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Has
flux’s concept ancient roots? An attempt at an approach
Christine Phili National technical University of Athens The
term flux, fluxion indicates motion. The idea of the fluxional calculus
developed from the concept that a geometrical magnitude was the result of
continuous motion of a point, line or plane. “This
motion, speaking of plane curves, could be considered, when referred to
coordinate axes, as the resultant of two motions, one in the direction of the
X-axis and the other in the direction of the Y-axis. The velocity of X-component
and the Y-component were called “fluxions” by Newton. He represented them by
The
problem of fluxions is, thus, dual in nature: 1.
The length of the space described being continually given; to find the
velocity of the motion at any time proposed. 2.
The velocity of the motion being continually given; to find the length of
the space described at any time proposed”[1],
[2]. In
the beginning of the 6th century the first kinematic ideas in the
Greek philosophy are: the eternally moving apeiron of Anaximander and the
Heraclitean doctrine. Anaximander’s apeiron[3],
inexhaustible and imperishable, encompasses and steers all things, a denial of
all kinds of limits, is in eternal motion and from it worlds come into being and
pass away. Heraclitus “holding that everything is in constant change – this
is his most famous doctrine, the one which makes him for all subsequent ages the
philosopher of flux[4]
- he naturally metamorphoses the static order to “the dynamic order which
marks the intertransformations of its elements”[5]. The
resonances of the Heraclitean doctrine penetrated ancient Greek thought. So, for
example Plato’s consideration of time as “a moving image of eternity”[6]
brings a dimension of ordered constancy to the incostancy of flux”[7].
In
the Theaetetus[8],
Plato presents a theory that “all physical so – called things are not
things, but slow motions that our sense organs also are slow motions and that
perception is the result of the meeting of these motions”[9]. Aristotle
in his Metaphysics says: A. 987 a 29: “After the systems we have named came the philosophy of Plato which in most respects followed these thinkers (i.e. Pythagoreans) but had peculiarities that distinguished it from the philosophy of the Italians. For in his youth he first became familiar with Cratylus and with the Heraclitean doctrines that all sensible things are ever in a state of flux”. The
same account is repeated in M 1086 a 37 – b 5: “They
(the believers in the Ideas) thought that the particulars in the sensible world
were in a state of flux and none of them was stable”. The
sequence point-line-surface-solid appeared already in the Pythagoreans[10], but there is no
indication of how this sequence is generated. The static concept of limit is the
basis of the explanation; but we can also find that the explanation of this
generation is motion. Let
us return again to Aristotle. Principally on the basis of De Anima 409 a 4:
“For they say that the movement of a line creates a plane and that of a point
a line; and likewise the movements of units will be lines”. Conford
in his book Plato and Parmenides[11]
argues for a fluxion theory from points, taking the fluxion from Sectus
Empiricus: “But
some assert that the body is constructed from one point, for this point when it
has flowed produces the line, and the line when it has flowed makes the plane,
and this, when it has moved towards depth generates the body … But this view
of the Pythagoreans differs from that of the earlier ones. For these latter
formed the numbers, from the principles, the one and the Indefined Dyad, and
then, from the numbers, the points and lines and both the plane and solid
forms”[12]. However
the kinematic approach appears again, in Proclus and he refers to those who call
a line “the flux of point”[13]
and finds this definition highly satisfactory. More precisely Proclus[14]
in his commentaries on Euclid gives a very interesting interpretation of the
three postulates of construction: 1.
to draw a straight line from any point to any point . 2.
to produce a finite straight line continuously in a straight line. 3.
To describe a circle with any center and distance. The fact to draw a straight line from any point to any point is the consquence that the line is the flux of a point; and a straight line is the flux of a plane … If we imagine that the point has a movement equal and short as possible, we arrive to the other point and the first postulate will be realised. But if the straight line is terminated by a point, we imagine that its extremity has a movement minimal and equal and so the second postulate will proceed easily … Finally, if we imagine that a finite straight line, fixed in one of its extremities, the third postulate will be born”. Proclus
here, uses another definition which does not belong to Euclid “the line is the
flux of a point” and the line is the flux of a plane. Proclus, tried to
elucidate the simplicity of the first three postulates with the concept of
movement. The demand of the postulates can be accorded the introduction of the
movement[15].
Nevertheless the introduction of the notion of movement is not so simple.
Proclus tried to explain his choice. If
anybody asks how we apply movements to geometrical beings which are not mobile
and how we move beings which have no parts (i.e. the point) all this is
absolutely impossible, we ask him not to be angry … [16].
We conceive the movement under an uncorporal form and imaginary[17].
We do not agree that the partless beings (i.e. the point) have corporal
movement, but they have imaginary movement. Because νούς
which is without part moves, but not in the locus. Similarly the imagination,
who is without part relatively to itself posses also its own movement. In our
view about corporal movements, we do not accept movements as being without
dimension. Many
centuries later, circa 1290, Petri Philomeni de Dacia in Algorismum vulgarum
Johannis de Sacrobosco Commentarius[18]
presents the sequence of generation point – line – surface by motion[19],
so line generates a surface, a surface generates a solid. In
the latin edition of Euclid’s Elements by Christoph Clavius (1574)[20]
we found the word fluere for the description of the origin of lines and
surfaces, by means of flowing points and lines, similar to Petrius. In
the fourtheenth century the study of mathematical sciences flourished in Oxford.
Oresme used some concepts and terminology of Newton’s theory of fluxions.The
Merton calculators[21]
used the terms fluxus and fluens. But for Oresme these terms were related to
geometrical representation. In Oresme’s exist two fundamental notions: firstly
the representation of a physical quality by a surface; secondly the concept of a
surface as the flux or motion parallel to itself. Oresme gave emphasis to the
description of lines, surfaces and solid by motion[22]. Napier
in 1614, in his Descriptio, employed the idea of the fluxion of a quantity to
represent by means of lines the relations between logarithms and numbers[23].
“Sit punctus A a quo decenda sit linea fluxu alterius puncti, qui sit B; fluat
ergo primo momento B ab B in C, secundo momento C in D etc.”[24] Cavalieri
followed this trend in holding that surfaces and volumes could be regarded as
generated by the flowing indivisibles. “Communes sectiones talis moti sive
fluntis plani et figurae”[25],
“planum motu seu fluxus”[26].
The flowing motion in Cavalieri plays a relatively minor role as he did not
develop this idea into geometrical method. This was done by his successor
Torricelli. Toricelli considered the curves generated by a point which moves
along a uniformly rotating line with a velocity, not necessarily uniform[27]. Roberval
also regarded every curve as the path of a moving point and accepted as an axiom
that the direction of motion is also that of the tangent[28]. Although
Torriceli’s results remained unpublished[29]
his pupils and associates Angeli (1623-97) and Ricci (1619-82) were able[30]
to continue his research. But the concept of flux still did not flourish. After
a long trip through France, Italy, Smyrni, Constantinople, Venice, Germany and
the Low Countries an English erudite became Regius Professor of Greek in the
University of Cambridge. His name was Isaac Barrow and he was the person who
culminated flux’s concept and led his pupil Issac Newton to establish his
theory of fluxions, his interpretation of the Infinitesimal calculus, in a
extended and systematic process. The Lectiones Geometricae of Barrow, although
it reflected the philosophical nature of the writer, it dealt with the study of
curves as generated by moving points and lines. The old derivation point –
line – surface – solid, the ancient sequence has appeared again intact and
fertile, and incorporated in the first official steps of the new born analysis. The
Geometrical Lectures appeared in 1670 and J. M. Child[31]
deduced the opinion that were for the most part evolved during Barrow’s
professorship at Gresham College (July 1662 – May 1664). In
the opening words of these Lectures, Barrow presents the prependerant role of
the motion: “I am now entering upon a new field of discource, whether more
pleasant or more fruitful, hardly know; but it is pleasant by reason of its
abundant variety and decidedly useful since it comprehends the origin of
mathematical hypotheses, from which its definitions are formed and its
properties emerge. I mean the generation of magnitudes or the ways whereby the
different kinds of magnitude may be conceived to be produced … The most
important are local movements … because without motion mothing can be
generated or produced. Therefore this must considered first. There is a well
known maxim of Aristotle’s “He that is ignorant of Motion must necessarily
know nothing about Nature””[32]. The
nature of Time and Motion, the geometric representation of these magnititudes,
resonances of Oresme’s and Galileo’s methods are the subject of the first
Lecture Generation of Magnitudes. “Since quantity of motion cannot be
discerned without Time, it is necessary first to discuss Time”[33] and he concludes “Time
implies Motion to be measurable; without motion we could not perceive the
passage of Time”[34]
Time, continues Barrow, has many analogies with a line and may be represented by
it, so the ancient idea of a line as a flux of points appears again. Time is
constituted either as a simple addition of successive instants or as a
continuous flow of one instant. Thus, a line is considered to be the trace of a
moving point[35]. The
germ of Geometrical Lectures found the appropriate ground. The student at
Cambridge, the student of Barrow, Isaac Newton, under the influence of his
tutor,[36]
gave the final development in the flux’s concept, formalizing and unifying the
algorithmus of the calculus. From
the manuscripts of Newton we know that his first thought about fluents and
fluxions dated from 1665 – 1666 when he was 23-24 years old. But the fear of
publication, a fear which remains present in all his life, did not permit the
appearance of his research. Thus, only in 1687, in the first edition of his
Principia Newton introduces his ideas. In the Book II, Section II, Lemma II
Newton presents the generation of magnitudes as flux: “These quantities I here
consider as variable and indetermined and increasing or decreasing as it were by
perpetual motion or flux; and I understand their momentaneous increments or
decrements by the name of Moments; … velocities of the increments and
decrements (which may also be called the motions, mutations and fluxions of
quantities)”[37]. The
latin edition of John Wallis’s Algebra which appeared in 1693, contains on
pages 390-396 a treatise on the Quadrature of Curves, which Newton had prepared
many years before. But the fear of the public remains alive. Newton speaks of
himself in the third person. Nevertheless the flowing quantities, entities of
mathematics, arrive in the world of sense perception. “By flowing quantities
he understands indeterminates, that is, those which in the generation of curves
by local motion are always increased or diminished, and by their fluxions he
understands the velocity of increase or decrease”.[38] In
his Introductio ad quadraturam curvarum (1704)[39],
Newton considered the generation of mathematical quantities in the same manner
as the Greeks. The sequence point – line – surface – solid remains the
same as we can find in Sextus Epiricus; the point when it has flowed produces
the line and the line when it has flowed makes the plane reappeared again[40]
in the logical procedures of the new calculus. The
genesis of the mathematical quantities “per motum continuum” concept which
has its roots in antiquity gave a “preliminary”[41]
framework to the first[42]
exposing system of the Infinitesimal Calculus. “… geneses in rerum natura
locum vere habent et in motu corporum quotidie cernuntur”[43]
does not constitute in the future a reasoning to understand flux or even the
variability. The concepts of the new calculus “purified” and free from any
metaphysics will be based upon the limit concept. The era of rigorous foundation
of analysis begins with the works of Bolzano, Cauchy and Weierstrass. [1] F. Cajori. A History of Mathematics. London. Mac Millan 1919. P. 193. [2] For more details see F. Wren & J. Garret, The Development of the fundamental concept of Infinitesimal Analysis. Amer. Math. Mon. Vol. XL 1933 pp. 269-281. [3] See R. Mondolfo, L’ infinito nel pensieri dell’ antichità classica. Firenze 1956; H. Kahn, Anaximander and the origins of Greek Cosmogony. New York. 1960; F. Solmsen, “Anaximander’s Infinite. Traces and Influences” Arch. Ges. Phil. 44.2 1962 pp. 109-131. [4] G. Vlastos: Plato’s Universe. Clarendon Press. Oxford 1975 p. 5. [5] Idem p. 7. [6] Plato’s Timaeus 37 d 5. [7] G. Vlastos op. cit p. 29. [8] 155 d 5 – 157 c 2, 181 b 8 – 183 c 7. [9] D. Ross. Plato’s Theory of Ideas. Clarendon Press. Oxford 6th ed. 1971 p. 157. [10] At the beginning of the De Caelo 268 a 11, Aristotle appears to be attributing to the Pythagoreans the line-plane-solid derivation usually attributed to Plato. [11] London 1939 pp. 11-13. [12] Sextus Empiricus: Against the Physicits. Against the Ethicists. Adversus Mathematicus ix. Xi trans. R. G. Bury. Cambridge Mass.; London 1936 pp. 346, 349. [13] Proclus: The Philosophical and Mathematical Commentaries of Proclus (transl. T. Taylor) 2 Vols. London. 1788 p.123. [14] A. Szabo in his book Anfänge der griechischen Mathematik [Les débuts des mathématiques grecques trad. M. Federspiel Vrin, Paris 1977 p. 303-306] presents Proclus’ interpretation to elucidate the notion of the term postulate as demand, with the very old sense of this term in dialectics. [15] A. Szabo op. cit. p. 304. [16]
“ει δε τις
αποροίη, πως
κινήσεις
επεισάγομεν
τοις
γεωμετρητοίς
οίσιν, πως δέ τά
αμερή
κινούμεν,
ταύτα γάρ
αδύνατα είναι
παντελώς” (185:25-186-2). [17]
“την δέ
κίνησιν μη τοι
σωματικήν
αλλά
φανταστικήν
νοήσωμεν καί
τα αμερή τάς
μέν σωματικάς
κινήσεις
κινείσθαι μή
συγχωρώμεν τά
τίς
φανταστικάς
διεξόδους
υπομένειν” (186:9-13). [18] Ed. M. Curtie. Kopenhagen 1897. [19] Op. cit. p. 72 lin 8 & 11. [20] We must quote the successive editions: 1589, 1591, 1603, 1607, 1612. [21] Bradwardine, Swinshead, Heytesbury, Dumbleton. [22] For more details cf. M. Clagett. The Science of Mechanics. Pergamon Press. Oxford. 1969. [23] H. Zeuthen. Geschichte der Mathematik in Altertum und Mittelalter. Kopenhagen. 1896 pp. 134-135. [24] Napier, Descriptio 1614 pp. 1-2. [25] Exercitationes geometricae sex. Bononiae. 1647 p. 12. [26] Idem p. 16. [27] De infinitis spiralibus. Opere I part 2. 1919 p. 349-59. [28] Roberval, G. P de “Divers ouvrages” Mémoires de l’ Académie Royale des Sciences depuis 1666 jusqu’ à 1699, VI (Paris, 1730) pp. 436-78. [29] Only in 1919 were published his works. [30] M. Ricci Exercitatio geometrica. Rome 1666; Angeli, S. degli. Problemata geometrica Venice 1658 etc. [31] cf. Lectiones Geometricae. Preface J. M. Child. Chicago and London 919. [32] Idem p.32. [33] id. p. 35. [34] Idem. [35] id. p. 37. [36] Leibniz stresses this influence in the following manner: “Si quelqu’un a profité de M. Barrow, ce sera plutôt M. Newton qui a étudié sous lui “cf. Lettre de 9 Avril 1716. Briefwechsel, … éd. Gerhardt I p. 281. [37] Principia, 1st ed. 1687, Book II, Section II, Lemma II. [38] J. Wallis. Algebra 1693 p. 391. [39] “I don’t here consider Mathematical Quantities as composed of parts extremely small but as generated by a continual motion. Lines are described and by describing are generated, not by any apposition of parts, but by a continual motion of points. Surfaces are generated by the motion of lines, solids by the motions of surfaces … Time by a continual Flux and so in the rest” Math. Works ed. D. T. Whiteside 1964 p. 141. [40] The continuous motion became foundamental in his system. [41] In his Philosophiae Naturalis Principia Mathematica (1687) Newton exposes the rules of the new calculus with none reference to word fluxion. [42] With respect to the truth priority. [43] Introductio ad quadraturam curvarum ed. Amsterdam 1723 p. 43. |
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