Apollonius de Pergé, Coniques. Tome 4: Livres VI et VII. Commentaire
historique et mathématique. Édition et traduction du texte arabe by
Roshdi Rashed
Berlin/New York: De Gruyter, 2009. Pp. xii + 572. ISBN 978–3–11–
019940–6. Cloth € 137.00
Reviewed by
Aldo Brigaglia
Università di Palermo
brig@math.unipa.it
In memoriam Hélène Bellosta
With the publication of the fourth volume of the new edition of the
Conics by Apollonius of Perga (ca 262–180 bc), Roshdi Rashed has
completed his very important work on the edition of the Arabic text,
its translation into French, and a vast mathematical commentary.
Apollonius’ treatise itself may well be considered one of the highest
achievements of Greek mathematics at its most brilliant. In fact,
together with the corpus of the mathematical work of Archimedes
(287–216 bc), the Conics constitute the greater part of Greek higher
mathematics.
Rashed’s edition of the text of the Conics is the latest episode
in the long and intriguing history of the transmission of this major
mathematical work to us. The first four books arrived to Western
mathematical culture through the edition by Eutocius (fifth century
ad), which was translated into Latin in the 16th century by Johannes
Baptista Memus, Francesco Maurolico, and Federico Commandino.
Books 5, 6, and 7 of the Conics arrived in Europe only through
the Arabic translations of the Greek text: the first text of the lost
Greek books was contained in an Arabic compendium of the Conics
written by Abu’l-Fath Mahmud al-Isfahani (second half of the 10th
century). This text was given to Cardinal Ferdinando I de’ Medici
(later grand duke of Tuscany) by the Patriarch of Antioch as early as
1578 but was edited and translated by the Maronite deacon Abraham
Ecchellensis (Ibrahim al-Haqilani) under the supervision of Giovanni
© 2012 Institute for Research in Classical Philosophy and Science
All rights reserved
issn 1549–4497 (online) issn 1549–4470 (print) issn 1549–4489 (cd-rom)
Aestimatio 9 (2012) 241–260
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Aestimatio
Alfonso Borelli only in 1661. This text was heavily manipulated by
its Arabic editor.
In the ninth century, the Banū Mūsā brothers (Muhammad,
Ahmad and al-Hasan) made great efforts to acquire, understand, and
obtain a translation into Arabic of the complete text of the Conics.
Books 1–4 were translated under the supervision of Ahmad by Hilal b.
abi Hilal al Himsi, and books 5–7 by Thabit b. Qurra. (Book 8 is now
considered to have been lost by this date). This Arabic translation
was brought to Holland by Jacobus Golius in 1629 (it now is in the
Bodleian Library in Oxford); but even though its existence was well
known in Europe, it was published in a Latin translation by Edmund
Halley only in 1710. Halley’s edition remained the main reference
for books 5-7 of Conics until recently and constituted the basis for
the first English translation of these books [Heath 1896] as well as of
the first French translation [ver Eecke 1923]. A more recent English
translation (with the Arabic text) of books 5–7 was published in 1990
by Gerald J. Toomer.
It is worth noting that from Halley’s edition on, the Banū Mūsā
version has been used to give us the translation of the last three books
only of the Conics, while the first four books were always published
on the basis of the edition of Eutocius directly from the Greek text.
As Rashed pointed out, this reveals some prejudices, among which I
may cite:
∘ the idea that the edition by Eutocius provides us with Apollonius’ exact text of the first four books of the Conics, and
∘ the idea that the Arabic translation of the first four books is
that of this same edition by Eutocius.
The complete edition of the entire corpus of the Banū Mūsā version
allows us to understand, for example, that there are many differences—and sometimes very profound ones—between the edition of
Eutocius and the Arabic translation, mainly in book 4. Rashed points
out some of these differences:
∘ in Eutocius’s edition, book 4 consists of 57 propositions but
there are only 53 in the Arabic translation;
∘ some propositions of Eutocius’s edition are missing from the
Arabic translation;1
1
An attentive examination of these propositions shows that they may be
ALDO BRIGAGLIA
243
∘ there are two propositions in the Arabic translation that do
not appear in Eutocius’s edition;
∘ the order of propositions differs;
∘ the figures and their letters differ in a certain number of
propositions; and
∘ there are different proofs and, moreover, some proofs are
erroneous.
In any case, the Banū Mūsā ’s edition of books 5–7 has always been
reputed to be the principal source for that part of Apollonius’ work
and Rashed’s edition makes a very important contribution to our
knowledge of it.
Book 6 is concerned with the problem of defining equality and
similarity between conic sections. The first part (up to proposition
27) treats what we can call the ‘criteria for equality and similarity’.
The second part (up to proposition 33) poses the main problem: how
to cut a given right cone so that the result is a section equal (or
similar) to another given one.
This poses an interesting conceptual problem: ‘What is really
meant by the terms “equality” and “similarity” between conic sections?’ Rashed’s commentary dedicates many pages to this matter.
With regards to equality, Apollonius resorts to the idea of ‘superposition’: two conics are equal if they can be superposed on one another
(by means of a motion). In the words of Apollonius as rendered in
Rashed’s translation,
les sections de cônes que l’on dit égales sont celles dont les
unes peuvent se superposer aux autres et dont aucune n’excède
l’autre. [90]
Toomer [1990, 1.264] uses the term ‘can be fitted’ for Rashed’s ‘peuvent se superposer’. In any case, this is a usage that goes far beyond
what was done by Euclid, who in his fourth common notion effectively says, ‘Things which coincide with one another are equal to one
another’ [Heath 1956, 1.153]. The phrase written in Greek is «καὶ τὰ
ἐφαρμόζοντα ἐπ᾽ ἀλλήλα ἴσα ἀλλήλοις ἐστίν». The word «ἐφαρμόζοντα»
defective. For example, proposition 4.7 depends on a hypothesis which is
supposed to have been given in the preceding proposition 4.6; but this hypothesis does not exist.
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Aestimatio
was translated by Heath as ‘coinciding’. The translation of al-Hajjai,
reproduced by Rashed, is:
Celles (les choses) qui se superposent les unes aux les autres
sont égales les unes aux autres. [10]
While in the Euclidean definition we can thus discern the use of
the term ‘superposition’, it is not at all clear how this is connected
to the idea of motion. How is this common notion (note that in
Euclid this is not a definition) to be used concretely? The absence
of any postulate regarding the use of this notion and, in particular,
the notion of the rigid motion that would lead the two figures to be
superposed on one another, makes verifying the congruence of the
two figures problematic.
Actually, Euclid prefers to make use of it in a very limited way.
In the second proposition of the Elements, he constructs a segment
equal to a given segment (thereby showing how to ‘move’ a segment)
but he does not make use of congruence, and the verification of the
equality of the two segments is entrusted to postulate 3 (‘All radii
of a given circle are equal to each other’) and to common notions
2 and 3 (‘The addition/subtraction of equals to/from equals result
in equals’). In the fourth and eighth propositions (criteria for the
equality of triangles), he actually uses equality by superposition: from
that point on, as far as the equality of polygons is concerned, no use
at all is made of superposition. It is another matter with regard
to the equality of arcs of circles, whose verification often requires
reasoning based precisely on equality by superposition of figures since
one cannot resort to equality between triangles as in the case of
polygons.
The relevance and the meaning of this definition and its connection with Euclid’s common notion has also been discussed by
Fried and Unguru [2001] in great detail; and although it might be
worthwhile to compare their point of view with that of Rashed, this is
beyond the scope of the present review. Still, it is perhaps worthwhile
to underline, as Rashed does, the fact that from the point of view
of modern criticism, in the absence of any postulate regarding rigid
motion, Euclid’s reasoning does not appear to be rigorous. On the
other hand, Hilbert showed that it is necessary to assume Euclid’s
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245
proposition 4 (the first criterion of equality of triangles) as a postulate, given that it is not at all a logical consequence of the Greek
mathematician’s postulates, axioms, and common notions.
Naturally, Apollonius, who is comparing segments of a conic,
is forced to resort more than once to the criterion of equality by
superposition. The differences in formulation between Apollonius
and Euclid regarding equality by superposition are highlighted in
Rashed’s commentary. The use that Apollonius makes of the concept
of superposition implies some idea (even though never explicated) of
motion:
La définition de l’égalité par superposition…peut encore se
dire ainsi: deux sections—ou portions—coniques sont dites
égales si elles coïncident parfaitement une fois que l’une est
amenée sur l’autre par un déplacement, de sort que leurs
contours s’identifient. [11]
In this definition:
(1) no concept of magnitude or measure is ever introduced;
(2) an idea of motion in the sense of a transformation is presumed
but never explicated by Apollonius;
(3) there are no operating concepts, a fact which thus necessitates
the integration of other properties whose use is more directly
operative (the symptoms); and
(4) it is necessary to insert other procedures that integrate the
concept of motion.
These observations will become clearer if we examine some of the
first propositions of book 6.
Propositions 6.1 and 6.2 concern the equality of two conic sections
(the parabola in the first and the hyperbola in the second). It is
proven, for example, that two parabolas are equal if and only if
they have the same latus rectum. Recall that the latus rectum of a
parabola has the following property [see Figure 1]: the latus rectum is
defined as that segment 𝑐, where 𝐵 is a point of the parabola, 𝐶 the
corresponding point on the axis, and 𝐴 the vertex of the parabola,
such that 𝐶𝐵 is the mean proportional between 𝐴𝐶 and 𝑐. In modern
terms, if we set 𝐵𝐶 = 𝑥 and 𝐴𝐶 = 𝑦, then we have 𝑥 2 = 𝑐𝑦. This is
the ‘symptom’ of the parabola.
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Figure 1
Apollonius’ reasoning can be summarized in this way [see Figure
1]. If 𝑐 = 𝑐 ′ and if we transport line 𝐴𝐶 so that it is superposed
on line 𝑍𝐿 such that 𝐴 is carried onto 𝑍 , and if we call the point
on 𝑍𝐿 where 𝐶 falls 𝐿 (and thus 𝑍𝐿 = 𝐴𝐶 ), we will have 𝐶𝐵 2 =
𝑐𝐴𝐶 = 𝑐 ′ 𝑍𝐿 = 𝐿𝐻 2 . Thus, 𝐵 too is superposed on 𝐻 and the two
parabolas are pointwise superposed. This reasoning can be clearly
inverted.
In analogous fashion, Apollonius proceeds to find the conditions
for the equality of two ellipses or two hyperbolas (the central conics),
except that in this case what comes into play in addition to the latus
rectum is either the axis [prop. 6.2] or an arbitrary diameter[corollary
to prop. 6.2]: two central conics are equal if and only if their respective
‘figures’—that is, the rectangles formed by the axis (or a diameter)
and the corresponding latus rectum—are equal.
It should be noted that, once these propositions have been proven,
Apollonius no longer needs to refer the equality of two conics to the
poorly defined concept of ‘superposition’ but can refer instead directly
to their ‘symptoms’, which in some way correspond to the equations
of analytical geometry. Thus, for example, to see that two ellipses
are equal it is sufficient to see that their latera recta and axes are
equal. Rashed rightly notes:
La tâche qui est celle d’Apollonius dans le livre VI est donc,
pour l’essentiel, de déterminer les conditions pour que les deux
sections soient superposables…à l’aide des symptomata, sans
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247
toutefois s’intéresser à la nature même de ces transformations
ponctuelles. [6]
In other words, Apollonius, like Euclid, defines equality by means
of superposition, which implies an idea, never explicated nor clearly
defined, of motion. But he then tries to rid himself of that onerous
condition through determining the equality of the conics by means
of a simple comparison of magnitudes (segments or surfaces). As
Rashed underlines,
de fait, au cours des démonstrations, l’égalité/superposition
est doublée de l’égalité des aires—ou des longueurs. Apollonius
recourt alors aux symptomata. [11]
This process is completely analogous to that followed by Euclid:
thanks to the theorems in the equality of triangles [Elem. 1.4 and
1.8], verification of the equality of two triangles (and, thus, of any
two polygons) is reduced to the equality of segments and angles. A
similar procedure is used in book 3 regarding circles, whose equality is
attributed (in this case starting from the definitions) to the equality
of the diameters.
It is interesting to note that, while this technique makes it
possible for Apollonius to free himself from having to resort to superposition any time that two conics must be compared in their entirety,
it loses its efficaciousness when he has to compare portions of conics:
in this case, it is necessary to go back to the original definition of
equality and thus to superposition. In this sense, the idea of superposition in book 6 of the Conics takes on a role and importance that it
never assumed in either Euclid or in the other books of the Conics.
This has been made clearly evident by Rashed [11]:
Malgré l’inspiration euclidienne patente, la définition de l’égalité/superposition recouvrira chez Apollonius plusieurs contenus.
One instance of the role played by equality by superposition in
this book can be found in the proof of what today we would call the
symmetry of conics with respect to the axes. For example, let us
examine prop. 6.4:
If there is an ellipse and a line passes through its center
such that its extremities end at the section [i.e. the line is a
diameter] then it cuts the boundary of the section into two
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equal parts, and the surface is also bisected. [Toomer1990,
276]
The proof, which in this proposition is limited to the case in which the
diameter is the axis, proceeds by reductio ad absurdum [see Figure 2].
Given axis 𝐴𝐵, it is supposed by way of reductio ad absurdum that,
after being turned over, the arc of ellipse ΑΓΒ does not coincide2
with arc ΑΕΒ, and that it is precisely point Γ where ΑΓΒ does not
superpose itself on arc ΑΕΒ. If from Γ we drop the perpendicular
ΓΔ to the axis and extend it until it meets arc 𝐴𝐸𝐵 in a point 𝐸 ,
we find, by the definition of axis, that ΓΔΕ and, further, ΔΕ are
perpendicular to ΑΒ; thus, after being turned over ΔΓ coincides with
ΔΕ and Γ coincides with Ε, contrary to the initial hypothesis.3
I believe it evident that such considerations of Apollonius’ proofs
lead us to imagine a superposition achieved by some motion. Yet, in
my opinion, it is not completely clear what Apollonius’ idea of that
motion was; but at the same time, there seems to be no doubt that,
as Rashed shows amply, the point of view expressed in book 6 had a
profound influence on later Arabic mathematicians. As Rashed puts
it, we are dealing with ‘proto-transformations ponctuelles, que les
mathématiciens ne cesseront d’exhiber et de développer à partir du
IXe siècle à Bagdad’ [11].
The definition of similarity, however, is quite different. Apollonius wrote:
And similar are such that, when ordinates are drawn in them
to fall on the axes, the ratios of the ordinates to the lengths
2
Toomer uses the term ‘coincide’ in his translation, while Rashed uses ‘tombe
sur’ (‘fall on’). The two translations are comparable if we take ‘coincide’ to
mean ‘coincide after being turned over’. In any case, Rashed’s translation
provides a much clearer idea of motion than that implied by Toomer’s.
3 It is also worthwhile observing that today we would have preferred an indirect proof rather than one by reductio ad absurdum. Such a proof might
have proceeded in this way:
given any point Γ on arc ΑΓΒ, we will show that after turning over
ΓΔ with respect to axis ΑΒ a point on arc ΑΕΒ is obtained.
But such a proof would have required considering an ellipse as being formed
of infinite points, something that was far from the way in which geometric
figures were conceived by the Greeks. However, considerations of this sort
would inevitably take us too far afield.
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ALDO BRIGAGLIA
Figure 2
they cut off from the axes from the vertex of the section are
equal to one another, while the ratios to each other of the
portions which the ordinates cut off from the axes are equal
ratios. [Toomer 1990, 264]
In this case, we are dealing with a functional definition: to equal ratios
between the abscissas correspond equal ratios between the relative
ordinates. Rashed points out that the concept of similarity between
conic sections is certainly present before Apollonius.4 Archimedes
stated that all parabolas are similar to each other and, thus, it is
entirely plausible that Apollonius was aware of this fact [Apollonius,
Con. 6.11]. But, as I believe, there is no difficulty in agreeing with
Rashed’s statement that
rien à notre connaissance ne permet d’affirmer qu’il y a eu
une étude réglée de la similitude des sections coniques avant
le livre VI. [23]
In this case as well, Apollonius moves immediately to substituting
the functional concept of similarity with his verification by means
of the ‘symptoms’. Two central conics are similar if and only if
their respective figures—that is, the rectangles formed by the axis
and latus rectum [Con. 7.12]—are similar. Thus, they are similar
when, given 𝑑 and 𝑑 ′ as the respective axes and the latera recta 𝑐
and 𝑐 ′ , 𝑑∶𝑑 ′ = 𝑐∶𝑐 ′ . The text continues with several generalizations
4
E.g., in his book On Conoids and Spheroids: see Heath 1897, 99–150.
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Aestimatio
(taking into account any given diameters instead of axes) and then
it is proved to be impossible for a conic to be similar to a conic of
a different name (for example, a parabola can never be similar to
a hyperbola, and so forth). Apollonius then deals with segments
that are similar or equal in conic sections, first for similar sections
and then for dissimilar sections. In this last case (dealt with in
propositions 6.23–25), there is a beautiful result: there cannot exist
similar segments in dissimilar sections. This result signifies, naturally,
that similarity is a local property: if two conics have two similar
segments (which are arbitrarily small, we would say), then they are
entirely similar. Concluding the part regarding similarity, Apollonius
proves that if a right cone is cut with two planes that are parallel
to each other, the conics obtained are similar. As Rashed points out,
what is in fact proven (using our terminology) is that in this case the
two conics are homothetic from the vertex of the cone, with the ratio
of homotheity equal to the ratio between the respective distances
from the vertex itself.
In contrast, the final part of book 6 presents problems:
∘ given a conic section and a right cone, cut the cone with a
plane so that the intersection is a conic equal to the given one;
and
∘ given a conic section, find a cone similar to a given cone such
that the given conic is a section of the cone found.
This kind of problem appears to be meaningful and may in some way
provide a clue to Apollonius’ aim in writing this book. In a certain
sense, it is an inversion of what was done in the first book: while
book 1 dealt with constructing the section of a cone as a plane curve,
book 6 deals with cutting a given cone according to a given conic
section in a plane.
Using the notations shown in Figure 3, where Α represents the
vertex of the cone, Θ the centre of the circle of the base, and the
end points of the diameter of the base Β and Γ, the condition under
which it is possible to carry out the proposed construction (with 𝑑
as usual as the diameter and 𝑐 the latus rectum), is that
𝑑
ΑΘ2
≥
𝑐
ΒΘ2
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Figure 3
If this condition is satisfied, the construction is done by inserting
between the extension of line ΑΠ and ΑΒ a segment (labelled ΝΠ in
Figure 3) parallel to ΑΘ whose length is equal to 𝑑. Rashed makes
two observations in this regard that I find particularly interesting.
(1) The first concerns the condition under which the construction
can be carried out. The author notes that this condition
est équivalent à la condition selon la quelle l’angle
entre les asymptotes de l’hyperbole ne doit pas être plus
grand que l’angle 2𝛼 au sommet du cône. La recherche
de la condition de possibilité dans le cas d’un cône
oblique aurait été plus difficile: elle ne s’exprime pas
en termes d’angle au sommet. [66]
This observation raises a question which has already been posed by
Zeuthen [1886], that is, ‘Why is it, having in all preceding books
set for himself the more general conditions of oblique cones, that
here Apollonius always refers to right cones?’ The answer is certainly
not that provided by Toomer, who wrote, ‘It is easy to see that his
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[Apollonius’] solutions in book VI can be extended to the oblique
cone’ [1990, lviii]. As Rashed notes [66], this is not true at all and the
solution becomes quite complicated, at least in this instance, in the
passage to the oblique case [see also Brigaglia 1997]. In my opinion,
the question remains open; but Rashed in any case provides, with
reference to the problems that follow, an interesting hypothesis about
this fact.
(2) As was seen earlier, the construction of the hyperbola requires
inserting a segment of a given length that is parallel to one
given line. This simple construction is completely absent in
Apollonius. Rashed provides a complete proof of this fact,
the absence of which appears not to have been noticed by the
Arabic translators, although Halley did.
As we said, the final three propositions regard the construction
of a right cone (similar to a given one) whose section is a given conic.
Here again we can ask why Apollonius limited himself to the case of
the right cone.5 Rashed notes that while in the right case the problem
is determinate; in the oblique case, it remains indeterminate. He
concludes:
C’est précisément par ce caractère d’unicité de la solution
que les propositions 31 à 33 diffèrent des propositions 49 et
50 du livre I. C’est ce même caractère qui semble expliquer le
choix d’Apollonius du cône droit.
It would be worthwhile to develop this interesting observation further.
Rashed’s presentation of book 6 of the Conics ends with a section
that is particularly original, ‘Le sixième livre et la géométrie prototransformationnelle’. He writes:
Le commentaire systématique du sixième livre révèle en effet
qu’il s’agit indubitablement d’une géométrie où l’on procède
pragmatiquement par mouvement et transformations ponctuelles. [77]
The word ‘pragmatiquement’ is especially interesting: in this book,
Apollonius makes ample use of concepts such as motion or transformation but without either defining them precisely or using them in
a way that is altogether self-conscious. In fact, to find a fully self5
See ‘Remarques sur le propositions 31 à 33’ [77].
ALDO BRIGAGLIA
253
conscious use of them, we will have to wait for the works of La Hire
and then, another two centuries later, of Felix Klein. With regard to
the works of La Hire, Rashed writes:
Ce regard, même s’il englobe celui d’Apollonius et l’éclaire,
n’est cependant pas le sien: ses concepts, ses instruments et
son langage sont en effet différents. Cependant, les objets
géométriques étudiés dans les Coniques possèdent bien ces
propriétés, qui ne seront appréhendées et révélées que par
les successeurs d’Apollonius…. C’est donc en restant fidèle à
la pensée du mathématicien alexandrin que l’historien peut
s’inspirer de ces propriétés, pour mieux pénétrer cette réalité
mathématique que celui-ci abordait avec les moyens de la
géométrie de son temps. Aussi pour compléter le commentaire
du sixième livre, allons-nous le considérer avec d’autres yeux
que ceux d’Apollonius, ceux d’un lointain successeur. [78]
This lointain successeur (distant successor) is, in fact, Felix Klein.
The lengthy digression [78–83] in which Rashed reconstructs the entire
sixth book from the point of view of projection and transformation
groups may appear at first to be out of place, but this is not the case.
The final lines [83] make the author’s motivations clear:
Une interprétation de ce type permette de mettre en évidence
les transformations ponctuelles sous-jacentes au travail d’Apollonius. Á partir du IXe siècle, ce livre VI, ainsi que les autres
travaux d’Apollonius sur les lieux plans, ont incité les géomètres à concevoir les transformations ponctuelles de courbe
à courbe (Thābit ibn Qurra et Ibn al-Haytham par exemple).
To my mind it is precisely here that we find one of the aspects of
greatest value in the new translation of the Arabic text of Apollonius,
which can be inserted into the imposing context of the Arabic tradition of translating mathematical texts. Interpreted in this light, we
can appreciate the work of the great Arabic mathematicians not only
as transmitters of Greek thought, but also as original interpreters of
the mathematics that was made available to them, interpreters who
were capable, through new ideas, of opening new roads—even though
a significant portion of them would receive their natural development
only much later and in a different culture.
Before going on to a brief look at Book 7, I should like to
go back to a central point in Rashed’s formulation. Book 6 has
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Aestimatio
traditionally been considered secondary in the context of Apollonius’
work. As evidence of this, I cite Zeuthen, who says that no real
geometric difficulty is overcome here. However, Apollonius himself
had something to say about this book:
We have enunciated more than what was composed by others
among our predecessors.…What we have stated on this is
fuller and clearer than the statements of our predecessors.
[Toomer 1990, 262]
Fried and Unguru [2001] have a different appreciation as they
say that the importance of book 6 lies in the fact that:
(1) equality and similarity of conic sections is, for Apollonius, a
far more subtle affair than we would like to think;
(2) the investigation of equality and similarity is necessary to
clarify what is meant by a conic section being ‘given’; and
(3) it does not merely elaborate ideas already elaborated in book
1 but complements those ideas somewhat in the way Euclid’s
Data complements the Elements.
Rashed, however, goes further. Indeed, without pretending to be
completely original, Apollonius does give himself credit for providing
a more complete and systematic organization of the material. This
is precisely what Rashed claims. For him, Apollonius is in search
of new means for extending the study of equality and similarity to
curved figures:
Il fallut trouver les moyens de faire correspondre une section
à une autre, différente, une portion à une autre, différente.
Thus, Apollonius’ aim was
trouver les moyens d’étendre aux sections coniques la recherche accompli pour les figures rectilignes et pour les arcs
de cercle, et déterminer les conditions requises par une telle
extension. [5]
The historical importance of book 6, then, lies in its having paved
the way later taken by numerous Arabic mathematicians:
les mathématiciens qui les premiers ont pris davantage de
distance à l’égard de la géométrie des figures et ont introduit
mouvement et transformations ponctuelles se sont précisément
ALDO BRIGAGLIA
255
référés à ce livre VI—ainsi Thābit ibn Qurra, al-Sijzi, Ibn
Hūd, Ibn Abi Jarrāda…. [7]
This is a point of view that I believe is novel, one which only someone
like Rashed, who truly knows the contributions of these mathematicians, could provide and which deserves to be examined in greater
depth.
The seventh book is quite another story. While the purposes
of book 6 are extremely clear, book 7 appears quite difficult to
read, not in the sense that it is mathematically difficult but in the
sense of trying to understand the aims of its author. In the general
introduction to book 1, Apollonius wrote, ‘another [scil. book 7] [deals]
with theorems concerning determinations.’6 In the accompanying
letter from Apollonius to Attalus, he also wrote:
Peace be with you…. In this book are many wonderful and
beautiful things on the topic of the diameters and the figures
constructed on them, set out in detail. All of this is of great
use in many types of problems, and there is much need for it
in the kind of problems which occur in conic sections which we
mentioned, among those which will be discussed and proven
in the eighth book of this treatise. [Toomer 1990, 382]
Here the word ‘diorismes’ (‘determinations’) signifies the determination of a problem’s conditions of solvability. Thus, we are dealing
with a book in which are determined the range of possible variation
for values relative to the diameters and latera recta of conic sections.
As Rashed rightly observes, it is very difficult to comprehend
fully the significance of the choices made by Apollonius without
having access to the eighth book (which, as mentioned, has been
lost definitively), because it in fact appears that we are dealing with
elements that are very closely tied to the solution of problems given
in that eighth book. All of this is clearly highlighted by Rashed:
Quant à l’usage qui serait fait de ces théorèmes au huitième
livre, nous l’ignorons puisque celui-ci est définitivement perdu
et qu’aucun témoignage fiable ne nous est parvenu à son
propos. [241]
6
‘determinations’: diorismes, in Rashed’s translation.
256
Aestimatio
Figure 4
Thus, book 7 has to be read on its own, since no references to
book 8 are possible. This is done in an exemplary way in the rest of
the text that follows. The hinge of Rashed’s interpretation is that it
effectively consists in the study of the variation of several magnitudes
tied to diameters and latera recta and, thus, to the determination
of the maximum and minimum values that these can reach. This
provides a point of continuity with book 5, which is dedicated to
the determination of maxima and minima of magnitudes such as the
distance of a point from the points of a conic section:
Le livre VII est dans une certaine continuité avec le livre V.
Nous avons en effet montré que, dans ce dernier, Apollonius
étudie la variation de la distance d’un point donné aux points
d’une section conique. Mais cette continuité s’observe aussi
dans la formulation des propositions…et dans la communauté
du lexique. [245]
It seems to me that this continuity is amply proven by Rashed’s
examination of the text, with perhaps one caveat: while book 5 is selfcontained and its purpose lies in the search for maxima and minima
of some magnitudes found in it, book 7 is completely oriented towards
book 8 and is, therefore, much more difficult for a modern reader
who does not have access to that last book, to grasp fully the beauty
and depth of the theorems that are contained in it.
It is precisely these characteristics that prevent me from going
into technical details. One central point of the second proposition
in this book is the introduction of a new magnitude that Rashed
translates as ‘segment semblable en proportion’ [251] (which Toomer,
257
ALDO BRIGAGLIA
following Halley, translated as ‘homologue’). This is what is at issue
[see Figure 4]:7
Let there be hyperbola Η with transverse axis 𝑑0 and latus
rectum 𝑐0 , and let Θ be on diameter ΑΓ such that there is
𝑑
ΘΓ
= 0 .
ΘΑ
𝑐0
We would call ΘΑ the segment ‘similar in proportion’.
Five lemma-like propositions are dedicated to this magnitude. Book 7
then goes on with a group of propositions (from 6.6 to 7.20) dedicated
to the determination of formulas relative to ratios between different
magnitudes like
𝑑0 2
.8
(𝑑 − 𝑑 ′ )2
The second part of the book is the more substantial one, and is dedicated to the study of the variation of magnitudes such as diameters,
associated latera recta, their sums, differences, products or ratios.
This is the part of book 7 that is most directly connected to book
5, itself dedicated to the determination of the maxima and minima
that are (presumably) necessary for determining the conditions of
solvability for the problems treated in the lost book 8. It is precisely
here that we see the important role played by the attempts to reconstruct the long lost book 8 in order to understand the kind of
interpretation that various readers have given to the theorems set
out in book 7. Famous among such attempts are those of Halley and
the 11th-century Arabic mathematician Ibn al-Haytham. Rashed
refers in particular to the latter in the reconstruction proposed in
this present volume.9 Rashed cites two examples that I believe it is
interesting to repeat here:
7
I will discuss only the case of the hyperbola, the analogous one for the ellipse
is given in 7.3.
8 Here 𝑑 represents as before the transverse axis, while 𝑑 and 𝑑 ′ represent
0
any diameter and its conjugate.
9 He had earlier proposed and commented on this reconstruction in Rashed
2000.
258
Aestimatio
(1) Given a central conic, find a point such that the diameter
𝑑 drawn from this point and the associated latus rectum 𝑐
satisfy the equation 𝑑𝑐 = 𝑘, with 𝑘 given.
(2) Given a central conic with transverse axis 𝑑0 10 and associated
latus rectum 𝑐0 , find a diameter 𝑑 and its associated latus
rectum 𝑐 such that 𝑑 + 𝑐 = 𝑘, with 𝑘 given.
From the use that the Arabic mathematician makes of the propositions of book 7 to solve these and other problems, Rashed draws
interesting conclusions:
Par « théorèmes relatifs aux diorismes », il semble donc que
Apollonius entende deux choses à la fois. Il s’agit de propositions qui d’une part renferment elles-mêmes des diorismes,
et qui d’autre parte interviennent dans la conception des diorismes lors de la construction des problèmes au moyen de
l’intersection des coniques. Tel est bien le cas pour un bon
nombre des propositions du septième livre. Or cette dualité de sens, seulement implicite, ne pouvait qu’intriguer les
commentateurs. [248]
To be sure, Rashed’s rich commentary made it possible for me to
retrace by following his text a magnificent itinerary through Greek
mathematics filtered through Arabic culture.
Before finishing with book 7, I should like to note that, as he did
in book 6, here too Rashed concludes an introductory commentary
with a section (‘Étude analytique de la variation des grandeurs associées à 𝑑, 𝑑 ′ , 𝑐, 𝑐 ′ ’) that translates the text of the mathematician
from Alexandria into modern language. This not only facilitates its
comprehension by a modern reader but also makes evident the thread
that ties the different mathematical languages together:
Grâce à ce modèle, la vérité des propositions se passe de l’appel
constant aux figures, ainsi qu’à l’imagination des constructions auxiliaires. Plus importante encore, ce commentaire fait
apparaître des liaisons entre les propositions, invisibles à la
pure géométrie, et met en évidence des idées majeures qu’on
ne pouvait saisir par la démonstration géométrique—ainsi
les idées qui président à l’étude de la variation. Cette fois
encore, et comme tous le géants qui jalonnent l’histoire des
10
Here I believe there is a typographical error because what is written is 𝑑0′ .
ALDO BRIGAGLIA
259
mathématiques, Apollonius n’œuvre pas seulement dans le
présent, mais dans le futur mathématique, avec les moyens
du présent. Situation éminemment féconde et extrêmement
subtile, qui exige pour être comprise qui soient multipliés les
commentaires. [348]
This is precisely where the fascination of this edition lies: it
unites philological rigor with a panoramic point of view which comprises successive developments of Apollonius’ ideas without leading to
anachronistic flights of fancy that depict mathematicians of classical
antiquity as improbable precursors of modernity, but which highlights the thread of continuity that makes the history of mathematics
a description of a fascinating adventure in search of those ‘hidden
harmonies’ (‘riposte armonie’) of which Federigo Enriques spoke so
convincingly.
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