Historians Of Science

"The efforts of a multitude of writers have rather been directed towards producing alternatives for Euclid which shall be more suitable, that is to say, easier, for schoolboys. It is of course not surprising that, in these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute "a royal road to geometry"; the marvel is that a book which was not written for schoolboys but for grown men (as all internal evidence shows, and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of "practical" geometry) should have held its own as a schoolbook for so long."

"While then for a long time everyone was at a loss, Hippocrates of Chios was the first to observe that, if between two straight lines of which the greater is double of the less it were discovered how to find two mean proportionals in continued proportion, the cube would be doubled; and thus he turned the difficulty in the original problem into another difficulty no less than the former. Afterwards, they say, some Delians attempting, in accordance with an oracle, to double one of the altars fell into the same difficulty. And they sent and begged the geometers who were with Plato in the Academy to find for them the required solution. And while they set themselves energetically to work and sought to find two means between two given straight lines, Archytas of Tarentum is said to have discovered them by means of half-cylinders, and Eudoxus by means of the so-called curved lines. It is, however, characteristic of them all that they indeed gave demonstrations, but were unable to make the actual construction or to reach the point of practical application, except to a small extent Menaechmus and that with difficulty."

"There has been a rush of competitors anxious to be first in the field with a new text-book on the more "practical" lines which now find so much favour. The natural desire of each teacher who writes such a text-book is to give prominence to some special nostrum which he has found successful with pupils. One result is, too often, a loss of a due sense of proportion... It is, perhaps too early yet to prophesy what will be the ultimate outcome of the new order of things; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more."

"Any satisfactory reproduction of the Conics must fulfil certain essential conditions: (1) it should be Apollonius and nothing but Apollonius, and nothing should be altered either in the substance or in the order of his thought, (2) it should be complete, leaving out nothing of any significance or importance, (3) it should exhibit under different headings the successive divisions of the subject, so that the definite scheme followed by the author may be seen as a whole."

"By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded pari passu with the completion of the body of the Elements."

"There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius... "Eratosthenes to King Ptolemy greeting. "There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,Too small thy plan to bound a royal tomb. Let it be double; yet of its fair form Fail not, but haste to double every side.But he was clearly in error; for when the aides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometers also continued to investigate the question in what manner one might double a given solid while it remained in the same form."

"Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood. The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day."

"The most probable view is that adopted by Nesselmann, that the works which we know under the three titles formed part of one arithmetical work, which was, according to the author's own words, to consist of thirteen Books. The proportion of the lost parts to the whole is probably less than it might be supposed to be. The Porisms form the part the loss of which is most to be regretted, for from the references to them it is clear that they contained propositions in the Theory of Numbers most wonderful for the time."

"It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. ...Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"Hippocrates himself is an example of the concurrent study of the two departments. On the one hand, he was the first of the Greeks who is known to have compiled a book of Elements. This book, we may be sure, contained in particular the most important propositions about the circle included in Euclid, Book III. But a much more important proposition is attributed to Hippocrates; he is said to have been the first to prove that circles are to one another as the squares on their diameters (= Eucl. XII., 2) with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance is said to have worked at the problem while in prison."

"An edition is... still wanted which shall, while in some places adhering... to the original text, at the same time be so entirely remodelled by the aid of accepted modern notation as to be thoroughly readable by any competent mathematician, and this want it is the object of the present work to supply."

"The idea that time may be an active factor in causation has the mathematical significance that ' t ' (for the system in question) must appear explicitly in the formulation of the law. ...Such law may claim to express the fact of historic, irreversible duration."

"The question of the reversibility of natural processes provides the key to a great intellectual struggle which is now behind the complexities of philosophic and scientific thought. The issue can be formulated thus: Is there a real temporal process in nature? Is the passage of irreversible time a necessary element in any view of the structure of nature? Or, alternatively, is the subjective experience of time a mere illusion of the mind which cannot be given objective expression? These are not metaphysical questions that can still be neglected with impunity. For just as Einstein made his advance by analysing conceptions such as simultaneity, which had been thought to be adequately understood for the purposes of experimental science, so the next development of physical theory will probably be made by carrying on the analysis of time from the point at which Einstein left it."

"Although the anti-causal inclinations of an Eddington (or a Jeans) are most pertinent... they were not characteristic for their milieu. Far more typical for British natural-philosophical thought in this period is that interpretation of the conceptual situation in physics advanced by Lancelot Law Whyte in 1927 in Archimedes, or the Future of Physics, namely that "in order to straighten out its atomic problems physics will have to take a hint from biology." This notion, casually stated in the language of the work-a-day world, had come to Whyte two years before as a most powerful experience, a veritable revelation. "...That just as the Solution of Relativity demanded a fundamental reconsideration of the so-called limits of Science & their absorption into Science & reconstruction & a new understanding of them, So the solution of the Relativity-Quantum problem might involve the problem of life in such a way as to throw real light on the relation of Religion, Art & Science." ...while Whyte anticipates a revolution in science, indeterminism receives no explicit attention...Whyte is simply unconcerned with that aspect of Weyl's and Eddington's views. And this seems characteristic... [of] how very far the British were from focusing on causality."

"L.L. Whyte, in his marvellous account of the way in which the duality of the human nervous system became the conflicting dualism of reason against instinct, writes: "Intellectual man had no choice but to follow the path which facilitated the development of his faculty of thought, and thought could only clarify itself by separating out static concepts which, in becoming static, ceased to conform to their organic matrix or to the forms of nature.""

"L. L. Whyte, in a short but prophetic essay, Archimedes or the Future of Physics, pointed out that in each of the two great new physical theories of this century the fundamental role was played by a particular constant of nature: in Relativity by c, the velocity of light in vacuo, and in Quantum Theory by h, Plank's constant. He suggested that the next great advance in our understanding of nature would be associated with a new fundamental constant, and he prophesied that this would be concerned with the flow of time."

"Whyte showed how at the core of Newtonian physics lies the assumption that the elementary processes of nature are reversible, or would be if they could be isolated, and hence in the system of Natural Philosophy time would not appear as an explicit factor. ...In the cosmological theories of Einstein, de Sitter and Lamaître new ideas were introduced concerning the character of universal space, but no corresponding advance was made in connection with the idea of time, except in so far as the idea of expansion pointed to a finite rather than an infinite past."

"The author... has known for that for several centuries freethinkers have led mankind. Only recently... new to him though perhaps long understood by others, possibly Kant and certainly Nietzsche, there emerged into his mind a clarity that will remain... the conception of transcendental divinity is damaging to man."

"Belief in a transcendental divinity arose from a misinterpretation of intimations from the less conscious levels of the mind. ...God is in the unconscious, is the unconscious, perhaps."

"A naturalistic reinterpretation renders all that is authentic about the Christian doctrine greater not less, for it makes it a part of a new and stronger man, not of some fancied "superman," but simply man as he is but less distorted by a dissociating tradition."

"The "divine" in man: creative bliss, the experience of perfection, the surprising joys of love all human, not divine. ...It is time that God was put in his place, that is, in man, and no nonsense about it. But, to prevent misunderstanding, instead of speaking of the "divine" in man I will call it the human sense of perfection or unity. ...Need I add that we may retain the Sermon on the Mount, Saint Paul's poem to charity, and much else, though we discard the Christian God?"

"To rob man of his noblest faculty, the experience of and aspiration to perfection and unity in himself , we can now see to have been a truly hellish surgery."

"Thought is born of failure. When action satisfies there is no residue to hold the attention; to think is to confess a lack of adjustment which we must stop to consider. Only when the human organism fails to achieve an adequate response to its situation is there material for the process of thought, and the greater the failure the more searching they become. (p. 1)"

"We are sick today for lack of simple ideas which can help us be what we want to be."

"The basic challenge to mankind is not population, poverty, war, technology, pollution, religious or racial intolerance, or blind nationalism, but an underlying nihilism promoting violence and frustrating sane policies on these issues. ...the only hope lies in the emergence of a potentially worldwide consensus of heart, mind, and will, appealing to all sane men and women everywhere ...The time has come for the west to speak to the world in universal terms. ...the consensus, if it comes, is likely to surprise by its suddenness, timeliness, and universality."

"This essay touches bottom for twentieth century man. ...it is one many signals marking the end of "Antiman," with his hopeless relativism, and announcing "Unitary Man," ...able to be more harmonious because he has become aware of the ordering processes at all levels in nature, without and within. ...here at last subject and object are potentially fused in a single insight."

"I consider that Curie's Principle has two major consequences:- First: It shows that the class of processes which can be isolated for causal representation, not requiring the inference of external causes, is wider than the class of energetically closed systems. One-way processes in which the system loses energy can be isolable, in the sense that they can be given complete representation without taking their environment into account. Second: It suggests the possibility of a geometrical physics treating 3D spatial relations , i.e., angles or lengths, as primary. Just as statistical mechanics, the theory of crystal symmetry, and Group theory in quantum mechanics, are useful without assumptions about forces, so Curie's principle, with an appropriate model, can determine the path of a one-way process without such assumptions..."

"Atomism originally stood for iconoclasm, impiety, and atheism, because the Greek atomists conceived a universe under the reign of chance."

"Two extreme interpretations of atomism have persisted through centuries: the näive assumption of objectively real indivisible pieces of matter, and the sophisticated view that "atom" is merely a name given to abstractions which it is convenient to assume in simplifying complex phenomena. The second perhaps stems from Ockham, who wrote in 1330 of "the fiction of abstract nouns"; from John Troland, who in 1704 interpreted material particles as mental fictions; and from countless others down to Ernst Mach, who after starting as a physical atomist came to regard atoms as "mental artifices" or "economical ways of symbolizing experience." Both views have advantages..."

"The idea of the unconscious mental processes was, in many of its aspects, conceivable around 1700, topical around 1800, and became effective around 1900, thanks to the imaginative efforts of a large number of individuals of varied interests in many lands."

"We know nothing about the Christian transcendental God except His total indifference both to individual suffering and to the collapse of the pseudo Christian civilization which the Church supported. The pretension to a transcendental authority... was a hypnotic given to children and remaining with them as adults. During one period in history it served a purpose... its early rational opponents... were strangely naive, for they imagined that a half-developed faculty called "reason" should, and could, lead the species. But reason... is not a prime mover."

"Every scientific generation, measured by its most vocal members, exaggerates the historical importance of its own members. ...there is a perpetual temptation to study the latest and to neglect the past."

"There is no doubt of the need for an up-to-date, balanced, and comprehensive work on the history of atomism, drawing ideas, mathematics, and experiment together into a single story. When available, it should become required reading for all students of the exact sciences."

"No one is so brilliant that he can afford to neglect what history can teach him."

"Systematic errors of theory can seldom be discovered by direct attack; it is easier to uncover them by studying how and why physical theory took the path it did. That is why a clue to the future can sometimes be found in the past, and this is my reason for studying the history of atomism."

"Discontinuity of its linguistic and logical terms is for the conscious analytical intellect psychologically and logically prior to notions of continuity. ...This functional priority... may not have been reflected in the history of the development of reason in all human communities. ...But it is relevant for the West that the Pythagoreans, with their discrete integers and point patterns, came before Euclid, with his continuous metrical geometry, and that physical atomism as a speculative philosophy preceded by some two thousand years the conception of a continuous physical medium with properties of its own."

"There are good reasons to expect... a return to a concreteness of basic ideas, to simpler fundamentals easily understood, to principles that will bring exact science closer to the human perspective."

"Faced by the dire nihilism of our time, we need a greater honesty... The Western search for unifying truth did not come to an end with Christianity, any more than with the physical theories of forty years ago."

"The most productive novelties often spring, in thought as in biological evolution, from more primitive and simpler forms, rather than from differentiated ones which, through their elaboration, have become too specialized to be adaptable to new tasks."

"No scientist has yet provided an acceptable definition of "mind" or "mental" that reveals the character of "unconscious mental processes," and no physicist a lucid definition of "elementary particles" that shows how they can appear or disappear, and why there are so many."

"Did ever the history of the intellect so little conceal so much?"

"A clue to the future must lie in the past... every scientist, and everyone with intellectual curiosity, can learn something useful from a brief study of the history of atomism."

"Physics and psychology are going somewhere, but where they do not know. But... they are traveling from: Democritan permanent particles and the Cartesian mind necessarily aware. ...they are both traveling away from the same point of origin and in the same general direction: from the isolation of supposedly permanent "substances" towards the identification of changing relations potentially affecting everything; briefly, from substance to changing relations and structures."

"By openly recognizing the inescapable rhythm of harmony and tension which is the form of all human processes unitary man achieves a far-reaching emancipation. Much that was concealed can now stand in the open. The neutrality and objectivity of the quantity symbolism seemed to dissociated man a guarantee of the liberation of the mind from anthropomorphic and subjective illusions. But at deeper level it expressed merely the desire to escape inner conflict in a harmony of static form. This escape was wholly illusory; the superficial neutrality of science left it open to abuse, and the spirit of man has been punished by its attempt to escape struggle in an intellectual harmony. Unitary man renounces such separation and partakes in the development of the whole. Man finds himself in the universal process, by finding the universal process within himself. Tension continues, but henceforward his struggle is with, not against, the process of nature. (p. 276)"

"We are indeed a blind race, and the next generation, blind to its own blindness, will be amazed at ours."

"The material particle or the conscious mind—has been discovered not to be sufficiently unchanging to be treated as a thing in isolation... but more often to be the opposite: a changing system in a changing environment."

"It is widely believed that only those who can master the latest quantum mathematics can understand anything of what is happening. That is not so, provided one takes the long view, for no one can see far ahead. Against a historical background, the layman can understand what is involved, for example, in the fascinating challenge of continuity and discontinuity expressed in the antithesis of field and particle."

"Unitary man escapes these confusions through his recognition that one factor is of supreme importance: the maturity proper to man can only come through the experience of adult unity. This experience may come in many ways, but it means that the individual has, for the moment, outgrown the sense of any division, either within himself of from others, through a mature relations to at least one other person. Tension is inherent in the process, but tension does not become frustrating conflict if the overriding unity is realized. (p. 255)"

"Security is an impostor; little can be achieved while each seeks his own freedom from want, from war, or from fear. The general nature of all fear is the awareness that development is threatened. Fear and its consequences can be eliminated only by action leading to continuous development. Action can bring the assurance of a development which is more welcome than either spiritual or material security. Only through the pursuit of a general development can the species acquire the unity of purpose which may, as one of its secondary consequences, eliminate unemployment and war. In the individual life the same is true: only in unitary development can fear be overcome. Every individual experiences countless shocks from his first breath to his last, and these challenges are necessary to his development. But a unitary tradition can assist the members of each maturing generation to turn these challenges to advantage and to retain their basic integrity. (p. 256)"