% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 2000.

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\centerline{\Largebf ON A THEORY OF QUATERNIONS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (British Association Report, 1844,
   Part~II, p.~2.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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{\largeit\noindent
On a Theory of Quaternions.  By\/}
{\largerm Sir} {\largesc William R. Hamilton},
{\largeit M.R.I.A.}

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\centerline{[{\it Report of the Fourteenth Meeting of the British
Association for the Advancement of}}
\centerline{{\it Science; held at York in September 1844}.}
\centerline{(John Murray, London, 1845), Part~II, p.~2.]}

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It has been shown, by Mr.~Warren and others, that the results
obtained by the ordinary processes of algebra, involving the
imaginary symbol $\sqrt{-1}$, admit of real interpretations, such
as those which relate to compositions of linear motions and
rotations in one plane.  Sir W.~Hamilton has adopted a system of
three such imaginary symbols, $i$,~$j$,~$k$, and assumes or
defines that they satisfy the nine equations
$$i^2 = j^2 = k^2 = -1,\quad
  ij = k = - ji,\quad
  jk = i = - kj,\quad
  ki = j = - ik,$$
which however are not purely arbitrary, and for the adoption of
which the paper assigns reasons.  He then combines these symbols
in a {\it quaternion}, or imaginary quadrinomial, of the form
$${\sc q} = w + ix + jy + kz,$$
in which $w$, $x$, $y$, $z$, are four real quantities; and states
that he has established rules for algebraical operations on
such expressions, and has assigned geometrical interpretations
corresponding; so as to form a sort of {\it Calculus of
Quaternions}, which serves as an instrument to prove old
theorems, and to discover new ones, in the geometry of three
dimensions, and especially respecting the composition of motions
of translation and rotation in space.

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