The Calculus Gallery: Masterpieces from Newton to Lebesgue
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The Calculus Gallery: Masterpieces from Newton to Lebesgue
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Product Details
| ISBN-13: | 9780691182858 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 11/13/2018 |
| Series: | Princeton Science Library , #60 |
| Edition description: | New |
| Pages: | 256 |
| Product dimensions: | 5.40(w) x 8.40(h) x 0.80(d) |
About the Author
Read an Excerpt
CHAPTER 1
Newton
Isaac Newton (1642–1727) stands as a seminal figure not just in mathematics but in all of Western intellectual history. He was born into a world where science had yet to establish a clear supremacy over medieval superstition. By the time of his death, the Age of Reason was in full bloom.
This remarkable transition was due in no small part to his own contributions.
For mathematicians, Isaac Newton is revered as the creator of calculus, or, to use his name for it, of "fluxions." Its origin dates to the mid-1660s when he was a young scholar at Trinity College, Cambridge. There he had absorbed the work of such predecessors as René Descartes (1596–1650), John Wallis (1616–1703), and Trinity's own Isaac Barrow (1630–1677), but he soon found himself moving into uncharted territory. During the next few years, a period his biographer Richard Westfall characterized as one of "incandescent activity," Newton changed forever the mathematical landscape. By 1669, Barrow himself was describing his colleague as "a fellow of our College and very young ...
but of an extraordinary genius and proficiency".
In this chapter, we look at a few of Newton's early achievements: his generalized binomial expansion for turning certain expressions into infinite series, his technique for finding inverses of such series, and his quadrature rule for determining areas under curves. We conclude with a spectacular consequence of these: the series expansion for the sine of an angle.
Newton's account of the binomial expansion appears in his epistola prior, a letter he sent to Leibniz in the summer of 1676 long after he had done the original work. The other discussions come from Newton's 1669 treatise De analysi per aequationes numero terminorum infinitas, usually called simply the De analysi.
Although this chapter is restricted to Newton's early work, we note that "early" Newton tends to surpass the mature work of just about anyone else.
Generalized Binomial Expansion
By 1665, Isaac Newton had found a simple way to expand — his word was "reduce" — binomial expressions into series. For him, such reductions would be a means of recasting binomials in alternate form as well as an entryway into the method of fluxions. This theorem was the starting point for much of Newton's mathematical innovation.
As described in the epistola prior, the issue at hand was to reduce the binomial (P + PQ)mn and to do so whether m/n "is integral or (so to speak) fractional, whether positive or negative". This in itself was a bold idea for a time when exponents were sufficiently unfamiliar that they had first to be explained, as Newton did by stressing that "instead of [MATHEMATICAL EXPRESSION OMITTED], etc. I write a1/2, a1/3, a5/3, and instead of 1/a, 1/aa, 1/a3, I write a-1, a-2, a-3".
Apparently readers of the day needed a gentle reminder.
Newton discovered a pattern for expanding not only elementary binomials like (1 + x)5 but more sophisticated ones like [MATHEMATICAL EXPRESSION OMITTED]. The reduction, as Newton explained to Leibniz, obeyed the rule
[MATHEMATICAL EXPRESSION OMITTED] (1)
where each of A, B, C, ... represents the previous term, as will be illustrated below. This is his famous binomial expansion, although perhaps in an unfamiliar guise.
Newton provided the example of [MATHEMATICAL EXPRESSION OMITTED]. Here, [MATHEMATICAL EXPRESSION OMITTED], m = 1, and n = 2. Thus,
[MATHEMATICAL EXPRESSION OMITTED]
To identify A, B, C, and the rest, we recall that each is the immediately preceding term. Thus, A = (c2)1/2 = c, giving us
[MATHEMATICAL EXPRESSION OMITTED]
Likewise B is the previous term — i.e., B = x2/2c — so at this stage we have
[MATHEMATICAL EXPRESSION OMITTED]
The analogous substitutions yield C = x4/8c3 and then D = x6/16c5.
Working from left to right in this fashion, Newton arrived at
[MATHEMATICAL EXPRESSION OMITTED]
Obviously, the technique has a recursive flavor: one finds the coefficient of x8 from the coefficient of x6, which in turn requires the coefficient of x4, and so on. Although the modern reader is probably accustomed to a "direct" statement of the binomial theorem, Newton's recursion has an undeniable appeal, for it streamlines the arithmetic when calculating a numerical coefficient from its predecessor.
For the record, it is a simple matter to replace A, B, C, ... by their equivalent expressions in terms of P and Q, then factor the common Pm/n from both sides of (1), and so arrive at the result found in today's texts:
[MATHEMATICAL EXPRESSION OMITTED] (2)
Newton likened such reductions to the conversion of square roots into infinite decimals, and he was not shy in touting the benefits of the operation. "It is a convenience attending infinite series," he wrote in 1671,
that all kinds of complicated terms ... may be reduced to the class of simple quantities, i.e., to an infinite series of fractions whose numerators and denominators are simple terms, which will thus be freed from those difficulties that in their original form seem'd almost insuperable.
To be sure, freeing mathematics from insuperable difficulties is a worthy undertaking.
One additional example may be helpful. Consider the expansion of [MATHEMATICAL EXPRESSION OMITTED], which Newton put to good use in a result we shall discuss later in the chapter. We first write this as (1 - x2)-1/2, identify m = - 1, n = 2, and Q = - x2, and apply (2):
[MATHEMATICAL EXPRESSION OMITTED] (3)
Newton would "check" an expansion like (3) by squaring the series and examining the answer. If we do the same, restricting our attention to terms of degree no higher than x8, we get
[MATHEMATICAL EXPRESSION OMITTED]
where all of the coefficients miraculously turn out to be 1 (try it!). The resulting product, of course, is an infinite geometric series with common ratio
x2 which, by the well-known formula, sums to 1/1-x2. But if the square of the series in (3) is 1/1-x2, we conclude that that series itself must be [MATHEMATICAL EXPRESSION OMITTED]. Voila!
Newton regarded such calculations as compelling evidence for his general result. He asserted that the "common analysis performed by means of equations of a finite number of terms" may be extended to such infinite expressions "albeit we mortals whose reasoning powers are confined within narrow limits, can neither express nor so conceive all the terms of these equations, as to know exactly from thence the quantities we want".
Inverting Series
Having described a method for reducing certain binomials to infinite series of the form z = A + Bx + Cx2 + Dx3 + ···, Newton next sought a way of finding the series for x in terms of z. In modern terminology, he was seeking the inverse relationship. The resulting technique involves a bit of heavy algebraic lifting, but it warrants our attention for it too will appear later on. As Newton did, we describe the inversion procedure by means of a specific example.
Beginning with the series z = x - x2 + x3 - x4 + ···, we rewrite it as
[MATHEMATICAL EXPRESSION OMITTED]
and discard all powers of x greater than or equal to the quadratic.
This, of course, leaves x - z = 0, and so the inverted series begins as x = z.
Newton was aware that discarding all those higher degree terms rendered the solution inexact. The exact answer would have the form x = z + p, where p is a series yet to be determined. Substituting z + p for x in (4) gives
[MATHEMATICAL EXPRESSION OMITTED]
which we then expand and rearrange to get
[MATHEMATICAL EXPRESSION OMITTED]
Next, jettison the quadratic, cubic, and higher degree terms in p and solve to get
[MATHEMATICAL EXPRESSION OMITTED]
Newton now did a second round of weeding, as he tossed out all but the lowest power of z in numerator and denominator. Hence p is approximately , so the inverted series at this stage looks like x = z + p = z + z2.
But p is not exactly z2. Rather, we say p = z2 + q, where q is a series to be determined. To do so, we substitute into (5) to get
We expand and collect terms by powers of q:
[MATHEMATICAL EXPRESSION OMITTED]
As before, discard terms involving powers of q above the first, solve to get [MATHEMATICAL EXPRESSION OMITTED], and then drop all but the lowest degree terms top and bottom to arrive at q = z3/1. At this point, the series looks like x = z + z + q = z + z + z.
The process would be continued by substituting q = z3 + r into (6). Newton, who had a remarkable tolerance for algebraic monotony, seemed able to continue such calculations ad infinitum (almost). But eventually even he was ready to step back, examine the output, and seek a pattern. Newton put it this way: "Let it be observed here, by the bye, that when 5 or 6 terms ... are known, they may be continued at pleasure for most part, by observing the analogy of the progression".
For our example, such an examination suggests that x = z + z2 + z3 + z4 + z5 + · · · is the inverse of the series z = x - x2 + x3x4 + ··· with which we began.
In what sense can this be trusted? After all, Newton discarded most of his terms most of the time, so what confidence remains that the answer is correct?
Again, we take comfort in the following "check." The original series z = x - x2 + x3 - x4 + ··· is geometric with common ratio - x, and so in closed form z = x/1 + x. Consequently, x = z/1 - z, which we recognize to be the sum of the geometric series z + z2 + z3 + z4 + z5 + ···. This is precisely the result to which Newton's procedure had led us. Everything seems to be in working order.
The techniques encountered thus far — the generalized binomial expansion and the inversion of series — would be powerful tools in Newton's hands. There remains one last prerequisite, however, before we can truly appreciate the master at work.
Quadrature Rules from the De Analysi
In his De analysi of 1669, Newton promised to describe the method "which I had devised some considerable time ago, for measuring the quantity of curves, by means of series, infinite in the number of terms". This was not Newton's first account of his fluxional discoveries, for he had drafted an October 1666 tract along these same lines. The De analysi was a revision that displayed the polish of a maturing thinker. Modern scholars find it strange that the secretive Newton withheld this manuscript from all but a few lucky colleagues, and it did not appear in print until 1711, long after many of its results had been published by others. Nonetheless, the early date and illustrious authorship justify its description as "perhaps the most celebrated of all Newton's mathematical writings".
The treatise began with a statement of the three rules for "the quadrature of simple curves." In the seventeenth century, quadrature meant determination of area, so these are just integration rules.
Rule 1. The quadrature of simple curves: If y = axm/n is the curve AD, where a is a constant and m and n are positive integers, then the area of region ABD is [MATHEMATICAL EXPRESSION OMITTED] (see figure 1.1).
A modern version of this would identify A as the origin, B as (x, 0), and the curve as y = atm/n Newton's statement then becomes [MATHEMATICAL EXPRESSION OMITTED], which is just a special case of the power rule from integral calculus.
Only at the end of the De analysi did Newton observe, almost as an afterthought, that "an attentive reader" would want to see a proof for Rule 1. Attentive as always, we present his argument below.
Again, let the curve be AD with AB = x and BD = y, as shown in figure 1.2. Newton assumed that the area ABD beneath the curve was given by an expression z written in terms of x. The goal was to find a corresponding formula for y in terms of x. From a modern vantage point, he was beginning with [MATHEMATICAL EXPRESSION OMITTED] and seeking y = y(x). His derivation blended geometry, algebra, and fluxions before ending with a few dramatic flourishes.
At the outset, Newton let ß be a point on the horizontal axis a tiny distance o from B. Thus, segment Aß has length x + o. He let z be the area ABD, although to emphasize the functional relationship we shall take the liberty of writing z = z(x). Hence, z(x + o) is the area Aßd under the curve. Next he introduced rectangle BßHK of height v = BK = ßH, the area of which he stipulated to be exactly that of region BßdD beneath the curve. In other words, the area of BßdD was to be ov.
At this point, Newton specified that [MATHEMATICAL EXPRESSION OMITTED] and proceeded to find the instantaneous rate of change of z. To do so, he examined the change in z divided by the change in x as the latter becomes small. For notational ease, he temporarily let c = an/(m + n) and p = m + n so that z(x) = cxp and
[MATHEMATICAL EXPRESSION OMITTED] (7)
Now, z(x + o) is the area Aßd, which can be decomposed into the area of ABD and that of BßdD. The latter, as noted, is the same as rectangular area ov and so Newton concluded that z(x + o) = z(x) + ov. Substituting into (7), he got
[MATHEMATICAL EXPRESSION OMITTED]
and the binomials on the left and right were expanded to
[MATHEMATICAL EXPRESSION OMITTED]
Applying (7) to cancel the leftmost terms on each side and then dividing through by o, Newton arrived at
[MATHEMATICAL EXPRESSION OMITTED] (8)
At that point, he wrote, "If we suppose Bß to be diminished infinitely and to vanish, or o to be nothing, v and y in that case will be equal, and the terms which are multiplied by o will vanish". He was asserting that, as o becomes zero, so do all terms in (8) that contain o. At the same time, v becomes equal to y, which is to say that the height BK of the rectangle in figure 1.2 will equal the ordinate BD of the original curve. In this way, (8) transforms into
[MATHEMATICAL EXPRESSION OMITTED] (9)
A modern reader is likely to respond, "Not so fast, Isaac!" When Newton divided by o, that quantity most certainly was not zero. A moment later, it was zero. There, in a nutshell, lay the rub. This zero/nonzero dichotomy would trouble analysts for the next century and then some. We shall have much more to say about this later in the book.
But Newton proceeded. In (9) he substituted for z(x), c, and p and solved for
[MATHEMATICAL EXPRESSION OMITTED]
Thus, starting from his assumption that the area ABD is given by [MATHEMATICAL EXPRESSION OMITTED], Newton had deduced that curve AD must satisfy the equation y = axmn He had, in essence, differentiated the integral. Then, without further justification, he stated, "Wherefore conversely, if axmn = y, it shall be [MATHEMATICAL EXPRESSION OMITTED]." His proof of rule 1 was finished.
This was a peculiar twist of logic. Having derived the equation of y from that of its area accumulator z, Newton asserted that the relationship went the other way and that the area under y = axm is indeed [MATHEMATICAL EXPRESSION OMITTED]. Such an argument tends to leave us with mixed feelings, for it features some gaping logical chasms. Derek Whiteside, editor of Newton's mathematical papers, aptly characterized this quadrature proof as "a brief, scarcely comprehensible appearance of fluxions". On the other hand, it is important to remember the source. Newton was writing at the very beginning of the long calculus journey. Within the context of his time, the proof was groundbreaking, and his conclusion was correct. Something rings true in Richard Westfall's observation that, "however briefly, De analysi did indicate the full extent and power of the fluxional method".
Whatever the modern verdict, Newton was satisfied. His other two rules, for which the De analysi contained no proofs, were as follows:
Rule 2. The quadrature of curves compounded of simple ones: If the value of y be made up of several such terms, the area likewise shall be made up of the areas which result from every one of the terms.
Rule 3. The quadrature of all other curves: But if the value of y, or any of its terms be more compounded than the foregoing, it must be reduced into more simple terms ... and afterwards by the preceding rules you will discover the [area] of the curve sought.
Newton's second rule affirmed that the integral of the sum of finitely many terms is the sum of the integrals. This he illustrated with an example or two. The third rule asserted that, when confronted with a more complicated expression, one was first to "reduce" it into an infinite series, integrate each term of the series by means of the first rule, and then sum the results.
(Continues…)
Excerpted from "The Calculus Gallery"
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Copyright © 2005 Princeton University Press.
Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
Illustrations, ix,
Acknowledgments, xiii,
Preface to the Princeton Science Library Edition, xv,
Introduction, 1,
Chapter 1 Newton, 5,
Chapter 2 Leibniz, 20,
Chapter 3 The Bernoullis, 35,
Chapter 4 Euler, 52,
Chapter 5 First Interlude, 69,
Chapter 6 Cauchy, 76,
Chapter 7 Riemann, 96,
Chapter 8 Liouville, 116,
Chapter 9 Weierstrass, 128,
Chapter 10 Second Interlude, 149,
Chapter 11 Cantor, 158,
Chapter 12 Volterra, 170,
Chapter 13 Baire, 183,
Chapter 14 Lebesgue, 200,
Afterword, 220,
Notes, 223,
Index, 233,
What People are Saying About This
Bill Dunham has done it again. The Calculus Gallery is a masterly journey through the works of thirteen mathematicians who formulated, formalised, and reformed the calculus into the modern analysis we learn today. Readers of his earlier books have learned to expect a clarity of exposition that few others can attain: they will not be disappointed.
— Robin Wilson, author of "Four Colors Suffice"
Pedagogically excellent and extremely well written, The Calculus Gallery bridges the gap between general histories and detailed studies of individual mathematicians. Dunham has described mathematical developments in an engaging style rarely found in literature of this kind.
— Annette Imhausen, Trinity Hall, Cambridge
The Calculus Gallery is one of the best efforts at mathematical exposition I have ever read! Dunham presents in detail and in his own words the sequence of ideas of classical giants of mathematics, but each new idea is described in modern terms and notation, so I had absolutely no trouble following along. Furthermore—and this is an astounding achievement—the entire work has a tightly woven development. If it were a detective story I would say it had a plot with no loose ends. An amazing feat. I wish I could plan a single lecture, never mind a course or a book, that well!
— Henry Pollak, Teachers College, Columbia University
What a fine resource! All of the famous functions that have shaped calculus and analysis parade before the reader in the original words of their creators. Bill Dunham has produced an excellent volume that teachers and students will enjoy and appreciate.
— Thomas Banchoff, Brown University
This is an excellent book—an amazing mathematical page-turner. William Dunham has done the seemingly impossible: he has taken some difficult, advanced mathematics and, without sacrificing the technical details, written a lively, readable book about it.
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A welcome addition to the literature. The idea of presenting a 'museum of mathematics' is new. It allows the author to present a nonstandard selection of theorems, so that even mathematicians with a strong historical background will learn a few things.
— Franz Lemmermeyer, Bilkent University, author of "Reciprocity Laws: From Euler to Eisenstein"