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Just when this discovery took place and whether it was made by Pythagoras himself, by the early Pythagoreans, or by later members of the school are moot points in the history of mathematics. The rnethod of application would suggest as a form of proof the geometrical equivalent of the process of finding the greatest common divisor, but there is another aspect of Pythagorean thought which points to a different sort of reasoning. A prevailing belief in the unity and harmony of nature and knowledge had led the Pythagoreans not only to explain different aspects of nature by various mathematical abstractions, as already suggested, but also to attempt to identify the realms of number and magnitude.

Jahrhunderts" : "Zur Entdeckungsgeschichte des Irrationalen. Heath History of Greek Mathe- matics, I, would place it "at a date appreciably earlier than that of Democritus. If geometrical abstractions were the elements of actual things, number was the ultimate element of these abstractions and thus of physical bodies and of all nature. This doctrine could not be applied to the diagonal of a square, however, for no matter how small a unit was chosen as a measure of the sides, the diagonal could not be a "progression of multitude" beginning with this unit.

The proof of this fact, as given by Aristotle and which possibly is that of the Pythagoreans , 31 is based on the distinction between the odd and the even, which the Pythagoreans themselves had emphasized. The incommensurability of lines remained ever a stumbling block for Greek geometry. That it made a strong impression on Greek thought is indicated by the story, repeated by Proclus, that the Pythagorean who disclosed the fact of incommensurability suffered death by shipwreck as a result.

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  • It is demonstrated also, and more reliably, by the prominence given to the doctrine of irrationals by Plato and Euclid. It never occurred to the Greeks to invent an irra- tional number 32 to circumvent the difficulty, although they did develop as a part of geometry found, for example, in the tenth book of Euclid's Elements the theory of irrational magnitudes.

    Failing to gen- eralize their number system along the lines suggested later by the development of mathematical analysis, the only escape for Greek mathematicians in the end was to abandon the Pythagorean attempt to identify the realm of number with that of geometry or of con- tinuous magnitude. The effort to unite the two fields was not given up, however, before intuition had sought another way out of the difficulty. Conceptions in Antiquity 21 finite line segment so small that the diagonal and the side may both be expressed in terms of it, may there not be a monad or unit of such a nature that an indefinite number of them will be required for the diagonal and for the side of the square?

    We do not know definitely whether or not the Pythagoreans them- selves invoked the infinitely small. We do know, however, that the concept of the infinitesimal had entered into mathematical thought, through a doctrine elaborated in the fifth century b. After the failure of the early Ionian attempts to find a fundamental element out of which to construct all things, there arose at Abdera the materialistic doctrine of physical atomism, according to which there is no one physis, not even a small group of substances of which everything is composed. The Abderitic school held that everything, even mind and soul, is made up of atoms moving about in the void, these atoms being hard indivisible particles, qualitatively alike but of countless shapes and sizes, all too small to be perceived by sense impressions.

    There is nothing either logically or physically inconsistent in this doctrine, which is a crude anticipation of our own chemical thought; but the greatest of the Greek atomists, Democritus, did not stop here: he was also a mathematician and carried the idea over into geometry. As we now know from the Method of Archimedes, which was discovered as a palimpsest in , Democritus was the first Greek mathematician to determine the volumes of the pyramid and the cone. How he derived these results we do not know.

    The formula for the volume of a square pyramid was probably known to the Egyp- tians, 33 and Democritus in his travels may have learned of it and generalized the result to include all polygonal pyramids. The result for the cone would then be a natural inference from the result of increasing indefinitely the number of sides in a regular polygon forming the base of a pyramid.

    This explanation would correspond to others involving similar infinitesimal conceptions, which we know Democritus entertained and which later influenced Plato. That he was interested in other mathematical problems bearing on the infinitesimal we know from the titles of works now lost, but which are referred to by Diogenes Laertius. One of these seems to have been on horn angles the angles formed by curves which have a common tangent at a point , and another on irrational incommen- surable lines and solids. It has been maintained 41 that Democritus was too good a mathematician to have had anything to do with such a theory as that of indivisible lines; but it is difficult to imagine how a mathematical atom is to be conceived if not as an indivisible.

    At all events, whatever his conception of the nature of infinitesimals may have been, the influence of Democritus has persisted. The idea of the fixed infinitesimal magnitude has clung tenaciously to mathe- matics, frequently to be invoked by intuition when logic apparently failed to offer a solution, and finally to be displaced in the last century by the rigorous concepts of the derivative and the integral.

    Simplicii commentarii in octo Aristotelis physicae auscultationis libros, p. Heath, History of Greek Mathematics, 1, Conceptions in Antiquity 23 That the infinitesimal was not eagerly welcomed into Greek geom- etry after the time of the Pythagoreans and Democritus may have been due largely to a school of philosophy that had risen at Elea, in Magna Graecia.

    The Eleatic school, although not essentially mathe- matical, was apparently familiar with, and probably influenced by, Pythagorean mathematical philosophy; but it became an opponent of the chief tenet of this thought. Instead of proclaiming the constitution of objects as an aggregate of units, it pointed out the apparent con- tradictions inherent in such a doctrine, maintaining against the atomic view the essential oneness and changelessness of the world. This stultifying monism was upheld by Parmenides, the leader of the school, with perhaps a touch of skepticism derived from his iconoclastic predecessor, the poet-philosopher Xenophanes.

    In an indirect defense of this doctrine, the Eleatics proceeded to demolish, with skillful dialectic, the basis of opposing schools of thought. The most damaging arguments were offered by Zeno, the student of Parmenides. After presenting the obvious objection to the Pythagorean indefinitely small monad — that if it has any length, an infinite number will constitute a line of infinite length; and if it has no length, then an infinite number will likewise have no length — he added the following general dictum against infinitesimals: "That which, being added to another does not make it greater, and being taken away from another does not make it less, is nothing.

    That they were intended merely as dialectical puzzles may perhaps be indicated by the passage in Plutarch's life of Pericles: Also the two-edged tongue of mighty Zeno, who, Say whp. It is not improbable that Zeno, although he was neither a mathematician nor a physicist, propounded the paradoxes to point out the weakness in the Pythagorean definition of a point as unity having position, and in the resulting Pythagorean multiplicity which did not distinguish clearly betw r een the geometrical and the physical.

    The arguments would hold equally well, of course, against mathematical atomism. The first two paradoxes the dichotomy and the Achilles 46 are directed against the opposite conception, that of the infinite divisibility of space and time, and are based upon the impos- sibility of conceiving intuitively the limit of the sum of an infinite series. Zenon d'Elee et Georg Cantor. See further, Cajori, "Purpose of Zeno's Arguments. Therefore the arrow does not move. See The Works of Aristotle, Vol. II, Physica VI. The argument in the stade, as given by Aristotle, is obscure because of brevity , but is equivalent to the following: Space and time being assumed to be made up of points and instants, let there be given three parallel rows of points, A, B, and C.

    Let C move to the right and A to the left at the rate of one point per instant, both relative to B; but then each point of A will move past two points of C in an instant, so that we can subdivide this, the smallest interval of time; and this process can be continued ad infinitum, so that time can not be made up of instants.

    Therefore, since the regression is infinite, motion is impossible, inasmuch as the body would have to traverse an infinite number of divisions in a finite time. The argument in the Achilles is similar. Assume a tortoise to have started a given distance ahead of Achilles in a race. Then by the time Achilles has reached the starting point of the tortoise, the latter will have covered a certain distance; in the time required by Achilles to cover this additional distance, the tortoise will have gone a little farther; and so ad infinitum.

    Since this series of distances is infinite, Achilles can never overtake the tortoise, for the same reason as that adduced in the dichotomy. Conceptions in Antiquity 25 the concepts of the differential calculus. There is no logical difficulty in the dichotomy or the Achilles, the uneasiness being due merely to failure of the imagination to realize, in terms of sense impressions, the nature of infinite convergent series which are fundamental in the precise explanation of, but not involved in our obscure notion of, continuity.

    The paradox of the flying arrow involves directly the conception of the derivative and is answered immediately in terms of this. The argument in this paradox, as also that in the stade, is met by the assumption that the distance and time intervals contain an infinite number of subdivisions. Mathematical analysis has shown that the conception of an infinite class is not self-contradictory, and that the difficulties here, as also in the case of the first two paradoxes, are those of conceiving intuitively the nature of the continuum and of infinite aggregates.

    It is clear that the answers to Zeno's paradoxes involve the notions of continuity, limits, and infinite aggregates — abstractions all related to that of number to which the Greeks had not risen and to which they were in fact destined never to rise, although we shall see Plato and Archimedes occasionally straining toward such views.

    That they did not do so may have been the result of their failure, indicated above in the case of the Pythagoreans, clearly to separate the worlds of sense and reason, of intuition and logic. Thus mathematics, instead of being the science of possible relations, was to them the study of situations thought to subsist in nature. The inability of Greek mathematicians to answer in a clear manner the paradoxes of Zeno made it necessary for them to forego the attempt to give to the phenomena of motion and variability a quanti- tative explanation. Only the static aspects of optics, mechanics, and astronomy found a place in Greek mathematics, and it remained for the Scholastics and early modern scientists to establish a quantitative dynamics.

    Zeno's argu- ments and the difficulty of incommensurability had also a more general effect on mathematics: in order to retain logical precision, it was necessary to give up the abortive Pythagorean effort to identify the domains of number and geometry, and to abandon also the pre- mature Democritean attempt to explain the continuous in terms of the discrete.

    It is, however, impossible satisfactorily to interpret the world of nature and the realm of geometry spheres which for the Greeks were not essentially distinct without superimposing upon them a framework of discrete multiplicity; without ordering, by means of number, the heterogeneity of impressions received by the senses; and without at every point comparing nonidentical elements. Thought itself is possible only in terms of a plurality of elements.

    As a conse- quence, the concept of discreteness cannot be excluded completely from the study of geometry. The continuous is to be interpreted in terms of successive subdivision, that is to say, in terms of the discrete, although from the Greek point of view the former could not be logically identified with the latter. The clever manner in which the method of successive subdivision was applied in Greek geometry, without the loss of logical rigor, will be seen later in the method of exhaustion — a procedure which was developed, not in Italy, but in and around the Greek mainland, whither many Pythagoreans wandered, on the breaking up of the school, toward the beginning of the fifth century b.

    Zeno likewise lived for a time in Athens, the rising center of Greek culture and mathematics. Here Pericles, the political leader of that city in its Golden Age, is said to have been one of his listeners. He may not have contrib- uted much original work in mathematics, but he advanced the sub- ject, nevertheless, through his great enthusiasm for it. He is said to have paid particular attention to the principles of geometry — to the " Plutarch, Lives, p.

    Conceptions in Antiquity 27 hypotheses, definitions, methods. In his dialogues he considered the Pythagorean problem of the nature of number and its relationship to geometry, 53 the difficulty of incom- mensurability, 64 the paradoxes of Zeno, 66 and the Democritean ques- tion of indivisibles and the nature of the continuum. If Plato made an attempt to arithmetize mathematics in this sense, he was the last of the ancients to do so, and the problem remained for modern mathe- matical analysis to solve. The thought of Aristotle we shall find diametrically opposed to any such conceptions.

    It has been suggested that Plato's thought was so opposed by the polemic of Aristotle that it was not even mentioned by Euclid. Certain it is that in Euclid there is no indication of such a view of the relation of arithmetic to geometry; but the evidence is insufficient to warrant the assertion 68 that, in this connection, it was the authority of Aristotle which held back for two thousand years a transformation which the Academy sought to complete.

    A sound basis for either mechanics or arithmetic must be built upon the limit concept — a notion which is not found in the extant works of Plato nor in those of his successors. The Platonists, on the contrary, attempted to develop the misleading idea of indi- visibles or fixed infinitesimals, a notion which the modern arith- metization of analysis has had cause to reject.

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    H To Plato are ascribed, among other things, the formulation of the analytic method and the restriction of Euclidean geometry to constructions possible with ruler and compass only. Heath History of Greek Mathematics, I, would place it earlier. M See, in particular, Theaetetus ; Laws dc. He was strongly influenced by both of these schools, but apparently felt that their views were too much the result of sense experience. Plato's criterion of reality was not consistency in experi- ence but reasonableness in thought. For him, as for the Pythagoreans, there was no necessary distinction between mathematics and science; both were the result of deduction from clearly perceived first principles.

    The Pythagorean monad and the Democritean mathematical atom- ism,, which gave every line a thickness, perhaps appealed too strongly to materialistic sense experience to suit Plato, so that he had recourse to the highly abstract apeiron or unbounded indeterminate.

    This was the eternally moving infinite of the Ionian philosopher, Anaximander, 60 who had suggested it in opposition to Thales' less subtle assertion that the concrete material element, water, was the basis of all things. According to Plato, the continuum, could better be regarded as gen- erated by the flowing of the apeiron than thought of as consisting of an aggregation however large of indivisibles.

    This view represents a fusion of the continuous and the discrete not unlike the modern intuitionism of Brouwer. Mathematics has found it necessary to discard both views in making the infinitesimal subor- dinate to the derivative in the logical foundation of the calculus. However, the notion of the infinitesimal proved very suggestive in the early establishment of the calculus, and, as Newton remarked more than two thousand years later, the application to it of our intuitions of motion removes from the doctrine much of the harshness felt in the mathematical atomism of Democritus and later of Cavalieri.

    This, however, necessarily led to a loss both of precise logical defi- 69 See T. Heath, History of Greek Matliematics, I, Conceptions in Antiquity 29 nition and of clear sensory interpretation, neither of which Plato supplied. The conjunction of mathematics and philosophy, as found, for ex- ample, in Plato, Descartes, and Leibniz, has been perhaps as valuable in suggesting new advances as has the blending of mathematical and scientific thought illustrated by Archimedes, Galileo, and Newton.

    The disregard in Platonic thought of any basis in the evidences of sense experience has not unjustly been regarded, from the scientific point of view, as an "unmitigated misfortune. One may therefore say, in a very general sense, that "we know from Plato's own writings that he was thinking out the solution of problems that lead directly to the discovery of the calculus. That the doctrines of the continuous and the infinitesimal did not develop along the abstract lines vaguely indicated by Plato was probably the result of the fact that Greek mathematics included no general concept of number, 67 and, consequently, no notion of a con- tinuous algebraic variable upon which such theories could logically have been based.

    It is difficult, however, to perceive on what grounds such a thesis is to be defended. M Ibid. Aristotelian thought, while not destroying the rigorously deductive character of Greek geometry, may have preserved in Greek mathematics that strong reasonable and matter-of-fact cast which one finds in Euclid and which operated against the early development of the calculus, as well as against the Platonic tendency toward speculative metaphysics. Although Plato did not solve the difficulties which the Pythagoreans and Democritus encountered, he urged their study upon his asso- ciates, inveighing against the ignorance concerning such problems which prevailed among the Greeks.

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    • In this con- nection the demonstrations which Eudoxus gave of the propositions previously stated without proof by Democritus on the volumes of pyramids and cones led to his famous general method of exhaustion and to his definition of proportion. The achievements of Eudoxus are those of a mathematician who was at the same time a scientist, with none of the occult or mystic in him.

      In method and spirit the later work of Euclid will be found to owe much more to Eudoxus than to Plato. We have seen that the Pythagorean theory of proportion could not be applied to all lines, many of which are incommensurable, and that the Democritean view of infinitesimals was logically untenable. Eudoxus proposed means by which these difficulties could be avoided.

      The paths he indicated, in his theory of proportion and in the method of exhaustion, were not the equivalents of our modern conceptions of number and limit, but rather detours which obviated the necessity of using the latter. They were significant, however, in that they made it possible for the Greek mind confidently to pursue its attack upon prob- lems which were to eventuate much later in the calculus.

      The Pythagorean conception of proportion had been the result of the identification of geometrical magnitudes and integral numbers. M See Laws dc. Conceptions in Antiquity 31 Two lines, for example, were to each other as the ratio of the integral numbers of units in each. With the discovery of the incommensura- bility of some lines with others, however, this definition could no longer be universally applied. Eudoxus substituted for it another which was more general, in that it did not require two of the terms in the propor- tion to be integral numbers, but allowed all four to be geometrical entities, and required no extension of the Pythagorean idea of number.

      Euclid 70 states Eudoxus' definition as follows: "Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. It is interesting to see that after the development of mathematical analysis, the con- cept of proportion resembles the arithmetical form of the Pythag- oreans rather than the geometrical one of Eudoxus.

      Even when the ratio is not expressible as the quotient of two integers, we now sub- stitute for it a single number and symbol such as v or e. Although Eudoxus did not, as we do, regard the ratio of two incommensurable quantities as a number, 72 nevertheless his definition of proportion ex- presses the ordinal idea involved in the present conception of real number.

      The assertion that it is "word for word the same as the general definition of number given by Weierstrass" 73 will be found, however, to be incorrect, both literally and in its implications. The formulation of Eudoxus was, on the contrary, a means of avoiding the need of such an arithmetic definition as that of Weierstrass. The method of exhaustion of Eudoxus shows the same abandon- ment of numerical conceptions which we have seen in his theory of proportion.

      Length, area, and volume are now carefully defined numerical entities in mathematics. After the time of the Pythagoreans, classic Greek mathematics did not attempt to identify number with 70 Book V, Definition 5. Heath, II, As a result no rigorous general definitions of length, area, and volume could then be given, the meaning of these quantities being tacitly understood as known from intuition.

      The question, "What is the area of a circle? But the query, "What is the ratio of the areas of two circles? Obviously the old Pythagorean method of the application of areas cannot be employed in the case of circles, so Eudoxus had recourse to an idea which had been advanced sometime before by Antiphon the Sophist and again a generation later by Bryson. These men had inscribed within a circle a regular polygon, and by successively doubling the number of sides they seem to have hoped to reach a polygon which would coincide with the circle and so "exhaust" its area. It should, however, be borne in mind that we do not know just what Antiphon and later Bryson said.

      The method of Antiphon has been described 75 as equivalent at one and the same time to the method of Eudoxus as given in Euclid XII, 2 , and to our conception of the circle as the limit of such an inscribed polygon, but merely expressed in different terminology. This cannot be strictly correct. If Antiphon had considered the process of bisection as carried out to an infinite number of steps, he would not have been thinking in the terms of Eudoxus and Euclid, as we shall see. If, on the other hand, he did not regard the process as continued indefinitely but only as carried out to any desired degree of approximation, he could not have had our idea of a limit.

      Furthermore, our conception of the limit is numerical, whereas the notions of Antiphon and Eudoxus are purely geometrical. The suggestive idea of Antiphon, however, was adopted by Bryson, who is reputed not only to have inscribed a polygon within the circle 74 Cf. Vogt, Der Grenzbegrijf in der Elementar-mathematik, p. Conceptions in Antiquity 33 but also to have circumscribed one about it as well, saying that the circle would ultimately, as the result of continued bisection, be the mean of the inscribed and circumscribed polygons.

      Again we do not know exactly what he said, and cannot tell clearly what he meant. However, the idea which he suggested was developed by Eudoxus into a rigorous tyDe of argument for dealing with problems involving two dissimilar, heterogeneous, or incom- mensurable quantities, in which intuition fails to represent clearly the transition from one to the other which is necessary to make a com- parison possible.

      The procedure which Eudoxus proposed has since become known as the method of exhaustion. The principle upon which this method is based is commonly called the lemma, or postulate, of Archimedes, although the great Syracusan mathematician himself ascribed it 78 to Eudoxus and it is not improbable that it had been formulated still earlier by Hippocrates of Chios.

      From the fact that, on continuing the process indicated in the axiom of Archimedes, the magnitude remaining can be made as small as we please, the procedure introduced by Eudoxus came much later to be 74 Ibid. See T. HI, p. It is to be remarked, however, that the word exhaustion was not applied in this connection until the seventeenth century, 81 when mathematicians somewhat ambiguously and uncritically employed the term indifferently to designate both the ancient Greek procedure and their own newer methods which led immediately to the calculus and which truly "exhausted" the mag- nitudes.

      The Greek mathematicians, however, never considered the process as being literally carried out to an infinite number of steps, as we do in passing to the limit — a concept which allows us to interpret the area or volume as truly exhausted, or at least as defined as the limit of the infinite numerical sequence obtained in this manner. There was always, in the Greek mind, a quantity left over although this could be made as small as desired , so that the process never passed beyond clear intuitional comprehension. A simple illustration will per- haps serve to make the nature of the method clear.

      The proposition, in Euclid XII, 2, that the areas of circles are to each other as the squares on their diameters will suffice for this purpose. The substance of this proof is as follows: Let the areas of the circles be A and a, and let their diameters be D and d respectively. If a' is smaller than a, then in the circle of area a we can inscribe a polygon of area p such that p is greater than a' and smaller than a.

      This follows from the principle of exhaustion Euclid X, 1 — that if from a magnitude such as the difference in area between a' and a we take more than its half, and from the difference more than its half, and so on, the difference can be made less than any assignable magnitude. In a similar manner it can be shown that the supposition a! Vincent, Opus geometricum, pp. Euclid, Elements, Heath trans. Ill, pp. Conceptions in Antiquity 35 to the type of argument now employed in proving the existence of a limit in the differential and the integral calculus, does not represent the point of view involved in the passage to the limit.

      The Greek method of exhaustion, dealing as it did with continuous magnitude, was wholly geometrical, for there was at the time no knowledge of an arithmetical continuum. This being the case, it was of necessity based on notions of the continuity of space — intuitions which denied any ultimate indivisible portion of space, or any limit to the divisibility in thought of any line segment.

      The inscribed polygon could be made to approach the circle as nearly as desired, but it could never become the circle, for this would imply an end in the process of subdividing the sides. However, under the method of exhaustion it was not neces- sary that the two should ever coincide. By an argument based upon the reductio ad absurdum, it could be shown that a ratio greater or less than that of equality was inconsistent with the principle that the difference could be made as small as desired.

      The argument of Eudoxus appealed at every stage to intuitions of space, and the process of subdivision made no use of such unclear con- ceptions as that of a polygon with an infinite number of sides — that is, of a polygon which should ultimately coincide with the circle. No new concepts were involved, and the gap between the curvilinear and the rectilinear still remained unspanned by intuition.

      Eudoxus, however, had most ingeniously contrived to demonstrate — without resort to the logically self -contradictory infinitesimal previously invoked by vague imagination — the truth of certain geometrical propositions requiring a comparison of the curvilinear with the rectilinear and of the irrational with the rational. There is no logical difficulty to be found in the argument used in the method of exhaustion, but the cumbersomeness of its application led later mathematicians to seek a more direct approach to problems in which the application of some such procedure would have been indicated.

      The method of exhaustion has, most misleadingly, been characterized as "a well-established algorithm of the differential cal- culus. Nevertheless, it is not incorrect to say that the procedure involved actually directed 83 Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p.

      In fact the ancients never made the first step in this direction: they did not formulate the principle of the method as a general proposition, reference to which might serve in lieu of the argument by the ubiquitous double reductio ad absurdum. To trace this development is the purpose of this essay; but at this point it may not be amiss to anticipate the final formulation to the extent of comparing the nature of the basic concept of the calculus — that of the limit of an infinite sequence — with the view indicated in the method of exhaustion.

      The limit of the infinite sequence P h P 2 ,. The spatial intuition of the method of exhaustion, with its application of areas, unlimited subdivision, and argumentation by a reductio ad absurdum, here gives way to definition in terms of formal logic and number, i. The method of exhaustion corre- sponds to an intuitional concept, described in terms of mental pic- tures of the world of sensory perception. The notion of a limit, on the other hand, may be regarded as a verbal concept, the explication of which is given in terms of words and symbols — such as number, infinite sequence, less than, greater than — with regard not to any mental visualization, but only to their definition in terms of the primary undefined elements.

      The limit concept is thus by no means to be considered ineffable; nor does it imply that there is other than 84 Cf. Brunschvicg, Les tiapes de la philosophic mathtonatique, pp. Heath, p. Conceptions in Antiquity 37 empirical experience. It simply makes no appeal to intuition or sensory perception.

      It resembles the method of exhaustion in that it allows our vague instinctive feeling for continuity to shift for itself in any effort that may be made to picture how the gap between the curvilinear and the rectilinear, or between the rational and the irra- tional, is bridged, for such an attempt is quite irrelevant to the logical reasoning involved. The limit C is not for this reason to be regarded as a sophistic or inconceivable quantity which somehow nevertheless enters into real relations with other similar quantities, nor is it to be visualized as the last term of the infinite sequence.

      It is to be considered merely as a number possessing the property stated in the definition. It is to be borne in mind that although adumbrations of the limit idea appear in the history of mathematics in ancient times, nevertheless the rigorous formulation of this concept does not appear in work before the nineteenth century — and certainly not in the Greek method of exhaustion.

      He wrote a work now lost On the Pythagoreans, dis- cussed at some length the paradoxes of Zeno, mentioned Democritus frequently in mathematics and science although always to refute him , was intimately familiar with Plato's thought, and was ac- quainted with the work of Eudoxus. In spite of his competence in mathematics and of his frequent use of geometry in his constructions, 89 Aristotle's approach to the problems involved was essentially scientific, in the inductively descriptive sense.

      Furthermore, for Plato's mathe- matical intellectualism he substituted a grammatical intuitionism. As a consequence, he did not think of a geometrical line, as had Plato, as an idea which is prior to, and independent of, experience of the concrete. Neither did he regard it, as does modern mathematics, as an abstraction which is suggested, perhaps, though not in any way defined, by physical objects. He viewed it rather as a character- istic of natural objects which has merely been separated from its irrelevant context in the world of nature.

      Since a straight line is what it is, it is necessary that the angles of a triangle should equal two right angles. In fact, with the exception of Plato's successors in the Academy and, perhaps, of Archimedes, they were those accepted by the body of Greek mathematicians after Eudoxus.

      Only in the case of the indivisible, however, do Aristotle's views coincide with the present notions in mathematics. Modern science has opposed, modern mathematics upheld, Aristotle in his vigorous denial of the indivisible, physical and mathematical, of the atomic school. Recent physical and chemical theories of the atom have furnished a description of natural phenomena which offers a higher degree of consistency within itself and with sensory impressions than had the Peripatetic doctrine of continuous substantiality.

      Science has conse- quently, under Carneades' doctrine of truth, accepted the atom as a physical reality. Modern mathematics, on the other hand, agrees with Aristotle in his opposition to minimal indivisible line segments; not, however, because of any argument from experience, but because it has 91 T.

      Conceptions in Antiquity 39 been unable to give a satisfactory definition and logical elaboration of the concept. However, the mathematical indivisible, in spite of the opposition of Aristotle's authority, was destined to play an important part in the development of the calculus, which in the end definitely excluded it. That the concept enjoyed an extensive popularity even in Aristotle's day, Greek logic notwithstanding, is seen by the fact that a Peripatetic treatise formerly ascribed to Aristotle but now thought to have been written by Theophrastus, or Strato of Lampsacus, or perhaps by someone else , the De lineis insecabilibus, u was directed against it.

      This presents many arguments against the assumption of indivisible lines and concludes that "it conflicts with practically every- thing in mathematics. It has been asserted 96 that neither Aristotle nor Plato's successors understood the infinitesimal concept of their master, and that only Archimedes rose to a correct appreciation of it. It is to be remarked, however, that such an assump- tion is wholly gratuitous. Plato in his extant works offered no clear definition of the infinitesimal. Archimedes, moreover, made no men- tion of any indebtedness to Plato in this matter, and, as will be-seen, explicitly disclaimed any intention of regarding infinitesimal methods as constituting valid mathematical demonstrations.

      As in the case of the infinitesimal, so also with respect to the infinite the views of Aristotle constitute excellent illustrations of his abiding confidence in the ultimate interpretability of phenomena in terms of distinctly clear concepts derived from sensory experience.

      From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics

      VI, Opuscula. K De lineis insecabilibtts a. For one of the most recent philosophical discussions of this subject, see Edel, Aristotle's Theory of the Infinite. However, no one of Aristotle's predecessors had made quite clear his position with respect to the infinite. Anaxag- oras, at least, seems to have realized that it is only the imagination which objects to the infinite and to an infinite subdivision. In consequence he denied altogether the existence of the actual infinite and restricted the use of the term to indicate a potentiality only.

      His refusal to recognize the actual infinite was in keeping with his fundamental tenet that the unknowable exists only as a potentiality: that anything beyond the power of comprehension is beyond the realm of reality. Such a methodological definition of existence has led investigators in inductive science to continue to the present time the Aristotelian attitude of negation toward the infi- nite; but such a view, if adopted in mathematics, would exclude the concepts of the derivative and the integral as extrapolations beyond the thinkable, and would, in fact, reduce mathematical thought to the intuitively reasonable.

      That the Aristotelian doctrine of the infinite was abandoned in the mathematics of the nineteenth century was largely the result of a shift of emphasis from the infinite of geometry to that of arithmetic; for in the latter field assumptions appear to be less frequently dictated by experience.

      For Aristotle such a change of view would have been impossible, inasmuch as his conception of number was that of the Pythagoreans: a collection of units. Conceptions in Antiquity 41 smallest number in the strict sense of the word 'number' is two," said Aristotle. When, then, Aristotle distinguished two kinds of potential infinite — one in the direction of successive addition, or the infinitely large, and the other in the direction of successive subdivision, or the infinitely small — we find the behavior of number to be quite different from that of magnitude: Every assigned magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude Number on the other hand is a plurality of "ones" and a certain quantity of them.

      Hence number must stop at the indivisible. But in the direction of largeness it is always possible to think of a larger number. Hence this infinite is potential, With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. They pos- tulate only that the finite straight line may be produced as far as they wish.

      Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitude. This method assumes in the proof only that the bisection can be continued as far as one may wish, not car- ried out to infinity. How far it lies from the point of view of modern analysis is indicated by the fact that the latter has been called "the symphony of the infinite.

      After having considered place and time in the fourth book of the Physica, and change in the fifth, Aristotle turned in the sixth book to the continuous. His account is based upon a Physica IV. The nature of continuous magnitude has been found to lie deeper than Aristotle believed, and it has been explained on the basis of concepts which require a broader definition of number than that held during the Greek period. The Aristotelian dictums on the subject were not unfruitful, however, for they led to speculations during the medieval period which in turn aided in the rise of the calculus and the modern doctrine of the continuum.

      In connection with the study of continuous magnitude, Aristotle attempted also to clarify the nature of motion, criticizing the atomists for their neglect of the whence and the how of movement. In the light of modern scientific method, this lack of mathematical expression gives to his treatment of motion and varia- bility the appearance of a dialectical exercise, rather than of a serious effort to establish a sound basis for the science of dynamics.

      This is evidenced by his definition of motion as "the fulfillment of what exists potentially, in so far as it exists potentially," and by the further remark, "We can define motion as the fulfillment of the movable qua movable. Mach, The Science of Mechanics, p. Conceptions in Antiquity 43 of impetus developed by the Scholastics, in Hobbes' explanation of velocity and acceleration in terms of a conatus, and even in meta- physical interpretations of the infinitesimal of the calculus as an intensive quantity — that is, as a "becoming" rather than a "being.

      However, his influence was in another sense quite adverse to the development of this concept in that it centered attention upon the qualitative description of the change itself, rather than upon a quantitative interpretation of the vague instinctive feeling of a continuous state of change invoked by Zeno. The calculus has shown that the concept of continuous change is no more free from that of the discrete than is the numerical continuum, and that it is logically to be based upon the latter, as is also the idea of geometrical magnitude.

      As long as Aristotle and the Greeks con- sidered motion continuous and number discontinuous, a rigorous mathematical analysis and a satisfactory science of dynamics were difficult of achievement. The treatment of the infinite and of continuous magnitude found in the Physica of Aristotle has been regarded as presenting the ap- pearance of a veritable introduction to a treatise on the differential calculus. He asserted that "Nothing can be in motion in a present.

      Nor can anything be at rest in a present. Aristotle's denial of instantaneous velocity, as realized in the world described by science, is, to be sure, in conformity with the recognized limitations of sensory perception. Only average velocities, — , are At recognizable in this sense. In the world of thought, on the other hand, it has been found possible — through the calculus and the limit con- cept — to give a rigorous quantitative definition of instantaneous.

      The failure of Aristotle to distinguish sharply between the worlds of experience and of mathematical thought resulted in his lack of clear recognition of a similar confusion in the paradoxes of Zeno. Aristotle refuted the arguments in the stade and the arrow by an appeal to sensory perception and the denial of an instantaneous velocity. Modern mathematics, on the other hand, has answered them in terms of thought alone, based on the concept of the derivative.

      Lacroix and the Calculus (Science Networks. Historical Studies)

      In the same manner Aristotle resolved the paradoxes in the dichotomy and the Achilles by the curt assertion, suggested by experience, that although one cannot traverse an infinite space in finite time, it is possible to cover an infinitely divided space in finite time because of the infinite divisibility of the latter.

      If one demands that Zeno's paradoxes be answered in terms of our vague instinctive feeling for continuity — as essentially different from the discrete — no answers more satisfying than those of Aristotle to whom we owe also the statement of the paradoxes, since we do not have Zeno's words have been given. The unambiguous demonstration that the difficulties implied by the paradoxes are simply those of visualization and not those of logic was to require more precise and adequate definitions than any which Aristotle could furnish for such subtle notions as those of continuity, the infinite, and instantaneous velocity.

      Such definitions were to be given in the nineteenth century in terms of the concepts of the calculus; and modern analysis has, upon the basis of these, clearly dissented from the Aristotelian pronouncements in this field. Conceptions in Antiquity 45 as is all too frequently and uncritically maintained — as gross mis- conceptions which for two thousand years retarded the advancement of science and mathematics. They were, rather, matured judgments on the subject which furnished a satisfactory working basis for later investigations which were to result in the science of dynamics and in the mathematical continuum.

      Nevertheless, there is apparent in the work of Aristotle the cardinal weakness of Greek logic and geometry: a naive realism which regarded thought as a true copy of the external world. It has been said that the fifth book of Euclid's Elements and the logic of Aristotle are the two most unobjectionable and unassailable treatises ever written.

      The two men were roughly contemporaries: Aristotle lived from to b. There is in Euclid none of the metamathematics which played such a prominent part in Plato's thought, nor do metaphysical speculations on mathematical atomism enter. Mathematics was regarded by Euclid neither as a necessary form of cosmological intelligibility, nor as a mere tool of See, for example, Mayer, "Why the Social Sciences Lag behind the Physical and Bio- logical Sciences"; cf.

      For him it had entered the domain of logic, and in this connection Proclus tells us that Euclid subjected to rigor- ous proofs what had been negligently demonstrated by his prede- cessors. This latter confidence was adopted likewise by Euclid. Greek geometry was not formal logic, made up of hypothetical propositions, as mathematics largely is today; but it was an idealized picture of the world of actuality. Just as Aristotle seems not to have clearly recognized the tentative character of scientific knowledge thus leaving himself open to the attacks of the Skeptics , so also he failed to appreciate that although the conclusions drawn by mathematics are necessary inferences from the premises, nevertheless the latter are quite arbitrarily selected, subject only to an inner compatibility.


      Aristotle considered hypotheses and postulates as statements which are assumed without proof, but which are nevertheless capable of demonstration. As such, it excluded any notions the nature of which was not clearly and compellingly "felt" through intuition. The infinite was never invoked in the demonstrations, true to Aristotle's statement that it was unnecessary, its place being taken by the method of exhaustion which had been developed by Eudoxus.

      The limitation of the concept of number to that of positive integers apparently was Cf. Proclus Diadochus, In primum Euclides elementorum librum commentariorum. Conceptions in Antiquity 47 continued, a broader view being made unnecessary by the Eudoxian theory of proportion. Furthermore, the axioms, postulates, and definitions of Euclid are those suggested by common sense, and his geometry never loses contact with spatial intuition.

      Such purely formal, logical concepts as those of the infinitesimal and of instantaneous velocity, of infinite aggregates and the mathematical continuum, are not elab- orated in either Euclidean geometry or in Aristotelian physics, for common sense has no immediate need for them.

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      The ideas which were to lead to the calculus had not in Euclid's time reached a stage' at which a logical basis could have been afforded; mathematics had not attained the degree of abstraction demanded for symbolic logic. Although the origin of the notions of the derivative and the integral are undoubtedly to be found in our confused thought about varia- bility and multiplicity, the rigorous formulation of the concepts in- volved, as we shall find, demanded an arithmetical abstraction which Euclid was far from possessing. Even Newton and Leibniz, the inven- tors of the algorithmic calculus, did not fully recognize the need for it.

      The logical foundations of the calculus are much further removed from the vague suggestions of experience — much more subtle — than those of Euclidean geometry. Since, therefore, the ideas of variability, continuity, and infinity could not be rigorously established, Euclid omitted them from his geometry. The Elements are based on "refined intuition," and do not allow free scope to the "naive intuition" which was to be especially active in the genesis of the calculus in the seventeenth century. Barry, The Scientific Habit of Thought, pp. The work of Euclid represents the final synthetic form of all mathematical thought — the elaboration by deductive reasoning of the logical implications of a set of premises.

      Back of his geometry, how- ever, stood several centuries of analytical investigation, carried out often on the basis of empirical research, or on uncritical intuition, or, not infrequently, on transcendental speculation. It was to be largely from indagation of a similar type, rather than from the rigorously precise thought of Euclid, that the development of the concepts of the calculus was to proceed. This in its turn was necessarily to give way, in the nineteenth century, to a formulation as eminently deduc- tive — albeit arithmetic rather than geometric — as that found in the Elements.

      The greatest mathematician of antiquity, Archimedes of Syracuse, displayed two natures, for he tempered the strong transcendental imagination of Plato with the meticulously correct procedure of Euclid. Lacroix wrote texts in arithmetic, algebra, geometry, trigonometry, and probability, as well as a book on the teaching of mathematics and many biographical entries for mathematicians in a volume Biographie Universelle. The first is not a text but a thorough summary of the subject, while the second was intended to be used in schools. Both were extremely successful, going thorough many editions and being translated into many languages.

      They were the standard works for more than fifty years. In the course of this he treats the works of Euler and other earlier writers. As a result, the book gives a good view of how calculus was thought of, and written about, in the eighteenth century. English mathematics was thus freed to progress, which it commenced to do. No doubt some other text, translated by someone else, could have had the same effect, but we cannot know when that would have happened.

      Lacroix and the Calculus is an admirable book, admirably produced. I noticed only one typographical error. He helpfully translates into English all the passages in French and Latin that he quotes. Skip to main content. Search form Search. Login Join Give Shops.