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(AB + BC) : AC? = (DE+EF): DF?; therefore
(AB + BC): AC = (DE + EF): DF. Componendo, (AB + BC + AC): AC = (DE + EF + DF): DF,
2 AB: AC = 2DE: DF; that is,
AB: AC = DE: DF.
Summary of results.
If AB be any straight line divided at C in extreme and mean ratio, AC being the greater segment, and if we have a cube, a dodecahedron and an icosahedron inscribed in one and the same sphere, then: (1) (side of cube) : (side of icosahedron) = Į (AB” + AC?): V (AB+ + BC?); (2) (surface of dod.): (surface of icos.)
= (side of cube) : (side of icosahedron); (3) (content of dod.): (content of icos.)
= (surface of dod.): (surface of icos.); and (4) (content of dodecahedron) : (content of icos.)
= (AB+ AC%): (AB + BC?).
II. NOTE ON THE SO-CALLED “BOOK XV.”
The second of the two Books added to the genuine thirteen is also supplementary to the discussion of the regular solids, but is much inferior to the first,“ Book xiv." Its contents are of less interest and the exposition leaves much to be desired, being in some places obscure and in others actually inaccurate. It consists of three portions unequal in length. The first (Heiberg, Vol. v. pp. 40--48) shows how to inscribe certain of the regular solids in certain others, (a) a tetrahedron ("pyramid ") in a cube, (6) an octahedron in a tetrahedron (“pyramid”), (c) an octahedron in a cube, (d) a cube in an octahedron and (e) a dodecahedron in an icosahedron. The second portion (pp. 48–50) explains how to calculate the number of edges and the number of solid angles in the five solids respectively. The third (pp. 50—66) shows how to determine the angle of inclination between faces meeting in an edge of any one of the solids. The method is to construct an isosceles triangle with vertical angle equal to the said angle of inclination ; from the middle point of any edge two perpendiculars are drawn to it, one in each of the two faces intersecting in that edge; these perpendiculars (forming an angle which is the inclination of the two faces to one another) are used to determine the two equal sides of an isosceles triangle, and the base of the triangle is easily found from the known properties of the particular solid. The rules for drawing the respective isosceles triangles are first given all together in general terms (pp. 50–52); and the special interest of the passage consists in the fact that the rules are attributed to “ Isidorus
our great teacher." This Isidorus is no doubt Isidorus of Miletus, the architect of the Church of St Sophia at Constantinople (about 532 A.D.), whose pupil Eutocius also was; he is often referred to by Eutocius (Comm. on Archimedes) as ο Μιλήσιος μηχανικός Ισίδωρος ημέτερος διδάσκαλος. Thus the third portion of the Book at all events was written by a pupil of Isidorus in the sixth century. Kluge (De Euclidis elementorum libris qui feruntur XIV et XV, Leipzig, 1891) has closely examined the language and style of the three portions and conjectures that they may be the work of different authors; the first portion may, he thinks, date from the end of the third century (the time of Pappus), and the second portion too may be older than the third. Hultsch however (art. “Eukleides ” in Pauly-Wissowa's Real-Encyclopädie der classischen Altertumswissenschaft, 1907) does not think his arguments convincing
It may be worth while to set out the particulars of Isidorus' rules for constructing isosceles triangles with vertical angles equal respectively to the angles of inclination between faces meeting in an edge of the several regular solids. A certain base is taken, and then with its extremities as centres and a certain other straight line as radius two circles are drawn; their point of intersection determines the vertex of the particular isosceles triangle. In the case of the cube the triangle is of course right-angled; in the other cases the bases and the equal sides are as shown below.
Base of isosceles triangle
Equal sides of isosceles triangle
For the tetrahedron
the side of a triangular face
the perpendicular from the
vertex of a triangular face to its base
For the octahedron
the diagonal of the square
on one side of a triangular
For the icosahedron
For the dodecahedron
the chord joining two non
consecutive angular points
consecutive angular points
the perpendicular from the
middle point of the chord joining two non-consecutive angular points of a face to the parallel side of that face (HX in the figure of Eucl. XIII. 17)
ADDENDA ET CORRIGENDA.
Frontispiece. This is a facsimile of a page (fol. 45 verso) of the famous Bodleian ms. of the Elements, D'Orville 301 (formerly x. i inf. 2, 30), written in the year 888. The scholium in the margin, not very difficult to decipher, though some letters are almost rubbed out, is one of the scholia Vaticana given by Heiberg (Vol. v. p. 263) as . No. 15: Ala Toù Kévtpov oủow oủx nv ζητήσεως άξιον, ει δίχα τέμνουσιν αλλήλας: το γαρ κέντρον αυτών η διχοτομία, ομοίως και η εί της ετέρας διά του κέντρου ούσης ή ετέρα μη δια του κέντρου είη, ότι ου δίχα τέμνεται η διά του κέντρου. The ή before εί in the last sentence should be omitted. PFVat. read y without ei. The marginal references lower down are of course to propositions quoted, (1) Slà tò a' roll y, "by nl. 1,” and (2) Slà tò y toll aútoù, "by 3 of the same.”
Vol. 1. p. 20. I am aware that the assumption that the reference in the Mechanics (1. 24, p. 62, ed. Nix and Schmidt) is to Posidonius of Rhodes is disputed. It is pointed out that the context seems to show that the Posidonius referred to lived before Archimedes. Hoppe considers that the reference is to Posidonius of Alexandria, who was a pupil of Zeno the Stoic in the third century B.C. (cf. Meier, De Heronis aetate, pp. 19-21). The passage of the Mechanics in the German translation is as follows: “Posidonius, ein Stoiker, hat den Schwer- und Neigungspunkt in einer natürlichen (physikalischen ?) Definition bestimmt und gesagt: der Schwer- oder Neigungspunkt ist ein solcher Punkt, dass, wenn die Last in demselben aufgehängt wird, sie in zwei gleiche Teile geteilt wird. Deshalb haben Archimedes und seine Anhänger in der Mechanik diesen Satz spezialisiert und einen Unterschied gemacht zwischen dem Aufhängepunkt und dem Schwerpunkt." This passage may certainly indicate that Posidonius' definition “represents a more imperfect standpoint than that of Archimedes” (Eneström in Bibliotheca Mathematica Vill3, p. 177). But I do not feel certain that “deshalb” necessarily means so much as that it was the particular definition given by Posidonius personally which suggested to Archimedes the necessity for a distinction between the “Aufhängepunkt” and the “Schwerpunkt." I agree however with Meier (p. 21) that the doubt as to the reference makes it impossible to build upon the passage for the purpose of determining the date of Heron. Vol. 1. pp. 32–33.
As bearing on the question whether Proclus continued his commentary beyond Book 1., I should have referred to the scholium published by Heiberg in Hermes xxxvIII., 1903, p. 341, No. 17. It begins with the heading "Scholium on the scholium of Proclus on the oth proposition
where he says...” the words then quoted being taken from the last five lines of the long scholium x. No. 62 (Heiberg, Vol. v. pp. 450—2), one of the scholia Vaticana ; and similar words lower down are accompanied by the parenthetical remark, “as the scholium of the divine Proclus says.” If Proclus was really the author of the scholium, this is a point in favour of those who maintain that Proclus did write commentaries on the other Books (cf. Meier, De Heronis aetate, pp. 27-28). Heiberg however points out that, while the scholium shows that a Byzantine scholar took the collection of scholia Vaticana to be the work of Proclus, it does not prove more than this, and certainly it is not conclusive evidence that Proclus' commentaries covered all the Books. That this is possible cannot be denied; the scholia Vaticana to the other Books may, like those to Book 1., have been extracted from Proclus, as also may the fragments which they contain of the commentary of Pappus, though it is not easy to explain why Proclus should have included extracts from Pappus which had already been put into the text by Theon. But it is much more probable, Heiberg thinks, that a Byzantine mathematician who had in his ms. of Euclid the collection of scholia Vaticana, and knew that those on Book 1. came from Proclus, himself attached the name of Proclus to the rest of the collection; and this hypothesis seems to be confirmed by the fact that none of the other, older, sources of the scholia Vaticana have Proclus' name in x. No. 62.
Vol. 1. pp. 64–66. Hultsch has some valuable remarks on the origin of the scholia (Bibliotheca Mathematica V1113, pp. 225 sqq. and art. “Eukleides ” in Pauly-Wissowa's Real-Encyclopädie der classischen Altertumswissenschaft, 1907). Theodorus, Plato's teacher, is quoted in Plato's Theaetetus 147 D as having proved the irrationality of 13, 15 etc. up to 117; and the expressions used to describe such square roots, evidently Theodorus' own, are δύναμις ποδιαία, δύναμις τρίπους, δύναμις πεντάπους etc., the “square root “side” of “
one, three, five etc. square feet.” The same phraseology survives in the scholia x. Nos. 52, 94, 149, where we have the expressions η τρίπους, η τετράπους, η πεντάπους, η εξάπους, η επτάπους, η οκτάπους, η εννεάπους etc. Hultsch concludes that the sources go back as far as Theodorus. As regards the extracts from Geminus, Hultsch observes that the scholia to Book 1. contain a considerable portion of Geminus' commentary on the definitions. They 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 considers it probable that the scholium at the beginning of Book v. (No. 3) attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid represents the tradition going back to Geminus; similarly he regards scholium xiii. No. I as having the same origin.
Vol. 1. p. 71.
The scholium numbered 17 on page 341 in Hermes Xxxviii. is taken from a MS, which was written in the with cent. Since the Arabic figures in it are in the first hand, it follows that the acquaintance of the Byzantines with these figures dates 100 years further back than the date given (12th cent.).
Vol. I. p. 71.
In the numerical illustrations of Euclid's propositions sexagesimal fractions are often used ; e.g. approximations to the values of
surds are expressed as so many units, so many of the fractions 1/60, so many of the fractions 1/60” etc., going as far as “fourth-sixtieths” or the fractions 1/604. Hultsch wrote a short paper on the sexagesimal fractions in the scholia to Book x (Bibliotheca Mathematica V3, pp. 225-233). He shows that numbers expressed in these fractions are handled with skill and sometimes include results of surprising accuracy, as when 727 is given (allowing a slight correction of the last fraction by means of the context) as 5° 1' 46" 10"", where represents units and dashes the successive sexagesimal fractions, which gives for 13 the approximation 1° 43' 55" 23"", being the same result as that given by Hipparchus in his tables of chords reproduced by Ptolemy and correct to the seventh decimal place. Similarly 18 is given as 2° 49' 42" 20"" 10"", which is equivalent to V2 = 1'4142135. Hultsch gives instances of the various operations, addition, subtraction, multiplication and division, carried out in these fractions, and shows how the extraction of the square roots was effected, after the method which Theon of Alexandria in his commentary on Ptolemy's gúvtatis applies to the evaluation of 14500, and which evidently goes back to Hipparchus.
Vol. 1. p. 101. In the Bibliotheca Mathematica IX3, 1908, p. 76, A. Sturm notes that the preface to Camerarius' Euclid was not by Rhaeticus but by Camerarius himself, since the printer of the Steinmetz edition, Johann Steinmann, says, in a short preliminary notice, that Camerarius had written the preface 28 years before “sub alieno nomine."
Vol. 1. p. 116. The date given for Eudoxus is that arrived at by Susemihl, “ Die Lebenszeit des Eudoxos von Knidos” in Rheinisches Museum für Philologie, Lill., 1898, pp. 626-8. Hultsch however shows cause for rejecting this conjecture and for adhering to the earlier determination of the date as 408—355 B.C. Vol. 1. pp. 249, 370.
The statement that Euclid does not use the expression ai Bar, “the straight lines BAC,” for “the straight lines BA, AC” is not accurate. Although I have not found it in the early Books, it is somewhat common in Books x, xi and will. Thus, e.g., in Book x “the rectangle (contained) by BD, DC” is often written tò ÚTÒ TÔ BAT or tò ÚTÒ BAT, and in one place (x. 59) we find tò ovykcijevov ék TÔ årò Tơv MNE for “the sum of the squares on MN, NO.” In Book xi the contracted form is used in expressions for the plane through two straight lines, e.g. tò dià Tô BAA étrimedov, “the plane through BD, DA.” In xill. 11 we have ovvaupótepos ♡ ATM for “the sum of the two straight lines DC, CM," where DC, CM form an angle.
Vol. 1. pp. 343–4, 351; Vol. 11. p. 97; Vol. III. pp. 1-3, etc. Heinrich Vogt’s paper “Die Geometrie des Pythagoras” in the Bibliotheca Mathematica IX, (September, 1908), pp. 14-54, unfortunately appeared too late to be noticed in the proper places. I do not think it would have enabled me to modify greatly what I have written regarding the supposed discoveries of Pythagoras and the early Pythagoreans, because I have throughout endeavoured to give the traditions on the subject for what they are worth and no more, and not to build too much upon them. Vogt's paper is however a valuable piece of criticism, deserving of careful study; and it requires notice here so far as considerations of space allow. G. Junge had in his paper Wann haben die Griechen das Irrationale entdeckt? mentioned above (Vol. 1. p. 351, Vol. III. p. i n.)