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made by Eudemus in the summary of the history of geometry, so frequently quoted in these pages:

'Plato, who came next after them [Hippocrates of Chios, and Theodorus of Cyrene], caused the other branches of knowledge to make a very great advance through his earnest zeal about them, and especially geometry: it is very remarkable how he crams his essays throughout with mathematical terms and illustrations, and everywhere tries to rouse an admiration for them in those who embrace the study of philosophy.'

17

The way in which Plato is here spoken of is in striking contrast to that in which Eudemus has, in the summary, written of the promoters of geometry.

1 Πλάτων δ' ἐπὶ τούτοις γενόμενος, μεγίστην ἐποίησεν ἐπίδοσιν τὰ τε ἄλλα μαθήματα καὶ τὴν γεωμετρίαν λαβεῖν διὰ τὴν περὶ αὐτὰ σπουδήν, ὅς που δῆλός ἐστι καὶ τὰ συγγράμματα τοῖς μαθηματικοῖς λόγοις καταπυκνώσας καὶ πανταχοῦ τὸ περὶ αὐτὰ θαῦμα τῶν φιλοσοφίας ἀντεχομένων ἐπεγείρων. Proclus, ed. Friedlein, p. 66.

1

NOTES AND ADDITIONS.

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CONTINUATION OF BIBLIOGRAPHICAL NOTICES.*

INCE the publication of the concluding part of this work in Hermathena (July, 1887), M. Paul Tannery has collected his Papers, which appeared in the Bulletin des Sciences mathématiques et astronomiques, since April, 1885, and published them in a volume entitled: La Géométrie Grecque comment son histoire nous est parvenue et ce que nous en savons. Essai Critique. Première partie. Histoire générale de la géométrie élémentaire. Paris, 1887.

M. Paul Tannery has also published a volume on the origin of science in general-Pour l'Histoire de la Science Hellène de Thalès à Empedocle. Paris, 1887. This work is founded on articles which were published by M. Tannery in the Revue philosophique.

Dr. Heiberg has completed his edition of the Elements of Euclid by the publication of vol. v.-Continens Elementornm qui feruntur Libros XIV-XV. et Scholia in Elementa cum Prolegomenis criticis et Appendicibus. Lipsiae, 1888.

The first part of a Monograph on Eudoxus by Herr Hans Künsberg has recently appeared-Der Astronom, Mathematiker und Geograph EUDOXOS, von Knidos, I. Theil: Lebensbeschreibung des Eudoxos, Ueberblick über seine astronomische Lehre und geometrische Betrachtung der Hippopede von Hans Künsberg, kgl. Reallehrer. (Programm zum Jahresbericht der vierkursigen königl. Realschule Dinkelsbühl pro 1888.) Druck von C. Fritz in Dinkelsbühl.

There has also been recently published: A Short Account of the History of Mathematics, by Walter W. Rouse Ball, Fellow and Assistant-Tutor of Trinity College, Cambridge; and of the Inner Temple, Barrister-at-law. London, Macmillan and Co., 1888. This book is for the most part a transcript of some lectures delivered this year by Mr. W. W. Rouse Ball.

* See pp. 1, 52, 150, 180, and 206.

PAGES 11, 12.

The passage of Geminus referred to here is taken from his Review of Mathematics, and is given in extenso in Chapter VI., pp. 164, 165.

PAGES 16, 80.

Harpedonaptae. See Cantor (Vorlesungen über Geschichte der Mathematik, pp. 55-57), who points out the Greek origin (åpredóvn, a rope, and arreu, to fasten), previously overlooked, of this name, and shows from inscriptions on the Egyptian temples that the duty of these rope-fasteners' consisted in the orientation of the buildings by reference to the constellation of the Great Bear. The meridian being thus found, the line at right angles to it was probably determined by the construction of a triangle with ropes measuring 3, 4 and 5 lengths respectively. We have seen (p. 29) that the Egyptians knew that such a triangle would be rightangled. The operation of rope-stretching, Cantor adds, was one of unknown antiquity, being noticed in a record of the time of Amenemhat I., which is preserved in the Berlin Museum.

PAGES 29-32.

In connection with this passage of Plutarch, and the observations thereon, it is interesting to note that M. Paul Tannery (la Géométrie Grecque, p. 105) has found in G. Pachymeres (MSS. de la Bibliothèque nationale) the expression τὸ θεώρημα τῆς νύμφης, to designate the 'theorem of Pythagoras' (Euclid 1. 47). In a letter to me, of July 3, 1886, M. Tannery mentions that the Arabs call it the theorem of the bride.' This name for the theorem seems to point to the old Egyptian idea as handed down by Plutarch.

PAGE 37, NOTE 58.

I have since found in Billingsley's Euclide1 the following note on I. 43:—

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This proposition Pelitarius calleth Gnomicall, and misticall for that of it (sayeth he) spring infinite demonstrations and uses in Geometry.' (Fol. 54.)

On referring to Peletarius, however, it will be found that he only calls the figure Gnomic: not the proposition. After the demonstration of I. 43, he says: 'Vix enim usquam in toto opere Geometrico occurrit Figuratio magis fœcunda quam haec Gnomica: hoc est, quae uno parallelogrammo et Gnoma confla

tur

. . Nam hîc Gnomonis explicandi locus est maximè oportunus: licet Euclides ad secundum librum distulerit.

'Hanc ego Figuram mysticam soleo vocare: Ex ea enim, velut ex locupletissimo promptuario, innumerabiles exeunt Demonstrationes. Quod cum magna voluptate perspiciet qui re Geometrica seriò se exercebit.' (Iacobi Peletarii Cenomani, in Euclidis Elementa Geometrica Demonstrationum Libri Sex, p. 41. Lugduni, apud Ioan. Tornæsium et Gul. Gazeium, 1557.)

PAGE 43, NOTE 64.

This proposition-Euclid X. 117-is an interpolation, and is recognised as such by August, who gives it in Appendix V., pars. ii.,

The Elements of Geometrie of the most auncient Philosopher EUCLIDE of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London.

Whereunto are annexed certaine Scholies, Annotations, and Inventions, of the best Mathematiciens, both of time past, and in this our age.

With a very fruitfull Præface made by M. I. Dee, specifying the chiefe Mathematicall Scieces, what they are, and whereunto commodious: where, also, are disclosed certaine new Secrets Mathematicall and Mechanicall, untill these our daies, greatly missed.

Imprinted at London by John Daye, 1570.

De Morgan (Smith's Dictionary of Greek and Roman Biography, Eucleides, vol. II., p. 73) states that Henry Billingsley "was a rich citizen and was mayor (with knighthood) in 1591."

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