Pièce 115 - Letter from Thomas Perronet Thompson

4 pp.

Thanks WW for a copy of his Mechanical Euclid ['The Mechanical Euclid', 1837]: 'If I were allowed to criticise, I should feel disposed to submit, that geometry can stir more than 'four steps' without resting her foot upon an axiom. Up to the 28th Prop. of the first book of Euclid...I should be disposed to say there was nothing which it was not mere 'idleness and avoiding of labour' to refer to an axiom. Whether the great sticking-point about Parallels will ever be really got over, time must show; but if anybody should ever be fortunate enough to light on such a discovery, I apprehend there would be an end of Axioms in Euclid'. It is because we are unable to get over this point we retain the axioms. TPT believes 'that the mystery of parallels is lodged, or to be sought for, in what for shortness I will call the Platonic property of the sphere, namely that by which any sphere or spheres can be turned about the centre without change of place' [see his 'Geometry without Axioms. Or the First Books of Euclid's Elements', 1830, in which he tries to establish the theory of parallel lines without recourse to any principle not founded on previous demonstration].