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found it in Latin in a MS. which was then in his own possession but was about 20 years afterwards stolen or destroyed in an attack by a mob on his house at Mortlake1. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible MS. had caused him much trouble and (in a later passage) speaks of “the actual, very ancient, copy from which I wrote out...” (in ipso unde descripsi vetustissimo exemplari). The Latin translation of this tract from the Arabic was probably made by Gherard of Cremona (1114-1187), among the list of whose numerous translations a “liber divisionum” occurs. The Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adaptation of it; it contains mistakes and unmathematical expressions, and moreover does not contain the propositions about the division of a circle alluded to by Proclus. Hence it can scarcely have contained more than a fragment of Euclid's work.

But Woepcke found in a MS. at Paris a treatise in Arabic on the division of figures, which he translated and published in 18512. It is expressly attributed to Euclid in the MS. and corresponds to the description of it by Proclus. Generally speaking, the divisions are divisions into figures of the same kind as the original figures, e.g. of triangles into triangles; but there are also divisions into “unlike” figures, e.g. that of a triangle by a straight line parallel to the base. The missing propositions about the division of a circle are also here: “to divide into two equal parts a given figure bounded by an arc of a circle and two straight lines including a given angle” and “to draw in a given circle two parallel straight lines cutting off a certain part of the circle.” Unfortunately the proofs are given of only four propositions (including the two last mentioned) out of 36, because the Arabic translator found them too easy and omitted them. To illustrate the character of the problems dealt with I need only take one more example: “To cut off a certain fraction from a (parallel-) trapezium by a straight line which passes through a given point lying inside or outside the trapezium but so that a straight line can be drawn through it cutting both the parallel sides of the trapezium.” The genuineness of the treatise edited by Woepcke is attested by the facts that the four proofs which remain are elegant and depend on propositions in the Elements, and that there is a lemma with a true Greek ring: “to apply to a straight line a rectangle equal to the rectangle contained by AB, AC and deficient by a square.” Moreover the treatise is no fragment, but finishes with the words “end of the treatise,” and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is not only Euclid's own work but the whole of it. A restoration of the work, with proofs, was attempted by Ofterdinger3, Who however does not give Woepcke's props. 30, 31, 34, 35, 36. We have now a satisfactory restoration, with ample notes

1 R. C. Archibald, Euclid's Book on the Division of Figures with a restoration based on Woepck's text and on the Practica geometriae of Leonardo Pisano, Cambridge, 1915, pp. 4-9.

2 Fournal Asiatique, 1851, p. 233 sqq.

3 L. F. Ofterdinger, Beiträge zur Wiederherstellung der Schrift des Euklides über die Theilung der Figuren, Ulm, 1853.

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    • William Watson Goodwin, Syntax of the Moods and Tenses of the Greek Verb, Chapter II
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