Quaternions in Paradise Lost

Last night I checked a few books out from a library. One was Milton’s Paradise Lost and another was Kuipers’ Quaternions and Rotation Sequences. I didn’t expect any connection between these two books, but there is one.

photo of books mentioned here

The following lines from Book V of Paradise Lost, starting at line 180, are quoted in Kuipers’ book:

Air and ye elements, the eldest birth
Of nature’s womb, that in quaternion run
Perpetual circle, multiform, and mix
And nourish all things, let your ceaseless change
Vary to our great maker still new praise.

When I see quaternion I naturally think of Hamilton’s extension of the complex numbers, discovered in 1843. Paradise Lost, however, was published in 1667.

Milton uses quaternion to refer to the four elements of antiquity: air, earth, water, and fire. The last three are “the eldest birth of nature’s womb” because they are mentioned in Genesis before air is mentioned.


Dot, cross, and quaternion products

This post will show that

quaternion product = cross product – dot product.

First, I’ll explain what quaternions are, then I’ll explain what the equation above means.

The complex numbers are formed by adding to the real numbers a special symbol i with the rule that i2 = -1. The quaternions are similarly formed by adding to the real numbers i, j, and k with the requirement that

i2 = j2 = k2 = ijk = -1.

A quaternion is a formal sum of a real number and real multiples of the symbols i, j, and k. For example,

q = w + xi + yj + zk.

In this example we say w is the real part of q and xi + yj + zk is the vector part of q, analogous to the real and imaginary parts of a complex number.

The quaternion product of two vectors (x, y, z) and (x´, y ´, z´) is the product of q = xi + yj + zk and q‘ = x’i + y’j + z’k as quaternions. The quaternion product qq´ works out to be

– (xx´ + yy´ + zz´) + (yz´ – zy´)i +(zx´ – xz´)j + (xy´ – yx´)k

The real part is the negative of the dot product of (x, y, z) and (x´, y´, z´) and the vector part is the cross product of (x, y, z) and (x´, y´, z´).

This relationship is an interesting bit of algebra on its own, but it is also historically important. In the 19th century, a debate raged regarding whether quaternions or vectors were the best way to represent things such as electric and magnetic fields. The identity given here shows how the two approaches are related.

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Footnote: To multiply two quaternions, you need to know how to multiply i, j, and k by each other. It is not immediately obvious, but you can derive everything from i2 = j2 = k2 =ijk = -1. For example, start with ijk = -1 and multiply both sides on the right by k. So ijk2 = –k, and since k2 = -1, ij = k. Similar manipulations show jk = i and ki = j.

Next, (jk)(ki) = ij, but it also equals jk2i = –ji, so ij = –ji = k. Similarly kj = –jk = –i and ik = –ki = –j and this completes the multiplication table for i, j, and k.

The way to remember these products is to imagine a cycle: i -> j -> k -> i. The product two consecutive symbols is the next symbol in the cycle. And when you traverse the cycle backward, you change the sign.