Theory of Parallels — Lobachevski's Introduction

Summary

In geometry, I have identified several imperfections, which I hold responsible for the fact that this science, apart from its translation into analysis, has taken no step forward from the state in which it came to us from Euclid. I consider the following to be among these imperfections: vagueness in the basic notions of geometricmagnitudes, obscurity in the method and manner of representing the measurements of such magnitudes, and finally, the crucial gap in the theory of parallels. Until now, all mathematicians’ efforts to fill this gap have been fruitless. Legendre's labors in this area have contributed nothing. He was forced to abandon the one rigorous road, turn down aside path, and seek sanctuary in extraneous propositions, taking pains to present them—in fallacious arguments—as necessary axioms.

I published my first essay on the foundations of geometry in the “Kazan Messenger“ in the year 1829. Hoping to provide an essentially complete theory, I then undertook an exposition of the subject in its entirety, publishing my work in installments in the “Scholarly Journal of the University of Kazan” in the years 1836, 1837, and 1838, under the title, “New Principles of Geometry, with a Complete Theory of Parallels”. Perhaps it was the extent of this work that discouraged my countrymen from attending to its subject, which had ceased to be fashionable since Legendre. Be that as it may, I maintain that the theory of parallels should not forfeit its claim to the attentions of geometers. […]