History

The 1970s

Posted on by John

Here’s a perspective on the 1970s I found interesting: The decade was so embarrassing that climbing out of the ’70s was a proud achievement.

The 1970s were America’s low tide. Not since the Depression had the country been so wracked with woe. Never — not even during the Depression — had American pride and self-confidence plunged deeper. But the decade was also, paradoxically, in some ways America’s finest hour. America was afflicted in the 1970s by a systemic crisis analogous to the one that struck Imperial Rome in the middle of the third century A.D. … But unlike the Romans, Americans staggered only briefly before the crisis. They took the blow. For a short time they behaved foolishly, and on one or two occasions, even disgracefully. Then they recouped. They rethought. They reinvented.

Source: How We Got Here: The 70’s: The Decade That Brought You Modern Life—For Better or Worse

An algebra problem from 1798

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The Lady’s Diary was a popular magazine published in England from 1704 to 1841. It contained mathematical puzzles such as the following, published in 1798.

What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?

From Benjamin Wardhaugh’s new book A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing.

See also my brief review of How to Read Historical Mathematics by the same author.

Reading historical math

Posted on by John

I recently received review copies of two books by Benjamin Wardhaugh. Here I will discuss How to Read Historical Mathematics (ISBN 0691140146). The other book is his anthology of historical popular mathematics which I intend to review later.

Here is the key passage, located near the end of How to Read Historical Mathematics, for identifying the author’s perspective.

But not all historical mathematics is significant. And perhaps there is a second kind of significance, where something can be historically significant without being mathematically significant. Some historians (I’m one of them) delight in investigating mathematical writing that contains little or no important or novel mathematics: popular textbooks, self-instruction manuals, … or old almanacs and popular magazines with mathematical news or puzzles in them. These kinds of writing … are certainly significant for a historian who wants to know about popular experiences of mathematics. But they’re not significant in the sense of containing significant mathematics.

Wardhaugh’s perspective is valuable, though it is not one that I share. My interest in historical math is more on the development of the mathematical ideas rather than their social context. I’m interested, for example, in discovering the concrete problems that motivated mathematics that has become more abstract and formal.

I was hoping for something more along the lines of a mapping from historical definitions and notations to their modern counterparts. This book contains a little of that, but it focuses more on how to read historical mathematics as a historian rather than as a mathematician. However, if you are interested in more of the social angle, the book has many good suggestions (and even exercises) for exploring the larger context of historical mathematical writing.

Life off the clock

Posted on by John

There was a lot of work to do a few generations ago, but the work wasn’t regulated by a clock.

With the growth of industrial capitalism during the post-Civil War years, more and more Americans were feeling pressure to be “on time.” (The phrase itself was a colloquialism which did not appear until the 1870s.) The corporate drive for efficiency … reinforced the spreading requirement that people regulate their lives by the clock. … And though there was much resistance, especially among workers from a preindustrial background, the triumph of clock time seemed assured by 1890, when the time clock was invented. [1]

Clocks had been around for centuries, but no one punched a time clock until 1890. People had regular schedules, some more so than others, but in general their schedules were not rigid or synchronized.

Increasing numbers of people now enjoy flexible work schedules. This is not something new but a return to something old. Industrialism made synchronization necessary. Post-industrial work is partially returning to pre-industrial norms.

[1] “No Place of Grace: Antimodernism and the Transformation of American Culture, 1880-1920” by T. J. Jackson Lears.

Why the symbol for magnetic field is ‘B’

Posted on by John

I asked on Twitter the other day

What is the historical reason for denoting magnetic filed “B”?

Eric Eekhoff sent me an answer and with his permission I’m copying his email below:

Hi John,

I saw your question on your GrokEM twitter account about why magnetic field was denoted as B. I recall that Maxwell just used the letters A through H for vectors in his Treatise on Electricity and Magnetism and some of them stuck and some of them didn’t. A is still used for vector potential, B for magnetic field (or magnetic induction or flux density, depending who you ask), H for magnetic intensity, etc. Maxwell used C and G for other vectors that I don’t recall at the moment. They, for some reason, never stuck.

Hope that helps. Have a good day,

Eric

Stigler’s law and Avogadro’s number

Posted on by John

Stigler’s law says that no scientific discovery is named after its original discoverer. Stigler attributed his law to Robert Merton, acknowledging that Stigler’s law obeys Stigler’s law.

Avogadro’s number may be an example of Stigler’s law, depending on your perspective. An episode of Engines of our Ingenuity on Josef Loschmidt explains.

The Italian, Romano Amadeo Carlo Avogadro, had suggested [in 1811] that all gases have the same number of molecules in a given volume. Loschmidt figured out [in 1865] how many molecules that would be.

You could argue that Avogadro’s constant should be named after Loschmidt, and some use the symbol L for the constant in honor of Loschmidt. Jean Perrin came up with more accurate estimates and proposed in 1909 that the constant should be named after Avogadro. Loschmidt made several important contributions to science that are now known by other’s names.

As I’d mentioned in an earlier post, there are some fun coincidences with Avogadro’s number.

  1. NA is approximately 24! (i.e., 24 factorial.)
  2. The mass of the earth is approximately 10 NA kilograms.
  3. The number of stars in the observable universe is 0.5 NA.

Never a time so completely parochial

Posted on by John

“There never was a time when those that read at all, read so many books by living authors rather than books by dead authors. Therefore there was never a time so completely parochial, so completely shut off from the past.” — T. S. Eliot

Posts related to T. S. Eliot:

Posts on old books:

Five interesting things about Mersenne primes

Posted on by John

A Mersenne prime is a prime number that is one less than a power of 2. These primes are indexed by the corresponding power of two, i.e. Mp = 2p − 1. It turns out p must be prime before 2p − 1 can be prime.

Here are five things I find interesting about Mersenne primes.

1. Record size primes

The largest known prime number is a Mersenne prime, M82,589,993, proved prime in 2018. And ever since M521 was proven prime in 1952, the largest known prime has always been a Mersenne prime (with one exception in 1989). See a history of prime number records.

One reason for the prevalence of Mersenne primes in the record books is that there is a special algorithm for testing whether a number of the form 2p − 1 is prime, the Lucas-Lehmer test.

2. Finiteness

There may only be a finite number of Mersenne primes. Only 51 are known so far. There are conjectures that predict there are an infinite number of Mersenne primes, but these have not been settled.

3. Connection with perfect numbers

Euclid proved that if M is a Mersenne prime, M(M + 1)/2 is a perfect number. Two millennia later, Euler proved that if N is an even perfect number, N must be of the form M(M + 1)/2 where M is a Mersenne prime. (More details here.)

Since we only know of 51 Mersenne primes at the moment, and we don’t know of any odd perfect numbers, there are only 51 known perfect numbers.

4. Connection with random number generation

The Mersenne twister is a popular, high-quality random number generator. The generator is so named because its period is a Mersenne prime, M19,937.

5. History

Mersenne primes are named after the French monk Marin Mersenne (1588–1648) who compiled a list of Mersenne primes. Mersenne wasn’t the first to be aware of such primes. As mentioned above, Euclid connected these primes with perfect numbers.

Marin Mersenne is one of my academic ancestors. I studied under Ralph Showalter, who studied under Tsuan Ting, and so forth back to Frans van Shooten Jr., who studied under Marin Mersenne.

What I find fascinating about this is not my particular genealogy, but that adequate records exist to construct such genealogies. The Mathematics Genealogy Project has over 150,000 records, some reaching back to the Middle Ages.