Archimedes in the Middle Ages. III. The Fate of the Medieval Archimedes. Part iv

170 Pages·1978·7.88 MB·English

by  Marshall Clagett

ARCHIMEDES
in the Middle Ages
Memoirs of the
VOLUME THREE
AMERICAN PHILOSOPHICAL SOCIETY
held at Philadelphia
THE FATE OF THE MEDIEVAL ARCHIMEDES
for Promoting Useful Knowledge
1300 to 1565
Volume 125 Part C
Part IV: Appendixes, Bibliography, Diagrams and
Indexes

marshall clagett
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1978
Copyright © 1978 by The American Philosophical Society
PART IV
Appendixes, Bibliography, Diagrams
and Indexes
Library of Congress Catalog Card Number 76-9435
International Standard Book Number 0-87169-125-6
US ISSN 0065-9738
APPENDIX I
Corrections and Additions
to Volume One
1. Short Corrections and Additions
These corrections and additions embrace the few Addenda et Corrigenda
already noted on page 720 of Volume One and, in addition, numerous
others that have been uncovered in my continuing search of the manu­
scripts and literature of medieval geometry. I have reserved complete
texts for other parts of this appendix.
Page xiv, line 10: For Theorum read Theorem.
P. xxiii, item 19: P. Treutlein, “Der Traktat des Jordanus Nemorarius
'De numeris datis,’ ” Abhandlungen zur Geschichte der Mathematik,
2. Heft (1879), p. 131 (whole article pp. 125-66), dates MS H as falling
between 1350 and 1380. Also at line 13 for 153r read 153v.
P. xxiv, line 2: For Q.150 read Q.510. Also at line 4: For 89r-v read
89r-90r.
P. xxv, line 8: For 92r-v read 93r.
P. xxv, line 10: For 92v read 93r.
P. xxviii, line 15: For Alkiudi read Alkindi.
P. xxix, line 11: For Millas read Millas.
P. 5, line 26: For page 605 read page 60.
P. 7, note 8: For Sines read Chords.
P. 12, paragraph 2: for an expanded consideration of the use of the
Moerbeke translations at Paris, see Volume 3, Part I.
P. 12, last line: For autorship read authorship.
P. 13, lines 20-21: Delete in . . . reprinted.
P. 13, n. 28: For translation read translation.
P. 14, n. 29: For O. Kristeller read P. O. Kristeller.
P. 14, n. 30: For Averianum read Averanium.
P. 28, comments 12-13: For Section 2 read page 60.
P. 33, end of first paragraph: We can also note that Robert Grosseteste
briefly cites the De quadratura circuli in his Commentarius in VIII
libros physicorum Aristotelis, ed. of R. Dales (Boulder, Colorado,
1963), Bk. VII, p. 128.
1249
1250 ARCHIMEDES IN THE MIDDLE AGES CORRECTIONS AND ADDITIONS 1251
P. 33, line 5 from bottom: R. Gunther, Early Science in Oxford, Vol. 2 P. 143, n. 4, line 7: For .Igitur hec read G. Hec.
P. 145, n. 5: The word maiuratura that appears three times in this note
(Oxford, 1923), p. 52, notes a tract of one of Bradwardine’s con­
temporaries: Simon Bredon’s Conclusiones quinque de numero should be inauratura. It is an early term for the “surface of a sphere”
quadrato (MS Oxford, Bodl. Libr. Digby 178, llv-13r), written in and appears in the gromatic work ascribed to Epaphroditus. See above
Bredon’s own hand. At the end of 13r we read: “Has conclusiones Part II, Chap. 3, Sect. I.
recommendo ego Simon de Bredone volenti circa quadraturam circuli P. 158, line 138: For prius read prius..
P. 163, comment 31: For Birt. read Brit..
laborare.” Gunther prints a plate including 12v and 13r, from which I
have corrected his reading of quadraturas to quadraturam. P. 169, line 2 from bottom: For texts read texts..
P. 34, n. 11: There is a free version of the section on quadrature from P. 170, title: For Collegu read Collegii.
Bradwardine’s Geometry (lines 39-86 in my note) in MS Cracow, Bibl. P. 174, line 72: For B read B..
Jag. 1918, 15c., 40r, where Bradwardine’s name is given as Magister P. 194, line 26: For . [1] read [1]..
Henricus Brangburdinus. P. 203, line 12 from bottom: For middel read middle.
P. 36, ns. 12-13: See above, Part II, Chapter 3, Sections II-III for various P. 223, title: For filorum read filiorum.
manuals of practical geometry. P. 228, line 11: For accasion read occasion.
P. 37, line 14 from bottom: For broken letter T read J and line 13: For PP. 232-33: I now believe that the marginal notes are introduced by
Q. 150, 89r-v read Q.510, 89r-90r. the word alibi rather than by the phrase in alio, as I formerly thought.
P. 38, line 10: For BE 8.y. 18 read Gen. 1115. Glasgow MS Gen. 1115 P. 234, line 12 from bottom: For 55v read 55r.
is dated 1480 on folio 172v. P. 237, line 3: For second read first.
P. 38, line 5 from bottom: For 1948 read 1947. P. 238, variant 1-2: Change the Arabic “h” to “d” in the Arabic name
P. 48, line 90: Vat. lat. 4275, 82v, adds after ”153” the following: “Et for “Ahmad.”
P. 249, v. 21: The term “taksir” appears in the sense of mensura through­
proportio EG ad GZ est maior quam 265 ad 153.” This is missing in
the early manuscripts. It is a correction of the statement in the Greek out the Geometry of al-Khwarizmi. See the text of Rosen reprinted in
text that EG/GZ = 265/153. It seems to reflect the correction of the H. Schapira, “Mischnath ha-Middoth,” Abhandlungen zur Geschichte
translation believed to be by Plato of Tivoli (see Vol. 1, p. 57, com­ der Mathematik, 3. Heft (1880), pp. 36-44.
ment 90). P. 263, lines 1, 2, 5, 7, 9, 10, 11: For square read rectangle. The Latin
P. 48, variant 94, line 10: For longitunine read longitudine. text uses quadratum, but this is another instance of Gerard’s ambiguous
P. 55, v. 146-48, first line of translation: For since read when. use of that term (See Vol. 1, p. 233). In Fig. 38, KHT should be drawn
P. 61, line 10: For Incidently read Incidentally. as a rectangle. Furthermore, in the English translation of the Arabic
P. 66, line 1 from bottom: For 92r-v read 93r. variant below, “KT2” (used three times) should be altered to “area
P. 82, line 8 and p. 83, line 7 from bottom: For Z read Y. Y was the KT.” Incidentally, in the second line of that translation, the parenthesis
reading in the manuscript (see variant line 8) and is necessary for the before “Vi DE" should be moved to a position before “Vi BG.”
succeeding reference to X.l of the Elements. P. 281, line 9: For DE + EA = ZE + EA read DE = ZE and EA = EA.
P. 84, line 21: For tringulus read triangulus. P. 287, line 6: For trianle equals to read triangle equals.
P. 84, line 36: For 4a, perhaps read 41a. A case can be made for either P. 292, v. 24, line 3: For et read secundi.
reading, but the latter seems to apply more directly. P. 333, line 6 from bottom; and p. 335, line 10: For area read volume.
P. 91, line 4 from bottom: For Proposition II read Proposition III. P. 364, Prop. XII: For a proof of this proposition, see G. B. Benedetti,
P. 96, n. 6: For IV. 11 read IV.4. Diversarum speculationum et physicarum liber (Turin, 1585), p. 252.
P. 101, line 13 from bottom: For segments read four segments. His point of departure was G. Cardano, De subtilitate, Bk. 16, Chap. 1.
P. 108, line 19: For qua read qua-. P. 366, Prop. XVII, c. 1-60: This solution given by the Banii Mfisa
P. 108, lines 21-22: See the review of Volume One by G. J. Toomer, appears to have influenced a Byzantine author who labels it as a
Speculum, Vol. 42 (1967), p. 363. methodos arabike (see J. L. Heiberg, “Kleine Anecdota zur byzantin-
P. 114, lines 50-52: See Toomer’s review, p. 364. ischen Mathematik,” Zeitschrift fur Mathematik und Physik, Vol. 33
P. 115, line 19: For eights read eighths. [1888], Hist.-lit. Abth., pp. 161-63, whole article pp. 161-70).
P. 124, lines 185-86: See Toomer’s review, p. 363. P. 369, n.l : For a > b > c read a > c > b and for property read property
P. 142, line 7 from bottom : For 1440 read 1540. of.
P. 142, line 5 from bottom: For 1468 read 1568. P. 374, line 3: For demonstrabiles read demonstrabo.
v
1252 ARCHIMEDES IN THE MIDDLE AGES CORRECTIONS AND ADDITIONS 1253
P. 374, line 17: Delete quorum cuiuslibet and for quantitas read quantus. scilicet quod divisio procedit in infinitum: et ideo non debet ei contra­
P. 379, asterisk note: It is not proved directly in Bk. I of the Almagest, dicere.” (I have changed punctuation and capitalization slightly.)
but it can be proved by one of the proofs of Chapter 9 of Bk. I, as P. 429, line 29: A somewhat garbled description of the procedures of
was demonstrated in the so-called Aliud commentum de ponderibus in Antiphon and Bryson is found in a single 13c vellum folio (No. 63-29)
Liber Iordani . . . de ponderibus propositiones XIII, Prop. V (Nurem­ of the William Rockhill Nelson Gallery of Art and Atkins Museum of
berg, 1533), pp. 10-13 (cf. the new edition by Joseph Brown, The Fine Arts (see The Nelson Gallery and Atkins Museum Bulletin, Vol. 5,
“Scientia de Ponderibus” in the Later Middle Ages, Dissertation, No. 1 [1971], pp. 32-35). I read the Bryson text as follows: “brissonis:
University of Wisconsin, 1967, pp. 255-58). The proof shows that, if arguitur sic, ubicumque reperitur maius et minus ibi reperitur [equale].
one take similar segments ABC and EFG (see Fig. Ap.1.1.1) with EFG Sed ibi reperitur maior et minor circulus, ergo et equalis. Non valet
the arc of the larger circle, then (chord EG!chord AC) = (arc EFG! quia hoc est intelligendum de hiis que sunt eiusdem rationis et pertinet
arc ABC). Further, draw chord EF, in the larger circle, equal to AC. ad geometriam has figuras dissolvere quia non negat principia geome­
Then (chord EG/chord EF) = (arc EFG/arc ABC). But by the Almagest, tric.” The Antiphon text runs as follows: “antifon negavit principium
Bk. I, Cap. 9, (arc EFG/arc EF) > (chord EGIchord EF). Therefore, geometrie, scilicet quod continuum sit divisibile in infinitum, et voluit
arc EF < arc ABC. quod in illis figuris esset devenire per decisiones super triangulos ad
But these are arcs subtended by equal chords. minima, et illud minimum esset tunc equale circumferende et sic omnes
Hence, the arc of the larger circle is less than the arc of the smaller partes, et sic quadratum esset equale circulo, et sic illum non est dis­
circle. Q.E.D. The pertinent proof in the Almagestum, Bk. I, Cap. 9 putative quia negavit principia geometrie.”
(Venice, 1515), 6v, begins: “. . . et dicam: Si descripte sint in circulo P. 430, line 6 from bottom: For quorem read quorum.
due chorde diverse: erit proportio chorde longioris ad chordam P. 438, comment 13-15: While continentur is the reading of the Basel
breviorem minor proportione arcus chorde longioris ad arcum chorde text, it should be corrected to continetur (see MS New York, Columbia
brevioris. . . . ” Univ. Libr. Plimpton 156, 115r).
P. 387, comment 79: In view of the proof given in the preceding addendum, P. 440, line 14: For cornor read corner.
the whole discussion is beside the point since the reference is indeed P. 442-43: The authorship of this work stills bristles with difficulties. See
to Book I of the Almagest. Toomer’s review of Volume 1, p. 364. Tinemue naturally leads to a
P. 399, line 11: For consequent read consequens. supposition of Tynemouth in Northumberland. And indeed there is a well-
P. 401, line 14: For diamter read diameter. known canonist, Johannes de Tinemue, who died about 1221 (see Vol.
P. 404, line 13: For Z.5 read ZA 1, page 720). But I have been unable to find any trace of scientific
P. 411, line 25: For o- read of. interests on the part of the canonist. In his review of my first volume,
P. 415, line 11: For exemple read example. Guy Beaujouan suggests the identification (as indeed I had some years
P. 425, line 5 from bottom: For ° read ”. ago in Osiris, Vol. 11, 1954, p. 299) of Johannes de Tinemue with
P. 427, line 16: A description of Antiphon’s procedure was available to Johannes Gervasius of Exeter, and he notes that there is a Tingmouth
medieval readers in Averroes’ Commentary on the Physics of Aristotle, (or Teignmouth) in the neighborhood of Exeter. But there are difficul­
Aristotelis opera cum Averrois commentariis, Vol. 4, (Venice, 1562), ties in connection with this identification. The first one is that, although
1 lvH-K: “Antiphonti autem non debet contradicere, quoniam, cum ille two manuscripts of the De curvis superficiebus contain the name of a
fecit in circulo figuram aequalium laterum et angulorum, et fecit super Gervasius, he is stated to be of Essex (Essexta) and not of Exeter
quodlibet laterum triangulum isoscelem, dividendo arcus inspicientes (Exonia). This could of course be a scribal confusion. A second diffi­
latera in duo media, deducendo lineas ad extrema lateris, et sic fecit culty casting doubt on the possible authorship of Johannes Gervasius
in circulo figuram aequalium laterum et angulorum maiorem prima, cuius of Exeter, at least of the original text, is that Johannes Gervasius would
numerus laterum est duplus ad numerum laterum primae figurae, et presumably have written the work when he was in Italy in 1262 or
cum fecit etiam in hac figura illud, quod fecit in prima, invenit etiam thereafter, i.e., after he had some contact with the circle of Moerbeke.
in circulo figuram aliam maiorem secunda, et sic existimavit, cum fecerit (for it is of interest that Johannes Gervasius died in Viterbo in 1268
sic, quod divisio non procedit in infinitum, sed perveniat ad figuram and hence he could scarcely have escaped knowing Moerbeke who was
aequalem circulo; et cum fecit quadratum aequale huic figurae, existi­ in Viterbo at the same time). But the original Latin De curvis super­
mavit quod illud quadratum erat aequale circulo. Sed istud peccatum ficiebus would have been written long before the 1260s since it was
non erat essentiale Geometriae, quia Geometer habet principium, already cited by Robert Grosseteste in his Commentarius in VIII libros
1254 ARCHIMEDES IN THE MIDDLE AGES CORRECTIONS AND ADDITIONS 1255
physicorum Aristotelis (ed. of Dales, p. 128; Dales following the man­ P. 512, line 5: For traingle read triangle.
uscripts has the erroneous reading of eternis for curvis \ cf. MS Venice, P. 530, line 9: For 92r read 93r.
Bibl. Naz. Marc. VI. 222, 26r, c. 2), a work written in all likelihood P. 531, line 3 from bottom: For DEF read DLF.
before ca. 1232 (ed. Dales, xiv) and of course certainly before his death P. 541, line 1: For two, surfaces read two surfaces.
in 1253: “Dubitat autem aliquis utrum omnis omni comparabilis aut P. 548, line 4: For to read do.
non. Contra in libro De quadratura circuli et De eternis (! curvis) super- P. 558, lines 2 and 5: For remanents read remnants.
ficiebus.” The De curvis superficiebus was also known to Gerard of P. 567, title: For Non-Archimedian read Non-Archimedean.
Brussels and cited in his De Motu under the title of De piramidibus P. 569, end offirst paragraph: There are two “existence” proofs in Arabic
(see Vol. 1, p. 440, n. 4). The De motu was composed considerably that ought to be mentioned in this context. The first is by Ibn al-Haitham
earlier than the 1260’s. Note that the Biblionomia of Richard de (Alhazen) in his Quadrature of the Circle. He concludes that a circle
Foumival (certainly written before 1260 and probably much earlier) inscribed in the lune on the side of a square is some determinate part
describes a codex that contains a De piramidibus (L. Delisle, Le Cabinet of that lune. With that part assumed as found—for he could in no way
des manuscrits de la Bibliotheque Nationale, Vol. 2 [Paris, 1874], find that part by construction—the proof is successfully completed (see
p. 526: “42. Dicti Theodosii liber de speris, ex commentario Adelardi. H. Suter, “Die Kreisquadratur des Ibn el-Haitam,” Zeitschriftfur Math-
Item Archimenidis Arsamithis liber de quadratura circuli. Liber de ematik und Physik, Vol. 44 (1899), Hist.-lit. Abteilung, pp. 32-47).
piramidibus. Liber de ysoperimetris. Item libri de speculis, de visu et The second proof is attached to the end of Alhazen’s tract in two
de ymagine speculi. In uno volumine cuius signum est littera D.”) Berlin manuscripts. Since it is similar to the Jordanus existence proofs
Cf. A. Birkenmajer, Etudes d’histoire des sciences et de la philosophie I present here the substance of it (the Arabic text and German trans­
du moyen age (Wroclaw, Warszawa, Krakow, 1970), p. 166. It is of lation are given by Suter, pages 47, 41): (1) Construct square BG on
interest that theD^ piramidibus follows upon theDe quadratura circuli. given line AB [see Fig. Ap.1.1.2]. (2) Inscribe circle DE in square BG,
P. 443, line 8: For Exter read Exeter. with diameter DE = AB. (3) Since circle DE is a determinate part of
P. 443, line 9: For Winton read Winchester and for 1261 read 1262. square BG, the one has a fixed ratio to the other which we let be
P. 443, line 11: For 1261 read 1262. BZ/AB. (4) We lengthen AB to H so that BH is the mean proportional
P. 443, n. 15: Add A. B. Emden, A Biographical Register of the University between AB and BZ; and thus ABIBH =BH!BZ. [(5) AB/BZ = BH2/BZ2
of Oxford to A. D. 1500, Vol. 2 (Oxford, 1958), p. 757. = AB2/BH2—constituting steps omitted by the author.] (6) Therefore,
P. 445, n. 23: Note that after De curvis superficiebus, Francischus adds from (5), (3) and (2), square BG!circle DE = square BG/square BT. (7)
an interesting phrase: “et est ut allegat iste conclusio quinta, licet Hence it follows that circle DE = square BT.
secundum aliam cotationem sit tertia. ’ ’ This is clear evidence that Fran­ P. 578, lines 15-17: See the review of Volume One by G. J. Toomer,
cischus had access to the original text as well as to the tradition of Speculum, Vol. 42 (1967), p. 365.
manuscript D. P. 583, line 3 of Latin poem: For Olim read Olim licet.
P. 445, n. 24: The De curvis superficiebus was also cited in the Aliud P. 583, line 2 of English poem: For Once read Although once.
commentum de ponderibus, Prop. V, ed. of J. Brown in The “Scientia P. 587: Another manuscript is that of New York, Columbia Univ. Libr.
de ponderibus” in the Later Middle Ages (Dissertation, University of Smith West. MS Add. 1 (formerly Voynich MS no. 10), 138r-39r, 14c.
Wisconsin, 1967), p. 255. Cf. an earlier citation of the same proposition P. 610, line 15: For most read more.
in Proposition I, lines 133-34 (ibid., p. 190). Similar citations to this P. 611, line 17: For DZa read Z.
same proposition are found in De proportionibus velocitatum in motibus P. 612: For other treatments of the quadrature by lunes, see below
of Symon de Castello written in the third quarter of the 14c, ed. of J. Appendix II.
McCue, The Treatise “De proporcionibus velocitatum in motibus” P. 617, line 4: For whit read wit.
Attributed to Nicole Oresme (Dissertation, Univ. of Wisconsin, 1961), P. 625, comment 55, line 3: For a read the.
pp. 44, 119; MS Paris, Bibl. de ΓArsenal 522, 133r, c. 2, 147r, c. 2. P. 633, line 1: For Archimede read Archimedes.
P. 445, n. 25: For Archimenisis read Archimenidis. P. 634, line 4 from bottom: For e.d. read ed..
P. 447, line 11 from bottom: For Q. 150 read Q. 510. P. 634: An additional citation to Archimedes occurs in the late thirteenth-
P. 447, line 8 from bottom: For collected read collated. century tract of Bernard of Verdun, Tractatus super totam astrologiam,
P. 448, line 19: For 153r read 153v. Distinctio III, Cap. 2 (ed. P. Hartmann, O. F. M., Werl-Westf., 1961,
P. 448, lines 3-2 from bottom: For 92v. . . . -tion) read 93r-96v. p. 110): “Et quia ex libro Archimedis (/ Archimenidis?) et aliorum scitur
P. 467: Note to Figure 66 should appear under Figure 67, page 471. proportio cuiusque proportionis (/ portionis?) circuli ad sphaeram, et ex
1256 ARCHIMEDES IN THE MIDDLE AGES CORRECTIONS AND ADDITIONS 1257
scientia cordarum et sinuum sciuntur quantae sunt proportiones (/ por­ P. 708: For triangulus rectanglus read triangulus rectangulus.
tiones?) eclipticae luminarium notis punctis diametri eclipticis, ideo P. 711: Under Berlin read Q.510 for Q.150. Under Glasgow read BE
patet scientia faciendi tabulam quantitatis (/ quantitatum?) tenebra­ for Be.
rum. . . I suspect Hartmann’s text is in error. Even with correc­ P. 713: For Alverny, M. -T. d read Alverny, M. -T. d’.
tions suggested in the parentheses, it is not clear what Bernard is P. 714: Under Aristotle add at line 3 his Mechanica, 166-67. Also add
citing from Archimedes. If we assume Xh&Xad sphaeram merely means a separate entry: Aristotelis rota, 166-67.
ad totum circulum, then he is saying that Archimedes tells how to deter­ P. 717, col. 1, line 1: After 274-89 add 353 v.
mine the ratio of any segment of the circle to the whole circle. In the P. 717: Under Johannes de Muris change 162 to 164.
De mensura circuli of Archimedes in the Gerard of Cremona translation, P. 717: For al-Kharkhi read al-Karkhl.
we are given the area of a circle and in Cor. II to that proposition P. 717: Add al-Khazini, 353v.
the area of a sector of the circle. Then the area of the segment cut P. 717: For Kristeller, O. read Kristeller, P. O.
off by the chord to the arc of the sector is equal to the difference P. 717: For Liber de curvis superociebus, Chap. 4 read Liber de curvis
between the sector and the triangle whose base is the chord and whose superficiebus, Chap. 6.
altitude is the cosine of half the center angle of the sector. Hence, P. 719: For Shrader read Schrader.
with tables of sines we could determine the cosine and thus the area
[Corrections to Volume Two
of the triangle. Thus we could establish a ratio between any segment
P. 7, n. 28, line 2: For Reg. lat. read Reg. Vat.
and the whole circle.
P. 27, n. 21, line 3: “pre-Apollonian terms”. The terms given there are
P. 637, line 10: Similar references to the Philotegni are found also in
not precisely the pre-Apollonian terms, though I believe they were
MSS Utrecht, Bibl. Univ. 725, 107v; Edinburgh, Crawford Observatory
dependent on them. See my forthcoming History of Conic Sections in
Library 1.27, 24r, and Venice, Bibl. Naz. Marc. VIII.8, 7r.
the Latin Middle Ages, Chap. 3.
P. 639, line 2 from bottom: For bais read basis.
P. 33 n. 3 : For Becker read Bekker.
P. 640: I have found five other manuscripts of Version I: Edinburgh,
P. 56, line 30: For Rinucci di Castiglione read Rinuccio da Castiglione.
Crawford Observatory Library 1.27, 24r-v, middle 13c; Milan, Bibl.
P. 71, line 17: For 153vc read 153ve.
Ambros. T.91 sup., 54v-55r, 2nd half of 13c; Escorial N.II.26,42v-43r,
P. 497, 37vU: For Ad1·2 read AD1·2.]
16c; Utrecht, Bibl. Univ. 725, 107v-08r, 15c; Erfurt, Amplon. Q.376,
149v-50r. Two other manuscripts (Paris, BN lat. 7215, 107v, 14c;
Venice, Bibl. Naz. Marc. VIII 8, 7r) contain the enunciation without
2. A Variant Form of the Naples Version
proof. I had not realized that Curtze had already published Version II
in his “Eine Studienreise,” Centralblatt fiir Bibliothekswesen, XVI. of the De mensura circuli
Jahrgang, 6. u. 7. Heft (1899), pp. 297-301, whole article, pp. 257-306.
P. 648, line 4: For propostiti read propositi. One of the earliest of the reworked versions of Gerard of Cremona’s
P. 661, line 19: For O read Oa. translation of the De mensura circuli, Proposition I, was that one which
P. 666, n. 2: For some further solutions of the trisection problem in I published in Volume One (pages 80-91) under the title of the Naples
Arabic works, see C. Schoy, “Graeco-Arabische Studien,” Isis, Vol. 8 Version. After Volume One had gone to press, I discovered a free adaptation
(1926), pp. 27-35, whole article, pp. 21-40. of that version in MS Vat. lat. 4275, 81v-83v. This Vatican adaptation
P. 669, n. 7: Cf. Regiomontanus who uses “modum Alhacen in quinto contains all three propositions of XheDe mensura circuli. Like the Naples
perspective” to trisect an angle: See above, Part III, Chap. 2, Sect. Version, it follows the Gerard of Cremona translation very closely in
II, n. 15. Proposition III (see my description below), but it presents a somewhat
P. 678, n. 3; line 6: For scans read secans. freer version of Proposition II, which in the Naples Version was simply
P. 678, n. 3, line 8: For qide read vide. a copy of the Gerard translation. I have accordingly presented here the
P. 687, line 10: For 82-83 read 182-83. text of Propositions I and II of the Vatican copy.
P. 691, line 8: For play play read play. In comparing the Naples and Vatican copies of Proposition I, we notice
P. 692: For adhere read adherere. that on occasion they are verbally identical but that there is, on the whole,
P. 695: For conteneri read contineri. considerable diversity. In the enunciation of Proposition I, the Vatican
P. 699: Add inauratura: 145n-46n. copy returns to the wording as given by Gerard rather than the altered
P. 701: Delete maiuratura: 145n-46n. wording of the Naples copy.
P. 701: For meditas diametri read medietas diametri. As in the Naples copy, the word lunula is misapplied to a segment.
1258 ARCHIMEDES IN THE MIDDLE AGES THE NAPLES VERSION 1259
Notice that in the second half of the proof of Proposition I, the Vatican the numbers (for example, line 114, “14608” instead of “14688”; line
copy uses it simultaneously for “segment” (lines 76, 78) and for the 133, “59241/2” instead of “1823”; line 149, “86” instead of “71”). The
figure bounded by an arc and any pair of sides of the circumscribed numbers are given in a curious system of combined Roman and Indo-
octagon, a figure which, like the Naples copy, it calls a lunula exterior Arabic numerals. Thus, for “4673” in lines 104, 111, and 114, the Vatican
(lines 83, 84). In the Naples Version, its use for segment does not appear scribe has in the genitive case, “4 milium et dctorum et 73.” The “dc”
in the second half of the proof. is the Roman numeral representation of 600.
The Vatican proof of Proposition I makes virtually the same citations to the As in my other editions, I have capitalized the words of the enunciations
Elements of Euclid. However, in the slightly differing section of lines and also the letters that designate points and geometrical quantities, al­
36-39, the Vatican Version alters the citation from 1.3 and 1.8 simply though small letters are used in the manuscript. No figures are given
to 1.4. On the other hand, it has specific and proper references to 1.41, in the manuscript and so I have added the figures for Proposition I
which are missing in the Naples copy, unless (as is probable) the from the Naples copy (which are labeled in Volume One as Figures 11
reference in the Naples copy in line 36 to 1.4 should rather be to 1.41. and 12), making the few letter changes demanded by the text of the
Note further in line 66 the Vatican copy has changed III. 17 to III. 18. Vatican copy. In the first figure L replaces I of the Naples copy; in the
This is probably an error, although it is barely possible that the author second figure, H, G, B and F replace respectively B, C, P and N of
of the Vatican Version could have seen the medieval translation of the the Naples copy. Further, I have omitted letters D, E, E', G and M
Elements from the Greek, where the proposition is numbered as III. 18. from the Naples figure and added E at the center. Figure Ap.I.2.3 is taken,
The only point worthy of notice in connection with the Vatican copy’s unchanged, from the Gerard translation (Vol. 1, page 47).
adaptation of Proposition II is that, like the Cambridge Version (Vol.
1, page 74, lines 87-88), the author has stressed that the proof is based
on the conclusion that the circumference is equal to Vh the diameter. [De circuli quadratura]
I have not included here the text of Proposition III, which is also in the
Vatican manuscript, since it is a close copy of the Gerard translation in 8iv, c.2 [I ] /OMNIS CIRCULUS TRIANGULO ORTHOGONIO EST EQUALIS
the second tradition of manuscripts Bl (see Vol. 1, pages 37, 48-55). CUIUS UNUM DUORUM LATERUM RECTUM CONTINENTIUM
However, the author of this version has made some slight additions worth ANGULUM MEDIETATI DIAMETRI CIRCULI EQUATUR ET
noting. For example, after “153” in line 90 he adds: “et proportio EG ad ALTERUM IPSORUM LINEE CIRCULUM CONTINENTI.
GZ est maior quam 265 ad 153.” This is a correction of the Greek text 5 Sit circulus AC[Fig. Ap. 1.2.1]; ex eius semidiametro et circum­
which indicated that the ratio of EG to GZ was equal to that of 265 ferentia includentibus angulum rectum constituatur triangulus Z quem dico
to 153. As I pointed out in Volume One (page 57, comment 90), this esse equalem AC circulo, quoniam non est maior vel minor.
ratio was missing in Gerard’s translation but present in the Plato of Tivoli Sit igitur primo circulus maior et sit quantitas Y excessus per quem
translation from the Arabic. I do not know the source of the Vatican AC circulus excedit Z. Est igitur circulus AC maior Y. Cum autem per
manuscript’s addition. It could have been the Plato translation or even 10 primam decimi propositis duabus quantitatibus inequalibus a maiori maius
the treatment of Johannes de Muris in theDe arte mensurandi (see above, est medietate abscindere et a residuo maius eius medietate et sic
Part I, Chap. 4). A similar addition appears after “153” in line 102: “et deinceps donec minore positarum quantitatum minor quantitas relinqua­
EK est plus 2339 et quarta,” which is also found in the treatment by tur, subtrahamus ab AC circulo maius eius medietate et ita deinceps
Johannes de Muris. My conclusion that the Vatican copy was made from donec relinquatur quantitas minor Y et hoc isto modo.
a manuscript in Tradition II is based on a careful collation which 15 Inscribatur circulo AC quadratum ABCD quod probo esse maius
showed that the variant readings of the Vatican manuscript were almost medietate AC circuli partim cum ex una parte A et partim ex alia duco
all those of Tradition II as indicated in my text in Volume One. A strange lineam equalem et equedistans BD linee, que sit linea EAE. Et produco RE
addition from Tradition II is found in the Vatican manuscript between et DE. Est igitur ABD triangulus subduplus ad ED parallelogrammum per
the two halves of the proof of Proposition III: “Omnis trianguli in semi­ 41<am) primi< Sunt igitur AEB et AER trianguli equales ABD triangulo.
circulo cadentis unius duorum laterum in alterum multiplicatio est equalis 20 Igitur ABD maior est BKA et AQD lunulis et eadem ratione BCD
multiplicationi dyametri in perpendicularem que cadit super basim tri­
anguli.” This proposition was included as a fourth proposition in some
of the manuscripts of Tradition II (see Vol. 1, page 54, variant reading Prop. I
for line 157). Finally, we should note in connection with the Vatican 2 cuius corr. ex eius?
copy of Proposition III, that the editor makes occasional errors in writing 10 post -equalibus scr. Vat. et delevi restat
1260 ARCHIMEDES IN THE MIDDLE AGES THE NAPLES VERSION 1261
triangulus maior est BSC et CRD lunulis. Igitur ABCD quadratum maius Ab E [Fig. Ap. 1.2.2] centro ad unum angulorum qui sit F ducatur
est illis 4 lunulis. Cum igitur illud quadratum et ille lunule constituunt 65 linea Secans circumferentiam in puncto G a quo puncto duc HGK lineam con­
circulum, patet quod illud quadratum maius est medietate circuli. tingentem circulum super quam erit linea EF perpendicularis per 18^am^
82r, c.l /Si autem ille 4 lunule sunt maiores Y, secetur AKB arcus per equalia in (/ 17?) 3‘. Subtendantur etiam due corde AG et GB. Et arguo sic: HGE
25 puncto K et fiat AKB triangulus qui (sub)duplus est ad ABPL angulus est rectus per 13<am> primi quia HGF est rectus. Similiter HAE
parallelogrammum, ut prius, ducta lineaPKL equali et equedistantiBA et est rectus. Sed EAG et EGA sunt equales, per 5^am^ primi; igitur residui
protractis lineis AL et BP. Igitur ut prius AKB maior est 2 lunulis AK et 70 partiales, scilicet GAH et AGH, sunt equales. Igitur per 6am primi
KB; et sic arguo de ceteris triangulis et ceteris lunulis. Igitur 4 lunularum respiciunt equa latera, AH videlicet etHG. SedHG minores tHF, quoniam
primarum abscisa est maior medietate. Relinquuntur igitur 8 lunule, que si HF rectum angulum respicit in triangulo HFG. Igitur AH minor est quam
30 non sunt minores Y fac ut prius donec occurrat quantitas minor Y. Sit igitur 82v, c.l HF. Sed que est proportio AH ad HF eadem / est AGH trianguli ad HFG
gratia exempli quod ille 8 lunule sint minores Y. Cum igitur AC circulus est triangulum ex prima sexti quoniam illorum est altitudo una in puncto G.
equalis Z, Y, et ille 8 lunule sunt minores Y, patet quod octogonium quod 75 Igitur AHG triangulus minor est HFG triangulo. Igitur a multo fortiori
est residuum circuli extractis 8 lunulis est maior Z. Igitur duplum octogonii dempta lunula ab AHG triangulo residuum erit minus HFG triangulo. Et
maius est duplo trianguli Z, quod probatur esse impossibile. Dividantur eadem rationeBKG triangulus minor est AEG triangulo et per multo fortius
35 enim latera octogonii per equalia, protractis lineis a centro ad illa puncta, dempta lunula erit residuum minus. Igitur illud quod relinquitur ab AHG
que per 3am tertii sunt perpendiculares super latera octogonii, sed etiam triangulo dempta lunula et illud quod relinquitur a BKG dempta eius lunula
probantur esse equales per 4^am^ primi, arguendo primo de triangulis 80 minora sunt HFK triangulo. Ab illa igitur parte quadrati que est extra AB
quorum latera octogonii sunt bases et postea de minoribus ductis a centro portionem abscinditur maior medietate per lineam HGK. Et similiter
lineis ad angulos. Fiunt igitur ibi 16 partiales trianguli quorum quilibet faciam de aliis angulis quadrati et iterum de residuo donec remaneat
40 subduplus est ad illud quod fit ex ductu unius linearum ductarum ad media minus Y. Sed gratia exempli sunt ille 8 lunule exteriores minores Y. Igitur
puncta laterum octogonii in medium basis octogonii et patet per 41 iste circulus et hec lunule sunt minus Z triangulo. Sed iste circulus et ille
primi. Erunt ergo duo triangulorum subduplo ad id quod fit ex ductu 85 lunule componunt illud octogonium. Igitur illud octogonium minus estZ
eiusdem linee in unam basim octogonii. Et per consequens omnes trianguli triangulo. Igitur duplum octogonii minus est duplo Z trianguli, quod sic
erunt subdupli ad id quod fit ex ductu unius predictarum linearum in omnia improbo. Duplum trianguli est quod fit ex ductu semidiametri in circum­
45 latera octogonii. Sed illi trianguli equales sunt octogonio. Igitur quod fit ex ferentiam. Duplum vero octogonii est quod fit ex ductu semidiametri in
ductu linee predicte in omnia latera octogonii duplum est octogonio. latera octogonii que sunt maiora circumferentia, ut prius probasti. Igitur
Cum igitur quelibet linearum mediantium latera octogonii minor est 90 duplum octogonii maius est duplo trianguli. Igitur non est minus.
semidiametro et latera octogonii minora sunt circumferentia circuli quia Cum igitur Z triangulus non potest esse maior circulo nec minor,
c.2 quelibet corda minor est suo / arcu, igitur duplum octogonii quod fit ex relinquitur quod sit equalis.
50 ductu linee minoris semidiametro in latera octogonii que minora sunt
[II] PROPORTIO ARE (E) OMNIS CIRCULI AD QUADRATUM
circumferentia circuli minus eo est quod fit ex ductu semidiametri in
DIAMETRI EIUS EST SICUT PROPORTIO UNDECIM AD
circumferentiam. Sed illud est duplum Z trianguli per 41 ^am^ primi. Igitur
QUATUORDECIM.
illud octogonium non est maius Z. Et 8 lunule sunt minores Y. Igitur
Hec conclusio fundatur super hac propositione, quod circumferentia
circulus non est maior Z.
5 circuli est tripla sesquiseptima ad eius diametrum.
55 Sit minor illa quodZ. Sit equalis circulo et Y quantitati. Circulo igitur
Quod si illa fuerit, sit AB diameter circuli et circumscribam quadratum
circumscribatur quadratum ABCD quod maius est Z triangulo quoniam
c-2 HG [Fig. Ap. 1.2.3]. Sit itaque DG medietas DE et sit linea EZ / septima
duplum eius maius est duplo Z trianguli. Duplum enim quadrati est quod
DG, et quia proportio trianguli AGE ad triangulum AGD est sicud pro­
fit ex ductu semidiametri circuli in omnia latera quadrati que maiora sunt
portio 21 ad 7 per primam 6li et proportio AEZ trianguli ad AGD est
circumferentia. Duplum vero trianguli quod fit ex ductu semidiametri in
circumferentiam. Cum igitur circulus et Y sunt equales Z et ABCD quad­
60
ratum maius Z, erit quadratum maius circulo et Y. Igitur pars quadrati 69 EGA corr. ex EAG
residua a circulo maior est Y. Dematur ideo maior residui medietate et 75 AHG corr. ex ABG?
76 AHG corr. ex AGH, AHG
cetera donec supersit minus Y et hoc sic.
77 KFG corr. ex BFG
81 portionem corr. ex proportionem
22 illud mg. Vat. ,83 8 corr. ex 13
33, 35 octogonii corr. ex octogoni Prop. U
39 angulos corr. ex angulis 2 diametri corr. ex diameter
62 medietate corr. ex medietas 6 diameter corr. ex diametri
THE NAPLES VERSION 1263
1262 ARCHIMEDES IN THE MIDDLE AGES
BP having been protracted. Therefore, as before, AKB is greater than
10 sicud proportio 1 ad 7 per primam 6li, et igitur proportio AGZ ad AGD est
the two segments AK and KB. And I argue in the same way concern­
sicut proportio 22 ad 7. Cum igitur AGD triangulus sit quarta (!) quadrati,
ing the other triangles and other segments. Therefore, more than half
quoniam medietas medietatis eius, igitur proportio AGZ ad illud quadratum
has been cut away from the first four segments with eight segments remain­
est sicut proportio 22 ad 28. Igitur econtra proportio quadrati ad tri­
ing. If these are not less than Y, proceed as before until a quantity less
angulum AGZ est sicud proportio 28 ad 22. Sed cum eadem sit proportio
than Y results. But, for the sake of an example, let those eight segments
15 14 ad 11 que est 28 ad 22, patet quod illud quadratum ad tri­
be less than Y. Therefore, since circle AC = Z + Y and since those eight
angulum AGZ est sicud proportio 14 ad 11. Sed cum GZ linea sit tripla
segments are less than Y, it is evident that the octagon which is the
sesquiseptima ad diametrum et per consequens equalis circumferende
remainder of the circle after the eight segments have been extracted is
circuli ut ponit hec conclusio et AG est semidiameter, includentes angulum
greater than Z. Therefore, double the octagon is greater than double
rectum, ille triangulus est equalis illi circulo. Et per consequens
ΔΖ. This is proved to be impossible, for let the sides of the octagon be
20 illud quadratum se habet ad illum circulum sicud 14 ad 11, quod fuit
probandum. bisected, and in addition lines are protracted from the center to those
points [bisecting the sides], which lines are perpendicular to the sides of
[On the Quadrature of the Circle] the octagon by III.3 [of the Elements]. But these lines are also proved to
be equal by 1.4 [of the Elements], arguing first concerning the triangles
[1] EVERY CIRCLE IS EQUAL TO A RIGHT TRIANGLE, ONE OF whose bases are the sides of the octagon and afterward concerning the
WHOSE TWO SIDES CONTAINING THE RIGHT ANGLE IS EQUAL lesser [triangles], lines having been drawn from the center to the angles
TO THE RADIUS OF THE CIRCLE WHILE THE OTHER OF [of the octagon]. Therefore, 16 partial triangles are constructed each of
THEM IS EQUAL TO THE CIRCUMFERENCE. which is one-half the product of one of the lines drawn to the
Let there be a circle AC [see Fig. Ap. 1.2.1]. From its radius and midpoints of the sides of the octagon and one-half a side of the octagon,
circumference as lines including a right angle let ΔΖ be constructed, and this is evident by 1.41 [of the Elements]. Therefore, two of the tri­
which triangle I say is equal to circle AC since it is neither greater nor angles will equal one-half the product of the same line [drawn from
less. the center to the midpoint of a side of the octagon] and one side of the
In the first place let the circle be greater [than ΔΖ], and let quantity octagon. Consequently, all of the triangles [together] will equal one-half
Y be the excess by which circle AC exceeds Z. Therefore, circle AC is the product of one of the aforesaid lines and all the sides of the octagon
greater than Y. But since, with two unequal quantities proposed, it is k [added together]. But those triangles are equal to the octagon. There­
[possible] by X.l [of the Elements of Euclid] to cut more than half from fore the product of the aforesaid line and all the sides of the octagon
the greater, and [again to cut] more than half from its remainder and is double the octagon. Therefore, since any of the lines bisecting the
to repeat this process successively until a quantity less than the sides of the octagon is less than the radius and the sides of the octagon
lesser of the proposed quantities remains, let us subtract from circle [together] are less than the circumference of the circle, because any chord
AC more than its half and repeat the process successively until a quantity is less than its arc, therefore double the octagon (which arises from the
less than Y remains and we do this in the following fashion. product of (1) a line less than the radius and (2) the sides of the octagon
Let square ABCD be inscribed in circle AC. This square I prove to be which are less than the circumference of the circle) is less than the product
greater than half of circle AC after drawing a line extending on each side of the radius and the circumference. But that is double ΔΖ by 1.41 [of
of A that is equal and parallel to line BD. Let this line be EAF. And I the Elements]. Therefore, that octagon is not greater than Z. And the
draw BE and DF. Therefore AABD = XA rectangle ED, by 1.41 [of the Ele- ' eight segments are less than Y. Therefore, the circle is not greater thanZ.
ments of Euclid]. Therefore ΔΑΕΒ + AAFD = AABD. Therefore ABD is Let it [the circle] be less than Z. Let it be equal to the circle plus
greater than segments BKA and AQD. By the same argument ABCD quantity Y. Therefore, let the square ABCD be circumscribed about the
is greater than segments BSC and CRD. Therefore, the square ABCD is circle. This square is greater than ΔΖ, since doubled it is greater
greater than the four segments. Hence, since the square and those seg­ than double ΔΖ. For the double of the square is the product of the
ments comprise the circle, it is obvious that the square is greater than radius of the circle and all of the sides of the square which are greater
half the circle. than the circumference, while double the triangle arises from the product
Now if these four segments are greater than Y, let arc AKB be bisected at of the radius and the circumference. Therefore, since the circle and Y
point K and let triangle AKB be drawn and it is half rectangle ABPL as before, are equal to Z and square ABCD is greater than Z, hence the square
line PKL having been drawn equal and parallel to BA and lines AL and * is greater than the circle plus Y. Therefore, the part of the square remain­
\