How to Read Historical Mathematics - Hardcover

Wardhaugh, Benjamin

Hardcover

ISBN 10: 0691140146 ISBN 13: 9780691140148

Publisher: Princeton University Press, 2010

Synopsis

Writings by early mathematicians feature language and notations that are quite different from what we're familiar with today. Sourcebooks on the history of mathematics provide some guidance, but what has been lacking is a guide tailored to the needs of readers approaching these writings for the first time. How to Read Historical Mathematics fills this gap by introducing readers to the analytical questions historians ask when deciphering historical texts. Sampling actual writings from the history of mathematics, Benjamin Wardhaugh reveals the questions that will unlock the meaning and significance of a given text--Who wrote it, why, and for whom? What was its author's intended meaning? How did it reach its present form? Is it original or a translation? Why is it important today? Wardhaugh teaches readers to think about what the original text might have looked like, to consider where and when it was written, and to formulate questions of their own. Readers pick up new skills with each chapter, and gain the confidence and analytical sophistication needed to tackle virtually any text in the history of mathematics. * Introduces readers to the methods of textual analysis used by historians * Uses actual source material as examples * Features boxed summaries, discussion questions, and suggestions for further reading * Supplements all major sourcebooks in mathematics history * Designed for easy reference * Ideal for students and teachers

About this title

About the Author

Benjamin Wardhaugh is a postdoctoral research fellow at All Souls College, University of Oxford. He is the author of "Music, Experiment, and Mathematics in England, 1653-1705".

From the Back Cover

"How to Read Historical Mathematics is definitely a significant contribution. There is nothing similar available. It will be a very important resource in any course that makes use of original sources in mathematics and to anyone else who wants to read seriously in the history of mathematics."--Victor J. Katz, editor of The Mathematics of Egypt, Mesopotamia, China, India, and Islam

"Wardhaugh guides mathematics students through the process of reading primary sources in the history of mathematics and understanding some of the main historiographic issues this study involves. This concise handbook is a very significant and, as far as I know, unique companion to the growing corpus of sourcebooks documenting major achievements in mathematics. It explicitly addresses the fundamental questions of why--and more importantly how--one should read primary sources in mathematics history."--Kim Plofker, author of Mathematics in India

From the Inside Flap

"How to Read Historical Mathematics is definitely a significant contribution. There is nothing similar available. It will be a very important resource in any course that makes use of original sources in mathematics and to anyone else who wants to read seriously in the history of mathematics."--Victor J. Katz, editor of The Mathematics of Egypt, Mesopotamia, China, India, and Islam

"Wardhaugh guides mathematics students through the process of reading primary sources in the history of mathematics and understanding some of the main historiographic issues this study involves. This concise handbook is a very significant and, as far as I know, unique companion to the growing corpus of sourcebooks documenting major achievements in mathematics. It explicitly addresses the fundamental questions of why--and more importantly how--one should read primary sources in mathematics history."--Kim Plofker, author of Mathematics in India

Excerpt. © Reprinted by permission. All rights reserved.

HOW TO READ Historical Mathematics

By Benjamin Wardhaugh

PRINCETON UNIVERSITY PRESS

Copyright © 2010 PRINCETON UNIVERSITY PRESS
All right reserved.
ISBN: 978-0-691-14014-8

Contents

Preface..............................................viiCHAPTER 1 What Does It Say?.........................1CHAPTER 2 How Was It Written?.......................21CHAPTER 3 Paper and Ink.............................49CHAPTER 4 Readers...................................73CHAPTER 5 What to Read, and Why.....................92Bibliography.........................................111Index................................................115

Chapter One

What Does It Say?

When the cube and the things together Are equal to some discrete number, Find two other numbers differing in this one. Then ... their product should always be equal Exactly to the cube of a third of the things. The remainder then as a general rule Of their cube roots subtracted Will be equal to your principal thing. —From Niccolò Tartaglia's account of the solutions to the cubic equation (1539) in Fauvel and Gray, The History of Mathematics: A Reader, pp. 255–56.

That's quite a mouthful. In your study of the history of mathematics, you'll quite often come across things like this. They can be baffling at first sight. On the other hand, the same piece of mathematics might be presented like this:

To solve x³ + cx = d, find u, v such that u – v = d and uv = (c/3)³. Then x = [cube root of u] - [cube root of v].

This looks much more straightforward: it's in a mathematical language which we can understand without much difficulty, and we can easily check whether it is true or not.

But it's not really obvious that the two versions say the same thing. Let's look in detail and see if we can trace how you get from one to the other. Before we start, pause for a moment and see how much of it you can make out yourself.

How far did you get? Give yourself a pat on the back if you managed to translate all eight lines into algebra and got some thing that made sense. Here's how it goes.

When the cube and things together

That's pretty cryptic, for a start. But I've told you that this is about solving cubic equations, so it's fair to assume that there's an unknown quantity—call it x—involved, and that "the cube" means x³.

What about these "things"? Well, if this is a cubic equation, they can only be (1) a multiple of x², (2) a multiple of x, or (3) a constant. If Tartaglia meant a multiple of x², he would surely say something about "squares" or "the square," so we can rule out (1). There seems to be no way to tell whether he means a multiple of x or a constant for the moment, so let's leave that and look at the next line.

Are equal to some discrete number,

That makes things a bit clearer. "Some discrete number" sounds pretty much like a constant—let's call it d. That means that "things" is most likely a multiple of x, not another constant. Let's call it cx. So putting the first two lines together gives us this: "x³ and cx together are equal to d." Or, to put it another way: x³ + cx = d.

We're getting somewhere. The first two lines state the problem; the rest of the quote presumably tells us how to solve it.

Find two other numbers differing in this one.

Suddenly we're lost again. Find two numbers—find u and v, say—differing in "this one." This what? Tartaglia means "this number": that is, the "discrete number" from the previous line, the constant that we called d. So this line means "find u and v differing by d" or "find u, v such that u – v = d."

Then ... their product should always be equal Exactly to the cube of a third of the things.

"Their product" is the product of u and v. It's meant to be equal to "the cube of a third of the things." The last time the word "things" was mentioned it meant cx. Here that would give us uv = (cx/3)³, right?

Wait a moment. If x is our unknown, we can't have it in our definition of u and v. What else can "things" mean?

Perhaps it means the coefficient: not cx but just c. That gives us uv = (c/3)³, which makes a lot more sense. Now,

The remainder then as a general rule Of their cube roots subtracted

"The remainder ... of their cube roots subtracted"—this has to mean "the remainder when their cube roots are sub tracted from each other." If we subtracted them from anything else, we wouldn't get one remainder, but two. So these lines mean [cube root of u] - [cube root of v].

Will be equal to your principal thing.

There's no prize for guessing that "your principal thing" is the unknown quantity we are looking for, x. So these final lines mean x = [cube root of [u] - [cube root of [v].

If you go back and look at what we've done, you'll see that the first two lines tell us that x³ + cx = d; then the next six lines go on to tell us how to solve this equation: in line 3 we're told to find u and v such that u – v = d, and in lines 4 and 5 Tartaglia says we must also have uv = (c/3)³. Then, in lines 6 through 8 he reveals that this gives us a solution: x = [cube root of [u] - [cube root of [v].

You've just seen how to translate sixteenth-century words into modern algebra: not a trivial task, but not an impossible one either. If you want some practice, there's another example similar to this one at the end of the chapter.

Here's another example, with some harder mathematics in it.

Quantities ... which in any finite time constantly tend to equality, and which before the end of that time approach so close to one another that their difference is less than any given quantity, become ultimately equal.

—Isaac Newton, Philosophiæ naturalis principia mathematica, Book 1, Lemma 1, translated by I. Bernard Cohen and Anne Whitman, 1999.

Once again, you might like to pause before reading on and have a try at translating this passage into modern notation, just as we did with the piece from Tartaglia. See how far you can get, and don't worry if you get stuck.

Here's how we might translate this passage into modern notation.

"Quantities," the extract begins. Newton is talking about two quantities: let's call them X and Y. They change over time, and we're interested in their behavior over time, so we'll consider them functions X(t) and Y(t) of time t. In particular, we are interested in their behavior "in a given time." Assuming that time is finite, we can call it the period from t = 0 to t = t₁.

During that time, Newton says, the two quantities "constantly tend to equality." That means the difference between them always gets smaller; in other words, |X(t) – Y(t)| is de creasing during our time interval.

Next, Newton gives a second condition on the two quantities, a more demanding one. During the specified time period they "approach so close to one another that their difference is less than any given quantity." If we call the "given quantity" e, what that means is that sooner or later the difference will be less than e. That is to say, we can find a time t' for which |X(t') – Y(t')| < e. And that's true for any "given quantity": any value of e. In modern terms, for all e there exists t' in [0, t₁] such that |X(t') – Y(t')| < e.

The last part of Newton's sentence tells us the consequence of these two conditions. If the conditions are met, he says, the two quantities "become ultimately equal." "Equal" obviously means that X = Y; "ultimately" means that this happens at the end of the time interval we're considering, at t = t₁. So, X(t₁) = Y(t₁).

So we've found that Newton's statement can be rewritten in modern notation like this:

Given X(t) and Y(t) with |X(t) – Y(t)| decreasing from t = 0 to t = t₁, if [for all]e [there exists]t'[member of][0, t₁] such that |X(t') – Y(t')|< e, then X(t₁) = Y(t₁).

A bit harder than Tartaglia, but again, not an impossible task.

Spotting the difference

But now let's mess things up. Do you think this modern version is really—strictly—equivalent to what Newton wrote? Be as picky as you can, and see if you can find some places where the two are not quite the same. Go back through our process of translation if you want, and check whether everything is absolutely watertight. You might find that it's not: you might have noticed even as we made our translation that we were introducing some changes that are not just matters of notation or of style.

Can you spot any differences? You might get some hints by comparing our modern version of Newton with the definition of a limit that you find in a modern textbook.

What points did you come up with? These are really picky things, and I'd like to consider just four of them, though you might have found more than that.

First: What exactly does Newton mean by "constantly tend"? We've said that |X(t) – Y(t)| is "decreasing," but for us that could mean that X(t) – Y(t) is constant in places, or even constant everywhere, making X(t) and Y(t) equal across the whole interval. I doubt that's what Newton means—"constantly tend" gives the impression of things changing, not staying the same. Perhaps the idea of two quantities "constantly" tending to be come equal might be better expressed by saying explicity that (1) they're not equal to begin with and (2) the difference between them is strictly (i.e., always) decreasing. Should we put that into our modern version?

Second: What exactly does Newton mean by a "quantity"? A real number? Yes and no. When were the real numbers first rigorously defined? Not until a long time after Newton: the late 1800s, in fact. That doesn't stop him from intuitively using them, and it seems obvious that he's thinking about quantities which change continuously here. On the other hand, it's not clear that the result is still true if the "quantities" and the "given quantity" are rationals, yet it seems a bit cheeky to foist onto Newton the condition that everything in sight is a real number when he wouldn't actually have known precisely what that meant. Should we have said that X, Y, t, e, and t' are all real numbers, or shouldn't we?

Third: Can e be negative? No, it can't; that wouldn't make sense—and it can't be zero either. So if we want our modern version to be strictly rigorous, we would have to specify that e > 0. Newton doesn't say that; he just says "any given quantity" without pointing out that it has to be a positive nonzero quantity. Maybe for him a "quantity" is necessarily positive. Maybe he just thought it was obvious. Should we leave our version as it is, or should we put in that e > 0?

Fourth: These are not the only differences you might have spotted, but it's time to move on—suppose that X(t) = 1/(t₁ – t) and Y(t) = X(t) – t₁ + t. Think about it for a moment. The difference between them behaves as it should: it's t₁ – t in the interval, and it satisfies the second condition too. But neither X nor Y is defined at t₁, so the result we're interested in—X(t₁) = Y(t₁)—is false in this case. What can we do?

There are ways to fix this up—we could add a condition that X and Y be bounded in the interval, for example, or that they be defined at t₁—but once again we'd be putting in something that Newton didn't say. Apparently he either didn't know about badly behaved functions like these or didn't care—or maybe he just thought it was obvious that they were not what he was talking about. Is it better to fix our modern version so it excludes cases like these, or to leave it closer to what's in Newton's version?

For each of the changes I've suggested making to our modern version, you could argue it either way, and I'll leave it up to you to decide what's best to do. The point is that there are some differences between what Newton said and our translation of it into modern mathematical language, and it's not easy to eradicate them without straying from what Newton really wrote. We'll come back to this again and again later in this chapter and throughout this book.

You probably found this exercise a bit picky. But think what we've learned about Newton's mathematics that we might have missed otherwise. His idea of "constantly tending" says more than our idea of "decreasing." When he says "quantities," he might be talking about the real numbers, but we can't really be sure exactly what he has in mind. He assumes that "any given quantity" is greater than zero, without saying so. And he assumes that his "quantities" are finite and generally well behaved in the interval he considers.

You've now seen two ways of finding out about a piece of old mathematics. The first is to translate it into modern terms; the second is to look closely at how the translation isn't exactly the same as the original. You can practice them as you go through the rest of this book, and on any other bits of historical mathematics you meet—and you'll find you can learn an awful lot about a piece of old mathematics like this. The box gives a summary of this second way. It gives a selection of questions you can ask to help you spot the difference between a modern version and the original; but those are not the only questions you can ask. See if you can think of some others of your own. Throughout this book we'll see that being a historian of mathematics is about learning to ask your own questions.

Translation

There's something else, too. Newton didn't say "quantity," and Tartaglia didn't say "some discrete number" or anything like it. They wrote in foreign languages: Newton in Latin and Tartaglia in Italian. Does that matter?

In a word, yes. Very often in the history of mathematics you'll read texts that have been translated from one language to another. A good translation will turn the words into English but should leave the mathematics and its notation more or less alone. If you suspect that the mathematics itself has been tampered with, or if you're lucky enough to be able to read the original language, you can always try to find a copy of the original version and have a look at it.

We'll spend some time in chapter 2 thinking about how to find sources for the history of mathematics in libraries and on the web. For the moment I'll just say that the first (Latin) edition of Newton's Principia is on at least one open-access website. In chapter 3 you'll learn about some of the other things you can find out by looking at the original source, and we'll also think in depth about some of the ways that sources can be tampered with before they get to us.

If you don't, or can't, have a look at the original, it might be possible to find more than one English translation of it. There aren't a great many pieces of historical mathematics that have been translated into English more than once, but not surprisingly Newton's Principia is one of them. Here's the same passage we looked at above, but from an older translation. (In fact, this was the first English translation of Newton to be published—in 1729—and it's on Google Books).

(Continues...)

Excerpted from HOW TO READ Historical Mathematicsby Benjamin Wardhaugh Copyright © 2010 by PRINCETON UNIVERSITY PRESS. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Edition details

• Publisher: Princeton University Press
• Publication date: 2010
• Language: English
• ISBN 10: 0691140146
• ISBN 13: 9780691140148
• Binding: Hardcover
• Number of pages: 136