"Let us for the present admit, that a new work were written on a plan different from that of Euclid, constructed upon different principles, built upon different data, and exhibiting the leading results of geometrical science of a different order. Let us wave also the great improbability, that even an experienced instructor should execute a work superior to that which has been stamped with the approbation of ages, and consecrated, as it were, by the collected suffrage of the whole civilised globe. Still it may be questioned whether, on the whole, any real advantage would be gained. It is certain that all would not agree in their decision on the merits of such a work. Euclid once superseded, every teacher would esteem his own work the best, and every school would have its own class book. All that rigour and exactitude, which have so long excited the admiration of men of science, would be at an end. These very words would lose all definite meaning. Every school would have a different standard; matter of assumption in one, being matter of demonstration in others; until, at length, Geometry, in the ancient sense of the word, would be altogether frittered away, or be only considered as a particular application of Arithmetic and Algebra."