Über die Autorin bzw. den Autor

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.

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"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

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"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

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HOW TO READ Historical Mathematics

PRINCETON UNIVERSITY PRESS

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

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...