So the Pythagorean Geometry of the Fifth Century B.C. Was Based on the Discrete Number Concept

The calculus had its origin in the logical difficulties encountered by the ancient Greek mathematicians in their attempt to express their intuitive ideas on the ratios or proportionalities of lines, which they vaguely recognized as continuous, in terms of numbers, which they regarded as discrete. So the Pythagorean geometry of the fifth century B.C. was based on the discrete number concept. During the latter part of that century it has discovered that there exist pairs of line segments such as the edge and diagonal of a square, that are not commensurable – they cannot be subdivided as integral multiples of segments of the same length, and hence the ratio of their lengths is not equal to the ratio of two integers. This conclusion has a consequence, in the sense that the process of comparison of two magnitudes may have no end (Infinite process). It is easy enough to recognize that the resolution of the first two paradoxes of zone, that of Dichotomy and that of Achilles and the tortoise, is depended on the summation of certain geometric series and involves the notions of limit, convergence, continuity and other concepts which are scarcely placed upon A satisfactory mathematical basis until 19th century because Zeno was the first to emphasize the difficulties inherent in such considerations Bertrand Russell, among others, has claimed that Zeno anticipated Weierstrass. As is well known, the discovery of the Immensurability of edge and diagonal of the square resulted in the first crisis in mathematics. The Pythagorean theory of proportion could not be applied to all lines, many of which are incommensurable, and that the Democritean view of infinite smalls has 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 conception of number and limit, but rather detours which obviated the necessity of using the later. 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 number of sides they seem to have hoped to reach A polygon which would coincide with the circle and so “ exhaust” its area. The method of Antiphon has been described 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

1 polygon, but merely expresses in different terminology. The procedure which Eudoxus proposed has since become know as the method of exhaustion. This axiom (as given in Euclid X.1) states that, given two unequal magnitudes if from the grater there be substracted a magnitude greater its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. From the fact that, on continuing the process indicated in the axiom of Archimedes, the magnitudes remaining can be made as small as we please, the procedure introduced by Eudoxus came much later to be called the method of Exhaustion. 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 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. The method of exhaustion, although equivalent in many respects 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 method of exhaustion has, most misleadingly, been characterized as “ a well established algorithm of the differential calculus”. It is indeed true that the problems to which the method has applied were those which led toward the calculus. In modern terminology: if a,e are two homogenous magnitudes with a greater than e (we) substract from a a magnitude a1 greater than its half and from the remainder substract a magnitude a2 greater than its half and so on. We denote by (Sn) neN the sequence with s1=a1,

th s2=a1+a2,…, sn= a1+a2+a3+…an. The remainder obtained of the n step will be the magnitude a-sn. From the exhaustion principle for ε there exists some no such that a-sn will be less than ε for all n驰 n0 limn s n = a . Unlike the Greeks Simon Stevin (1548-1620) was prepared to make use of the general consideration that any two magnitudes, the difference between which can be shown to be less than any assigned magnitude, are equal to one another.

2 Luca Valerio (1552-1618) sets out with the intention of constructing a geometric theory of limits which will enable him to dispense altogether with reduction proofs. Without the benefit of any sort of functional notation Valerio could achieve little in this direction but there is no doubt that he was seeking primarily to lay down conditions which would enable him to conclude that the limit of the quotient of two variables is equal to the quotient of their limits. In modern

f1(n ) f3 (n ) notation, if = and if by suitable choice of n, it is possible to find l1,l2,l3 f2(n ) f 4 ( n )

and l4 such that f1(n )- l 1 < e , f 2 ( n ) - l 2 < e , f 3 ( n ) - l 3 < e , f 4 ( n ) - l 4 < e ,

l l where e is any arbitrarily chosen quantity , however small, then 1 = 3 i.e. l2 l 4

limf (n ) limf (n ) f (n ) 1 f (n ) 3 lim1 =n = lim 3 = n  . n n  f2(n ) lim f 2 ( n ) f 4 ( n ) lim f 4 ( n ) n n  If therefore, a relation of this form be established between the areas (or volumes) of certain inscribed and circumscribed figures this relation also holds between the areas (or volumes) of the figures themselves. Notable as are all these attempts to modify the proof structure of Archimedes another stream of development was already in motion which has to lead to the temporary eclipse of the exhaustion method.

3 Newton (1642-1727) in the Tractatus de quadratura Curvavum (1676) gave the method of prime and ultimate ratio. He considered the function y=x ν to find the fluxion of y or xν let x by “ flowing” be come x+0 Then xν becomes

v2 - v (x+ 0)n = x n + v 0 xv-1 + 0 2 x v - 2 + ...... The increase of x and y, namely “o” and 2

v2 - v (x+ 0)n = x n + v 0 xv-1 + 0 2 x v - 2 + ...... are to each other as 1 to 2

v2 - v vxv-1+ x v - 2 +...... ” let now the increment vanish and their last proportion 2 will be” 1 to vxv-1. Then the fluxion of x is to the fluxion of xn as 1 to vxv-1. This is the Prime ratio of the nascent increment. f( x+ h ) - f ( x ) Newton: Consider a ratio of finite quantities (in modern notation h as h goes to zero). The ratio eventually becomes what Newton called an “ultimate ratio”. Ultimate ratios are “limits to which the ratios of quantities decreasing without limit do always converge, and to which they approach nearer than by any given difference, but never go beyond nor ever reach until the quantities vanish. Newton; the limit had been though as a bound which could be approached closer and closer though not surpassed. By 1800, with the work of L’ Huilier and Lacroix on alternating sense, the restriction that the limit be one-sided had been abandoned. Maclaurin had said that the sum of a series was the limit of the partial sums. Newton realized the difficulties involved in a naïve view of infinitesimals is indicated, however, in his further statement in the Principia that: “ Ultimate ratios in which quantities vanish, are not, strictly speaking, ratios of ultimate quantities, but limits to which the ratios of these quantities decreasing without limit, approach, and which, though they can come nearer than any given difference whatever, they can neither pass over not attain before the quantities have diminished indefinitely. This is the clearest statement of Newton gave as to the nature of ultimate ratios, but we shall find that, in continuing this argument in a lemma in the second book of his principia, his exposition again took on more strongly the dependence upon the idea of infinitely small quantities with the concept of limit somewhat implied as basic.

4 Leibniz (1646-1716) defined the integral as the sum of all the values of a magnitude, or the sum of an infinite number of infinitely narrow rectangles, or as modern mathematics would express it, as the limit of a certain characteristic sum. Benjamin Robins, a self-educated mathematician, lays the foundation of calculus upon the concept of limit. He speaks of a limit as a magnitude to which a varying magnitude can approach within any degree of nearness whatever, though it can never be made absolutely equal to it”. Here for the first time is the stand taken openly, clearly, explicitly, that a variable can never reach the limit. The use of infinitely small quantities is rigidly excluded. James Jurin, Newton’s student, gave a response to Robinson. Jurin argues that the word fiunt ultimo aequales used by Newton in lemma I of Book I in the principia (the inscribed and circumscribed polygon) do at last become actually perfectly, and absolutely equal . In modern phraseology, the limit is reached. Jurin conceptions were quite broad for his time. He said : “Now whether a quantity or a ratio shall arrive at its limit or shall not arrive at it, depends entirely upon the supposition we make of the time during which the quantity or ratio is conceived constantly to tend or approach towards its limits”. In other words, whether a variable reaches its limits or not is a matter choice. We may impose conditions, so that the variable reaches its limit, or conditions under which it does not reach its limits. Thus Jurin was perhaps the first consciously to modify and generalize the limit concept. A serious difficulty in permitting variables of the kind ordinarily arising in geometry to reach their limits lay in the fact that the imagination is not able to follow the variable though an infinite of steps that lead into the limit. The imagination exhausts itself in the effort. In all discussions of limits during the 18th century the question of the existence of a limit of a given sequence was never raised. The theory of irrational number caused no great anxiety to 18th century workers operations applicable to rational numbers were extended without scruble to a domain of numbers which embraced both rational and irrational. There was no careful exposition of the number system used.

5 D’ Alembert was probably led by two earlier works to regard the method of limits as fundamental in the calculus. D’ Alembert interpreted Newton’s phrase “prime and ultimate ratio” not literally, as a first or last ratio of two quantities just springing into being, but as a limit one quantity he called the limit of another if the second can approach the first nearer than by any given quantity, or so that the difference between them is absolutely in assignable properly speaking, however, he felt that the varying quantity never coincides with , or is equal to, its limits. D’ Alembert thus agreed essentially with Robins’ s interpretation of Newton’s meaning. He said that the basis of the differential calculus, like that method of fluxions was to be found in the idea of a limit. In 1787 Simon L’ Huilier was published his essay. In this work L’ Huilier proposed to show that “ the method of the ancients know under the name of Method of Exhaustion, conveniently extended suffices to establish with certainly the principles of the new calculus. In conformity with this purpose he modified the method of exhaustion to interpret it in terms of limits. Making the limit concept basic. As in modern textbooks L’ Huilier made the differential ratio or quotient fundamental defining this as the limit of the ratio of the increment in the function to that in the independent variable. He focused attention upon a single number (the derivative) as the limit of a single variable (the ratio of increments) rather than upon an ultimate ratio of two evanescent quantities, or of two fluxions or of any two quantities which have the same ratio these. Although L’ Huilier’ s definition of the differential quotient is in most respects that to be found in present day elementary textbooks on the calculus he does not seem to have been aware of the possible difficulties involved in the limit concept. He had awided the mysticism of the infinite small, the vagueness of the ultimate

0 ratio, and the inanity as the symbol but he failed to appreciate that the subtlety 0 of the limit concept was to make an extremely careful definition essential. His variable was always less than or greater than his limit. “ Given a variable quantity always smaller or greater than a proposed constant quantity but which can by less than any proposed quantity however small; this constant quantity is called the limit in greatness or in smallness of the variable quantity”

6 Sylvestre Lacroix proved that the limit of a product is the product of the limits. Let p be the limit of P, q of Q . In general P=p+A, Q=q+b, where a and b vanish together after passing though “every stage of successive diminution”. Since PQ=(p+a)(q+b), PQ=pq+pb+qa+qb. Thus we see that “the difference PQ-pq may be made as small as we please by assigning appropriate values to a and b”, But Lacroix did not be make explicit computations of how small a and b must be to ensure that PQ –pq be less than any given quantity. Nevertheless, the limit arguments he gave are important because they exemplify translations of A verbal limit concept into algebraic language, however simple Lagrange tried to avoid all limiting process by defining the derivatives of a function in terms of the coefficients of its expansion as a Taylor series. During the 18th century a great deal of works was done on particular sequences and series, and sophisticated methods for approximating and finding limits were development. D’ Alembert had studied the binomial expansion of (1+x)n where n is rational (1768) and established the boundedness of the series by comparison with two geometric series when x <1. It was Augustin Louis Cauchy, the most prolific mathematician of the nineteenth century, who first established the calculus on the basis of the limit concept so familiar to day. Although the notion of limits had been discussed much earlier, even by Newton, Cauchy was the first to translate the somewhat vague notion of a function approaching a particular value into arithmetic terms by means of which one could actually prove the existence of limits. Cauchy used his notion of limit in defining continuity (in the modern sense) and convergence of sequence, both of numbers and of functions. Cauchy’ s notion of convergence, first published in 1821, was also developed in essence by the Czech mathematician Bernhard Bolzano in 1817.

7 Cauchy on variables and their limits.

Augustin Louis Cauchy(1789-1897) implemented d’ Alembert’ s program in his two textbooks of analysis: i. Cours d’ Analyse de l’ Ecole Royale Polytechnique, Paris 1821 ii. Resume des lecons donees a L’ Ecole Royale Polytechnique, Paris 1823 In both books Cauchy begins by attempting to give a mathematical description of what is conceived as a limiting process on page 4 of (ii) he explains the term “variable quantity” The name variable is given to a quantity of which one assumes that it can take on successively several different values, one after the other. One denotes such a quantity by a letter taken from the last letters of the alphabet. Cauchy’ s definition of limit occurs near the beginning of his Cours d’ Analyse: If the successive values attributed to the same variable approach indefinitely a fixed value, such that finally they differ from it by as little as one wishes, this latter is called the limit of all the others. As an example, Cauchy noted that an irrational number is the limit of the various fractions that approach it. He also defined an infinitely small quantity to be a variable whose limit is zero. Note that Cauchy is

limf ( x )= b , not defining the modern concept x a for that concept involves two different variables. He seems to suppress the role of the independent variable entirely. Furthermore, it may appear that Cauchy’ s definition of a limit is little different from that of d’ Alembert. To see, however, what Cauchy meant by his verbal definition and to discover the difference between it and the definitions of his predecessors, it is necessary to consider his use of the definition to prove certain specific results on limits. In fact, Cauchy not only deals with both the dependent and independent variables, but also translates his statement arithmetically by use of the language of inequalities. As an example, consider Cauchy’s proof of the following: THEOREM If, for increasing values of x, the difference f(x+1)-f(x) converges to a certain limit k, the fraction f(x)/x converges at the same time to the same limit. Cauchy begins by translating the hypothesis of the theorem into an arithmetic statement:

8 Given any value ε, as small as one wants, one can find a number h such that if x h, then k-ε< f(x+1)-f(x)

f( h+ n ) - f ( h ) n

f( h+ n ) - f ( h ) It follows that =k + a n f( x )- f ( h ) where –ε1) has limit .

9 The suffix notation for sequences stems from Lagrange (1759) who actually used exponents and not suffices. The suffix notation proper is due to S.F. Lacroix (1800), but it was in Cauchy’s hand that it became the powerful tool we know today. Most of the limits which were studied during the 18th century were one-sided. The “lim” notation was due to L’ Huilier in 1786, and mid- nineteenth century mathematicians including

lim n2 = Weierstrass, were content with expressions such as n= In 1905 the Cambridge mathematician J.G. Leathen proposed the use of “ n  ”

10 What is a Limit?  Leibniz(1684) : If any continuous transition is proposed terminating in a certain limit, then it is possible to form a general reasoning, which covers also the final limit.  Newton (1687) : The ultimate ratio of evanescent quantities…[are] limits towards which the ratio of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.  Mauclaurin (1742) : The ratio of 2x+0 to α continually decreases while 0 decreases and is always greater than the ratio of 2x to α while 0 is any real increment, but it is manifest that it continually approaches to the ratio of 2x to α as its limit.  D’ Alembert (1754): This ratio [ a: 2y+z] is always smaller than a : 2y +z, but the smaller z is, the greater the ratio will be and, since one may choose z as small as one pleases, the ratio a : 2y + z can be brought as close to the ratio a: 2y as we like. Consequently, a :2y is the limit of the ratio a :2y + z.

 Lacroix (1806) : The limit of the ratio (u1-u)/h… is the value towards which this ratio tends in proportion as the quantity h diminishes, and to which it may approach as near as we choose to make it.  Cauchy (1821) : If the successive value attributed to the same variable approach indefinitely a fixed value, such that they finally differ from it by as little as one wishes, this latter is called the limit of all the others.

11 Continuity of a Function

Throughout the 17th and 18th centuries, and for many mathematicians well on into 19th century, the only functions which were under consideration were polynomials, rational functions, trigonometric and exponential functions, and combinations of these. Although the verbal definitions of a function used during the 18th century sound like modern definitions, only functions defined by a formula were under consideration. Such functions are continuous and infinitely differentiable except possibly at isolated points (e.g. at x=0 for the function

1 x ). Even continuous functions given by different formula on different x segments of their domain, such as the equation of a plucked string were only admitted with discomfort. When Euler and Lagrange use the term “continuous function they are referring to a function given by a single analytical expression; “continuity means continuity of algebraic form.”

12 Given the definition of a limit, Cauchy could now define the crucial concept of continuity. Recall that for Euler a continuous function was one expressed by a single expression while a discontinuous one was expressed in different parts of its domain by different expressions. Cauchy realized that such a definition was contradictory. For example, the function f(x) which is equal to x for positive values and –x for negative values would appear to be discontinuous according to Euler’s definition. On the other hand, one can write this same function using the

2 x2 dt single analytic expression f( x ) = 2 2 so f(x) is continuous. The geometric p 0 t+ x notion of a continuous curve, one without any breaks, was generally understood, but Cauchy sought to find an analytic definition expressing this idea for functions. Lagrange had attempted such a definition earlier in the specific case of a function “continuous at 0” and having value 0 there: “ we can always find an abscissa h corresponding to an ordinate less than any given quantity; and then all smaller values of h correspond also to ordinates less than the given quantity.” Having read Lagrange’s work, Cauchy generalized Lagrange’s idea and gave his own new definition: “The function f(x) will be between two assigned values of the variable x, a continuous function of this variable if for each value of x between these limits, the numerical [absolute] value of the difference f(x+a)-f(x) decreases indefinitely with a. In other words, the function f(x) will remain continuous with respect to x between the given values if, between these values, an infinitely small increment of the variable always produces an infinitely small increment of the function itself”. Note that Cauchy presents both an arithmetic definition and one using the more familiar language of infinitely small quantities. But because Cauchy had already defined such quantities in terms of limits, the two definitions mean the same thing. Cauchy demonstrated how to use his definition by showing, for example, than sinx is continuous (on any interval). For

1 1 sin(x+a)-sinx=2sin acos(x+ a), and the right side clearly decreases indefinitely 2 2 with a. It is interesting that Bernhard Bolzano (1781-1848), A Czech mathematician also familiar with the work of Lagrange, had some four years earlier given definition of continuity virtually identical o Cauchy’s. As part of his plan to prove

13 rigorously the intermediate value theorem, “that between any two values of the unknown quantity which give results of opposite sign [when substituted in a continuous function f(x)] there must always lie at least one real root of the equation [f(x)=0], Bolzano needed to give a clear definition of the type of functions for which the theorem would hold. Noting that others had given definitions of continuity in terms of non mathematical concepts such as time and motion, Bolzano gave what he called a “correct definition”: “A function f(x) varies according to the law of continuity for all values of x inside or outside certain limits… if [when] n is some such value, the difference f(x+n)-f(x) can be made smaller than any given quantity provided ω can be taken as small as we please.” Although the two definitions are very similar, there is no convincing evidence that Cauchy had read Bolzano’s work when he produced his own definition. Both men, being interested in giving a definition from which certain “obvious” results could be proved, came up with essential the same idea. From a modern point of view, of course, neither man defined continuity at a point. They both defined continuity over an interval. It would appear, however, that one can read each definition as defining continuity at each point in the interval. Namely, given a particular value x and an ε>0, one can find a δ>0 such that

f( x+ a ) - f ( x ) < e whenever α<δ. As we will see in the next section, though, Cauchy was not entirely clear on what quantities δ depended. This lack of clarity would lead him to an incorrect result. After beginning this text with the same definition of continuity as in the earlier one, Cauchy proceeded in Lesson 3 to define the derivative of function, what he called the fonction derivee, as the limit of [f(x+i)-f(x)]/i as i approaches the limit of 0, as long as this limit exists. Just as he did in his definition of continuity, Cauchy defined the concept of a derivative over an interval, in fact an interval in which the function f is continuous. He noted that his limit will have a definite value for each value of x, and therefore is a new function of that variable , a function for which he used Lagrange’s notation f’(x). The definition itself, although it can be thought of as expressing the quotient of infinitesimal differences as in Euler’s work, is more directly taken from that section of Lagrange’s Analytic Functions in which Lagrange, as part of his power series expansion of f, showed that f(x+i)=f(x) + if’(x)+iV where V is some function that

14 goes to zero with i. Cauchy was able to translate this theorem about derivatives into a definition of the derivative. He then calculated the derivatives of several elementary functions. For example, if f(x)=sinx, then the quotient of the definition

1 sin( )i 1 reduces to 2 cos(x+ ( ) i ) whose limit f’(x) is seen to be cosx. 1 ( )i 2 2

15 The distinction between continuity at a point and continuity on an interval was not clear in Cauchy’ s work. In this proof of the integrability of continuous functions (1823), Cauchy assumed continuity but used uniform continuity. Dirichlet proved that a function which was continuous proved on a closed interval was uniformly continuous on that interval in his lectures in Berlin in 1854. The first proof of this result in A Weierstrassia context was published by E. Heine in 1872.

16 Definitions of Continuity

 Euler (1784): A continuous curve is one such that its nature can be expressed by a single function of x. If a curve is of such a nature that for its various parts… different functions of x required for its expression…, then we call such a curve discontinuous.  Bolzano (1817): A function f(x) varies according to the law of continuity for all values of x inside or outside certain limits… if [when] x is some such value, the difference f(x+ω)-f(x) can be made smaller than any given quantity provided ω can be taken as small as we please.  Cauchy (1821): The function f(x) will be, between two assigned values of the variable x, a continuous function of this variable if for each value of x between these limits, the [absolute] value of the difference f(x+a)-f(x) decreases indefinitely with a.  Dirichlet (1837): One thing of a and b as two fixed values and of x as a variable quantity that can progressively take all values lying between a and b. Now if to every x there corresponds a single, finite y in such a way that, as x continuously passes through the interval from a to b, y=f(x) also gradually changes, then y is called a continuous function of x in this interval.  Heine (1872): A function f(x) is continuous at the particular value x =X if from every given quantity ε, however small, there exists a positive number

ηo with the property that for no positive quantity η which is smaller than ηo

does the absolute value of f(X η)-f(X) exceed ε. A function f(x) is continuous from x=a to x=b if for every single value x=X between x=a and x=b, including x=a and x=b, it is continuous.

17 The construction of the real number system was the principal step in the arithmetization of analysis during the closing third of the nineteenth century. The final loose end tied by Weierstrass in his purely arithmetical formulation of the limit concept, which previously had been tinged with connotations of

limf ( x ) = L continuous motion-it was said that x a provided that f(x) approaches L as x approaches α . Weierstrass objected to this “dynamic” description of limits, and replaced it with a “static” formulation involving only real numbers, with no

limf ( x ) = L appeal to motion or geometry: x a provided that given ε>0, there exists a number δ>0 such that f( x )- L