Lacock Abbey Chipp^{m}

Wilts – Lacock Abbey

^{d}

9 Jan^{y} 63

Thanks for y^{r} explan^{n} – In the quartern^{n} system, some of the rules of common Algebra are subverted & others not, so that a beginner is easily perplexed –

On looking at Hamiltons book, I found his demonst^{n} that αβγ = γβα (these being vectors) and ~~also~~ he gives also the following extended theorem

If αβγδε &c be any __odd__ n^{o} of vectors their product may be inverted so that α·β·γ·δ·ε = ε·δ·γ·β·α And But if their n^{o} be __even__, this eq^{n} is not true. And on consider^{g} this theorem I found out a v^{y }simple demonstrat^{n} of it for myself, w^{ch} I think must be correct.

Can you inform me whether the follow^{g} theorem is correct, w^{ch} I deduce f^{m} the quatern^{n} doctrines? It looks like one of the ordinary theorems about coordinates, but I have got no books here, & my memory fails me on the subject.

__Theorem__. If ~~O~~ <illegible deletion> OX OY OZ are 3 rectang^{r} axes of coordinates, & OP, OQ are 2 given lines fixed in posit^{n} issuing from the origin O – and if a b c are the project^{ns} of OP on the 3 axes, & a′b′c′ those of OQ Then aa′ + bb′ + cc′ is a cons^{t} quant^{y}, however the situat^{n} of the axes is changed.

<expanded version>

Lacock Abbey Chippenham

Wiltshire – Lacock Abbey

9 January 1863

Thanks for your explanation – In the quarternion system, ^{<1>} some of the rules of common Algebra are subverted & others not, so that a beginner is easily perplexed –

On looking at Hamiltons book, ^{<2>} I found his demonstration that αβγ = γβα (these being vectors) and ~~also~~ he gives also the following extended theorem

If αβγδε &c be any __odd__ number of vectors their product may be inverted so that α·β·γ·δ·ε = ε·δ·γ·β·α And But if their number be __even__, this equation is not true. And on considering this theorem I found out a very simple demonstration of it for myself, which I think must be correct.

Can you inform me whether the following theorem is correct, which I deduce from the quaternion doctrines? It looks like one of the ordinary theorems about coordinates, but I have got no books here, & my memory fails me on the subject.

__Theorem__. If ~~O~~ <illegible deletion> OX OY OZ are 3 rectangular axes of coordinates, & OP, OQ are 2 given lines fixed in position issuing from the origin O – and if a b c are the projections of OP on the 3 axes, & a′b′c′ those of OQ Then aa′ + bb′ + cc′ is a constant quantity, however the situation of the axes is changed.

#### Notes:

**1**. Quaternion theory is an anticommutative algebra invented by Sir William Rowan Hamilton (1805–1865), Irish mathematician. According to the story, he scratched the fundamental formula (i^{2} = j^{2} = k^{2} = ijk = −1) into the Brougham bridge on the Royal Canal, Dublin. Quaternion algebra was later found to have applications to quantum mechanics.

**2**. Probably William Rowan Hamilton, *Lectures on Quaternions: containing a systematic statement of a new mathematical method* (Dublin: Hodges and Smith, 1853).