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In the paper "On the Relative Forms of Quaternions" (CP 3.323), we observe Peirce providing the following sorts of explanation:
In the paper "On the Relative Forms of Quaternions" (CP 3.323), we observe Peirce providing the following sorts of explanation:
{| align="center" cellpadding="6" width="90%"
| If X, Y, Z denote the three rectangular components of a vector, and W denote
|
|
| 1 = (W:W) + (X:X) + (Y:Y) + (Z:Z)
<p>If <math>X, Y, Z\!</math> denote the three rectangular components of a vector, and <math>W\!</math> denote numerical unity (or a fourth rectangular component, involving space of four dimensions), and <math>(Y:Z)\!</math> denote the operation of converting the <math>Y\!</math> component of a vector into its <math>Z\!</math> component, then</p>
|-
| align="center" |
<math>\begin{matrix}
1 & = & (W:W) & + & (X:X) & + & (Y:Y) & + & (Z:Z)
\\
i & = & (X:W) & - & (W:X) & - & (Y:Z) & + & (Z:Y)
\\
j & = & (Y:W) & - & (W:Y) & - & (Z:X) & + & (X:Z)
\\
k & = & (Z:W) & - & (W:Z) & - & (X:Y) & + & (Y:X)
\end{matrix}</math>
|-
|
|
<p>In the language of logic <math>(Y:Z)\!</math> is a relative term whose relate is a <math>Y\!</math> component, and whose correlate is a <math>Z\!</math> component. The law of multiplication is plainly <math>(Y:Z)(Z:X) = (Y:X),\!</math> <math>(Y:Z)(X:W) = 0,\!</math> and the application of these rules to the above values of <math>1, i, j, k\!</math> gives the quaternion relations</p>
|-
| align="center" |
<math>i^2 = j^2 = k^2 = -1, \quad ijk = -1, \quad \text{etc}.</math>
|-
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|
<p>The symbol <math>a(Y:Z)\!</math> denotes the changing of <math>Y\!</math> to <math>Z\!</math> and the multiplication of the result by <math>a.\!</math> If the relatives be arranged in a block</p>
|-
| align="center" |
<math>\begin{matrix}
W:W && W:X && W:Y && W:Z
\\
X:W && X:X && X:Y && X:Z
\\
Y:W && Y:X && Y:Y && Y:Z
\\
Z:W && Z:X && Z:Y && Z:Z
\end{matrix}</math>
|-
|
|
| k = (Z:W) - (W:Z) - (X:Y) + (Y:X)
<p>then the quaternion <math>w + xi + yj + zk\!</math> is represented by the matrix of numbers</p>
|-
| align="center" |
<math>\begin{array}{*{7}{r}}
w && -x && -y && -z
\\
x && w && -z && y
\\
y && z && w && -x
\\
z && -y && x && w
\end{array}</math>
|-
|
|
<p>The multiplication of such matrices follows the same laws as the multiplication of quaternions. The determinant of the matrix = the fourth power of the tensor of the quaternion.</p>
<p>The imaginary <math>x + y \sqrt{-1}</math> may likewise be represented by the matrix</p>
|-
| align="center" |
<math>\begin{array}{rr}
x & y
\\
-y & x
\end{array}</math>
|-
|
|
<p>and the determinant of the matrix = the square of the modulus.</p>
<p>C.S. Peirce, ''Johns Hopkins University Circulars'', No. 13, p. 179, 1882. Reprinted, ''Collected Papers'', CP 3.323.</p>
|}
</pre>
This way of talking is the mark of a person who opts to multiply his matrices "on the rignt", as they say. Yet Peirce still continues to call the first element of the ordered pair <math>(I:J)\!</math> its "relate" while calling the second element of the pair <math>(I:J)\!</math> its "correlate". That doesn't comport very well, so far as I can tell, with his customary reading of relative terms, suited more to the multiplication of matrices "on the left".
This way of talking is the mark of a person who opts to multiply his matrices "on the rignt", as they say. Yet Peirce still continues to call the first element of the ordered pair <math>(I:J)\!</math> its "relate" while calling the second element of the pair <math>(I:J)\!</math> its "correlate". That doesn't comport very well, so far as I can tell, with his customary reading of relative terms, suited more to the multiplication of matrices "on the left".
