The Calculus: A Genetic Approach

by Otto Toeplitz, David Bressoud (Foreword by)

Overview

When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus.  In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus.

Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.

Product Details

ISBN-13:	9780226806693
Publisher:	University of Chicago Press
Publication date:	07/02/2018
Sold by:	Barnes & Noble
Format:	eBook
Pages:	192
File size:	15 MB
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About the Author

Otto Toeplitz (1881-1940) was a leading scholar in linear algebra and functional analysis. He was the author of many scholarly articles and coauthor of The Enjoyment of Mathematics.

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CHAPTER 1

THE NATURE OF THE INFINITE PROCESS

Two subjects, analytic geometry and differential and integral calculus (or infinitesimal calculus, as it is more comprehensively and appropriately called), form the principal courses for beginners in our university program of mathematical instruction. The distinction between the two subjects seems clear enough from their names alone; the one deals with geometry, the other with calculation. In reality, however, the principle which this distinction expresses is not sound; infinitesimal processes tie in as well with geometric objects as with calculational ones, and the geometry of figures of the second and third degree in the plane or in space can culminate in a purely calculational determinant theory. Therefore, the true distinction between these two subjects is that infinitesimal calculus uses infinite processes, whereas analytic geometry avoids them. That distinction extends, far beyond the beginning courses, throughout mathematics and offers the only serious basis from which to proceed to a classification of the whole science. As mathematics develops, that distinction becomes ever clearer, while that between geometry and calculation fades away. Our first task, therefore, will be to delineate the essential nature of the infinite process. In later chapters we will consider such particular kinds of infinite processes as differentiating, integrating, and summing infinite series.

1. THE BEGINNINGS OF GREEK SPECULATION ON INFINITESIMALS

It bespeaks the true greatness of a new idea if it appears absurd to contemporaries encountering it for the first time. The paradoxes of Zeno are the first indication to us of the emerging idea of the infinite process at a time about whose intellectual life we otherwise know but little. To their author they were doubtless not the quasi-punning puzzles which are reported to us, an impression which is accentuated by the form given to the argument. In the report to which we largely owe our knowledge of the paradoxes, Aristotle certainly discounts their disguise by arguing: "I cannot go from here to the wall! To do so, I would first have to cover half the distance, and then half the remaining distance, and then again half of what still remains; this process can always be carried on and can never be brought to an end." It is unreasonable to suppose that Zeno was unaware that the times needed to traverse these successive halves themselves become shorter and shorter. He is protesting only against the antinomy of the infinite process which we encounter in proceeding along a continuum. And this protest, expressed with youthful exuberance but recorded almost against his will, indicates that mathematicians had then first dared to undertake the summation of infinitely many, but ever decreasing, bits of time, like

1 + 1/2 + 1/4 + 1/8 + ....

It is interesting to compare this report of Aristotle's with one of the fragments, likewise from the fifth century B.C. (which has come down from Anaxagoras): "There is no smallest among the small and no largest among the large; but always something still smaller and something still larger." These words seem trivial to us today; they certainly were not so in the Age of Atomism. This was not the atomism of which we were thinking in imagining discrete material atoms distributed in space but a theory which anticipated discontinuity of space itself and considered the possibility that a line segment might not be indefinitely divisible. Zeno's paradoxes go far beyond a flat rejection of this atomism as expressed in the statement of Anaxagoras. Though we do not know much about the mutual relations of Eleatics, Pythagoreans, and other philosophical schools, we cannot doubt that Zeno's criticism is directed against some first, uncertain claims of a new mathematics trying to replace the naïve atomistic view based on intuition with laws found by systematic reasoning.

The conflict to which Zeno's paradoxes give expression comes to the open with the "Pythagoreans" at the moment when they discover the "irrational," thereby facilitating the rise of the idea of the infinite process and laying its base, which is valid to this day. Just what is this "irrational"? It is contained in the discovery of a side and the diagonal of one and the same square being "incommensurable," that is, lacking a common measure.

The "carpenter" rule for constructing a right angle had long been known: Make the two sides of a triangle 3 and 4 ells long and incline them toward each other in such a way that the line segment connecting the two ends measures exactly 5 ells; then the angle is a right angle, and the triangle a right triangle (Fig. 1). Mathematicians kept trying to find an analogous whole-number relationship among the sides for the much more obvious case of a right triangle with equal sides (Fig. 2). They divided each leg into five equal parts and then laid off one of these parts on the hypotenuse. This part seemed to be contained in the hypotenuse seven times, yet not quite exactly; the hypotenuse was a bit too long. They tried the same with twelve divisions of the legs; the hypotenuse this time fitted with seventeen such parts much more exactly than before but still not entirely. All such efforts to discover a "common measure" for both segments — legs and hypotenuse — were fruitless; finally, it was recognized that this quest must remain vain — that there exists no such common measure.

There are two proofs of this impossibility. The first proof is based on easy observations about even and odd numbers:

1. The square of an even number is always divisible by 4:

(2n)2 = 4n2.

2. The square of an odd number is always odd:

(2n + 1)2 = 4n2 + 4n + 1 = 2(n2 + 2n) + 1.

From these two observations follow two more:

3. If the square of a number is even, then the square is divisible by 4.

4. If the square of a number is even, then the number itself is even.

The impossibility proof itself is indirect. Suppose the side a and diagonal d of a square had a common measure, e, and d = pe and a = qe. By the Pythagorean theorem

d2 = a2 + a2 = 2 a2;

therefore,

(pe)2 = 2(qe)2,

so that

Here it may be assumed that p and q have no common factor except 1, since otherwise the common measure e would have been chosen too small and could be so enlarged.

Since the right side of equation (1.1) is evidently even, the left side is too. According to Observation 4, p must then be even. On the other hand, since p is even, it is divisible by 4 according to Observation 3. But then the right side of equation (1.1) is divisible by 4, so that q must be divisible by 2 and therefore even. Once more application of Observation 4 shows q itself even.

Therefore, p and q would both be even, contrary to the express assumption that p and q have no common factor except 1. The initial supposition of the proof — that the side and diagonal of the square had a common measure — has led to a contradiction and is, consequently, disproved.

The second impossibility proof uses an elementary geometrical consideration instead of facts about even and odd numbers: In the square in question (Fig. 3), lay off on the diagonal beginning at B a segment BD of the same length as side AB; at D erect a perpendicular meeting side AC in B'; join BB' and B. Triangles ABBB' and DBBB' are congruent, since two pairs of corresponding sides are equal and the angles opposite the larger side are equal; therefore, ABB' = DBB'. The angle ACB is half a right angle; therefore, B'CD is an isosceles right triangle, and DBB' = DC. It has been established that

AB' = B'D = DC. (1.2)

Now erect at C the perpendicular to CD and draw through B' the parallel to CD which meets that perpendicular at A'. A square A'B'CD is obtained which is smaller than the original, ABCD, since the diagonal B'C is already covered by one of the sides of the original. To this new square there is now to be applied the same procedure that was applied to the original; mark off a segment B'D' on the diagonal equal to the length of side A'B' and at D' erect the perpendicular to the diagonal meeting side AB'C in B?. Then, as before,

A'B" = B"D' = D'C. (1.3)

It is clear that the procedure continues indefinitely and never terminates; instead, each time there remains a piece of the diagonal smaller than the previous remainder

CD > CD' > CD" > CD'" > .... (1.4)

Each of these remainders is the difference of the diagonal and side of one of the successive squares:

CD = CB - AB, CD' = CB' - A'B', CD" = CB" - A"B", .... (1.5)

This elementary geometrical consideration is the necessary preliminary to the proof; the proof itself is indirect. Suppose the side and diagonal of the square to be commensurable; that is, suppose that there is a common measure of the two — an interval E a certain exact multiple of which would equal the side of the square and a certain other exact multiple of which would equal the diagonal. Then it is only necessary to observe (Fig. 4) that the difference of any two intervals which are both exact multiples of E is likewise an exact multiple of E. So, if CB and AB are exact multiples of E, then from equation (1.5) CD is too. And as A'B' = CD, A'B' is then an exact multiple of E. The diagonal CB' of the square A'B'CD is such that CB' = CA - AB' = AB – CD — this last by equation (1.2) — and CB', being the difference of two exact multiples of E, is therefore an exact multiple of E. This property having been proved for the side and diagonal of the squareA'B'CD, it follows by the same kind of argument for all later squares.

The indirect proof can now be completed by arguing to a contradiction. The intervals appearing in (1.4), on the supposition that there is a common measure for the side and diagonal of the original square ABCD, must all be exact multiples of E. On the other hand, equation (1.4) asserts that the multiples of E continually decrease without either terminating or ever becoming zero. This is impossible for multiples of a fixed interval, since, if the initial term were 1,000 times E, then CD' would be a smaller exact multiple of E — at most 999 times E. At the very latest the 1,000th member in this chain would have to be smaller than E and yet still a multiple of E, and, therefore, zero times E, contrary to what has already been proved. This is the contradiction to which we have been led by the supposition that there is a common measure of the side and diagonal of a square; the supposition is therefore untenable.

Complete darkness covers the origins of this first impossibility proof. This great discovery, more than anything else, inaugurated the character of modern mathematics. The oldest and least ambiguous evidences are found in Plato and Aristotle. The latter, repeatedly referring to the subject, alludes to the first-mentioned proof, which later appears again with Euclid. Plato puts considerable emphasis on the fundamental nature of this discovery. In the Laws, at the point where he assigns that mathematical discovery a place in higher school instruction, he mentions that he first learned of it when he was a comparatively old man and that he had felt ashamed, for himself and for all Greeks, of this ignorance which "befits more the level of swine than of men." Especially in the dialogue dedicated to the memory of one of the greatest Greek mathematicians, Theaetetus, who had just fallen in battle, he gives an account of these matters. There he tells how Theodorus, the teacher of Theaetetus, a well-known Sicilian mathematician who was born about 430 B.C., had lectured to his students on the proof that the side of a square of area 3 square feet is incommensurable with an interval 1 foot long, and similarly for squares of area 5 up to 17 square feet, 9 and 16 square feet excepted, of course. From this citation it is quite clear that the teacher Theodorus was already in possession of a well-developed theory of such facts, the case of 2 square feet not even being mentioned, although Plato contrasts him with his student Theaetetus, who introduced a more abstract and general approach to this theory.

The essential content of Greek mathematics is found in the Elements of Euclid, about 300 B.C., and in the writings of mathematicians and commentators who came after him; but there is only the content, not the history of its development. Nothing but such fragments as those cited above permits us a fleeting glimpse into the beginnings of this mathematics, which is by no means the work of Euclid. One of the most important and comprehensive of these fragments consists of a few pages from what is probably the oldest textbook of Greek mathematics, or at least from the hand of Hippocrates, who lived about 450 B.C. It shows us the next stage in the development of the theory of infinite processes. Hippocrates halves a circle (Fig. 5) by diameter AB; then, with the midpoint of the lower semicircle as center, he draws a circle passing through A and B. He asserts that the crescent (shaded in Fig. 5) has the same area as the square on the radius BM of the original circle.

The proof rests on a lemma, the basis of which unfortunately is not given in the extant pages. It claims: The areas of two circles are to each other as the squares of their radii (Fig. 6). From this it follows that segments of two different circles which subtend equal angles (Fig. 7) are to each other as the squares of the radii. This is proved first for angles which are an aliquot part* of the full angle and later for arbitrary angles. Next Hippocrates connected the midpoint D of the upper semicircle (Fig. 8) with A and B and found that these lines touched the large circle at A and B without entering into it. The crescent under consideration consists, therefore, of three areas designated in the figure as a, ß, and γ. Area a is a segment of the given circle subtending one-fourth of the full angle; δ is a segment of the large circle subtending one-fourth of the full angle. In accordance with the above lemma, a and d are therefore to each other as the squares of the radii of the two circles, that is, as AM:2AD,2 which is evidently 1:2. Hence a and likewise ß are half of δ, which means a + ß = δ, and the area of the crescent a + ß + γ = δ + γ = area of triangle ABD = BM2.

The fundamental importance of this discovery lies in proving the possibility of areas bounded by curved lines being commensurable with areas bounded by straight lines. The problem of "squaring the circle" derived its great appeal from this discovery. For to accomplish for the very simplest curvilinear figure what had been accomplished for another curvilinear figure was surely a powerful challenge which attracted mathematicians for two millenniums; and even today, after the impossibility of its solution has long been demonstrated, non-mathematicians who do not understand this impossibility proof are still trying to "square the circle." Hippocrates clearly understood the problem and pursued his goal in a most methodical fashion. He tried to find other meniscuses (crescents, or lunulae, such as the changing phases of the moon present in all possible forms) which would have the same property as the first one, in order to build up eventually the whole circle from such crescents. He found two additional ones, very cleverly designed, which, while not commensurable to the square over the chord, had areas equal to that of a certain polygon. Transforming such figures with compass and ruler into a square of equal area — to "square" it — was a problem which the mathematicians of those days apparently were already mastering completely. The problem of squaring the circle in this manner had thus been well defined, but this first, broad attempt to solve it ended in failure because the circle could not be built up from the crescents which Hippocrates had constructed.

In those days, too, there were people who missed the point of the problem, for example, the Sophist Antiphon, who lived in Athens at about the same time as Hippocrates. Aristotle relates of him that he inscribed a square in a circle (Fig. 9), then constructed isosceles triangles over its sides forming a regular octagon inscribed in the circle, then similarly a regular polygon of sixteen sides, and so on. As any such rectilinear polygon could be transformed into a square, Antiphon believed that there must be a polygon of a sufficiently great number of sides which would be identical with the circle and that the square into which that polygon could then be transformed would be the solution.

(Continues…)

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Table of Contents

FOREWORD
PREFACE TO THE GERMAN EDITION
PREFACE TO THE ENGLISH EDITION

I . THE NATURE OF THE INFINITE PROCESS

1 . The Beginnings of Greek Speculation on Infinitesimals
2 . The Greek Theory of Proportions
3 . The Exhaustion Method of the Greeks
4 . The Modern Number Concept
5 . Archimedes’ Measurements of the Circle and the Sine Tables
6 . The Infinite Geometric Series
7 . Continuous Compound Interest
8 . Periodic Decimal Fractions
9 . Convergence and Limit
10 . Infinite Series

II . THE DEFINITE INTEGRAL

11 . The Quadrature of the Parabola by Archimedes 
12 . Continuation after 1. 880 Years
13 . Area and Definite Integral
14 . Non-rigorous Infinitesimal Methods
15 . The Concept of the Definite Integral 
16 . Some Theorems on Definite Integrals
17 . Questions of Principle

III . DIFFERENTIAL AND INTEGRAL CALCULUS

18 . Tangent Problems
19 . Inverse Tangent Problems
20 . Maxima and Minima 
21 . Velocity
22 . Napier
23 . The Fundamental Theorem
24 . The Product Rule
25 . Integration by Parts
26 . Functions of Functions
27. Transformation of Integrals
28 . The Inverse Function
29 . Trigonometric Functions
30 . Inverse Trigonometric Functions
31 . Functions of Several Functions
32 . Integration of Rational Functions
33 . Integration of Trigonometric Expressions
34 . Integration of Expressions Involving Radicals
35 . Limitations of Explicit Integration

IV . APPLICATIONS TO PROBLEMS OF MOTION

36 . Velocity and Acceleration
37 . The Pendulum
38 . Coordinate Transformations
39 . Elastic Vibrations
40 . Kepler’s First Two Laws 
41 . Derivation of Kepler’s First Two Laws from Newton’s Law
42 . Kepler’s Third Law

EXERCISES

BIBLIOGRAPHY
Works on the History of Mathematics
Special Works on the History of the Infinitesimal Calculus

BIBLIOGRAPHICAL NOTES
INDEX

Editorial Reviews

"Appropriate for students who have completed basic or high school calculus but have not yet stepped up to the rigors of advanced calculus. Here those students will find motivation for understanding techniques in response to the original problems that gave rise to them."

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