   Project Director: Professor Larry J Schaaf
 Back to the letter search > Result number 10 of 19:   < Back     Back to results list   Next >   Document number: 3858Date: 09 Jan 1863 Recipient: KELLAND PhilipAuthor: TALBOT William Henry Fox Collection: Wiltshire and Swindon History Centre, Chippenham Collection number: Lacock Abbey Deposit WRO 2664 Last updated: 7th July 2006

Lacock Abbey Chippm
Wilts – Lacock Abbey

Kelld

9 Jany 63

Thanks for yr explann – In the quarternn 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 demonstn that αβγ = γβα (these being vectors) and also he gives also the following extended theorem

If αβγδε &c be any odd no of vectors their product may be inverted so that α·β·γ·δ·ε = ε·δ·γ·β·α And But if their no be even, this eqn is not true. And on considerg this theorem I found out a vy simple demonstratn of it for myself, wch I think must be correct.

Can you inform me whether the followg theorem is correct, wch I deduce fm the quaternn 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 rectangr axes of coordinates, & OP, OQ are 2 given lines fixed in positn issuing from the origin O – and if a b c are the projectns of OP on the 3 axes, & a′b′c′ those of OQ Then aa′ + bb′ + cc′ is a const quanty, however the situatn of the axes is changed.

<expanded version>

Lacock Abbey Chippenham
Wiltshire – Lacock Abbey

Kelland

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 (i2 = j2 = k2 = 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).

 Result number 10 of 19:   < Back     Back to results list   Next >