Curves and Tangents

Smoryński, Craig

doi:10.1007/978-3-319-52956-1_2

Curves and Tangents

Craig Smoryński²

Chapter
First Online: 08 April 2017

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Abstract

The central concepts underlying the Mean Value Theorem are those of curves and tangents. The evolution of these concepts from the Greeks through the end of the 19th century is discussed, along with the crystallisation of the formal definitions of limit, continuity, and derivative. The chapter culminates with geometrically motivated derivations of the Mean Value Theorem itself.

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Notes

1.
Jacqueline Stedall (ed.), Mathematics Emerging: A Sourcebook 1540–1900, Oxford University Press, Oxford, 2008.
2.
Ibid., p. 10.
3.
Thomas Little Heath, The Thirteen Books of Euclid’s Elements, 3 volumes, Cambridge University Press, Cambridge, 1908. This translation has been reprinted a number of times. The edition put out by Dover Publications includes all the annotations. Two other editions currently in print but lacking the annotations are that in the series Great Books of the Western World and an attractively typeset single volume published by Green Lion Press.
4.
Ibid., vol. 1, p. 158.
5.
Ibid., pp. 158–165.
6.
Euclid first defined the point in Definition I as that which has no part.
7.
Proclus (Glenn R. Morrow , ed.), A Commentary on the First Book of Euclid’s Elements, Princeton University Press, Princeton, 1970, pp. 79–80.
8.
Heath, Elements,op. cit., vol. 1, pp. 158–159. I have omitted his parenthetical insertions of Greek terms and page references in Aristotle.
9.
Hans Hahn , “Die Krise der Anschauung”, in: Krise und Neuaufbau in den exakten Wissenschaften, F. Deuticke, Leipzig and Vienna, 1933. An English translation, “The crisis of intuition”, appears in: Hans Hahn (Brian McGuinness , ed.), Empiricism, Logic, and Mathematics: Philosophical Papers, D. Reidel Publishing Company, Dordrecht, 1980.
10.
Hahn , “The crisis of intuition”, op. cit., p. 88.
11.
Carl B. Boyer, History of Analytic Geometry, The Scholars Bookshelf, Princeton Junction (NJ), 1988, p. 20. This work was originally published in 1956 as numbers 6 and 7 of The Scripta Mathematica Studies. Incidentally, the numerical estimate given here is figurative, not literal: Boyer cites at least half a dozen curves known to the Greeks and on page 35 announces, “yet scarcely a dozen curves were familiar to the ancients”.
12.
Boyer, op. cit., p. 11.
13.
One can go a long way calculating with $\infty $ taking $\infty $ as an ideal element and applying rules like
$$\begin{aligned} a\pm \infty =\pm \infty ,\ \ a\cdot \infty =\infty ,\ \ a/\infty =0 \end{aligned}$$
for real a. Terms like $0\cdot \infty $, $\infty -\infty $, and $\infty /\infty $ are indeterminate and simple algebra doesn’t apply. In fact, I have cheated in writing $\infty ^{2}/\infty ^{4}=1/\infty ^{2}=0$. One should first manipulate (2.5) to express
$$\begin{aligned} \frac{e^{2}}{(1-e^{2})^{2}}=\frac{1}{(1-e^{2})^{2}/e^{2}}=\frac{1}{(1/e-e)^{2}} \end{aligned}$$
and only then plugging $\infty $ in for e:
$$\begin{aligned} \frac{1}{(1/\infty -\infty )^{2}}=\frac{1}{(0-\infty )^{2}}=\frac{1}{\infty ^{2}}=0. \end{aligned}$$
14.
Apologies to the reader: dt here denotes multiplication by d, not the differential.
15.
Lucas N.H. Bunt, Phillip S. Jones, and Jack D. Bedient, The Historical Roots of Elementary Mathematics, Prentice-Hall, Inc., Englewood Cliffs (NJ), 1976, pp. 105–106.
16.
To the Greeks, angles were between 0$^{\circ }$ and 180$^{\circ }$. As every obtuse angle is the sum of a right angle and an acute angle, and as the right angle is easily trisected, we need only concern ourselves here with acute angles.
17.
Julian Lowell Coolidge, A History of Geometrical Methods, Dover Publications, Inc., New York, 1963, pp. 46–47. This is a reprint of a volume originally published by Oxford University Press in 1940.
18.
The Law of Cosines, in what we might call a disguised form, appears as Propositions 12 and 13 in Book II of the Elements. To make this proof non-trigonometric and purely geometric requires merely a change in terminology.
19.
T.L. Heath, The Works of Archimedes Edited in Modern Notation with Introductory Chapters by T.L. Heath with a Supplement The Method of Archimedes Recently Discovered by Heiberg, Dover Publications, Inc., New York, no date given. Heath’s original edition was published in 1897 by Cambridge University Press, the supplement appearing subsequently in 1912. Cf. pp. cvi–cvii for his remarks on the conchoid.
20.
An English translation can be found in Heath’s book cited in the preceding footnote. The work On Spirals occupies pp. 151–188.
21.
Ibid., pp. 153–154.
22.
Nikolaus von Kues is often cited under variants of his name. The Latin form is Nicolaus Cusanus, though Cusanus often suffices. Other variants are Nikolaus von Cusa, Nicholas of Cusa, or simply Nicholas Cusa.
23.
Boyer, op. cit., p. 72.
24.
Nikolaus von Kues, Die mathematischen Schriften, 2nd. edition, Verlag von Felix Meiner, Hamburg, 1979, p. 220. The volume contains translations of Kues’s manuscripts from the Latin by Josepha Hofmann and an introduction and notes by Joseph Ehrenfried Hofmann . Footnote 37 on page 217 includes the remark, “The figure contained in the Oxford manuscript has led Wallis to the rash claim that Cusanus had aleady arrived at the construction of the cycloid”.
25.
John Martin, “The Helen of Geometry”, The College Mathematics Journal 41, no. 1 (2010), pp. 17–28; here: p. 17.
26.
I also suggest V. Frederick Rickey , “Build a brachistochrone and captivate your class” in: Amy Shell Gellasch (ed.), Hands on History. A Resource for Teaching Mathematics, Mathematical Association of America, 2007.
27.
t has a minus sign because the clockwise rotation is the reverse of the usual rotation.
28.
Cf., e.g., my exposition: Craig Smoryński, Adventures in Formalism, College Publications, London, 2012, pp. 99–104.
29.
Heath, Elements, op. cit., pp. 160–165.
30.
Boyer, op. cit. , p. 32. The bracketed insertion is Boyer’s.
31.
A brief word of explanation: For some time mathematicians viewed curves as the paths traced out by the intersection of two lines, eventually a vertical line moving along the x-axis and a horizontal one moving up and down the y-axis. With Fermat, however, the axes were not necessarily perpendicular but met at a given angle. The variables thus stood for the positions of the lines parallel to these axes. Viète had begun a short-lived practice of using vowels to denote variables and consonants to denote unspecified constants and Fermat adhered to this tradition.
32.
Mathematical historians distinguish 3 phases in the development of algebraic symbolism: rhetorical, in which everything is expressed in words; syncopated, in which some abbreviations are introduced; and symbolic, in which everything is expressed in abstract symbols and calculations follow strict term rewriting rules.
33.
Boyer , op. cit., pp. 75–76.
34.
This “analytic art” was the beginning of symbolical algebra. The adjective “analytic” here referred to the algebraic analysis of a problem — its expression in algebraic terms and the solution of the resulting equations. Except for “Analytic Geometry”, the adjective “analytic” today refers more generally to those areas of mathematics that the Calculus evolved into, Calculus itself having evolved from Analytic Geometry.
35.
Boyer , op. cit., pp. 88–89.
36.
Consider, e.g., the “curve” defined by the constant function $f(x, y)=0$.
37.
The only tricky part is recognising that
$$\begin{aligned} \gamma (.r_{0}9r_{1}9r_{2}9\ldots )&=\langle .r_{0}r_{1}r_{2}\ldots ,.999\ldots \rangle =\langle .r_{0}r_{1}r_{2}\ldots , 1\rangle \\ \gamma (.9r_{0}9r_{1}9r_{2}\ldots )&=\langle .999\ldots ,.r_{0}r_{1}r_{2}\ldots \rangle =\langle 1,.r_{0}r_{1}r_{2}\ldots \rangle . \end{aligned}$$
.
38.
Bertrand Russell, Principles of Mathematics, 2nd ed., W.W. Norton & Company, Inc., New York, no date given, p. 193. The first edition was published in 1903, the second originally in 1938. The printing I quote from is a paperback that I acquired new in the late 1960 s or early 1970 s and is thus a reprint of the second edition.
39.
The modern term for this is “dense”; “compact” has an altogether different meaning in mathematics.
40.
René Descartes (David Eugene Smith and Marcia L. Latham , trans.), The Geometry of René Descartes, Dover Publications, Inc., New York, 1954, pp. 42 (original French version) and 43 (English translation). The French original was published in 1637 as an appendix to Descartes’s philosophical work Discours de la Methode. The English translation was first published in 1925 by the Open Court Publishing Company.
41.
Ibid., pp. 90 (French) and 91 (English).
42.
Fermat followed Viéte in using vowels A, E, I, O, U for variables $x, y,\ldots $ Cf. p. 83f, below, for a more precise description of Fermat’s technique.
43.
Gert Schubring, Conflicts between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17–19th Century France and Germany , Springer Science+Business Media, Inc., New York, 2005, pp. 26–27. In quoting this I have omitted Schubring’s citations to the literature.
44.
Descartes and Fermat had introduced algebraic descriptions $f(x, y)=0$ for curves, where f was a polynomial; very quickly transcendental functions like sines, cosines, logarithms, etc., were introduced into the composition of f.
45.
Citation from Jean Itard , “Arbogast, Louis François Antoine”, in: Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography, vol. 1, Charles Scribner’s Sons, New York, 1970, p. 207. Itard adds, “The Academy was thus requesting a drastic settlement of the dispute between Jean d’Alembert, who adopted the second point of view, and Leonhard Euler, partisan of the first”.
46.
Judith Grabiner , “Cauchy and Bolzano: tradition and transformation in the history of mathematics”, in: Everett Mendelsohn (ed.), Transformation and Tradition in the Sciences: Essays in Honor of I. Bernard Cohen, Cambridge University Press, Cambridge, 1984, p. 112. A similar, earlier, discussion of the matter was given by Grabiner in: Judith V. Grabiner, The Origins of Cauchy’s Rigorous Calculus, The MIT Press, Cambridge (Mass.), 1981, pp. 91–92. This book was reprinted by Dover Publications, Inc., in 2005. Accessible fuller quotations from Arbogast can be found in: C.H. Edwards, Jr., The Historical Development of the Calculus , Springer-Verlag, New York, 1979, pp. 303–304; Umberto Bottazzini (Warren van Egmond (trans.)), The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, New York, 1986, pp. 34–35; and (in German) Klaus Volkert , Geschichte der Analysis, Bibliographisches Institut & F.A. Brockhaus AG, Zürich, 1988, pp. 170–171.
47.
Augustin Louis Cauchy, Cours d’analyse de l’École Royale Polytechnique; I. $^{{ re}}$ Partie. Analyse algébrique [Course in Analysis of the Royal Polytechnical School; Part I. Algebraic Analysis], de Bure, Paris, 1821. English translation: Robert E. Bradley and C. Edward Sandifer (eds. and trans.), Cauchy’s Cours d’analyse; An Annotated Translation, Springer Science+Business Media, LLC, New York, 2009. The function $\sin (\frac{1}{x})$ is cited on p. 12 of the Bradley/Sandifer edition.
48.
The word “limit” is used in two senses here. The first occurrence refers to what we now call limit points; the second refers to the endpoints of the interval $[-1, 1]$ on the y-axis.
49.
The manuscript is called “Functionenlehre” [“Theory of functions”] and can be found in: Steve Russ (ed.), The Mathematical Works of Bernard Bolzano, Oxford University Press, Oxford, 2004. For Bolzano’s counterexample, cf. pp. 471–472 (§§83–84, but see also §46, pp. 453–454).
50.
Ibid., p. 481, §102.
51.
Volkert, op. cit., p. 187. Cf. Lemma 3.1.5 on page 187, below.
52.
Gaston Darboux , “Mémoire sur les fonctions discontinues”, Annales scientifiques de l’École Normale Supérieure, 2nd series, vol. 4 (1875), pp. 57–112; here: p. 109.
53.
H. Turgay Kaptanoğlu, “In praise of $y=x^{\alpha }\sin (\frac{1}{x})$”, American Mathematical Monthly 108 (2001), pp. 144–150.
54.
Bernard Bolzano, Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewählen, wenigstens eine reelle Wurzel der Gleichung liege, Gottlieb Haase, Prague, 1817. The work also appeared the following year in volume 5 of the Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften, and was edited and reprinted by Philip E.B. Jourdain in 1905 as half of number 153 of Ostwalds Klassiker der exakten Wissenschaften. English translations by Steve Russ and William Ewald appeared first in 1980 and 1996, respectively. The most recent English version appears in Russ’s edition of The Mathematical Works, op. cit. Below, I shall refer to the Ostwald Klassiker reprint as “Bolzano, Klassiker” in what follows, but will also give references to The Mathematical Works for English translations. Thus, the above list of names can be found in: Bolzano, Klassiker, p. 6; Russ, op .cit., p. 253.
55.
Bolzano, Klassiker, pp. 3–4, Russ op. cit. , p. 256.
56.
Ibid. Presumably Bolzano intends “<” here to read “less in absolute value than”.
57.
Bolzano, Klassiker, p. 6; Russ, op. cit., pp. 256–257.
58.
A Cauchy sequence is a sequence $a_{0}, a_{1}, a_{2},\ldots $ of numbers satisfying: for any $\epsilon >0$ a number $n_{0}$ can be found such that for all $m, n>n_{0}$ one has $|a_{m}-a_{n}|<\epsilon $. The convergence of such sequences had been used without note by Euler . Bolzano drew attention to them and proved their convergence relative to his notion of real number as incompletely treated in a later work not published in his lifetime. Jacqueline Stedall finds the proof “incorrectly argued” (Stedall, op. cit., p. 496):

It turned out to be more difficult than it might seem, and Bolzano was forced to introduce [as] a fresh assumption the existence of a quantity X to which the terms of the series approach as closely as we please. Such a hypothesis, Bolzano claimed ‘contains nothing impossible’..., but it was precisely what he was trying to prove in the first place. The problem was deeper than Bolzano realized. Convergence of Cauchy sequences requires completeness of the real numbers or, simply speaking, that the number line is an unbroken continuum with no gaps. Convergence of Cauchy sequences is in fact mathematically equivalent to completeness: either must be assumed in order to prove the other. Without some such assumption, Bolzano was forced to introduce his hypothetical quantity X.

This is a fair criticism, but I give Bolzano full credit nonetheless as he later offered some justification for his variant of completeness on which his proof of the convergence of Cauchy sequences was based. I discuss this sort of thing in some detail in Smoryński, Formalism, op. cit., pp. 232–265.
59.
The Bolzano-Weierstrass Theorem asserts that any bounded sequence $a_{0}, a_{1}, a_{2},\ldots $ of numbers, i.e., any such sequence for which there is a bound $B>|a_{n}|$ for all n, has a convergent subsequence. It is a fundamental result of Analysis.
60.
Bolzano, Klassiker, p. 31; Russ , op. cit., p. 273.
61.
Bolzano is a little sloppy here: In his example cited above, the law of continuity is two-sided and does not apply to the endpoints $\alpha ,\beta $ of an interval, but his proof of the Theorem assumes the one-sided continuity of f and $\phi $ at the endpoints of the interval.
62.
Bradley and Sandifer , op. cit., p. 26. The editors explain that “solution of continuity” is to be read as “dissolving of continuity”, i.e., the breakdown of continuity is meant. Note again, as in footnote 48, the use of the word “limits” to mean “endpoints”.
63.
I.e., at every number $r+\eta $ in the interval, where r is real and $\eta $ is infinitesimal.
64.
Sic. This should read $f(x+h)$.
65.
Karl Weierstrass and Hermann Amandus Schwarz , Differential Rechnung, nach einer Vorlesung des Herrn Weierstrass im Sommersemester 1861, Hdschr. Koll. N 37 (Humboldt-Universität zu Berlin), pp. 2–3.
66.
Eduard Heine, “Die Elemente der Functionenlehre”, Journal für die reine und angewandte Mathematik 74 (1872), pp. 172–188.
67.
Ibid., p. 172.
68.
In the 1830s Bolzano offered a description of real numbers that nowadays one would treat as such a construction, but this went unpublished until the 20th century. At some later, undetermined, date (cf. pp. 334–335, below), Weierstrass offered such a construction treating real numbers as abstract sums of rationals. And in 1858 Richard Dedekind independently constructed the reals using sets of rationals. None of this was published until 1872 when several such constructions, new and old, simultaneously made it into print. Cantor’s , Charles Méray’s, and Heine’s constructions used Cauchy sequences .
69.
Heine, op. cit., p. 182.
70.
The reference is to his earlier definition cited on page 183, above.
71.
Ibid., p. 184.
72.
One way of visualising this is to imagine a rectangle $[a-\delta , a+\delta ]\times [f(a)-\epsilon , f(a)+\epsilon ]$ of fixed size $2\delta \times 2\epsilon $. As one moves $\langle a, f(a)\rangle $ along the curve, the graph over the interval $[a-\delta , a+\delta ]$ always remains inside the rectangle.
73.
His exposition is muddled and not everyone accepts it, but a correct proof was certainly within his grasp. Cf. pages 301–302, below, for details.
74.
Heine, op. cit., p. 182.
75.
In Craig Smoryński, A Treatise on the Binomial Theorem, College Publications, London, 2012, p. 138, I also credit Cauchy with a proof of this Theorem. In glancing over his two main textbooks I have not found the result proven although it is appealed to in the Résumé des leçons données a l’École Royale Polytechnique sur le calcul infinitesimal, de Bure, Paris, 1823. The nonstandard proofs of the Intermediate Value Theorem, which Theorem is proven in the Cours d’analyse, and the Extreme Value Theorem being virtually identical, I must have simply assumed Cauchy had proven the latter. It would naturally have fit into the projected second volume of the Cours. As this volume was intended as a textbook and policies at the École had changed, Cauchy did not include as much foundational material in the Résumé when he came to write this later. So he might have proven the result and simply neglected to include the proof in any of his textbooks.
76.
But see Sect. 6, below.
77.
Such a proof, long discredited, is nowadays acceptable thanks to the rigourous foundation and development of Nonstandard Analysis. . The reader unfamiliar with these modern developments may consider the proof merely heuristic. The curious reader who would like to know more is referred to Chapter II, Sect. 6, of Smoryński, Formalism, for an introduction to and some references on the subject.
78.
In Nonstandard Analysis, a set of nonstandard numbers is called $^{*}$finite if it can be put into one-to-one correspondence with an integer, finite or infinite, by an “internal” function. In simple terms, a $^{*}$finite set is a possibly infinite set that behaves like a finite set.
79.
H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904, p. 105.
80.
Y.R. Chao , “A note on ‘Continuous mathematical induction’ ”, Bulletin of the American Mathematical Society 26 (1919), pp. 17–18.
81.
Pete L. Clark, “The instructor’s guide to real induction”, online at http://arxiv.org/abs/1208.0973.
82.
The one-to-one correspondence $\gamma $ given at the end of the first section (p. 70, above) can be shown directly not to be continuous. The point $t_{0}=.01$ is mapped by $\gamma $ to the pair $\langle 0,.1\rangle $. The points
$$\begin{aligned} t_{n}=.00\underbrace{9999\ldots 99}_{2n} \end{aligned}$$
can be chosen as close to $t_{0}$ as one wishes by choosing n large enough, yet
$$\begin{aligned} |\gamma (t_{n})-\gamma (t_{0})|&=|\langle .0\underbrace{9\ldots 9}_{n}\,,.0\underbrace{9\ldots 9}_{n}\,\rangle -\langle 0,.1\rangle |\\&=\sqrt{(.09\ldots 9-0)^{2}+(.09\ldots 9-.1)^{2}}\\&>.09\ldots 9>.09, \end{aligned}$$
i.e., the points $\gamma (t_{n})$ are bounded away from $\gamma (t_{0})$.
83.
Cf., e.g., Hahn , op. cit., pp. 85–87, or Bernard R. Gelbaum and John M.H. Olmsted , Counterexamples in Analysis, Holden-Day, Inc., San Francisco, 1964, pp. 133–134. The publication of Gelbaum and Olmsted has been taken over by Dover Publications and the book is still in print. The authors also cite a couple of variant constructions.
84.
After making this translation, I was reminded by Ádám Besenyei that an excellent English translation can be found in: Hubert C. Kennedy (ed. and trans.), Selected Works of Giuseppe Peano, George Allen & Unwin Ltd, London, 1973. I bought a copy of this book decades ago and, being a logician, read some of the logical papers, storing the book on my general logic shelf. In my memory, the book was a selection of the logical papers of Peano and I thus neglected to consult it until receiving the reminder. Kennedy accompanies his translation with an excerpt from a later (1908) work of Peano in which a geometric construction is discussed.
85.
Peano writes “uniformes”, which I take to mean “well-defined”. Kennedy translates this as “single-valued”, which is perhaps a more felicitous choice.
86.
Kennedy uses the word “digit”, more in line with standard English usage. I tend to think of “digit” as referring to base 10 unless some modifier is added. In the present case this would result in “ternary digit”, which I didn’t like. So I stuck with the more literal “cipher”.
87.
The European fashion is to use commas and periods in decimal representations where Americans use periods and commas, respectively. I have followed Peano more closely in these small details than Kennedy, for better or for worse.
88.
More formally, Journal für reine und angewandte Mathematik. This journal was founded by August Crelle and is often called Crelle’s Journal in his honour.
89.
G. Peano, “Sur une courbe, qui remplit toute une aire plane”, Mathematische Annalen 36 (1890), pp. 157–160.
90.
Günther Frei (ed.), Der Briefwechsel David Hilbert – Felix Klein (1886–1918), Vandenhoeck & Ruprecht, Göttingen, 1985, pp. 70–71.
91.
David Hilbert, “Ueber die stetige Abbildung einer Linie auf ein Flächenstück”, Mathematische Annalen 38 (1891), pp. 459–460.
92.
Hubert C. Kennedy , Peano, Life and Works of Giuseppe Peano, D. Reidel Publishing Company, Dordrecht, 1980, p. 32.
93.
Whence, of course, follows Klein’s preference for Hilbert’s geometric presentation.
94.
Two examples are the paper of Hahn and the book of Gelbaum and Olmsted cited in footnote 83 a few pages back. Hahn accompanies the pictures of some of the curves in Hilbert’s sequence with the announcement, “It is now possible to give a rigorous proof that the successive motions considered here approach without limit a definite course, or curve, that takes the moving point through all the points of the large square in unit time”. Gelbaum and Olmsted give the parametrisation, but leave the details that the limit is a continuous function and that it fills the square as an exercise to the reader. E. Hairer and G. Wanner , Analysis by Its History, Springer-Verlag New York, Inc., New York, 1996, pp. 289–290 repeat Hilbert’s graphical presentation and give the parametric representation for a more general construction, proving the continuity of the limit, but leaving unproven the more intuitive fact that the range of the function is the entire square. They also present Peano’s construction geometrically as an exercise on page 298. A cursory check of my personal library found no fuller proof for the geometrical construction. Indeed, most of my textbooks on Analysis do not even mention the result.
95.
Heath, Elements , op. cit., vol. 2, p. 2.
96.
Ibid..
97.
Ibid., p. 3.
98.
I say this is “almost” a proof because we have not defined precisely what is meant by “crossing”. In algebraic terms we note that a line $Ax+By=C$ partitions the plane into three disjoint sets according as $Ax+By$ is $<C$, $=C$, or $>C$. A line may be said to cross the curve C at $P=\langle \alpha ,\beta \rangle $ if $A\alpha +B\beta =C$ and in any neighbourhood of P there are points of the curve in each of the sets $\{\langle x, y\rangle \,\big |\, Ax+By<C\}$ and $\{\langle x, y\rangle \,\big |\, Ax+By>C\}$. Can we give a precise, purely geometric definition of the notion? How about the notion of two curves crossing each other?
99.
Another problem is: how can we tell algebraically or analytically that a curve given by a continuous parametrisation $\gamma $ is smooth in the Greek sense?.
100.
Descartes, op. cit., p. 95.
101.
Ibid., pp. 95 ff. But see also Edwards , op. cit., pp. 125–127.
102.
The Latin original is “adæquentur”, later rendered into the French as “adégalera”. I suppose the most direct English translation would be “equate to”, but it is not clear that he really means “equate”. Thus historians of mathematics agree to keep the “ad”. The rest, i.e., what the term means, is hotly debated among the historians. Cf. Mikhail G. Katz , David M. Schaps , and Steven Shnider , “Almost equal: the method of adequality from Diophantus to Fermat and beyond”, arXiv:1210.7750v1.
103.
Pierre de Fermat, “Methodus ad disquirendum maximum & minimam”. Fermat did not publish the contents of this letter during his life, and it first appeared, in 1679 in Latin, in the Varia opera mathematica edited by his son Samuel de Fermat . A couple of centuries later, when it was translated into French for inclusion in the third volume (1896) of his collected works, Œuvres de Fermat, his antiquated notation was updated, the result being much more readable. In both these works, the letter to Descartes was accompanied by a number of later items on the method of maxima and minima. A translation into English of the modernised French translations of the letter to Descartes and its immediately following letter to Gilles Personne de Roberval (1602–1675) appeared in: Dirk Struik (ed.), A Source Book in Mathematics, 1200–1800, Harvard University Press, Cambridge (Mass.), 1969, pp. 222–227. The quotation reproduced above is from a more recent translation from the French edition by Jason Ross , which I found online. Ross translates all seven parts of Fermat’s method of maxima and minima.
104.
$\sim $ is the symbol used in the French translation to stand for ad-equality.
105.
“ III. On the same method”, p. 5 of Ross, op. cit..
106.
“ IV. The method of maximum and minimum”, Ross, op.cit. , p. 7.
107.
Ibid., p. 9.
108.
Judith V. Grabiner , “The changing concept of change: the derivative from Fermat to Weierstrass”, Mathematics Magazine 56 (1983), pp. 195–206; here: p. 197. Grabiner is, of course, using “E” where Fermat used “e”.
109.
Readable accounts of their contributions can be found in: Margaret E. Baron, The Origins of the Infinitesimal Calculus , Pergamon Press, Oxford, 1969, pp. 214–220; and Edwards , op. cit., pp. 127–132.
110.
Edwards , op. cit., pp. 131–132.
111.
Grabiner, “Changing concept...”, op. cit., p. 198.
112.
Scholars have identified this friend as Newton , who helped prepare the work for publication.
113.
Note that the character that looks like an $\ell $ is the Q of the text.
114.
I.e., multiplied by.
115.
The following is a bit opaque and the reader may wish to skip ahead to the modern explanation following this quotation.
116.
The G here is clearly a misprint for Q.
117.
There is a typo here: $-2rrmm$ should be $-2rrma$.
118.
There is a double typo here as the + between the two terms containing e accidentally changes to a −.
119.
Isaac Barrow (Edmund Stone trans.), Geometrical Lectures: Explaining the Generation, Nature and Properties of Curve Lines, London, 1735, pp. 171–175. This edition is available in facsimile online. The copy I downloaded, however, was very imperfectly done, some pages being repeated, and the fold-out plates scanned without being unfolded — whence not all the illustrations are available. One can, however, find all the illustrations online at ECHO (European Cultural Heritage Online) by searching, not for the Lectiones geometricæ of 1670, but for the larger work Lectiones opticæ & geometricæ of 1674 in which the former is incorporated. Figure 2.34, combines screen captures of pieces of one of the plates (indexed by thumbnail 361 at ECHO) cleaned up with photo-retouching software.
A more recent annotated, but abridged, translation by J.M. Child , The Geometrical Lectures of Isaac Barrow, was published in 1916 by the Open Court Publishing Company (Chicago and London). This translation is available in several reprinted editions and can also be found online.
Struik , op. cit., excerpts a couple of important passages from Barrow, including that portion of the above quotation omitting the Example.
Barrow illustrates his technique with five examples, of which I have cited the first, Child the fifth.
120.
His picture is imperfect here. The line TN is supposed to be tangent to the curve at M, not at N. The disposition of T, N, and M in his Fig. 2.35 is slightly better in that the tangent passes through M. Whether N lies on the curve or the tangent, however, is not discernible from these pictures. Fermat’s Fig. 2.32, separating N from I and V from O is clearer in this respect. Barrow himself did better in his later Fig. 2.36 — cf. p. 96. Indeed, Struik reproduces Fig. 2.36 in place of Fig. 2.34 in his excerpt cited in the preceding footnote.
121.
See the preceding footnote.
122.
Perhaps we should use Fermat’s adequality $\sim $ here.
123.
The parenthesis following ANq is merely a typographical error.
124.
Cf. his comment following Rule 3 on page 91, above.
125.
The other curves for which he finds tangents are two versions of the folium of Descartes, with equations $x^{3}+y^{3}=c$ and $x^{3}+y^{3}=cxy$, and the quadratrix .
126.
Cf. footnote 120.
127.
Figure 2.32. Fermat’s and Barrow’s labelling differ:
$$\begin{array}{l|c|c|c|c|c|c} \text {Barrow} &{} \text {M} &{} \text {N} &{} \text {R} &{} \text {P} &{} \text {T} &{} -\\ \hline \text {Fermat} &{} R &{} N &{} E &{} D &{} B &{} I \end{array}$$
128.
Baron, op. cit., p. 251.
129.
Ibid., pp. 251–252.
130.
Excerpts from Cavalieri’s work can be found in: David Eugene Smith (ed.), A Source Book in Mathematics, 1929 (reprinted: Dover Publications, Inc., New York, 1959, pp. 605–609); Struik, op. cit. , pp. 209–219; and Stedall (op. cit., pp. 62–65. Accounts can also be found in Edwards (op. cit., pp. 104–109) and Baron (op. cit., pp. 122–135). By far the most complete discussion in English however is Kirsti Andersen , “Cavalieri’s Method of Indivisibles”, Archive for History of Exact Sciences 31(1985), pp. 291–367.
131.
J.M. Child (ed. and trans.), The Early Mathematical Manuscripts of Leibniz, The Open Court Publishing Company, Chicago and London, 1920.
132.
Grabiner, “Changing concept...”, op. cit., p. 199.
133.
Baron, op. cit., p. 257.
134.
The fluxion is essentially the derivative of one of the variables with respect to time — more anon.
135.
Florian Cajori , “Newton’s fluxions”, in: David Eugene Smith (ed.), Sir Isaac Newton, 1727–1927; A Bicentenary Evaluation of His Work, The Williams & Wilkins Company, Baltimore, 1928, p. 193.
136.
Isaac Newton (John Colson ed. and trans.), The Method of Fluxions and Infinite Series with its Application to the Geometry of CURVE-LINES, 1736, p. 20.
137.
Ibid., p. 21.
138.
Ibid., pp. 24–25.
139.
Ibid., p. 44.
140.
For Newton the curve is traced out as the point of intersection of two non-rotating lines moving along a pair of axes. The axes need not meet at right angles and the lines, called the abscissa and ordinate, need not be vertical and horizontal, but must remain parallel to the axes. The abscissa is parallel to the y-axis and its coordinate is x, while the ordinate is parallel to the x-axis with coordinate y.
141.
Newton is, of course, being only approximate here. If cd is the moment, d lies on the curve infinitesimally close to the tangent line, but not on the tangent line. The line through D and d will cross the curve not touch it.
142.
Newton, op. cit., p. 46.
143.
The paragraph opens with a normal sized “1”, followed by a large drop cap “I”, and “consider” subsequently capitalised. I decided this required too much effort to duplicate completely.
144.
Newton is here taking a very Aristotelian view of the line.
145.
I.e., surfaces.
146.
“First ratio” and “nascent” are not exactly defined here. Their meaning will emerge when examples are discussed. In a couple of pages the “first ratio” will be called the “prime ratio”. My interpretation of this paragraph is that, for any fluents v, w, $\dot{v}/\dot{w}\approx \Delta v/\Delta w$, that is, $\dot{v}/\dot{w}\approx (\dot{v}o)/(\dot{w}o)$.
147.
This image is taken from: Florian Cajori, A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse , The Open Court Publishing Company, Chicago and London, 1919, p. 42, and is a clean reproduction of Newton’s original.
148.
In light of paragraph 3, he seems to be suggesting something like
$$\begin{aligned} \frac{d\int _{a}^{x}f(t)dt}{dx}\bigg /\frac{d\int _{a}^{x}g(t)dt}{dx}=\frac{f(t)}{g(t)}, \end{aligned}$$
where the curve ACc is given by $y=f(x)$ and GDa by $y=g(x)$. Paragraphs 3 and 4 are unhelpful in the extreme. He thus seems to be asserting the Fundamental Theorem of the Calculus via some clumsy reference to proportion.
149.
Thus, these will be our dx, dy, and ds.
150.
John Stewart , Sir Isaac Newton’s Two Treatises of the Quadrature of Curves, and Analysis by Equations of an infinite Number of Terms, explained, London, 1745, pp. 1–2. The second treatise referred to in the title is De analysi.
151.
This is remarkably close to our modern definition of “limit”.
152.
Smith , Source Book..., op. cit., pp. 617–618. Cf. also Struik , op. cit., pp. 299–300.
153.
That is, let x be a constant multiple of time, so that $y=x^{n}$, being a function of time, is in fact a function of x.
154.
Newton had extended the Binomial Theorem to the case of arbitrary rational exponents. For n not a positive integer, however, the expansion is an infinite series. Thus, Newton is here differentiating $x^{n}$ for arbitrary rational n. For more information, I refer the reader to Smoryński, Treatise.
155.
Stewart , op. cit., p. 4.
156.
Acta Eruditorum 3 (1684), pp. 467–473. A German translation of this and several further papers of Leibniz was published by Gerhardt Kowalewski as number 162 in Ostwald’s series of scientific classics in 1908. A partial English translation appears in Smith’s source book, op. cit., a full translation in Struik, op. cit. , and a nearly full translation in Stedall , op. cit..
157.
Newton had not yet published on the subject and consequently is not mentioned. It is not clear why Barrow is not mentioned. Child is of the opinion that Leibniz was hiding his dependence on Barrow, but it could also be the similarity of Barrow’s analytical method to Fermat’s .
158.
“Reductions” obviously refers to simplifying the equations for which tangents are sought. The process, especially in clearing surds, results in polynomials of higher degrees and the introduction of extra possibilities for the tangent. Presumably “depressions” refers at least in part to the elimination of false solutions. The point of this passage is that the method he is introducing is more direct and eliminates this excess work.
159.
Child , Early Mathematical Manuscripts..., op. cit., pp. 131–132.
160.
Ibid., pp. 132–133.
161.
Edwards, op. cit., p. 266. Edwards offers, incidentally, an excellent discussion of Leibniz’s early papers as well as the published ones.
162.
Think of AB and BC as the x-and y-axes, respectively, in Fig. 2.39.
163.
Smith, Source Book, op. cit., p. 617.
164.
Struik, op. cit., p. 276.
165.
Stedall, op. cit. , p. 66. Stedall includes (p. 69) an excerpt from Hobbes, Six lessons to the Professors of Mathematics, 1656, p. 46, criticising in the plainest language the use of infinitesimals by John Wallis (1616–1703) whose Arithmetica infinitorum of 1655 provided the spark that ignited Newton.
166.
Child, Early Mathematical Manuscripts..., op. cit., pp. 145–146.
167.
And not even this when courses are watered down and only drill is offered.
168.
Robert Woodhouse, The Principles of Analytical Calculation , University of Cambridge Press, Cambridge, 1803, pp. xvii–xviii.
169.
George Berkeley, The Analyst; or, a Discourse Addressed to an Infidel Mathematician., London, 1734, pp. 6–7.
170.
Ibid., pp. 8–9.
171.
Douglas M. Jesseph , Berkeley’s Philosophy of Mathematics, University of Chicago Press, Chicago, 1993, pp. 158–159. I confess to having given this book only a quick and superficial reading, but it strikes me as offering an excellent in-depth discussion of Berkeley’s criticism of the Calculus. Another source worthy of mention is Cajori , A History of the Conceptions of Limits..., op. cit..
172.
However “unduly hasty”, the rejection of Berkeley’s critique follows from Berkeley’s own Lemma cited in paragraph XII of The Analyst — cf. p. 117, below.
173.
Berkeley, op. cit., p. 10.
174.
Ibid., p. 12.
175.
This is the Latin original of Newton’s remark cited above that “The very smallest Errors in mathematical Matters are not to be neglected”.
176.
Berkeley, op. cit., pp. 14–16.
177.
Smith , Source Book, op. cit., p. 631.
178.
Ibid, pp. 19–21.
179.
I give a fairly complete account of the history of the Binomial Theorem in: Smoryński, Treatise, op. cit. It might be added that the two proofs are from two different works of Newton’s and thus their simultaneous existence, even should the results have been on equal footing, would not necessarily have been proof of anything more than variety.
180.
Child adds a footnote here explaining that “evanescent” should be read “vanishing into the far distance”.
181.
Child, Early Mathematical Manuscripts..., op. cit., pp. 146–148.
182.
Quoted in translation in H.J.M. Bos, Differentials, Higher Order Differentials and the Derivative in the Leibnizian Calculus , dissertation, Rijksuniversiteit te Utrecht, 1973, p. 73.
183.
Child , Early Mathematical Manuscripts..., op. cit., p. 150.
184.
Landen’s Discourse is available online and a print edition by Gale, 2010, of Book I of The Residual Analysis exists. Additionally, excerpts from Discourse are reproduced in Struik , op. cit., pp. 386–388 and Stedall , op. cit., pp. 398–401. I also refer to Smoryński, Treatise, op. cit., pp. 148–151, for a detailed account of Landen’s “proof ” of the Binomial Theorem .
185.
His term “derivative” and notation $f^{\prime }$ for the derivative are still used today.
186.
Both books by Lagrange are available online. English translations of excerpts from Théorie des fonctions can be found in Struik, op. cit. , pp. 389–391, and Stedall , op. cit., pp. 404–406. Other discussions of Lagrange’s approach can be found in Edwards , op. cit., pp. 296–299, and Smoryński, Formalism, op. cit., pp. 127–135. This last reference, incidentally, includes in Exercise 6.6 of Chapter II, pp. 184–185, an outline of Cauchy’s result mentioned above.
187.
J.L. Lagrange, “Sur la résolution des équations numériques, et additions au mémoire sur la résolution des équations numériques”, Mémoires de l’ Academie...Berlin 23 (1767), pp. 311–352 and 24 (1768), pp. 111 –180; reprinted in volume 2 of Oeuvres de Lagrange, Gauthier-Villars, Paris, 1867–1882.
188.
“Réflexions sur les suites et sur les racines imaginaires”, in: J. d’Alembert, Opuscules mathématiques, vol. 5, Briasson, Paris, 1768, pp. 171–215. An annotated English translation of the relevant portions can be found in Smoryński, Treatise, op. cit., pp. 182–188.
189.
English translation from: Stedall, op. cit. , pp. 297–298. Stedall includes also excerpts on limits from Wallis , Newton , Maclaurin , and Cauchy .
190.
Cf. p. 95, above.
191.
Judith V. Grabiner , Origins, op. cit. p. 95.
192.
Prague, 1816; English translation in: Russ , op. cit..
193.
Ibid., p. v; Russ , p. 158.
194.
Ibid., p. 15; Russ , p. 173.
195.
Ibid., p. 15; Russ , p. 173.
196.
Ibid., p. 20; Russ , p. 176.
197.
Cauchy, Résumé, op. cit., p. 9. After making many of my translations from the Résumé for this book, I learned of a complete translation of the work by Dennis M. Cates . There are two versions of this translation, an expensive annotated edition, A Guide to Cauchy’s Calculus; A Translation and Analysis of Calcul Infinitesimal, Fairview Academic Press, Walnut Creek (California), 2011, and a more affordable student edition, Cauchy’s Calcul Infinitesimal; A Complete English Translation, same publisher, 2012. In comparing my translations with his, I find the differences minor and have kept my own. Nonetheless, I shall give page references to the less expensive copy which is now in my possession. The reference in the present case is to p. 7.
198.
I.e., the absolute value.
199.
Cauchy, Résumé, op. cit., p. 27; Cates , op. cit., p. 23.
200.
Cf. pp. 301–304, below, for details.
201.
Particularly nice proofs of the Chain Rule and the Inverse Function Theorem can be found in Jan Mikusiński and Piotr Mikusiński, An Introduction to Analysis: From Number to Integral, John Wiley & Sons, New York, 1993, pp. 123–124 and 132–133, respectively.
202.
Russ, op. cit. , pp. 487–489, 507–508.
203.
In one course, at least, given in 1874, he said that the conditions could be relaxed to a being an integer $>1$ and $ab>1$, but that the proof was more difficult under these more general conditions.
204.
Cf. Sect. 2.2.3, above.
205.
Bibhutibhushan Datta and Avadesh Narayan Singh (Kripa Shankar Shukla , reviser), “Use of Calculus in Hindu mathematics”, Indian Journal of History of Science 19, No. 2 (1984), pp. 95–109; here: p. 98. Mediæval Hindu mathematicians, particularly in the Kerala region, were several centuries ahead of the Europeans in many areas, including the beginnings of the infinitesimal calculus. In the last few decades some primary sources have been published in English translation, but not enough yet for one to develop an accurate picture of their state of knowledge. The internet is rife with references to the Hindu origins of the above Lemma and the Mean Value Theorem, but they tend to offer no details. We discuss the matter in greater detail in Sect. 2.3 of Chap. 3.
206.
The use of the word “Classroom” here is a local one. The reader will not find it elsewhere in the literature and I introduce it merely to distinguish the theorem as stated from the myriad of forms of the Mean Value Theorem as the one familiar from the first year Calculus course. When the distinction is unimportant, I drop the adjective.
207.
R.B. Burckel and C. Goffman , “Rectifiable curves are of zero content”, Mathematics Magazine 44 (1971), pp. 179–180.
208.
Burckel and Goffman prove Theorem 2.3.46 for rectifiable curves. A curve is rectifiable just in case it is of bounded variation. Using (2.41) one easily shows continuously differentiable curves to be rectifiable, whence Theorem 2.3.46 is a special case of Jordan’s result. Not every differentiable function, however, is rectifiable. Gelbaum and Olmsted , op. cit., pp. 140–141, cite $x^{2}\sin (1/x^{2})$ as an example.

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Smoryński, C. (2017). Curves and Tangents. In: MVT: A Most Valuable Theorem. Springer, Cham. https://doi.org/10.1007/978-3-319-52956-1_2

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