From this moment a brilliant period opens for geometrical research of every kind. Analysts interpret all their results, and set to work to translate them by constructions. Geometers endeavour to discover in every question some general principle—in most cases impossible to prove without the aid of Analysis—so as to deduce a host of particular consequences, closely connected with each other and with the principle from which they are derived. Jacobi’s brilliant pupil, Otto Hesse, admirably develops the methods of homogeneous coordinates of which Plücker, perhaps, did not fully appreciate the value. Boole discovers in Bobillier’s polars the first notion of a covariant ; the theory of forms is created by the labours of Cayley, Sylvester, Hermite, and Brioschi. Aronhold, Clebsch, Gordan, and other geometers still with us, invent the definitive notation of the theory, establish the fundamental theorem relative to the limitation of the number of covariant forms, and thus succeed in giving to the theory its fullest extension and scope.

The theory of surfaces of the second order, constructed mainly by the school of Monge, is enriched by a large number of elegant properties—mostly due to Hesse, who later finds in Paul Serret a worthy rival and an investigator who will continue his work.

Were it not for Euclid’s Propositions iii. 26 (proved by superposition) and vi. 33, the statement that he nowhere defines the circumference of a circle or any curved line would surely be a truism. His definition of a circle is substantially only a locus definition. It may well be argued that neither does he define a straight line (or an angle) ; but in the case of a straight line he introduces sufficient ideas and axioms to make his subsequent propositions convincing, whereas he gives us nothing to bridge the interval between a straight line and a curved one.

And so, wise after the event, we may say that all the thought spent in old days over ‘squaring the circle’ had no adequate foundation, and was foredoomed to fail.

As to iii. 26 and vi. 33, they are only further examples of the truth with which we are familiar in the case of angles, that it is sometimes easier to prove that things are equal etc., than to say exactly what they are.