“prolegomena in Euclidis Elementorum geometriae libros”
(two definitions of geometry) and “Varia miscellanea ad geometriae cognitionem necessaria ab Isaaco Monacho collecta”
(mostly the same as pp. 252, 24-272, 27 in the Variae Collectiones included in Hultsch's Heron); lastly, a note of Dasypodius to the reader says that these scholia were taken “ex clarissimi viri Joannis Sambuci antiquo codice manu propria Isaaci Monachi scripto.”
Isaak Monachus is doubtless Isaak Argyrus, 14th c.; and Dasypodius used a MS. in which, besides the passage in Hultsch's Variae Collectiones, there were a number of scholia marked in the margin with the name of Isaak (cf. those in b under the name of Theodorus Cabasilas). Whether the new scholia are original cannot be decided until they are published in Greek; but it is not improbable that they are at all events independent arrangements of older scholia. All but five of the others, and all but one of the Greek scholia to Book v., are taken from Schol. Vat.; three of the excepted ones are from Schol. Vind., and the other three seem to come from F (where some words of them are illegible, but can be supplied by means of Mut. III B, 4, which has these three scholia and generally shows a certain likeness to Isaak's scholia).
Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c.) on Eucl. Book II., which finds a place in Appendix IV. to the Scholia in Heiberg's edition.
Hultsch has some remarks on the origin of the scholia1. He observes that the scholia to Book I. contain a considerable portion of Geminus' commentary on the definitions and are specially valuable because they contain extracts from Geminus only, whereas Proclus, though drawing mainly upon him, quotes from others as well. On the postulates and axioms the scholia give more than is found in Proclus. Hultsch conjectures that the scholium on Book v., No. 3, attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid, represents the tradition going back to Geminus, and that the scholium XIII., No. 1, has the same origin.
A word should be added about the numerical illustrations of Euclid's propositions in the scholia to Book X. They contain a large number of calculations with sexagesimal fractions2; the fractions go as far as fourth-sixtieths (1/60^{4}). Numbers expressed in these fractions are handled with skill and include some results of surprising accuracy3
1 Art. “Eukleides” in Pauly-Wissowa's Real-Encyclopädie.
2 Hultsch has written upon these in Bibliotheca Mathematica, V_{3}, 1904, pp. 225-233.
3 Thus \sgrt{(27)} is given (allowing for a slight correction by means of the context) as 5 II' 46'' 10''', which gives for \sgrt{3} the value 1 43' 55'' 23''', being the same value as that given by Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly \sgrt{8} is given as 2 49' 42'' 20''' 10'''', which is equivalent to\sgrt{2}=1.41421335. Hultsch gives instances of the various operations, addition, subtraction, etc., carried out in these fractions, and shows how the extraction of the square root was effected. Cf. T. L. Heath, Históry of Greek Mathematics, 1., pp. 59-63.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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