Royal Astronomer of Ireland, Prof. Charles Jasper Joly, details his mapping of quaternions onto projective geometry. Taken from the Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character Vol. 201, (1903), pp. 223327. Public Domain work.
[
223
VIII.
Quaternions
and
Projective
Geometry.
By
Professor
Charles
Jasper,
Joly,
Royal
Astronomer
of
Ireland.
Communicated
by
Sir
Robert
Ball,
F.R.S.
Received
November
27,
—
EeadDecember
11,
1902.
Introduction.
A
quaternion
q
adequately
represents
apoint
Q
to
which
a
determinate
weight
is
attributed,
and,
conversely,
when
the
point
and
its
weight
aregiven,
thequaternion
is
defined
without
ambiguity.
This
is
evident
from
the
identity
<?
=
(i
+
)sg
(A),
in
which
Qq
is
regarded
as
a
weight
placed
at
the
extremity
of
the
vector
drawn
from
any
assumed
srcin
o.
It
is
sometimes
convenient
to
employ
capitals
Q
concurrently
with
italics
q
to
represent
the
same
point,
it
being
understood
that
Q
=
<?;==
1
+
OQ
(C).
Thus
Q
represents
the
point
Q
affected
with
a
unit
weight.
The
point
o
may
be
called
the
scalar
point,
for
we
have
In
order
to
develop
the
method,
it
becomes
necessary
to
employ
certain
special
symbols.
With
one
exception
these
are
found
in
Art.
365
of
'Hamilton's
Elements
of
Quaternions/
though
in
quite
a
different
connection.
We
write
(a,
b)
=
bSa

aS6,
[a,
6]
=
V
.
VaVb
.(E);
and
in
particular
for
points
of
unit
weight,
these
become
(a,
b)
=
b
—
a,
[a,
b]
=
V.VaVb
=
V.
Va
.
(b
—
a)
.
.
.
(F).
Thus
(ab)
is
the
product
of
the
weights
SaS6
into
the
vector
connecting
the
points,
and
[ab]
is
the
product
ofthe
weights
into
the
moment
of
the
vector
connecting
the
points
with
respect
to
the
scalar
point.
The
two
functions
(ab)
and
[ab]
completely
define
the
line
ab.
VOL.
CCL—
A
338.
20.6.03
224
PROFESSOR
C.
J.
JOLY
ON
QUATERNIONS
AND
PROJECTIVE
GEOMETRY.
Affain
Hamilton
writes
[a
9
b,c]
=
(a,b,c)
—
[b,c]Sa
—
[c,a]S6
—
[a,&]Sc;
(a,&,c)
=
S[a,&, c]
=
SVaV&Vc
.
(G)
or
if
we
replace
a,
6,
c
by
(1
+
a)Sa,
(1
+
/2)S6,
(1
+
y)Sc,
where
a,
/3
and
y
are
the
vectors
from
the
scalar
point
to
three
points
a,
6
and
c,
we
have
[A,
B,
C]
=
8ot/3y
—
V(/3y
+
ya
+
a/3)
;
(a,
B,
c)
=
Sa/3y
.
.
,
'(H).
Hence
it
appears
that
[a,
6,
c]
is
the
symbol
of
theplane
a
9
b
9
c
;
for
—V
[a,
&,
c]
(a,
6,c)
1
is
the
reciprocal
of
the
vector
perpendicular
from
the
scalar
point
on
that
plane.
Also
(a,
b,
c)
is
the
sextupied
volume
of
the
tetrahedron
oabc.
Again,
Hamilton
writes
for
four
quaternions
(abed)
=
S
.
a[bcd]
.........
(I)
and
in
terms
of
the
vectors
this
is
seen
to
bethe
products
of
the
weights
into
the
sextupied
volume
of
the
pyramid
(abcd).
Other
notations
may
ol
course
be
employed
for
these
five
combinatorial
functions
oftwo,
three,
orfour
quaternions
or
points,
but
Hamilton's
use
of
thebrackets
seems
to
be
quite
satisfactory.
In
the
same
article
Hamilton
gives
two
most
useful
identities
connecting
any
five
quaternions.
These
are
a(bode)
+
b(cdea)
+
c(deab)
+
d(eabc)
+
e(abcd)
=
.
.
.
.
(J),
and
e(abcd)
=
[bcd]8ae
—
[acd]Sbe
+
[abd]$ce
—
[abc]$de
.
.
.
(K),
which
enable
us
to
express
any
point
in
terms
of
any
four
given
points,
or
in
terms
of
any
four
given
planes.
The
equation
of
a
plane
may
be
written
in
the
form
and
thus
Z,
any
quaternionwhatever,
may
beregarded
as
the
symbol
of
a
plane
as
well
as
of
a
point.
On
the
whole,
it
seems
most
convenient
to
take
as
the
auxiliary
quadricthesphereof
unitradius
S
a
s
=
(M)whose
centre
is
the
scalar
point.
With
this
convention
theplane
Blq
=
is
the
polarof
the
point
I
with
respect
to
the
auxiliary
quadric
;
or
theplane
is
the
reciprocal
of
the
point
Z.
Thus
the
principle
ofdualityoccupies
a
prominent
position.
The
formulae
of
reciprocation
([abc]
;
[abd])
=
[ah]
(abed)
;
[
[abc]
;
[abd]
]
=
—
(ab)
(abed)
.
.
.
(N)
connecting
any
four
quaternions
are
worthy
of
notice,
and
are
easily
proved
by
PROFESSOR
0.
J.
JOLY
ON
QUATERNIONS
AND
PROJECTIVE
GEOMETRY.
225
replacing
thequaternions
by
1
+
a
>
1
+
A
1
+
y,
and
1
+
§respectively.
In
complicated
relations
it
may
be
saferto
separate
thequaternions
as
in
theseformulas
by
semicolons,
but
generally
the
commas
or
semicolons
may
be
omitted
without
causing
any
ambiguity.
These
new
interpretations
are
not
in
the
least
inconsistent
with
any
principle
of
the
calculus
of
quaternions.
We
are
still
at
libertyto
regard
a
quaternion
as
the
separable
sum
ofa
vector
and
a
scalar,
oras
the
ratio
or
product
of
two
vectors,
oras
an
operator,
as
well
as
a
symbol
ofa
point
or
of
a
plane.
In
particular,
in
addition
to
Hamilton's
definition
of
a
vector
as
aright
line
of
given
direction
and
of
given
magnitude,
and
in
addition
to
his
subsequent
interpretations
ofa
vector
as
the
ratio
or
product
of
two
mutually
rectangular
vectors,or
as
a
versor,
we
may
now
consider
a
vector
as
denoting
the
point
at
infinity
in
its
direction,
or
theplane
through
the
centre
of
reciprocation.
For
the
vector
OQ
of
equation
(B)
becomes
infinitely
long
if
Sq
=
0,
and
the
plane
Blq
=
passes
through
the
scalar
point
if
81
=
0.
We
may
also
observe
that
the
difference
of
two
unit
points
A
—
b
is
the
vector
from
one
point
B
to
the
other
A,
and
this
again
is
in
agreement
with
the
opening
sections
of
the
Lectures.
Additional
illustrations
and
examples
may
be
found
in
a
paper
on
The
Interpre
tationof
a
Quaternion
as
a
Pointsymbol,
'
Trans.
Roy.
Irish
Acad./
vol.
32,
The
only
other
symbols
peculiar
to
this
method
are
the
symbols
for
quaternion
arrays.
The
five
functions
(a&),[a&],
[a&c],(a&c),
and
(abed)
areparticularcases
of
arrays,
being,
in
fact,
arrays
of
one
row.
In
general
thearray
of
m
rows
and
n
columns
CO}
C'o
eta
...
(%%
1
£
S
ããã
*'u
<
ã
»
ã
»
ã
ã
s
(O)
I
PiP%
Ps
ãã
ã
Pti
)
may
be
defined
as
a
function
of
mn
quaternion
constituents,
which
vanishes
if,
and
only
if,
the
groups
of
the
constituents
composing
the
rows
were
connected
by
linearrelations
with
the
same
set
of
scalarmultipliers.
In
other
words,
the
array
vanishes
if
scalars
t
l9
t
2
.
.
.
i
n
can
be
found
tosatisfy
the
m
equations
t
l
h
[
+
t^h
4
ããã
H
wii
:=
0?
aãã
hPi
+
hPz+
ã
ã
ã
+t»pH
=
o.
The
expansion
of
arrays
is
considered
in
a
paperon
Quaternion
Arrays/'
c
Trans.
Roy.
Irish
Acad./
vol.
32,
pp.
1730.
vol.
cci.
a,
2
G
226
PEOFESSOE
C.
J.
JOLY
ON
QUATERNIONS
AND
PEOJECTIYE
GEOMETEY.
SECTION
I.
Fundamental
Geometrical
Properties
of
a
LinearQuaternion
Function.
Art.
Page
1.
Definition
of
a
linear
quaternion
function
226
2.
The
general
linear
transformation
effected
by
a
linear
function
.
226
3.
Specification
of
a
function
by
four
quaternions
or
five
points
and
their
deriveds
.
..
227
4.
The
transformation
of
planes
effected
by
theinverse
of
the
conjugate
function
.
..
227
5.
Geometrical
interpretation
of
Hamilton's
method
of
inversion
227
6.
Geometrical
illustration
of
the
relations
connecting
Hamilton's
auxiliary
functions
.
228
7.
The
united
points
of
a
linear
transformation.
..............
229
8.
Eelations
connecting
the
united
points
of
a
function/
with
those
of
its
conjugate/'
.
230
9.
Introduction
of
the
functions
/o
=

(/+/'),
f,
=
\
(//')
ã
231
10.Scjf
Q
q
=
and
&$/,$>
—
representthegeneral
quadric
surface
and
the
general
linear
\J
VIJJi
L/XCA.
f
«
ff
ã
*s
ãa
a©
sb
8D
a
aa«e
eããas
«
iwl)
1
11.
The
pole
of
a
planeS#5
=
to
thequadric
is/o
1
^;
and
thepoint
of
concourse
oflinesof
the
complex
in
the
plane
is/
1
/;.
.
..
...
232
12.
The
united
points
of/
form
a
quadrilateral
on
the
sphere
of
reciprocation
....
232
1.
The
quaternion
equation
f(p
+
q)=fp+fq
.........
(1),
may
be
regarded
as
a
definition
of
the
nature
ofa
linear
quaternion
function
f
thequaternions
p
and
q
being
perfectly
arbitrary.
As
a
corollary,
if
x
is
any
scalar,
and
on
resolving
fq
in
terms
of
any
four
arbitrary
quaternions
a
1;
a.
2
,
a
3
,
a
4
.
we
must
have
an
expression
ofthe
form
fq
=
a
1
S6
1
g
f
+
oc^Bb^q
+
a
3
S&
3
g
r
+
aJ8&4#
...
..
.
(3),
because
the
coefficients
of
the
four
quaternions
a
must
be
scalar
and
distributive
functionsof
5.
Sixteen
constantsenter
into
thecomposition
of
the
function/,
being
four
for
each
of
thequaternions
6.
2.
When
a
quaternion
is
regarded
as
the
symbol
of
a
point,
the
operation
of
the
function
f
produces
a
linear
transformation
of
the
most
general
kind.
The
equations
f(xa
+
yb)
=
xfa
+
yfb
;
f(xa
+
yb
+
2c)
=
,r/a
+
?//&
+
zfc
.
.
(4),
show
that
the
right
line
a,
b
is
converted
into
the
right
line
fa,
fb
9
and
the
plane
containing
three
points
a,
6,
c
into
theplane
containing
their
correspondents,
fa,fb
andyb.
The
homographic
character
of
thetransformation
is
alsoclearly
exhibited.