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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf MEMORANDUM RESPECTING A NEW SYSTEM}

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\centerline{\Largebf OF ROOTS OF UNITY}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Philosophical Magazine, 12 (1856), p.~446.)}

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

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\centerline{\largerm 2000}

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\noindent
{\largeit Memorandum respecting a new System of Roots of Unity.
By\/} {\largerm Sir}
{\largesc William Rowan Hamilton}, {\largeit LL.D., M.R.I.A.,
F.R.A.S., \&c., Andrews' Professor of Astronomy in the
University of Dublin, and Royal Astronomer of
Ireland\/}\footnote*{Communicated by the Author.}.

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\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science,}}
\centerline{4th series, vol.~xii (1856), p.~446.]}

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I have lately been led to the conception of a new system, or
rather {\it family of systems}, of {\it non-commutative roots of
unity}, which are entirely distinct from the $ijk$ of the
quaternions, though having some general analogy thereto; and
which admit, even more  easily than the quaternion symbols do, of
{\it geometrical interpretation}.  In the system which seems at
present to be the most interesting one, among those included in
this new family, I assume three symbols,
$\iota$, $\kappa$, $\lambda$, such that
$$\left. \matrix{
\iota^2 = 1,\quad \kappa^3 = 1,\quad \lambda^5 = 1,\cr
\lambda = \iota \kappa;\cr}
   \right\}
   \eqno {\rm (A)}$$
where $\iota \kappa$ must be {\it distinguished\/} from
$\kappa \iota$, since otherwise we should have
$\lambda^6 = 1$, $\lambda = 1$.  As a very simple
{\it specimen\/} of the symbolical conclusions deduced from these
fundamental assumptions, I may mention that if we make
$$\mu  = \iota \kappa^2 = \lambda \iota \lambda,$$
we shall have also\footnote\dag{In fact, by (A),
$$\iota \kappa = (\iota \kappa)^{-4}
   = (\kappa^{-1} \iota^{-1})^4 = (\kappa^2 \iota)^4,$$
$$1 = \iota \mathbin{.} \iota \kappa \mathbin{.} \kappa^2
   = \iota (\kappa^2 \iota)^4 \kappa^2 = (\iota \kappa^2)^5;$$
also
$$\mu \iota \mu = \mu \kappa^2 = \iota \kappa^4
   = \iota \kappa = \lambda.$$}
$$\mu^5 = 1,\quad \lambda = \mu \iota \mu;$$
so that $\mu$ is a new fifth root of unity, connected with the
former fifth root~$\lambda$ by relations of perfect reciprocity.
A long train of such symbolical deductions is found to follow:
and every one of the results may be {\it interpreted}, as having
reference to the passage from {\it face to face\/} (or from
corner to corner) of the {\it icosahedron\/} (or of the
dodecahedron): on which account, I am at present disposed to give
the name of the ``Icosian Calculus,'' to this new system of
symbols, and of rules for their operation.  Some additional
remarks on this subject may soon be offered to the Philosophical
Magazine, under the title, already sanctioned by the Editors, of
``Extensions of the Quaternions.''

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Observatory of Trinity College, Dublin,

October~29, 1856.

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