History

Lissajous and Bowditch

Posted on by John

I was skimming through David Well’s book The Penguin Dictionary of Curious and Interesting Geometry and was surprised by this entry:

Lissajous figures or Bowditch curves First discussed by Nathaniel Bowditch in 1815, and later by Jules Antoine Lissajous in 1857 …

I wrote a couple posts about Lissajous curves years ago, and recently I wrote a couple posts about Nathan Bowditch, but I didn’t realize there was a connection between them.

I asked Claude to pin down Bowditch’s publication to see whether the curves came up in the course of his work on navigation. Here’s the answer I got.

Bowditch’s paper was “On the Motion of a Pendulum Suspended from Two Points,” published in the Memoirs of the American Academy of Arts and Sciences (Vol. 3, Part 2) in 1815.

He studied a compound pendulum hung from two points (a Y-shaped suspension, sometimes called a Blackburn pendulum). This lets the bob swing at different frequencies in two perpendicular directions, and he worked out mathematically the curves the bob traces. Those curves are what we now call Lissajous figures. Jules Antoine Lissajous studied them independently and much more visibly in 1857, using light beams reflected off vibrating tuning forks, and his name stuck. Because of Bowditch’s priority, the curves are sometimes called Bowditch curves.

Nathaniel Bowditch

Posted on by John

A couple days ago a friend told me about the book Carry On, Mr. Bowditch, a fictional account of the life of Nathaniel Bowditch (1773–1838). I’ve been listening to the book on Audible, and apparently it’s only lightly fictionalized.

Bowditch was a self-educated mathematician and astronomer, best known for his book The American Practical Navigator, first published in 1801. The book has been continually revised over the last two centuries and is still in print, available for download from the National Geospatial-Intelligence Agency. The latest edition begins with a brief account of Bowditch’s life, confirming the essential details of the fictional biography.

Two things stand out about Bowditch: his attention to detail and his desire to make ideas accessible. He taught himself Latin in order to read Newton’s Principia and followed the text so closely that he found a number of errors.

Bowditch’s navigation book grew out of the numerous corrections he made to error he found in John Hamilton Moore’s The Practical Navigator, the leading navigation text of the time.

At the beginning of the 19th century it was theoretically possible to determine time, and hence longitude, from lunar observation. However, the method required ideal observation conditions and laborious calculation. Bowditch developed a way to make the necessary measurements under more general conditions, and simplified the necessary calculations. According to the biographical preface mentioned above,

Bowditch vowed while writing this edition [of his navigation text] to “put down in the book nothing I can’t teach the crew,” and it is said that every member of his crew including the cook could take a lunar observation and plot the ship’s position.

After completing The American Practical Navigator, Bowditch began an English translation of Pierre Laplace’s encyclopedic Mecanique Celeste, filling in details to make the work accessible to a wider audience. He was able to translate four out of the five volumes by the end of his life.

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Impersonating an Edwardian math professor

Posted on by John

I’ve read some math publications from around a century or so ago, and I wondered if I could pull off being a math professor if a time machine dropped me into a math department from the time. I think I’d come across as something of an autistic savant, ignorant of what contemporaries would think of as basic math but fluent in what they’d consider more advanced.

There are two things in particular that were common knowledge at the time that I would be conspicuously ignorant of: interpolation tricks and geometry.

People from previous eras knew interpolation at a deeper level than citing the Lagrange interpolation theorem, out of necessity. They learned time-saving tricks have since been forgotten.

The biggest gap in my knowledge would be geometry. Mathematicians a century ago had a far deeper knowledge of geometry, particularly synthetic geometry, i.e. geometry in the style of Euclid rather than in the style of Descartes.

Sometimes older math books use notation or terminology that has since changed. I imagine I’d make a few gaffs, not immediately understanding a basic term or using a term that wasn’t coined until later.

If I had to teach a class, I’d choose something like real and complex analysis. Whittaker & Watson’s book on the subject was first published in 1902 and remains a common reference today. The only thing I find jarring about that book is that “show” is spelled “shew.” Makes me think of Ed Sullivan. But I think I’d have a harder time teaching a less advanced class.

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A couple more variations on an ancient theme

Posted on by John

I’ve written a couple posts on the approximation

\cos x \approx \frac{\pi^2 - 4x^2}{\pi^2 + x^2}

by the Indian astronomer Aryabhata (476–550). The approximation is accurate for x in [−π/2, π/2].

The first post collected a Twitter thread about the approximation into a post. The second looked at how far the coefficients in Aryabhata’s approximation are from the optimal approximation as a ratio of quadratics.

This post will answer a couple questions. First, what value of π did Aryabhata have and how would that effect the approximation error? Second, how bad would Aryabhata’s approximation be if we used the approximation π² ≈ 10?

Using Aryabhata’s value of π

Aryabhata knew the value 3.1416 for π. We know this because he said that a circle of diameter 20,000 would have circumference  62,832. We don’t know, but it’s plausible that he knew π to more accuracy and rounded it to the implied value.

Substituting 3.1416 for π changes the sixth decimal of the approximation, but the approximation is good to only three decimal places, so 3.1416 is as good as a more accurate approximation as far as the error in approximating cosine is concerned.

Using π² ≈ 10

Substituting 10 for π² in Aryabhata’s approximation gives an approximation that’s convenient to evaluate by hand.

\cos x \approx \frac{10 - 4x^2}{10 + x^2}

It’s very accurate for small values of x but the maximum error increases from 0.00163 to 0.01091. Here’s a plot of the error.

Ancient accurate approximation for sine

Posted on by John

This post started out as a Twitter thread. The text below is the same as that of the thread after correcting an error in the first part of the thread. I also added a footnote on a theorem the thread alluded to.

***

The following approximation for sin(x) is remarkably accurate for 0 < x < π.

\sin(x) \approx \frac{16x(\pi - x)}{5\pi^2 - 4x(\pi - x)}

The approximation is so good that you can’t see the difference between the exact value and the approximation until you get outside the range of the approximation.

Here’s a plot of just the error.

This is a very old approximation, dating back to Aryabhata I, around 500 AD.

In modern terms, it is a rational approximation, quadratic in the numerator and denominator. It’s not quite optimal since the ripples in the error function are not of equal height [1], but the coefficients are nice numbers.

***

As pointed out in the comments, replacing x with π/2 − x in order to get an approximation for cosine gives a much nicer equation.

\cos x \approx \frac{\pi^2 - 4x^2}{\pi^2 + x^2}

***

[1] The equioscillation theorem says that the optimal approximation will have ripples of equal positive and negative amplitude. This post explores the equioscillation theorem further and finds how far Aryabhata’s is from optimal.

New Twitter account: ElementFact

Posted on by John

I started a new Twitter account this morning: @ElementFact. I’m thinking the account will post things like scientific facts about each element but also some history around how the element was discovered and named and other lore associated with the element.

@ElementFact icon

We’ll see how this goes. I’ve started many Twitter accounts over the years. Some have taken off and some have not.

Six months ago I started @TensorFact as an experiment to see what would happen with a narrowly focused account. It did moderately well, but I ran out of ideas for content. This will be the last week for that account.

You can see a list of my current Twitter accounts here.

It seemed like a good idea at the time

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“Things are the way they are because they got that way … one logical step at a time.” — Gerald Weinberg

English spelling is notoriously difficulty. It is the result of decisions that, while often unfortunate, were not unreasonable at the time.

Sometimes pronunciation simplified but spelling remained unchanged. For example, originally all the letters in knife were pronounced. In software development lingo, some spellings were retained for backward compatibility.

Sometimes pronunciation was chosen to reflect etymology. This seems like a strange choice now, but it made more sense at a time when Latin and French were far more commonly known in England, and a time when loan words were pouring into English. These choices have turned out to be unfortunate, but they were not malicious.

For more on this story, see Episode 153: Zombie Letters from The History of English Podcast.

Aquinas on epicycles

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C. S. Lewis quotes Thomas Aquinas in The Discarded Image:

In astronomy an account is given of eccentricities and epicycles on the ground that if their assumption is made the sensible appearances as regards celestial motion can be saved. But this is not a strict proof since for all we know they could also be saved by some different assumption.

A warped perspective on math history

Posted on by John

Yesterday I posted on @TopologyFact

The uniform limit of continuous functions is continuous.

John Baez replied that this theorem was proved by his “advisor’s advisor’s advisor’s advisor’s advisor’s advisor.” I assume he was referring to Christoph Gudermann.

The impressive thing is not that Gudermann was able to prove this simple theorem. The impressive thing is that he saw the need for the concept of uniform convergence. My impression from reading the Wikipedia article on uniform convergence is that Gudermann alluded to uniform convergence in passing and didn’t explicitly define it or formally prove the theorem above. He had the idea and applied it but didn’t see the need to make a fuss about it. His student Karl Weierstrass formalized the definition and saw how generally useful the concept was.

It’s easy for a student to get a warped perspective of math history. You might implicitly assume that mathematics was developed in the order that you learn it. If as a student you learn about uniform convergence and that the term was coined around 1840, you might reasonably conclude that in 1840 mathematicians were doing what is now sophomore-level math, which is far from true.

Gudermann tossed off the idea of uniform convergence in passing while working on elliptic functions, a topic I wasn’t exposed to until sometime after graduate school. My mathematics education was more nearly reverse-chronological than chronological. I learned 20th century mathematics in school and 19th century mathematics later. Much of the former was a sort of dehydrated abstraction of the latter. Much of my career has been rehydrating, discovering the motivation for and application of ideas I was exposed to as a student.

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Proto-calculus

Posted on by John

David Bressoud has written a new book entitled Calculus Reordered: A History of the Big Ideas. He presents the major themes of calculus in historical order, which is very different from the order in which it is now taught. We now begin with limits, then differentiation, integration, and infinite series. Historically, integration came first and the rigorous definition of limits came last.

I wanted to quote a short excerpt from the book discussing a manuscript of Archimedes. In 1906, Johan Heiberg discovered that a medieval prayer book had recycled vellum that had contained an account of the methods used by Archimedes to compute areas and volumes, a sort of proto-calculus book. Researchers were able to reconstruct much of the original text that had been scraped off in order to reuse the vellum.

In 2003 a NOVA documentary speculated that calculus could have been developed centuries earlier had the manuscript not been lost, and that technology would be far ahead of where it is now, and that “We could have been on Mars today.”

Bressoud does not agree.

That is nonsense. As we shall see, Archimedes’ other works were perfectly sufficient to lead the way toward the development of calculus. The delay was not caused by an incomplete understanding of Archimedes’ methods but by the need to develop other mathematical tools. In particular, scholars needed the modern symbolic language of algebra and its application to curves before they could make substantial progress toward calculus as we know it.

Calculus Reordered book cover

Related post: History of the Central Limit Theorem