In the past few days I have added to my History of Mathematics
website a number of papers by Sir William Rowan Hamilton, in a
number of formats. The total amount of new material, which I have
been preparing over the last year, amounts to over 550 pages. (The
papers previously transcribed and made available at the site amounted
to approximately 250 pages.) It is my aim to make available,
eventually, all the published mathematical papers of Hamilton, with
the exception of the two books on quaternions and those items
from the Nachlass that were first published in this century. (I
estimate that I am about two-thirds of the way through this task,
though there are several major papers left to be transcribed and
edited.)
The URL for material related to Hamilton is the Hamilton 'Home
Page' at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/
This is part of the website at
http://www.maths.tcd.ie/pub/HistMath/
The recently added material is in the fields of optics, geometry
and real analysis.
The major addition in the field of optics is the 'Theory of Systems
of Rays' and its three supplements. (These are substantial papers,
and together account for 315 pages of the new material.) To these
I have added the shorter optics papers published in Hamilton's
lifetime, and as a result I believe that the online material now
covers all the writings on optics that Hamilton published in his
lifetime. These papers can be accessed by selecting 'Geometrical
Optics' on the Hamilton 'Home Page'.
Another substantial new item is the paper 'On Symbolical
Geometry' (72 pages). This paper was published in ten
installments in the first four volumes of 'The Cambridge and
Dublin Mathematical Journal' for the years 1846-9. It represents
what is essentially a 'coordinate-free' approach to the theory of
quaternions, with 'geometrical quotients' playing the role of the
quaternions. It can be accessed by selecting 'Geometry' on the
Hamilton 'Home Page'.
Another geometrical item is 'On Geometrical Nets in Space',
dating from 1862, which was Hamilton's last substantial published
paper (47 pages).
The only substantial paper in the field of real analysis that I
am aware of is 'On Fluctuating Functions' (47 pages). Hamilton
here seeks to explain the basic principles underlying the Fourier
Inversion Formula, by means of a 'Principle of Fluctuation'.
He also considers the convergence of Fourier series, and some
applications of Fourier analysis. By means of his 'Principle of
Fluctuation' he obtains some generalizations of the Fourier
Integral Formula. I found the first half, at least, of this
paper to be interesting. Hamilton appears to have been unaware
of the work of Dirichlet in this area: it would be interesting to
compare Hamilton's approach to that of Dirichlet - though I
suspect that, compared to Dirichlet, Hamilton's treatment would
be considered wanting in rigour. (For example, Hamilton does not
explicitly identify the requirement that the functions to which
the Fourier inversion formula is being applied should have only
a finite number of maxima and minima in any bounded interval.)
I have also included a few short papers in the area of real
analysis. In particular I have included a very short early paper
in which Hamilton observes that the function equal to exp(-1/x^2)
for positive x cannot be expanded as a power series in x (with
integer or fractional exponents), despite the fact that the
function tends to the limit zero as x tends to zero.
I have made these papers available in the following formats:
Plain TeX, DVI, PostScript, PDF. (Unfortunately, HTML does not
seem to me adequate at present for mathematical text.)
Finally, a word as to the text. I bear sole responsibility for
the typing and editing, such as it is. With one exception (a
short paper in French that appeared in Quetelet's 'Correspondance
mathematique et physique' which I cannot obtain), I have
consulted the original publications. In the case of those papers
that have been included in the volumes of 'The Mathematical
Papers of Sir William Rowan Hamilton' so far published, I have
also consulted that collected edition. There seem to be
comparatively few differences between the original publications
and the 'Mathematical Papers' - a few typographical errors not
noted by Hamilton have been corrected, and there are some
differences in punctuation and typography. I have tended to
follow the original publication in matters of punctuation and
typography, though I have adopted the corrections of certain
small typographical errors and inconsistencies of spelling or
punctuation from the 'Mathematical Papers'. For the substantial
papers, it is my standard practice to proof-read the text at
least five times. Unfortunately typos still seem to slip through
the net, no matter how many times I attempt to proof-read. I
would therefore be very grateful for any corrections,
discrepancies or comments on these texts.
David Wilkins
School of Mathematics, Trinity College, Dublin 2, Ireland.
dwilkins@maths.tcd.ie
http://www.maths.tcd.ie/~dwilkins/