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In giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematics1.

I will take first the works which are mentioned by Greek authors.

I. The Pseudaria.

I mention this first because Proclus refers to it in the general remarks in praise of the Elements which he gives immediately after the mention of Euclid in his summary. He says2: “But, inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms, and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry.”

The book is considered to be irreparably lost. We may conclude however from the connexion of it with the Elements and the reference to its usefulness for beginners that it did not go outside the domain of elementary geometry3.

1 See, for example, Loria, Le scienze esatte nell' antica Grecia, 1914, pp. 245-268; 1. L. Heath, History of Greek Mathematics, 1921, I. pp. 421-446. Cf. Heiberg, Litterargeschichtliche Studien über Euklid, pp. 36-153; Euclidis opera omnia, ed. Heiberg and Menge, Vols. VI.—VIII.

2 Proclus, p. 70, 1-18.

3 Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his commentary on Aristotle's Sophistici Elenchi (fol. 25 b): “Not only those (ἔλεγχοι) which do not start from the principles of the science under which the problem is classed...but also those which do start from the proper principles of the science but in some respect admit a paralogism, e.g. the Pseudographemata of Euclid.” Tannery (Bull. des sciences math. et astr. 2^{e} Série, VI., 1882, I^{e\re} Partie, p. 147) conjectures that it may be from this treatise that the same commentator got his information about the quadratures of the circle by Antiphon and Bryson, to say nothing of the lunules of Hippocrates. I think however that there is an objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson's quadrature did not start from the proper principles of geometry, but from some principles more general.

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