Archimedes in the Middle Ages, Vol. 4: A supplement on the Medieval Latin traditions of conic sections (1150-1566). Part i

234 Pages·1980·11.25 MB·English

by  Marshall Clagett

ARCHIMEDES
in the Middle Ages
VOLUME FOUR
Memoirs of the A SUPPLEMENT ON THE MEDIEVAL
AMERICAN PHILOSOPHICAL SOCIETY
LATIN TRADITIONS OF CONIC SECTIONS
held at Philadelphia
for Promoting Useful Knowledge (1150- 1566)
Volume 137 Part A
Part I: Texts and Analysis
MARSHALL CLAGETT
THE AMERICAN PHILOSOPHICAL SOCIETY
Independence Square
Philadelphia
1980
PREFACE
When I was preparing the edition of William of Moerbeke’s translations
of the works of Archimedes which appeared as Volume Two of my work,
I decided to investigate separately the question of whether the Archime­
dean and Eutocian treatment of conic sections exhibited in the translations
of Moerbeke exerted any influence on medieval mathematics. While I
found little or no influence of these translations on the medieval treat­
ment of conic sections, I did discover a number of medieval texts on
conic sections (largely but not exclusively a part of medieval optical tra­
ditions) that revealed a steadily increasing interest in the geometry of conic
sections. The results of my investigation constitute this fourth volume.
Copyright 1980 by the American Philosophical Society I have here followed the procedures of publication adopted in Volumes
Two and Three. Once more I have published the diagrams and indexes
in a separately bound fascicle. My system of numbering the diagrams
is obvious. The Arabic numeral before the period designates the chapter
for which the diagram is relevant and the succeeding Arabic numeral
or numerals the ordinal position of the diagram within the chapter. The
indexes of Latin mathematical terms, manuscripts cited, and names and
works are similar to those found in Volume Three. I have also added an
index of Apollonian definitions and propositions so that the reader may
quickly appreciate those parts of the Conics of Apollonius that were
known and used in the Latin Middle Ages and Early Renaissance.
Again I must thank the many libraries which provided me with access
to and microfilms of manuscripts and early printed editions, and most
especially the British Library where I spent many profitable hours on sev­
eral occasions. Furthermore, thanks are due to Professors David Lindberg
of the University of Wisconsin and Sabetai Unguru of the University of
Oklahoma for making available to me some of their microfilms. As usual,
I have learned much from discussions with my colleagues at The Institute
for Advanced Study of various problems of philology and mathematics,
and particularly from those with Professor Harold Cherniss and Dr. Her­
man Goldstine.
Library of Congress Catalog Card Number 62-7218 Thanks must also be given to my research assistants Mr. Glenn Sterr
International Standard Book Number 0-87169-137-X and Dr. Peter Marshall for their efforts in preparing the indexes and
US ISSN 0065-9738 reading proof, and to my secretary Mrs. Ann Tobias for typing and re­
typing this volume, for preparing the final diagrams, and for reading proof.
Needless to say, I am grateful to the American Philosophical Society for
undertaking the publication of these volumes, and especially to its edi­
torial staff for guiding them through the press. Finally, I must acknowledge
with special thanks the help and support of the staff of The Institute for
Advanced Study and its director Dr. Harry Woolf. Not only has the In­
stitute provided an ideal academic home but it has generously given finan­
cial support to these volumes.
Marshall Clagett
Contents
PAGE
Part I: Texts and Analysis
Chap. 1: The Latin Works of Alhazen and the Knowledge
of Conic Sections: De speculis comburentibus and
Perspectiva 3
Chap. 2: John of Palermo’s Translation of a Short Arabic
Tract on the Hyperbola 33
Text A: The Treatise on Two Lines Always
Approaching Each Other but Never Meeting 44
Text B: Another Version of the De duabus lineis 55
Chap. 3: William of Moerbeke and Witelo 63
Chap. 4: The Speculi almukefi compositio 99
Text: The Composition of the Parabolic Mirror 114
Chap. 5: Conic Sections in the Fourteenth and Fifteenth
Centuries: Jean Fusoris, Giovanni Fontana, and
Regiomontanus 159
Text A: Johannes Fusoris, The Booklet on the
Parabolic Section 185
Text B: Notes for Alhazen’s De speculis com­
burentibus as prepared by Giovanni Fontana 200
Text C: De speculis comburentibus (= Speculi al­
mukefi compositio) as prepared by Regiomontanus 203
Text D: Additions by Regiomontanus to the Speculi
almukefi compositio 222
Chap. 6: The Medieval Traditions of Conic Sections in the
Early Sixteenth Century: Giorgio Valla, Johann
Werner, and Albrecht Diirer 235
Text: Johann Werner, The Booklet on Twenty-two
Conic Elements 269
Chap. 7: The Last Stages of the Medieval Traditions of
Conic Sections 311
Text A: Orontius Finaeus, De speculo ustorio 360
Text B: Extracts from F. Barozzi’s Admirandam illud
geometricum problema 384
vii
Part II: Bibliography, Diagrams, and Indexes
Bibliography 465
Diagrams 471
Index of Latin Mathematical Terms 519
Index of Manuscripts Cited 553
Index of Apollonian Definitions and Propositions 557
Index of Names and Works 561
PART I
TEXTS AND ANALYSIS
viii
CHAPTER 1
The Latin Works of Alhazen and
the Knowledge of Conic Sections:
De speculis comburentibus
and Perspectiva
Before the twelfth century the knowledge of conic sections in the Latin
West was non-existent. It is true that occasionally the terms ellipsis,
hyperbola, and parabola had been used in earlier Latin texts but without
their mathematical meanings.1 The first traces of any knowledge of conic
sections in the West came as the result of the Latin translations of two
works of Alhazen (Ibn al-Haytham). The first was the translation by Gerard
of Cremona of Alhazen’s Liber de speculis comburentibus, a work on
the mathematical theory and construction of paraboloidal mirrors.2 To
this work, Gerard prefaced a short fragment translated from the Arabic
text of the introduction to Book I of Apollonius’ Conics and it is this
fragment that demands our initial attention. It begins in the best manuscript
(Paris, BN lat. 9335, 85v):3 “The things that follow are in the beginning
of the Liber de pyramidibus of Apollonius; they are axioms which he
1 See my Archimedes in the Middle Ages, Vol. 2 (Philadelphia, 1976), p. 433.
2 Gerard’s translation has been edited by J. L. Heiberg and E. Wiedemann, “Ibnal Haitams
Schrift uber parabolische Hohlspiegel,” Bibliotheca mathematica, 3. Folge, Vol. 10 (1909-
10), whole article, pp. 201-37; Latin text, pp. 218-31; German translation of the Arabic
text, pp. 205-18. An English translation of the Arabic text was published by H. J. J.
Winter and W. 'Arafat, “Ibn al-Haitham on the Paraboloidal Focussing Mirror,” Journal
of the Royal Asiatic Society of Bengal. Science, Vol. 15 (1949), pp. 25-40. I have used the
Arabic text in Majamu' al-rasa’il . . . ibn al-Haitham (Hyderabad, 1357 A.H.), 3rd tract.
For the latest list of the manuscripts of Gerard’s translation, see D. Lindberg, A Catalogue
of Medieval and Renaissance Optical Manuscripts (Toronto, 1975), pp. 20-21.
3Cf. J. L. Heiberg, ed., Apollonii Pergaei Quae graece exstant, Vol. 2 (Leipzig, 1893),
p. LXXV. Concerning the high quality of MS Paris, BN lat. 9335 for the texts of the
translations of Gerard of Cremona, see my Archimedes in the Middle Ages, Vol. 1 (Madison,
Wise., 1964) p. 227, η. 1. Experts now date the manuscript as early 13c. The quoted
statement in BN lat. 9335, 85v, runs: “Ista que sequuntur sunt in principio libri Apol[l]onii
de pyramidibus; sunt anxiomata (/) que premittit in libro illo.”
3
4 ARCHIMEDES IN THE MIDDLE AGES THE LATIN WORKS OF ALHAZEN 5
premises in that book.” Another manuscript adds: ”They are also valid straight line is produced in both directions, and the point is fixed so that it may
for the Liber de speculis comburentibus. ”4 Then follow the First Defini­ not move,H and the straight line revolves on the circumference of the circle
tions, to which I have added bracketed numbers that are equivalent to until it returns to the place from which it began: then 1 call each of the two
surfaces described by the line in revolution (and each of which surfaces is
those used by Heiberg in his Latin translation of the Greek text.5
opposite to its companion and susceptible to infinite extension since the
[1] When a straight line is drawn between some point and the circumference extension of the line is without end) the “surface of a cone.”7 And I call the
of a circle, and the circle and the point are not in the same plane, and the fixed point the “apex of each of the two surfaces of the two cones.” And I
call the straight line which passes through this point and through the center
4 See Heiberg, Apollonii . . . Quae graece exstant, Vol. 2, p. LXXV: '‘valent etiam
of the circle the “axis of the cone.”
ad librum de speculis comburentibus.”
[2] And I call the figure contained by the circle and the conic surface
5 Ibid., Vol. 1 (Leipzig, 1891), pp. 6-8 for the Greek text; and for the Latin text, Vol.
2, pp. LXXV-LXXVIII. The latter varies little from BN lat. 9335, 85v, which I now quote: between the apex and the circle a “cone.” And I again call the point which
”[1] Cum continuatur inter punctum aliquod et lineam continentem circulum per lineam is the apex of the surface of the cone the “apex of the cone.” And I call the
rectam, et circulus et punctum non sunt in superficie una, et extrahitur linea recta in ambas straight line drawn from the apex of the cone to the center of the circle the
partes, et figitur punctum ita ut non moveatur, et revolvitur linea recta super periferiam “axis of the cone.”8 And I call the circle the “base of the cone.”
circuli donec redeat ad locum a quo incepit, tunc ego nomino unamquamque duarum super- [3] And I call a cone “right” when its axis is erected at right angles to
ficierum quas designat linea revoluta per transitum suum, et unaqueque quarum est opposita its base. I call it “oblique” (declivem) when its axis is not erected orthogonally
sue compari et susceptibilis additionis infinite cum extractio linee recte est sine fine, super­
on its base.
ficiem piramidis. Et nomino punctum fixum caput cuiusque duarum superficierum duarum
[4] And when, from a point9 of any curved line (linee munani) which is
piramidum. Et nomino lineam rectam que transit per hoc punctum et per centrum circuli
in one plane, some straight line is drawn in that plane that bisects all lines
axem piramidis. [2] Et nomino figuram quam continet circulus et quod est inter punctum
drawn within the curved line and having their extremities on it and parallel10
capitis et inter circulum de superficie piramidis piramidem. Et nomino punctum quod est
caput superficiei piramidis caput piramidis iterum. Et nomino lineam rectam que protrahitur to some posited line; then I call the [bisecting] line the “diameter of the
ex capite piramidis ad centrum circuli axem piramidis. Et nomino circulum basim piramidis. curved line.” And I call the extremity of that line [i.e. the diameter] which
[3] Et nomino piramidem orthogoniam cum eius axis erigitur super ipsius basim secundum is at the curved line the “vertex of the curved line.” And I call the parallel
rectos angulos. Et nomino ipsam declivem (mailan) quando non est eius axis erectus ortho- lines which I have mentioned11 “lines of order [i.e. double ordinates] to
gonaliter super ipsius basim. [4] Et cum a puncto omnis linee munani (munhanin) que est that diameter.”
in superficie una plana protrahitur in eius superficie linea aliqua recta secans omnes lineas
que protrahuntur in linea munani et quarum extremitates ad eam, et est equidistans linee
alicui posite, in duo media et duo media, tunc ego nomino illam lineam rectam diametrum 6 The clause “so that it may not move” is a redundancy of the Arabic text. Cf. the
illius linee munani. Et nomino extremitatem illius linee recte que est apud lineam munani variant readings from the Arabic text reported by Heiberg, Gr 2, p. LXXI (1 p. 6, 5) and
caput linee munani. Et nomino lineas equidistantes quas narravi lineas ordinis (tartlbi) Nix, op. cit., Arabic text, p. 4, 1. 4 (I have numbered the lines beginning from the title).
illi diametro. [5] Et similiter iterum cum sunt due linee munani in superficie una, tunc From this note onward the two volumes of Heiberg’s edition of Apollonius will be abbre­
ego nomino quod cadit inter duas lineas munani de linea recta que secat omnes lineas viated respectively Gr 1 and Gr 2.
rectas egredientes in unaquaque duarum linearum munani equidistantes linee alique in duo 7 In the Greek text it is the double conical surface that is named the “conic surface”
media et duo media diametrum mugeniben (mujaniban). Et nomino duas extremitates dia­ while here each of the two surfaces is so named. Furthermore, here each surface is rather
metri mugenibi (mujanibi) que sunt super duas lineas munani duo capita duarum linearum improperly called the “surface of a cone” rather than a “conic surface. Similarly the apex
munanieni (munhaniain). Et nomino lineam rectam que cadit inter duas lineas munanieni and the axis are improperly described as being of a cone rather than of a conic surface.
et punctum super diametrum mugenib et secat omnes lineas rectas equidistantes diametro I say “improperly” for all three cases since the surfaces are considered indefinitely produced
mugenib, cum protrahuntur inter duas lineas munanieni donec perveniant earum extremi­ and the cones cut off by base circles have not yet been defined. Indeed in the last two
tates ad duas lineas munanieni, in duo media et duo media diametrum erectam (qutran cases the Greek text simply does not have the equivalent of “duarum superficierum duarum
qa’iman). Et nomino has lineas equidistantes lineas ordinis ad illam diametrum erectam. piramidum” and “piramidis” (cf. the Arabic variants in Heiberg, Gr 2, LXXI [I p. 6, 12]
[6] Et cum sunt due linee recte que sunt due diametri linee munani aut duarum linearum and Nix, op. cit., Arabic text, p. 4, lines 10 and 11).
munanieni, et unaqueque secat lineas equidistantes alteri in duo media et duo media, tunc 8 Again “piramidis” here and in the succeeding sentence are additions of the Arabic
nomino eas duas diametros muzdaguageni (muzdawijain). [7] Et nomino lineam rectam, text. Cf. Heiberg, Gr 2, LXXI (I. p. 6, 18 and 19) and Nix, op. cit., Arabic text, p. 4,
cum est diameter linee munani aut duarum linearum munanieni et secat lineas equidistantes line 16.
que sunt linee ordinis ei secundum angulos rectos, axem linee munani aut duarum linearum 9 The word “point” has been added in the Arabic text. Cf. Heiberg, Gr 2, p. LXXI
munanieni. [8] Et nomino duas diametros, cum sunt muzdaguageni et secat unaqueque (I p. 6, 24) and Nix, op. cit., Arabic text, p. 5, line 3.
earum lineas equidistantes alteri secundum rectos angulos, duos axes muzdaguageni linee 10 The Latin mistakenly has “est equidistans,” thus implying that it is the diameter
munani aut duarum linearum munanieni.” The Arabie text of these definitions has been that is parallel to the posited line rather than the bisected lines. The preceding “and having
published by L. M. L. Nix, Das fiinfte Buch der Conica des Apollonius von Perga in der their extremities on it” comes out of an Arabic addition. See Heiberg, Gr 2, p. LXXII
arabischen LIbersetzung des Thabit ibn Corrah (Leipzig, 1889), Arabic text, pp. 4-6. I have (I p. 6, 26) and Nix, op. cit., Arabic text, p. 5, lines 4-5.
also used the text in the beautiful Arabic codex of the Conics in MS Oxford, Bodleian 11 The expression “which I have mentioned” (narravi) is taken from the Arabic text.
Library, Marsh 667, lr-6r. See Heiberg, Gr 2, p. LXXII (I p. 6, 29) and Nix, op. cit., Arabic text, p. 5, line 8. The
THE LATIN WORKS OF ALHAZEN 7
6 ARCHIMEDES IN THE MIDDLE AGES
[5] And similarly again when there are two curved lines in one plane, then the Conics), let us pause to comment briefly on Gerard’s translation of
I call that part of the straight line which falls [i.e. is placed] between the two the First Definitions. The first point to notice is Gerard’s practice of
curved lines12 and bisects all the lines that are drawn within each of the curved transliterating the Arabic terms which were strange to him or were strange
lines and are parallel to some line the “transverse diameter" (diametrum to the Latin mathematical vocabulary that existed in the twelfth century.
mugeniben). And I call “vertices of the curved lines" the two extremities The curved line that is formed on the surface of the cone by a plane
of the transverse diameter that are on the two curved lines. And I call “erect that does not pass through the apex of the cone was rendered by Gerard
diameter" the straight line falling [i.e. placed] between the curved lines and
as linea munani (and at times in the dual linee munanieni), munani being
[intersecting] a point on the transverse diameter13 and bisecting all the lines
a transliteration of the Arabic munhani. The original Greek term was
parallel to the transverse diameter when they are drawn between the two
καμπύλης. The other transliterations from the Arabic used by Gerard
curved lines so that their extremities arrive at the curved lines.14 And I call
in this translation were diameter mugenib for a transverse diameter and
these parallel lines “lines of order [i.e. ordinates] to the erect diameter."15
diametri muzdaguageni for conjugate diameters, the transliterated words
[6] And when there are two straight lines which are diameters of a curved
line or of two curved lines and each of them bisects the lines parallel to the originating in the Arabic terms mujanib and muzdawijain respectively.
other, then I call them “conjugate diameters" (diametros muzdaguageni). The Greek expressions were διάμετρος πλαγία and συζυγείς διάμετροι.
[7] And I call the straight line that is a diameter of a curved line or of The only substantial point concerning the interpretation found in the
two curved lines and cuts all of the parallel lines which are lines of order Latin translation as compared with the Greek text is the move made in
[i.e. ordinate] to it16 at right angles the “axis" of the curved line or two paragraphs [4] and [5] toward the formal designation of the parallel bisected
curved lines.
lines as “lines of order’’ or, as they were later to be called, “ordinates.”
[8] And I call the two conjugate diameters each of which cuts the lines
Minor variations have been remarked on in the preceding footnotes.
parallel to the other at right angles “the two conjugate axes of the curved
Now let us return to the remainder of the Arabic introduction, as para­
line or two curved lines.”
phrased from the early propositions of Book I of Apollonius’ Conics.
I have added arbitrary passage numbers and I shall give brief comments
At this point the literal translation of Apollonius’ text ends.17 Before
to the passages.18
going on to examine Gerard’s translation of further definitions that appear
in the Arabic text of the Conics and were based on the early propositions And I now add a preface as an aid to the understanding of what is in
of Book I of Apollonius’ work (but which were not in the Greek text of this book.
[1] When a cone is cut by a plane that does not pass through the apex,
then the common section is a surface which a curved line contains; and when
succeeding “lines of order" (lineas ordinis) is simply ταταγμένως (ordinate-wise) in the
Greek text. the cone is cut by two planes, one of which passes through its apex and the
center of its base forming a triangular section and the other does not pass
12 The clause “which falls between the two curved lines” (quod cadit inter duas lineas
munani) is an expansion of the Greek. See Heiberg, Gr 2, p. LXXII (I p. 8, 2-3) and Nix,
op. cit., Arabic text, p. 5, lines 10-11.
18 MS Paris, BN lat. 9335, 85v (cf. Heiberg, Gr 2, pp. LXXVIII-LXXIX): “Et de eo
13 This phrase “and a point on the transverse diameter” (et punctum super diametrum
in cuius premissione scitur esse adiutorium ad intelligendum quod in isto existit libro est
mugenib) is not in the Greek text. Cf. Heiberg, Gr 2, p. LXXII (I p. 8, 7) and Nix, op.
quod narro. [1] Cum secatur piramis cum superficie plana non transeunte per punctum
cit., Arabic text, p. 5, line 21. The word “intersecting” is my addition.
capitis, tunc differentia communis est superficies quam continet linea munani, et quando
14 The phrase “and bisecting . . . transverse diameter” is much expanded beyond the
secatur piramis cum duabus superficiebus planis, quarum una transit per caput eius et per
Greek text. See Heiberg, Gr 2, p. LXXII (I p. 8, 8) and Nix, op. cit., Arabic text, p. 5,
centrum basis et separat eam secundum triangulum et altera non transit per caput ipsius,
lines 21-22. The Greek merely says that the erect diameter is the straight line lying between
immo secat eam cum superficie quam continet linea munani, et stat una duarum super-
the two curved lines which bisects all of the straight lines intercepted between the two
ficierum planarum ex altera secundum rectos angulos, tunc linea recta que est differentia
curved lines and drawn parallel to some straight line.
communis duarum superficierum planarum non evacuantur (/ evacuatur) dispositionibus
15 The expressions “these parallel lines” (has lineas equidistantes) and “erect diameter”
tribus, scilicet aut quin secet unum duorum laterum trianguli et equidistet lateri alteri,
{diametrum erectam) in the Greek text are “each of the parallels” and “diameter”. See
aut quin secet unum duorum laterum trianguli et non equidistet lateri alii (/ alteri), et cum pro­
Heiberg, Gr 2, p. LXXII (I p. 8, 10) and Nix, op. cit., Arabic text, p. 5, lines 24, 25.
ducatur ipsa et latus aliud secundum rectitudinem, concurrant in parte in qua est caput
16 The phrase “lines of order to it” is not in the Greek. See Heiberg, Gr 2, p. LXXII
piramidis, aut quin secet unum duorum laterum trianguli et non equidistet lateri alii (/
(I p. 8, 16) and Nix, op. cit., Arabic text, p. 6, line 3.
alteri), immo concurrant aut intra piramidem aut extra eam cum protrahuntur secundum
17 The First Definitions of Apollonius were not again translated into Latin from the Arabic
rectitudinem in parte alia in qua non est caput piramidis.” The Arabie text of all of the
until the translation of Christianus Ravius, Apollonii Pergaei Conicarum sectionum libri
additional paraphrased definitions translated by Gerard of Cremona and given in this note and
V. VI. et VII (Kilonii, 1669), pp. 8-13, where they are compared with the translations of
notes 21, 23, 26, 28, 31 accompanied the Arabic translation of the Conics made by Hilal
Federigo Commandino from the Greek as given in the edition of Claude Richard, Apollonii
ibn Abl Hilal al-Himsi found in the Marsh codex noted above in note 5, 6v-7r. It is evident
Pergaei Conicorum libri IV. Cum commentariis R. P. Claudii Richardi (Antwerp, 1655),
pp. 1-8. that Gerard made a very close translation of this text.
8 ARCHIMEDES IN THE MIDDLE AGES THE LATIN WORKS OF ALHAZEN 9
through the apex of the cone but rather cuts it by a plane which a curved [2] But if the straight line which is the common section of the two surfaces
of the [above-noted cutting] planes is parallel to one side of the triangle, then
line contains, and one of the two planes is at right angles to the other, then
the straight line which is the common section of the two planes is in one of the surface in which the cone is cut and which the curved line contains is
the following dispositions: (1) either it cuts one of the two sides of the triangle called a parabola (sectio mukefi).
[3] And if it [the aforementioned common section] is not parallel to the
and is parallel to the other side, or (2) it cuts one of the two sides and is not
parallel to the other—and when it and the other side are produced directly side of the triangle but rather meets it (when both are produced directly) in
they meet in the direction in which lies the apex of the cone, or (3) it cuts the direction in which lies the apex of the cone, then the surface in which
one of the two sides of the triangle, is not parallel to the other side but they the cone is cut and which the curved line contains is called a hyperbola
meet either inside or outside of the cone (when they are produced directly) (sectio addita).
in the direction that is not that of the apex. [4] And if it is not parallel to the side of the triangle but rather meets it
[when both are produced directly] in the other direction (that is, that in which
Of course, the “book” referred to in the preliminary sentence is the the apex of the cone does not lie), then the surface in which the cone is cut,
text itself of Apollonius’ Conics. Now let us examine passage [1], which if not a circle, is called an ellipse (sectio diminuta).
diverges from the Greek text of Apollonius. The paraphraser in this intro­
It will be immediately obvious to the reader that, as I have already
duction attempts as briefly as possible to set out the constructional proce­
suggested, only the constructional instructions for the three sections have
dures for his following definitions of the parabola, hyperbola, and ellipse.
been taken (in a loose fashion) from Apollonius’ propositions; the actual
In a sense, he is merely paraphrasing the conditional clauses of Apollonius’
properties of the sections in terms of the latus rectum and the proofs of
Propositions 1.11, 1.12 and 1.13. It is true that in doing this he vaguely
those propositions have been left out, as is evident from Apollonius’
implies at least two of the earlier propositions of Apollonius, the first
propositions, whose enunciations I now quote:22
of which is Proposition 1.2, which runs:19
11. If a cone is cut by a plane through its axis and is also cut by another
If on either one of the surfaces that are mutually placed with respect to
plane cutting the base of the cone in a straight line perpendicular to the base
the apex two points are taken, and the straight line joining the points does
of the axial triangle, and if further the diameter of the section is parallel to
not verge to the apex, then it will fall within the surface, but if produced
one side of the axial triangle, then any straight line which is drawn from the
it will fall outside.
section of the cone to its diameter parallel to the common section of the
It is this proposition, unspecified at this point of the introduction, that cutting plane and of the cone’s base will equal in square the rectangle con­
lurks in the background to guarantee that the line in the section specified tained by the straight line cut off by it on the diameter beginning from the
section’s vertex and by another certain straight line which has the ratio to
in passage [1] is indeed a curved line. The second proposition implied
the straight line between the angle of the cone and the vertex of the section
in the course of this passage is 1.3: “If a cone is cut by a plane through
that the square on the base of the axial triangle has to the rectangle con­
the apex, the section is a triangle.”20 The paraphrase of this proposition
tained by the remaining two sides of the triangle. And let such a section be
is however more restricted since it specifies that the plane passing through
called a parabola. . . .
the apex also passes through the center of the base of the cone. Missing
12. If a cone is cut by a plane through its axis and is also cut by another
from the paraphrase (but of course to follow in the Arabic text) is the plane cutting the base of the cone in a straight line perpendicular to the base
elegant march of Propositions 1.1 - 1.10 with their proofs which Apollonius of the axial triangle, and if the diameter of the section produced meets one
gives before presenting the key Propositions 1.11-1.13. These latter prop­ side of the axial triangle beyond the apex of the cone, then any straight
ositions the paraphraser then palely reflects in his passages [2]-[4].21 line which is drawn from the section to its diameter parallel to the common
section of the cutting plane and of the cone’s base will equal in square some
19 Heiberg, Gr 1, p. 10. See the translation of the Conics by R. C. Taliaferro in Great area applied to a straight line to which the straight line added along the
Books of the Western World, Vol. 11 (Chicago, 1952), p. 605, which I have used (but changed diameter of the section and subtending the exterior angle of the triangle has
as I have seen fit) in this and the succeeding quotations from the Conics. the ratio that the square on the straight line drawn from the cone’s apex to
20 Heiberg, Gr 1, p. 12; Taliaferro, op. cit., p. 606. the triangle’s base parallel to the section’s diameter has to the rectangle
21 BN lat. 9335, 85v-86r (cf. Heiberg, Gr 2, p. LXXIX): “[2] Quod si linea recta que est contained by the segments of the base which this straight line makes when
differentia communis duarum superficierum planarum equidistat lateri trianguli, tunc super­ drawn, this area having as breadth the straight line cut off on the diameter
ficies super quam secatur piramis et quam continet linea munani nominatur sectio mukefi.
beginning from the section’s vertex by this straight line from the section to
[3] Et si non equidistat lateri trianguli, immo concurrit ei quando protrahuntur secundum
the diameter and exceeding by a figure similar to and similarly situated to
rectitudinem in parte in qua est caput piramidis, tunc superficies super quam secatur piramis
the rectangle contained by the straight line subtending the exterior angle
et quam continet linea munani nominatur sectio addita. [4] Et si non equidistat lateri trianguli,
immo occurrit ei in parte alia in qua non est caput piramidis, tunc superficies super quam
secatur piramis, si non est circulus, nominatur sectio diminuta.” 22 Heiberg, Gr 1, pp. 36-48: Taliaferro, op. cit., pp. 615-18.
10 ARCHIMEDES IN THE MIDDLE AGES THE LATIN WORKS OF ALHAZEN 11
of the triangle and by the parameter. And let such a section be called a [5] When there are two hyperbolas with a common diameter and the gib­
hyperbola. . . . bosity of one of them follows the gibbosity of the other, then they are called
13. If a cone is cut by a plane through its axis and it is also cut by another two “opposite sections’’.
plane that on the one hand meets both the sides of the axial triangle and on
Again this is but a pale reflection of Apollonius (Proposition I.14):24
the other is extended neither parallel to the base nor subcontrariwise, and
if the plane in which the base of the cone lies and the cutting plane meet If the surfaces that are mutually placed with respect to the apex are cut
in a straight line perpendicular either to the base of the axial triangle or to by a plane not through the apex, the section on each of the two surfaces
it produced, then any straight line which is drawn from the section of the will be that which is called the hyperbola; and the diameter of the two sections
cone to the diameter of the section parallel to the common section of the will be the same straight line; and the parameters drawn to the diameter
planes will equal in square some area applied to a certain straight line to parallel to the straight line in the cone’s base are equal; and the transverse
which the diameter of the section has the ratio that the square on the straight side of the figure, that between the vertices of the sections, is common.
line drawn from the cone’s apex to the triangle’s base parallel to the section’s And let such sections be called opposite.
diameter has to the rectangle contained by the intercepts of this straight line
(on the base) from the sides of the triangle, an area having as breadth the Note that rather than following Apollonius’ statement that the param­
straight line cut off from the diameter beginning from the section’s vertex eters of the two branches of the hyperbola are equal the paraphrase merely
by this straight line from the section to the diameter, and deficient by a says that the gibbosity (i.e. convexity or curvature) of one is the same as
figure similar to and similarly situated to the rectangle contained by the that of the other, without specifying the measure of that gibbosity. Also
diameter and parameter. And let such a section be called an ellipse. . . . obscured by the paraphraser is the fact that the branches are formed by
passing a single plane through the opposite surfaces of a double cone
By translating only the introductory definitions and not the actual prop­
and that indeed the two branches constitute the curve. This consideration
ositions of Apollonius, Gerard has failed to present the parabola in terms
of the two branches as forming one curve (with one center) is usually
of the equality of the square of an ordinate and the rectangle composed
regarded as having originated with Apollonius; at any rate it seems likely
of the latus rectum and the ordinate’s abscissa. It is, of course, this key
that he was the first to investigate their properties completely.25 Our
property of the parabola that is fundamental to the proof of Proposition 1
paraphraser follows this definition of opposite branches with a definition
in the De speculis comburentibus, as we shall see. Its inclusion from
of the center of a hyperbola and of an ellipse:26
Proposition 1.11 of the Conics would have greatly aided the reader. The
reader of the Arabic text of the introductory definitions would have had [6] And between the two opposite sections there is a point through which
no such difficulty since the succeeding text of Book I of the Conics in­ all the lines that pass are diameters of the two opposite sections, and this
cluded the actual Propositions 1.11-1.13. Hence he would have been point is called the “center of the two sections.’’ And within the ellipse there
satisfied at this point of the introduction with these preliminary construc­ is a point through which all the lines that pass are diameters to it, and this
tional definitions. point is the “center of the section’’ [i.e. ellipse].
A word should be added concerning the Latin terminology employed
The introduction has thus given the substance of Apollonius, Def. 1
by Gerard for the three conic sections: sectio mukefi, sectio addita and
of the Second Definitions: “1. Let the midpoint of the diameter of both
sectio diminuta, which rendered respectively qatr mukafi, qafi zaid and
the hyperbola and the ellipse be called the center of the section.”27
qat' naqis. It is evident that the literal meanings of the Arabic terms, namely
Following this treatment of the centers of sections the introduction then
sections that are called “equivalent, augmented and diminished,’’ reflect
passes on to diameters:28
the literal meanings of the Greek terms παραβολή, υπερβολή and ελλειψις
as employed by Apollonius in his use of the technique of the application
24 Heiberg, Gr 1, p. 52; Taliaferro, op. cit., p. 620.
of areas for conic sections. It is obvious that mukafi puzzled Gerard of 25 T. L. Heath, Apollonius of Perga. Treatise on Conic Sections (Cambridge, 1896), p.
Cremona, for he settled for the transcription mukefi. As we shall see in lxxxiv.
Chapter 4 below, this transcription was often written as mukesi, the / 26 BN lat. 9335, 86r (cf. Heiberg, Gr 2, p. LXXIX): “[6] Et inter duas sectiones oppositas
est punctum per quod omnes linee que transeunt sunt diametri duarum sectionum oppo­
being misread as s.
sitarum. Et hoc punctum nominatur centrum duarum sectionum. Et intra sectionem dimi­
Passing on from the three basic sections, the Arabic introduction then
nutam est punctum per quod omnes linee que transeunt sunt ei diametri. Et hoc punctum
defines opposite branches:23 est centrum sectionis.”
27 Heiberg, Gr 1, p. 66; Taliaferro, op. cit., p. 626.
23 BN lat. 9335, 86r (cf. Heiberg, Gr 2, p. LXX1X): "[5] Et quando sunt due sectiones 28 BN lat. 9335, 86r (cf. Heiberg, Gr 2, pp. LXXIX-LXXX): ”[7] Et cum in sectione
addite quibus est diameter communis, et gibbositas unius earum sequitur gibbositatem al­ diminuta protrahuntur diametri, tunc ille ex illis diametris quarum extremitates perveniunt
terius, tunc ipse nominantur due sectiones opposite.’’ ad circumferentiam sectionis et non pertranseunt eam nec ab ea abbreviantur nominantur
12 ARCHIMEDES IN THE MIDDLE AGES THE LATIN WORKS OF ALHAZEN 13
[7] And when in an ellipse diameters are projected, those whose extremities Definitions of Apollonius to the three conic sections. Once more we find
are on the circumference of the section, not extending beyond it and not a promise of later treatment that remains unfulfilled in the Latin text.
falling short of it, are called “transverse diameters of the ellipse." Now it should be clear that this introductory fragment would hardly
[8] And the diameter whose beginning is from a point of the circumference
give the neophyte Latin reader much precise information that would be
of the section and whose other extremity falls short of the circumference
of significant use to him when he approached the very specific treatment
of the section or goes beyond it is simply called the "diameter."
of the parabola that followed in theDe speculis comburentibus of Alhazen,
[9] But the diameter which is called "second" is only in the two opposite
for, as we shall see, the fragment threw no light on those propositions
sections and it will pass through the center of both of them; and I shall
of Apollonius appealed to in Alhazen’s crucial first proposition. In fact,
describe it at the end of the sixteenth proposition of this tract [On Conics],
I shall limit my discussion of the tract to that first proposition since it
Passages [7] and [8] distinguish between the expressions "transverse contains the basic knowledge of the parabola that is repeated throughout
diameter” and "diameter.” The definition of the transverse diameter of the tract. However, it is worth noting that in his preface Alhazen refers
the ellipse was drawn by the paraphraser from a statement at the end of to the efforts of earlier investigators (“Archimedes, Anthemius and
Proposition 1.13 where ED, the diameter from which the latus rectum others”) to find mirrors which would reflect rays "to a point so that
is erected at a vertex, is so labeled as “transverse” (πλαγία).29 Passage combustion would be stronger.”32 It is then remarked that they discovered
[9] concerns the "second diameter” and the paraphraser specifically says that the reflection of the rays to a single point would be achieved from
that he (that is, Apollonius) will describe it at the end of Proposition 16, the surface of a concave paraboloid. But, according to Alhazen, the early
and indeed it is so defined as the third definition among Apollonius’ Second investigators did not sufficiently demonstrate the convergence property
Definitions.30 But Gerard translated no more of the actual text of Apol­ of the paraboloidal surface. Hence Alhazen will undertake this demon­
lonius and so the paraphraser’s promise of later treatment remains unful­ stration. Let us now turn to the first proposition.33
filled in Gerard’s translation.
In the final passage of the introduction the paraphraser shifts from
32 BN lat. 9335, 86r (cf. the text of Heiberg and Wiedemann cited in note 2, p. 219): "Et
diameters to axes of the conic sections:31
ex eis fuerunt quidam qui assumpserunt specula plurima sperica quorum radii converte­
rentur ad punctum unum ut combustio fortior esset, et illi qui invenerunt specula ista famosi
[10] The parabola has only one axis, while the ellipse has two axes within
fuerunt, sicut Archimenides et Anthinus (/ Anthemius) et alii ab istis duobus. Deinde accidit
it. But the hyperbola has one transverse axis and it is that which cuts the
eis cogitatio in proprietatibus figurarum ex quibus convertitur radius. Aspexerunt ergo in
lines of order [i.e. ordinates] at right angles whether it is within the section
proprietatibus sectionum piramidum et invenerunt radios qui cadunt super omnem plani­
or without, or whether it is partly within and partly without, and it has another ciem superficiei concave corporis mukefi converti ad punctum unum eundum. , . . Verum-
"erect axis" (axis erectus), and I shall demonstrate this in what follows. tamen ipsi non exposuerunt demonstrationem super hanc intentionem neque viam qua in­
And conjugate axes occur only in opposite sections [i.e., of the hyperbola] venerunt expositione sufficiente. Sed propter illud quod est in hoc de commoditatibus magnis
and in ellipses. However a line is called an "erect line” on which lines can et utilitatibus communibus vidi ut exponerem illud et explanarem quatinus contineret scien­
be drawn ordinate-wise to the diameter. tiam eius cuius voluntas est in cognitione veritatum et sciret illud cuius sollicitudo est in
velocitatibus rerum. Declaravi ergo in hoc tractatu et abbreviavi demonstrationem super
This is a descriptive passage composed by the paraphraser in order to scientia veritatis eius. . . . ”
apply the general definitions of axis and conjugate axis found in the First 33 Again I have translated the text in Paris, BN lat. 9335, 86r-87r, which differs only
slightly from Heiberg’s text in the article of his and Wiedemann’s, op. cit. in note 2, pp.
221-24. I have followed Heiberg’s practice of capitalizing the letters marking the points
diametri mugenibi sectionis diminute. [8] Et que ex eis est cuius principium est ex puncto and magnitudes. “In omni sectione mukefi cuius protrahitur sagitta et separatur ex ex­
circumferende sectionis et eius altera extremitas abreviata (/) est a circumferentia sectionis tremitate sagitte quantum est quarta lateris eius recti, omnis linea protracta equidistans
aut pertransit eam nominatur diameter absolute. [9] Diameter vero que nominatur secunda sagitte et perveniens ad sectionem et alia conversa ad punctum quod separat quartam con­
non est nisi in duabus sectionibus oppositis et transit per centrum ambarum, et narrabo tinent cum linea contingente sectionem super illud punctum (mg.\ scilicet, extremitatis) duos
illud in fine sextedecime figure huius tractatus.” angulos equales.
29 Heiberg, Gr 1, p. 52. [Fig. 1.1], “Verbi gratia, sit sectio ABG sectio mukefi, et sit eius sagitta AD et latus
30 Ibid., p. 66. eius rectum L. Et secabo ex AD lineam AE equalem quarte linee L, et producam lineam
31 BN lat. 9335, 86r (cf. Heiberg, Gr 2, p. LXXX): "[10] Et sectioni quidem mukefi TB equidistantem linee DA, et continuabo BE, et protraham KBH contingentem. Dico
non est nisi unus axis; sectioni vero diminute sunt duo axes intra ipsam; verum addite ergo quod angulus TBK est equalis angulo EBH.
est axis unus mugenib, et est ille qui secat lineas ordinis secundum rectos angulos, sive "Sit itaque in primis angulus BEH acutus. Ergo secundum semitam resolutionis ponam
ipse sit intra sectionem sive extra ipsam, sive pars eius intra sectionem et pars eius extra ut angulus TBK sit equalis angulo EBH. Et quoniam linea TB est equidistans linee DA,
ipsam, et est ei axis alter erectus, et ostendam illud in sequentibus. Et non cadunt axes erit angulus TBK equalis angulo BHE. Sed angulus TBK est equalis angulo HBE per posi­
muzdeguege nisi in sectionibus oppositis et in diminutis. Tamen et nominatur linea erecta tionem; ergo angulus EBH est equalis angulo BHE. Ergo linea BE est equalis linee EH\
linea super quam possunt linee protracte ad diametrum secundum ordinem.” ergo quadratum BE est equale quadrato EH. Et protraham BZ perpendicularem super sagit-