this concept to one of great importance. Since time and space are necessary to the realization of sensible objects and the phenomena of motion, the attributes of time and space adhere of necessity to the theory of these subjects, and hence the principles demonstrated in the abstract concerning time and space relations, are applicable to such phenomena. Herein lies the secret of the universal application of pure mathematical deductions to the many sciences that sum up human knowledge of the universe."

"Abstraction is the immediate ulterior result of analysis. We may speak of the analysis of the mathematical whole, and so of the abstraction of any of its parts. Wherever analysis may take place, abstraction likewise, is possible. ...The reason on account of which the analysis and abstraction of the mind are directed to the parts of the metaphysical whole as such, lies in the fact that the mental division of an object into its mathematical, or separable, parts, is not sufficient even for the ends of ordinary thought. We cannot, from such a division, adequately understand and express the nature of things. This purpose requires that we should consider and designate inseparable parts, such as powers, shapes, magnitudes, and attributes generally."

"They who are acquainted with the present state of the theory of Symbolical Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. ... It has happened in every known form of analysis, that the elements to be determined have been conceived as measurable by comparison with some fixed standard. The predominant idea has been that of magnitude, or more strictly of numerical ratio. The expression of magnitude, or of operations upon magnitude, has been the express object for which the symbols of Analysis have been invented, and for which their laws have been investigated. Thus the abstractions of the modern Analysis, not less than the ostensive diagrams of the ancient Geometry, have encouraged the notion that Mathematics are essentially, as well as actually, the Science of Magnitude. ... [T]his conclusion is by no means necessary. If every existing interpretation is shewn to involve the idea of magnitude, it is only by induction that we can assert that no other interpretation is possible. ...The history of pure Analysis is, it may be said, too recent to permit us to set limits to the extent of its applications. ...That to the existing forms of Analysis a quantitative interpretation is assigned, is the result of the circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis. It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its object and in its instruments it must at present stand alone."

"The chemist, in describing some mineral, may present all its attributes, color mass, density, volume, molecular construction, and its properties exhibited in relation to heat, light, sound, and electric waves, and the resulting conception will not be mathematical. On the other hand, in describing its form as cubical, he relies upon the purely mathematical abstraction involved in the definition of a cube, which includes that of the mathematical plane, which involves the geometric line, which, in turn, resolves into an assemblage of points. Not one of these conceptions finds its realization in physical phenomena, but at best an approximation. The types conceived by the mind remain, however, definite and unchangeable. The idea of the ultimate element, the "point," best illustrates the nature of these abstractions, and involves the first difficulty that lies in the way of the understanding. The point is defined as having but one positive attribute, and that is, position in space. ...The mathematical point cannot be realized by this process, for the resulting principle is this: all material things are indefinitely divisible, and, no matter how small, possess magnitude and occupy space. The negative of this thought is the point, which does not have magnitude, but position only. It is a negative thought concept, a mathematical abstraction..."

"The word element is a term which frequently occurs in philosophy. It signifies any of those parts of an object into which it is or may be separated by analysis; and which, therefore, may be separately considered by abstraction. ...A notion of a thing may be formed by the composition of mathematical parts, and such a composition in its relation to parts, and such a composition in relation to the object might be spoken of as mathematical conception."

"It was only in the nineteenth century that the winds of change started to blow. First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of "quantity" and of "measurement" beyond recoginition. Second, the rapidly multiplying of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and "existence" into the abstractions themselves."

"[M]ental Abstraction... is not [only the] Property of Mathematics, but is common to all Sciences. For every Science considers the Nature of its own Subject abstracted from all others; forms its own general Precepts and Theorems; and separates its own Properties from the Properties of others... For example, Physics considers the constitutive Principles, Matter, Form, &c. of Body in general; then the Affections common to all Bodies, viz. Quantity, Place, Motion, Rest, and the like; from whence it descends to the next lower Species, investigating their particular Natures and Properties; but meddles not with particular Bodies or Individuals, as well because they are innumerable and distinguished from one another by innumerable Differences... The same way Geometry proposes Magnitude for the Subject of its Enquiry, not the peculiar Magnitude of this or that Body, but the Magnitude taken universally; together with its general Affections, viz. Divisibility, Congruence, Proportionality, a Capacity of different Situation and Position, Mobility &c. declaring these to be inherent to it, and after what manner they are so: Next it defines the various Species of Magnitude, (viz. a Line, Superficies, and a Body or Solid) and particularly draws forth and demonstrates their distinct Properties; continually dividing these Species into others more contract, and searching and proving their Affections by universal Propositions, Rules and Theorems lawfully demonstrated, till it has wholly exhausted its Subject, and descended to the very lowest Species. And these Theorems however more or less general as to their Matter, may be truly and properly accommodated to Subjects particular to themselves. True Mathematical Abstraction then, is such as agrees with all other Sciences and Disciplines, nothing else being meant (whatsoever some do strangely say of it) than an Abstraction from particular Subjects, or a distinct Consideration of certain things more universal, others less universal being ommitted and as it were neglected."

"In a rigid body free to move in any manner there are six degrees of freedom, and the generalised coordinates most frequently chosen in this case are the three rectangular coordinates of some point in the body and three angular coordinates determining the orientation of the body about that point, generally the angles θ, φ, ψ of ordinary occurrence in rigid dynamical problems."

"When the body degenerates into a material straight line the number of degrees of freedom is reduced to five; and when this straight line is constrained to move parallel to some fixed plane the number of degrees of freedom is still further reduced to four."

"In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in the physical order of ideas. Perhaps the most striking example is to be found in the development of abstract Dynamics. The greatest treatise which the world has seen, on this subject, is Lagrange's Mécanique Analytique, published in 1788. ...conceived in the purely abstract Mathematical spirit ...Lagrange's idea of reducing the investigation of the motion of a dynamical system to a form dependent upon a single function of the of the system was further developed by Hamilton and Jacobi into forms in which the equations of motion of a system represent the conditions for a stationary value of an integral of a single function. The extension by Routh and Helmholtz to the case in which "ignored co-ordinates" are taken into account, was a long step in the direction of the desirable unification which would be obtained if the notion of potential energy were removed by means of its interpretation as dependent upon the kinetic energy of concealed motions included in the dynamical system. The whole scheme of abstract Dynamics thus developed upon the basis of Lagrange's work has been of immense value in theoretical Physics, and particularly in statistical Mechanics... But the most striking use of Lagrange's conception of generalized co-ordinates was made by Clerk Maxwell, who in this order of ideas, and inspired on the physical side by... Faraday, conceived and developed his dynamical theory of the Electromagnetic field, and obtained his celebrated equations. The form of Maxwell's equations enabled him to perceive that oscillations could be propagated in the electromagnetic field with the velocity of light, and suggested to him the Electromagnetic theory of light. Heinrich Herz, under the direct inspiration of Maxwell's ideas, demonstrated the possibility of setting up electromagnetic waves differing from those of light only in respect of their enormously greater length. We thus see that Lagrange's work... was an essential link in a chain of investigation of which one result... gladdens the heart of the practical man, viz. wireless telegraphy."

"Full use of Lagrange's own made the unification of the varied principles of statistics and dynamics possible—in statistics by the use of the principle of virtual velocities, in dynamics by the use of . This led... to generalized coordinates and to the equation of motion in their "Lagrangian" form... Newton's geometrical approach was now fully discarded; Lagrange's book was a triumph of pure analysis."

"The treatment of the kinetics of a material system by the method of generalised coordinates was first introduced by Lagrange, and has since his time been greatly developed by the investigations of different mathematicians. Independently of the highly interesting, although purely abstract science of theoretical dynamics which has resulted from these investigations, they have proved of great and continually increasing value in the application of mechanics to thermal, electrical and chemical theories, and the whole range of ."