Hamilton: two mysteries solved

23 views

John Baez

Nov 4, 2005, 1:13:13 PM11/4/05
to
Thanks to Dirk Schlimm, I've finally found an online map showing
the bridge where Hamilton carved his famous formula defining the
quaternions:

You can see where Broombridge Road crosses the Royal Canal -
that's the place. For pictures and bus directions, try this:

http://math.ucr.edu/home/baez/dublin/index.html#brougham_bridge

And speaking of Hamilton, Theron Stanford has sent me an answer to
one of life's persistent questions:

why is momentum denoted by the letter p?

Stanford writes:

While Googling various subjects, I came across the following from
your Quantum Gravity Seminar notes from 2001:

Again Oz was overcome with curiosity, so mimicking Toby's voice,
he asked, "Why do we call the momentum p?"

The Wiz glared at Toby. "Because m is already taken -- it stands
for mass! Seriously, I don't know why people call position q and
momentum p. All I know is that if you use any other letters,
people can tell you're not a physicist. So I urge you to follow

Well, I have an answer. Hamilton, the first physicist to actually
understand the importance of the concept of momentum, chose pi to
stand for momentum (actually, it's not the usual pi, but what TeX
calls varpi, a lower-case omega with a top, kinda like the top
of a lower-case tau). Jacobi changed this to p in one of his
seminal papers on the subject; he also used q in the same paper
to stand for position. In the 1800s (I want to say 1850s, though
it might have been a decade or two later) Cayley presented a paper
to the Royal Academy in which he says (and I paraphrase), "Well,
it seems that p and q are pretty well established now, so that's
what I'm going to use."

So, now we need to hold a seance to see why Hamilton chose the letter
"varpi" for momentum. In fact, we should have done this on Hamilton's
200th birthday, earlier this year!

Lee Rudolph

Nov 5, 2005, 6:35:52 AM11/5/05
to
ba...@galaxy.ucr.edu (John Baez) writes:

...

>And speaking of Hamilton, Theron Stanford has sent me an answer to
>one of life's persistent questions:
>
> why is momentum denoted by the letter p?
>
>Stanford writes:

...

> Well, I have an answer. Hamilton, the first physicist to actually
> understand the importance of the concept of momentum, chose pi to
> stand for momentum (actually, it's not the usual pi, but what TeX
> calls varpi, a lower-case omega with a top, kinda like the top
> of a lower-case tau).

...

>So, now we need to hold a seance to see why Hamilton chose the letter
>"varpi" for momentum. In fact, we should have done this on Hamilton's
>200th birthday, earlier this year!

Be careful. Unless there is evidence (not cited above) that Hamilton
intended the glyph that TeX calls "varpi" to be, in fact, a variant
of pi, it remains possible--and I think likely--that he intended that
glyph to be what a Russian Orthodox priest of my acquaintance (at a
time long before he was a priest; actually, at a time when he, I,
high school together) told me was called "ot": a ligature of "tau"
and "omega" (actually, a superposition of tau and omega, but apparently
that counts as a ligature in paleography if not in typography).
So: is there any Greek, English, Latin, or Gaelic word or phrase
that might plausibly suggest--or be suggested by--the word or concept
of momentum, and which could be abbreviated by "o" followed or preceded
by (or superposed with) "t"? (Same question for "p", of course.)

Lee Rudolph

John Baez

Nov 5, 2005, 11:20:19 AM11/5/05
to
In article <dki5eo$mdv$1...@panix2.panix.com>,
Lee Rudolph <lrud...@panix.com> wrote:

>>So, now we need to hold a seance to see why Hamilton chose the letter
>>"varpi" for momentum. In fact, we should have done this on Hamilton's
>>200th birthday, earlier this year!

>Be careful. Unless there is evidence (not cited above) that Hamilton
>intended the glyph that TeX calls "varpi" to be, in fact, a variant
>of pi, it remains possible--and I think likely--that he intended that
>glyph to be what a Russian Orthodox priest of my acquaintance (at a
>time long before he was a priest; actually, at a time when he, I,
>high school together) told me was called "ot": a ligature of "tau"
>and "omega" (actually, a superposition of tau and omega, but apparently
>that counts as a ligature in paleography if not in typography).

That's possible; unlike the LaTeX "varphi", which is an alternative
version the Greek phi, I'm not sure the LaTeX "varpi" is really
supposed to be a variant of "pi".

Let's look around....

There are a number of nonstandard Greek letters, of which
Wikipedia lists four: digamma, san, qoppa and sampi:

http://en.wikipedia.org/wiki/Greek_alphabet

The sampi is the only one here with any resemblance to the "varpi"...

But - aha! - "varpi" is called "pomega" here:

http://www.ams.org/STIX/stixfullr/stixfull-06.html

It plays a role in astronomy, where it stands for the "longitude of
the pericenter" (whatever the heck *that* is):

http://scienceworld.wolfram.com/physics/Pomega.html

I don't know how "pomega" is related to "ot".

>So: is there any Greek, English, Latin, or Gaelic word or phrase
>that might plausibly suggest--or be suggested by--the word or concept
>of momentum, and which could be abbreviated by "o" followed or preceded
>by (or superposed with) "t"?

Momentum says where a particle is going TO! :-)

PiSigma

Nov 5, 2005, 3:44:55 PM11/5/05
to

"John Baez" <ba...@galaxy.ucr.edu> wrote in message
news:dkim43$4sa$1...@glue.ucr.edu...

> In article <dki5eo$mdv$1...@panix2.panix.com>,
> Lee Rudolph <lrud...@panix.com> wrote:
>
> >>So, now we need to hold a seance to see why Hamilton chose the letter
> >>"varpi" for momentum. In fact, we should have done this on Hamilton's
> >>200th birthday, earlier this year!
>
> >Be careful. Unless there is evidence (not cited above) that Hamilton
> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
> >of pi, it remains possible--and I think likely--that he intended that
> >glyph to be what a Russian Orthodox priest of my acquaintance (at a
> >time long before he was a priest; actually, at a time when he, I,
> >high school together) told me was called "ot": a ligature of "tau"
> >and "omega" (actually, a superposition of tau and omega, but apparently
> >that counts as a ligature in paleography if not in typography).
>
> That's possible; unlike the LaTeX "varphi", which is an alternative
> version the Greek phi, I'm not sure the LaTeX "varpi" is really
> supposed to be a variant of "pi".

Actually, it's a variant of "pi": see
http://omega.enstb.org/yannis/pdf/boston99.pdf

Lee Rudolph

Nov 5, 2005, 7:13:37 PM11/5/05
to
"PiSigma" <psi...@tee.gr> writes:

...
>> Lee Rudolph <lrud...@panix.com> wrote:
...

>> >Be careful. Unless there is evidence (not cited above) that Hamilton
>> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
>> >of pi, it remains possible--and I think likely--that he intended that
>> >glyph to be what a Russian Orthodox priest of my acquaintance (at a
>> >time long before he was a priest; actually, at a time when he, I,
>> >high school together) told me was called "ot": a ligature of "tau"
>> >and "omega" (actually, a superposition of tau and omega, but apparently
>> >that counts as a ligature in paleography if not in typography).
>>
>> That's possible; unlike the LaTeX "varphi", which is an alternative
>> version the Greek phi, I'm not sure the LaTeX "varpi" is really
>> supposed to be a variant of "pi".
>
>Actually, it's a variant of "pi": see
>http://omega.enstb.org/yannis/pdf/boston99.pdf

That the glyph in question can be (and is) used as a variant
of pi does not rule out that it can be (and has been) used as
a ligature of omega and tau. And, in fact, as my friend told
me and Google has now verified, that is the case: "ot" is the
name (obviously, transcribed into Latin letters...) of just
such a ligature in the alphabet used for Church Slavonic.

The Orthodox Church Slavonic alphabet is shown in [4, 5].
There are the following differences between Church Slavonic
and Old Slavonic (ustav) writings:
...
* The order of the alphabet is changed, some letters changed
their names, some letters changed their shape. Some letters
became obsolete and are excluded from the alphabet, but as
a compensation the new letter _ot_ appears which is the
ligature between O (Greek omega) and T with three stems.
(In Old Slavonic the name _ot_ was reserved for Greek
\omega which is called _omega_ in Church Slavonic.)

(From http://www.uni-giessen.de/partosch/eurotex99/berdnikov2.pdf .
Unfortunately the scanned images of glyphs from actual Church
Slavonic documents, referenced above as "[4, 5]", were at URLs
that now seem to be inoperative; and the hacked-together glyph
included in a table at the end of that PDF document--which looks
like it wasn't even produced with Metafont, but is simply one standard
TeX glyph overstruck on another--is a very bad representation of
_ot_.)

The challenge is now to show that Hamilton was acquainted with
Church Slavonic. That isn't (perhaps) quite as absurd as it sounds;
he was already a polyglot at a very young age (according to the
biography, title long forgotten, from which I extracted the
information in ll. 40 _et seq._ of "Frobenius: A Sesquilogue",
http://black.clarku.edu/~lrudolph/poetry/frobenius/index.html,
to wit, that he knew Hebrew, Latin, and Greek by the age of 5,
and Sanskrit, Arabic, and Persian by the age of 10).

Lee Rudolph

Dik T. Winter

Nov 5, 2005, 8:14:49 PM11/5/05
to
In article <dkim43$4sa$1...@glue.ucr.edu> ba...@galaxy.ucr.edu (John Baez) writes:
> In article <dki5eo$mdv$1...@panix2.panix.com>,
> Lee Rudolph <lrud...@panix.com> wrote:
>
> >>So, now we need to hold a seance to see why Hamilton chose the letter
> >>"varpi" for momentum. In fact, we should have done this on Hamilton's
> >>200th birthday, earlier this year!
>
> >Be careful. Unless there is evidence (not cited above) that Hamilton
> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
> >of pi, it remains possible--and I think likely--that he intended that
> >glyph to be what a Russian Orthodox priest of my acquaintance
...

> That's possible; unlike the LaTeX "varphi", which is an alternative
> version the Greek phi, I'm not sure the LaTeX "varpi" is really
> supposed to be a variant of "pi".

I have also looked around. It appear that the distinction "phi" and "varphi"
just depends on font. But, there appears to have been quite some discussion
about the variant symbols in Unicode. See for instance:
<http://omega.enstb.org/yannis/pdf/amendments2.pdf>
where the status of "varpi" is said to be uncertain, but where it is stated
to be an alternative for the standard pi (in that document it is also not
suggested not to use it in Greek text, which is done for quite a few other
symbols). And:
<http://ptolemy.tlg.uci.edu/~opoudjis/unicode/letters.html>
states that "varpi" dates from the 8-th century and was used alongside
the standard form until the mid 20-th century.

> There are a number of nonstandard Greek letters, of which
> Wikipedia lists four: digamma, san, qoppa and sampi:
>
> http://en.wikipedia.org/wiki/Greek_alphabet

I can not view that article properly, but those are archaic letters, not
used in Greek texts. (San has been added with Unicode 4.0, as Sho,
which Wikipedia does not mention.) And another not mentioned letter
is the stigma...

> The sampi is the only one here with any resemblance to the "varpi"...

There is no relation.

> But - aha! - "varpi" is called "pomega" here:
> http://www.ams.org/STIX/stixfullr/stixfull-06.html

To me that appears to be techspeak.

I think that Hamilton used the pi as appeared frequently in hand-written
material. "varpi" was a cursive form (i.e. handwritten). That it
deviates quite far from the standard form is not so very surprising.
In the Latin script we have the a and g with in general deviations
between handwritten and printed material. In Cyrillic it is quite a
bit stronger. The lower case d, which in print looks like a small delta
with two descenders on both sides, is written as "g". And it took me
some time when I was in Bulgaria to determine that some neon advertisement
that wrote "onmuka" actually was displaying what in a normal transcription
would be "optika".
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

john_r...@sagitta-ps.com

Nov 5, 2005, 10:08:56 PM11/5/05
to

Dik T. Winter wrote:
>
> > http://en.wikipedia.org/wiki/Greek_alphabet
>
> I can not view that article properly, but those are archaic letters, not
> used in Greek texts. (San has been added with Unicode 4.0, as Sho,
> which Wikipedia does not mention.) And another not mentioned letter
> is the stigma...

There's also the digamma

The page http://ptolemy.tlg.uci.edu/~opoudjis/unicode/numerals.html
includes he following:

"If you're using letters of the alphabet as numerals, you want to have
some
device for differentiating numbers from words. The main such device in
antiquity and the middle ages was the overbar, which was also used to
cite words."

Hero

Nov 6, 2005, 2:47:01 PM11/6/05
to

John Baez wrote:
> Thanks to Dirk Schlimm, I've finally found an online map showing
> the bridge where Hamilton carved his famous formula defining the
> quaternions:
>
>
> You can see where Broombridge Road crosses the Royal Canal -
> that's the place. For pictures and bus directions, try this:
>
> http://math.ucr.edu/home/baez/dublin/index.html#brougham_bridge
>
> And speaking of Hamilton, Theron Stanford has sent me an answer to
> one of life's persistent questions:
>
> why is momentum denoted by the letter p?
Thanks to David R. Wilkins Hamilton's writings are online at:
http://www.maths.tcd.ie/pub/HistMath
May be one can find a hint to the letter p in it.

> So, now we need to hold a seance to see why Hamilton chose the letter
> "varpi" for momentum. In fact, we should have done this on Hamilton's
> 200th birthday, earlier this year!
David R. Wilkins wrote an article, celebrating this event. And he shows
the bridge too.
By the way, he was talking to his wife about knitting a pullover, and
she explained to him Aaran knitting style.
Hero

Dave Rusin

Nov 6, 2005, 4:42:16 PM11/6/05
to
In article <IpIE4...@cwi.nl>, Dik T. Winter <Dik.W...@cwi.nl> wrote:

>And it took me
>some time when I was in Bulgaria to determine that some neon advertisement
>that wrote "onmuka" actually was displaying what in a normal transcription
>would be "optika".

And you have to get used to going to the pectopah for something to eat.

Dik T. Winter

Nov 6, 2005, 9:46:51 PM11/6/05
to
In article <dkjhrh$nf$1...@panix2.panix.com> lrud...@panix.com (Lee Rudolph) writes:
...
> >> Lee Rudolph <lrud...@panix.com> wrote:
> >> >Be careful. Unless there is evidence (not cited above) that Hamilton
> >> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
> >> >of pi, it remains possible--and I think likely--that he intended that
> >> >glyph to be what a Russian Orthodox priest of my acquaintance (at a
> >> >time long before he was a priest; actually, at a time when he, I,
> >> >and John Baez's future undergraduate thesis advisor were in junior
> >> >high school together) told me was called "ot": a ligature of "tau"
> >> >and "omega" (actually, a superposition of tau and omega, but apparently
> >> >that counts as a ligature in paleography if not in typography).
...

> That the glyph in question can be (and is) used as a variant
> of pi does not rule out that it can be (and has been) used as
> a ligature of omega and tau.

Can be true, of course. I distrust the reference you give:
> http://www.uni-giessen.de/partosch/eurotex99/berdnikov2.pdf
but it will not work with the version of acrobat reader I have.
But I will look into Peter T. Daniels monumental work on world
scripts.

> The challenge is now to show that Hamilton was acquainted with
> Church Slavonic.

There are quite a few reasons to suspect a Greek source. "varpi"
has been used in Greek from about the 8-th century to well into
the 20-th century. The "omega" was still part of Cyrillic in 1708,
but was gone quite early (btw, the oldest Cyrillic dates from the
10-th century, and indeed, that one had an omega-like symbol (or
w-like) called "otu"). And while a short-hand 't' might have been
written as a tilde (in cursive form it looks like an 'm'), I would
expect such ligatures to be uncommon. Anyhow, your quote:

> * The order of the alphabet is changed, some letters changed
> their names, some letters changed their shape. Some letters
> became obsolete and are excluded from the alphabet, but as
> a compensation the new letter _ot_ appears which is the
> ligature between O (Greek omega) and T with three stems.
> (In Old Slavonic the name _ot_ was reserved for Greek
> \omega which is called _omega_ in Church Slavonic.)

is unreliable. In 10-th century the symbol was called "otu" and
in 1708 the same symbol was called "omega".

John Baez

Nov 7, 2005, 1:09:39 AM11/7/05
to
In article <dkjhrh$nf$1...@panix2.panix.com>,
Lee Rudolph <lrud...@panix.com> wrote:

>"PiSigma" <psi...@tee.gr> writes:

>>> Lee Rudolph <lrud...@panix.com> wrote:

>>> >Be careful. Unless there is evidence (not cited above) that Hamilton
>>> >intended the glyph that TeX calls "varpi" to be, in fact, a variant
>>> >of pi, it remains possible--and I think likely--that he intended that

>>> >glyph to be what a Russian Orthodox priest of my acquaintance [...]

>>> >told me was called "ot": a ligature of "tau" and "omega"

>>Actually, it's a variant of "pi": see
>>http://omega.enstb.org/yannis/pdf/boston99.pdf

>That the glyph in question can be (and is) used as a variant
>of pi does not rule out that it can be (and has been) used as
>a ligature of omega and tau.

For the purposes of figuring out how "p" got to be used as the
standard symbol for "momentum", the question is what Hamilton
intended by using the character "varpi" (or "ot") to stand for
momentum - and what the people who read it thought they were reading!

Since Jacobi changed this symbol to "p", it seems he considered
it to be a version of the letter "pi", not the letter "ot".

And, remark #5 on page 8 here:

http://omega.enstb.org/yannis/pdf/boston99.pdf

suggests that Hamilton could easily have used it as a version of "pi".
To quote:

The "round" pi (varpi) (U+03D6 in UNICODE). This letter has
been a variant of pi from the very first days of Greek typography:
in Figure 2, the reader can see an excerpt of [Pol], a book
printed in 1489 (!). In this excerpt both versions of pi have
been used, apparently randomly.

Once again, [Imp, p. 95] sugggests that the round pi be used only
at the beginning of a word. According to [Tre], this rule has
been applied in certain French journals (for example the Memoires
de la Societe de Linguistique in the late 19th and early 20th
century); [Tre] appliesit in all of his writings [....]

[Imp] again, but in a different chapter (the one on the typesetting
of mathematics, p. 108) calls this letter "Doric pi".

From this it seems quite possible that Hamilton wrote (or at least
typeset) the letter pi in this "rounded" or "Doric" form.
It seems less plausible to me that he would have used the character
"ot" from Orthodox Church Slavonic in his mathematical writings.
Even if he knew this language (and you're right, he was a famous
polyglot), why would he have used it in his mathematical writings?
Or: what evidence do we have for this?

What's really tantalizing is the reference above to appearance of
the round pi in "the typesetting of mathematics"! Can anyone get
ahold of this and see what it says:

Lexique des regles typographiques en usage a l'Imprimerie nationale,
Impremerie Nationale, 1990, page 108. [Look for "Doric pi", or something
like that in French.]

?

Peter T. Daniels

Nov 7, 2005, 9:06:32 AM11/7/05
to
[from sci.lang]

Have you tried looking in Cajori's *History of Mathematical Notations*?
(2 vols., 1928-29; Dover reprint in 1 vol., 1993.)

--
Peter T. Daniels gram...@att.net

Timothy Murphy

Nov 7, 2005, 6:44:01 PM11/7/05
to
John Baez wrote:

> For the purposes of figuring out how "p" got to be used as the
> standard symbol for "momentum", the question is what Hamilton
> intended by using the character "varpi" (or "ot") to stand for
> momentum - and what the people who read it thought they were reading!

My colleague David Wilkins, who is our resident expert on Hamilton,
and has probably spent more hours poring over his works,
both hand-written and printed, than anyone else -
but who unfortunately is not a devotee or sci.math -
writes as follows:

"Hamilton tended to be meticulous with regard to typography, proof-reading
etc. of his mathematical papers. He would have used the typefaces available
and in regular use by printers of academic works, and regularly used for
mathematical works.

I would suggest that the glyph \varpi was not infrequently
used in published mathematical works in the nineteenth century to denote
pi. For example, note Riemann's major paper on the Theory of Abelian
functions. When reprinted in the standard Weber 19th century edition
of Riemann's works, the printers (Teubner, if I remember correctly off
the top of my head) used \omega in the latter parts of the paper to
denote quantities denoted by \pi in the earlier part of the same paper.
But if one looks at the original printed text of this paper, published
by the Berlin Academy, one finds that the printers used \varpi to
denote pi in the latter part of the paper, and this seems not to
have been noticed by the typesetters of the collected works.

I myself would take it for granted that Hamilton might well use
\varpi as representing the Greek letter pi. I think it unlikely
that he would think in terms of some glyph in handwritten Greek
like the 'ot' thing that has been mentioned. I am not aware of any interest
in early manuscripts (as opposed to printed texts) on the part
of Hamilton.

(To help settle the matter, one might look at a variety of nineteenth
century publications of standard Greek texts, to see how often the
glyph \varpi is in regular use by academic printers at that time.)

In any case, Hamilton would, I am sure, give particular attention
on the *printed* appearance of his papers, and the systematic use
of Greek, roman, italic and small capitals fonts within them.
(On the occasions when he attempted to introduce new symbols, he
did this by rotating existing ones through one or two right angles.)

For my transcription of relevant Hamilton papers, see

Hamilton uses \varpi in conjunction with \eta, see in particular pp. 4-5
of the first of these papers.

...

Hamilton probably didn't think of his quantities \varpi_1, varpi_2 etc.
as 'momenta' but as partial derivatives of his characteristic and
principal functions; analogous to the quantities \sigma, \tau and
\upsilon that appear, in the analogous optical context, in his
famous Third Supplement (the paper containing the prediction of
conical refraction)."

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland

John Baez

Nov 9, 2005, 2:57:29 AM11/9/05
to
In article <7FRbf.18344\$R5....@news.indigo.ie>,
Timothy Murphy <t...@birdsnest.maths.tcd.ie> wrote:

>John Baez wrote:

>My colleague David Wilkins, who is our resident expert on Hamilton,
>and has probably spent more hours poring over his works,
>both hand-written and printed, than anyone else -
>but who unfortunately is not a devotee or sci.math -
>writes as follows:

>For my transcription of relevant Hamilton papers, see
>
>http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/SecEssay.pdf
>http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/BARep34A.pdf
>
>Hamilton uses \varpi in conjunction with \eta, see in particular pp. 4-5
>of the first of these papers.

Great! I like equation 12 in the first paper.

>Hamilton probably didn't think of his quantities \varpi_1, varpi_2 etc.
>as 'momenta' but as partial derivatives of his characteristic and
>principal functions; analogous to the quantities \sigma, \tau and
>\upsilon that appear, in the analogous optical context, in his
>famous Third Supplement (the paper containing the prediction of
>conical refraction)."

But of course the modern, sophisticated concept of "momentum" is
precisely the thing whose components are the partial derivative of the
Lagrangian, or equivalently of Hamilton's principal function,
with respect to velocity. So, Hamilton's \varpi is indeed the
ancestor of the modern p.

I looked at the first paper to try to guess why Hamilton chose the
letter \varpi, but I didn't see any obvious reason - it just seemed
to be one among many Greek letters he used for various quantities
(as mentioned above). He may not have considered it to be particularly
important.

So, the answer "why is momentum called p" may be "because Hamilton
chose \varpi for this variable" - no better reason.

tdp...@gmail.com

Nov 29, 2005, 7:41:35 AM11/29/05
to

John Baez wrote:
> Thanks to Dirk Schlimm, I've finally found an online map showing
> the bridge where Hamilton carved his famous formula defining the
> quaternions:
>
>
> You can see where Broombridge Road crosses the Royal Canal -
> that's the place. For pictures and bus directions, try this:
>
> http://math.ucr.edu/home/baez/dublin/index.html#brougham_bridge

Open the "Placemark"
at the URL below to see a closeup of the bridge on Google Earth.