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 extensions or developments) from the plan laid down by Euclid.

— Augustus De Morgan, “Short Supplementary Remarks on the First Six Books of Euclid’s Elements” (1848; published in the Companion to the Almanac for 1849)

And now that Euclid’s proofs and arrangement are no longer required from candidates at examinations, there has been a rush of competitors anxious to be first in the field with a new text-book on the more “practical” lines which now find so much favour. The natural desire of each teacher who writes such a text-book is to give prominence to some special nostrum which he has found successful with pupils. One result is, too often, a loss of a due sense of proportion; and, in any case, it is inevitable that there should be great diversity of treatment. It was with reference to such a danger that Lardner wrote in 1846:

“Euclid once superseded, every teacher would esteem his own work the best, and every school would have its own class book.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 another; 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.”

It is, perhaps, too early yet to prophesy what will be the ultimate outcome of the new order of things; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more.

— From the Preface to Sir Thomas L. Heath’s translation of the Elements of Euclid

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