"THERE never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensi(Jns or developments) from the plan laid down by Euclid." De Morgan wrote thus in October 1848 (Short supplementary remarks on "the first six . Books 0.1 Euclid's Elements in the Companion to the Almanac for 1849); and I do not .think that, if he had been living to-day, he would have seen reason to revise the opinion so deliberately pronounced sixty years ago. I t is true that in the interval much valuable work has been done on the continent in the investigation of the first principles, including the formulation and classification of axioms or postulates which are necessary to make good the deficiencies of Euclid's own explicit postulates and axioms and to justify the further assumptions which he tacitly makes in certain propositions, content apparently to let their truth be inferred from observation of the figures as drawn; but, once the first principles are disposed of, the body of doctrine contained in the recent textbooks of elementary geometry does not, and from the nature of the case cannot, show any substantial differences from that set forth in the Elements. In England it would seem that far less of scientific value has been done; the efforts of a multitude of writers have rather been directed towards producing alternatives for Euclid which shall be more suitable, that is to say, easier, for schoolboys.