Pièce 117 - Letter from Thomas Perronet Thompson

4 pp.

WW does TPT's 'little books (as the French say) too much honour, in expecting from it any novel presentations of the fundamental relations of space [see his 'Geometry without Axioms. Or the First Books of Euclid's Elements', 1830]...It professes only to stand upon the fact, that the intercepted portion of the radius vector in the Equiangular Spiral is finite (or as has been preferred to call it, limited), for any limited number of revolutions; and this in defiance of the truth or untruth of Euclid's axiom on Parallels'. However the method has 'no more claim to be a novel presentation of the fundamental relations of space, than if the fact had been that, through some oversight of everybody else, it had turned out to be as easy to prove that the three angles of a triangle cannot be less than two right angles, as it is to prove they cannot be greater' [see his 'Theory of Parallels: the Proof that the Three Angles of a Triangle are Equal to Two Right Angles', 1853]. TPT made the distinction between higher and lower geometry 'merely as what exists in language. But I suppose the distinction intended is, that the higher geometry involves the method of infinitesimals or infinitely small parts, which Euclid has ably and successfully evaded by his introduction of a sum to be approached to within less than any magnitude assigned. At the same time I cannot conceal my wish, that the demonstration (supposing it always to be one) had been brought within the narrower pale'.