Ian Mueller est mort en 2010 à l’âge de 72 ans. Professeur au département de philosophie de l’université de Chicago, il a été l’un des principaux spécialistes de l’histoire des mathématiques anciennes dans leurs rapports avec la philosophie, comme on le voit dans son grand ouvrage, Philosophy of Mathematics and Deductive Structure in Euclid’s ‘Elements’ (Dover Publications, 1981). C’est cet intérêt, ainsi que celui qu’il portait au problème de la réception de Platon et d’Aristote dans l’Antiquité tardive, qui le conduisirent à étudier de près les commentaires anciens d’Aristote. Il en traduisit un nombre conséquent dans la grande collection des commentateurs grecs d’Aristote dirigée par Richard Sorabji (ancientcommentators.org.uk). Venu à Paris en juin 2008 pour participer à un colloque sur la notion et l’histoire de la démonstration, il y présenta une communication dont le texte que voici est une version révisée par lui. Nous remercions Stephen P. Menn (McGill) et Bernard Vitrac (CNRS) d’y avoir introduit quelques précisions, indiquées entre crochets […] avec leurs initiales respectives (BV ou SPM), pour le rendre pleinement intelligible.
1 In book 3 chapter 8 of De Caelo, beginning at 306b1, Aristotle offers a series of criticisms of what I call Plato’s geometrical chemistry in the Timaeus. In that chemistry, each of what came to be called the four elements is associated with one of the five regular solids which are the subject of the last book of Euclid’s Elements, fire with the tetrahedron or, as it came to be called and will be called in this paper, pyramid, air with the octahedron, water with the icosahedron, and earth with the cube. Many later discussions also suppose that Plato assigned the fifth regular solid, the dodecahedron, to Aristotle’s fifth element, ether. At the beginning of chapter 8 Aristotle says:
In general trying to assign figures to the simple bodies is unreasonable, first of all because it will follow that the universe is not filled up; for it is agreed that there are three plane figures which fill a space (topos), the triangle, the square, and the hexagon and that there are only two solid figures < which do so >, the pyramid and the cube. But it is necessary for them to assume more than these < two solids > because they make there be more elements. [1]
3 For the most part, early discussions of this claim take it that a regular figure fills a space if some number of such figures can be joined with their vertices at a point without overlapping or leaving any empty space around the point. In more recent discussions plane figures are said to fill space if, to speak intuitively, they can be used to fill an infinite flat floor without overlapping; and similarly solid figures are said to fill space if they can be combined in such a way as to fill an arbitrarily large volume while leaving no gaps. [2] The two notions are not equivalent, and I will be concerned only with the first.
4 The sixthcentury Greek commentator Simplicius [3] gives a clear representation of the argument for Aristotle’s claim about plane figures. It rests on two points:
5 (i) if any number of straight lines in a plane meet at a point they make angles equal to four right angles; [4]
6 (ii) the sum of the angles in an nagon is 2(n2)90^{o}.
7 Using these propositions, it is easy to show that six equilateral triangles, four squares, and three regular hexagons fill a space, and that no other nagon does so. [5]
8 It is clear enough that eight cubes fill a space, and it is reasonable to think that the eight cubes with their solid angles can somehow provide a criterion for solids filling a space in the way that making the angles at a point equal to four right angles is the criterion for filling a space for planes. The problem for using this criterion is the difficulty of explaining how one adds and subtracts solid angles and compares the results in size. We will see how Themistius and, following him, others attempted to overcome this difficulty, which was not solved until the development of spherical trigonometry.
9 There is no trace of reasoning based on equality of solid angles in Simplicius’ remarks, although he does, without explanation, make the claim that twelve pyramids fill a space, a claim which we also find in other authors beginning with Themistius. [6] In his more detailed discussion of solids, Simplicius relies entirely on a certain Potamon, [7] whom he reports (via Alexander of Aphrodisias [ca. 200]) as saying,
In the case of the solid figures what need is there even to say that the cube fills out the space? For if one lays four cubes together along their sides he will fill out the space. To argue in another way:
the role (logos) which the square has among plane figures (en epipedois), the cube has among solid figures;
but among plane figures the square fills out the space;
therefore, among solid figures the cube will fill the space.
You will see < this > clearly if on the four squares which have been constructed together at one point you erect cubes having the squares as bases. For, in place of that point there will be the straight line which is drawn at the point perpendicular < to the plane of the four squares >; the four cubes will touch one another at that line and together fill the solid space. [8]
11 Potamon’s argument here clearly does not concern cubes filling a space around a point, but as we might put it their filling a space around a straight line. Solids fill a space in this sense if they share a common edge and leave no empty space around it. This notion is not found again in discussions of spacefilling solids until Francesco Maurolico, who distinguishes between solids’ filling a space at an edge (angulariter) and their filling a space at a vertex (verticaliter). [9]
12 A number of later writers invoke the principle comparing the logos of squares among plane figures and that of cubes among solid ones, a principle which Potamon reasserts for triangles and pyramids and which I shall call the dimensional analogy. [10] It is clearly not a reliable principle of proof.
13 I believe that Potamon is relying on the notion of filling a space angulariter in his discussion of pyramids, but what he says is not transparent:
And it is clear that the pyramid also < fills the space >, since a pyramid is nothing other than the angle of a cube. Consequently, since the angles of a cube filled up the space, the pyramid will also fill it up. To prove this another way, the cube itself has been completed from two pyramids. So if eight pyramids having their apexes at the centre of a sphere are combined they will fill out the space. … And it is evident through perception alone. For if someone were to combine eight pyramids, making their apexes incline toward one another, which act like wedges, and have their apexes inclining toward one another, he would not leave an empty space. [11]
15 It seems to me very likely that Potamon is here identifying pyramids with the triangular prisms formed by drawing parallel diagonals in two opposite faces of a cube. [12]
16 Themistius’ paraphrase of De Caelo survives only in a Hebrew translation, made in 1284, based on the Arabic translation of the original Greek, and a 1574 translation of the Hebrew into Latin, on which I have relied. The discussion of 306b39 is very problematic, [13] and I am not sure that I have interpreted it correctly, but I believe that Themistius is our earliest source for a line of argumentation taken over by Averroes (see below), which came to be thought of as the correct account of what Aristotle had in mind. In treating cubes Themistius proceeds by imagining four squares meeting at a point and forming a square, and four cubes above and four below them. For the pyramid, he imagines six equilateral triangles meeting at a point and forming a hexagon and six regular pyramids above and six below them. Apparently, he is not able to see (or at least not willing to admit) that, unlike the eight cubes, the twelve pyramids do not fill a space, but instead he gives an abstract argument that they do. The argument depends on identifying the size of a solid angle with the sum of the plane angles containing it. [14] Since a pyramidal angle is contained by three 60^{o} angles and a cubic angle by three 90^{o} angles, three pyramidal angles are equal to two cubic angles, and since eight cubes fill a space, twelve pyramids do as well. Themistius confirms this claim by citing the dimensional analogy in a somewhat obscure way, which I paraphrase as “the proportio which a triangle and square have as plane figure to plane figure is also had by what is constructed on them as solids, and so the proportio which the triangle has among planes, the pyramid has among solids”. Themistius completes his discussion by saying that no other solid figures fill a space; his explanation involves the fact remarked on in Elements 13.18. that pyramid, cube, octahedron, dodecahedron, and icosahedron are the only regular solids, but his formulation is again not clear, and he has done nothing to even suggest that a space could not be filled with octahedra, dodecahedra, or icosahedra.
17 I turn now to the first known proof that pyramids do not fill a space, given by Ibn alSalah [15] (d. ca. 1150). Ibn alSalah spends a good deal of time discussing Themistius’ argument in ways which do not always correspond to our text of Themistius, [16] and are not always clear to me. But his proof [17] is a beautiful example of a formal argument relying on nothing but complex results proved in book 13 of Euclid’s Elements, namely the remark after 13.18 [18] that the only regular solids are the five Platonic solids and results proved in 13.1317 about the relation of the length of an edge of one of these solids to that of the diameter of a circumscribing sphere. [19] Ibn alSalah also offers analogous arguments that regular octahedra, icosahedra, and dodecahedra cannot fill a space. [20] These proofs run along the same lines as the proof that Aristotle is wrong about pyramids, but I shall present that proof separately because it is the only one which recurs later and it is the simplest:
Suppose n regular pyramids with common vertex O and edges of length e filled a space; then they would form a regular solid with triangular faces, edges of length e, and inscribable in a sphere with centre O and radius of length e; by the remark after Elements 13.18 (cf. supra note 18) such a solid could only be a tetrahedron, octahedron, or icosahedron, but in none of these cases is the edge of the solid equal to the radius of the sphere in which it is inscribable (Elements 13.13, 14, and 16; cf. supra note 19).
19 For the other arguments, if n regular mhedra with equal edges OA_{1}, … , OA_{n} meeting at O fill the space around O, then A_{1}, …, A_{n} will define a regular nhedron P, with square faces if m = 8 (octahedron), with pentagonal faces if m = 20 (icosahedron), and with triangular faces if m = 12 (dodecahedron). In all cases P will be inscribed in a sphere S with center O and radius OA_{1}. In the case of the octahedron (icosahedron) P will be a cube (dodecahedron) with edges equal to the radius OA_{1}, contradicting Elements 13.15 (13.17; cf. supra note 19).
20 For n contiguous dodecahedra filling the space around O, the solid P will be either a pyramid or an octahedron or an icosahedron. Let it be a pentagonal face of one of the dodecahedra (with common vertex O) and let A_{1}A_{2} be its diagonal; then A_{1}A_{2} will be an edge of P. For pyramid (octahedron) one argues that, since A_{1}A_{2 }is the edge of a pyramid (octahedron) inscribed in a sphere with radius OA_{1}, A_{1}A_{2} and OA_{1} are commensurable in square by Elements 13.13 (13.14), but since A_{1}A_{2} is the diagonal of a pentagon with side OA_{1}, A_{1}A_{2} and OA_{1} are not commensurable in square [21], a contradiction.
21 The icosahedron case is more complicated and requires more details of Euclid’s Book 10 treatment of irrational straight lines than I can go into here. Since A_{1}A_{2} is the diagonal of a pentagon of which OA_{1} is a side, if A_{1}A_{2} is rational, OA_{1} is an irrational straight line called an apotome (cp. supra note 18). Ibn alSalah asserts and says he proves elsewhere, that since A_{1}A_{2} is the edge of an icosahedron inscribed in a sphere with radius the OA_{1}, if A_{1}A_{2 }is rational, OA_{1} is an irrational straight line called a major. [22] Hence OA_{1} is simultaneously an apotome and a major; this outcome contradicts the remark after Elements 10.111, according to which a straight line cannot be two different types of irrational.
22 Ibn alSalah’s essay appears to have had no effect on the subsequent history of the problem created by Aristotle’s remark. It certainly was unknown to Averroes (Ibn Rushd; 112698), who in his long commentary on the De Caelo [23] repeats in a clearer form the argument I have ascribed to Themistius. He first says that three straight lines intersecting at right angles make eight cubic angles, each of which is three right angles, so that intersecting straight lines which make eight cubic angles or angles equal to eight cubic angles fill a space and those which make angles equal to more or less do not. He then says that eight cubes meeting at a point will necessarily fill a space, and adds that twelve pyramids will as well, explaining:
For six pyramidal angles are equal to four angles of cubes because the solid angle of a pyramid is composed of two right angles and the angle of a cube of three. Therefore three angles of a pyramid will be equal to two angles of a cube, since they are equal to six right angles. For the ratio of the angle of the pyramid to four cubical angles is as the ratio of the angle of the triangle to four right angles [i.e., 180/1080 = 60/360], since the angle of the triangle is two thirds of a right angle. [24]
24 Averroes goes on to claim that regular octahedra, dodecahedra, and icosahedra do not fill a space. He does only the last case, pointing out (as we might express it) that the solid angles of a regular icosahedron are contained by five 60^{o} angles, and for no n does n(5 ^{.} 60) = 8(3 ^{.} 90). [25] He then cites Euclid for the proof that there are no other (regular) solids, so that there are certainly no other ones to fill a space. At this point in the story Themistius drops out of the picture, and Averroes becomes the acknowledged explainer of Aristotle’s position on pyramids.
25 I now turn to the Latin West and first to two approximate contemporaries, who are directly dependent on Averroes. In his commentary on the De Caelo, Albert the Great (Albertus Magnus; d. 1280) gives a version of Averroes’ argument that pyramids fill a space, but does not repeat his argument against regular solids other than cube and pyramid filling a space. [26] In his Opus Tertium [27] Roger Bacon (d. ca. 1292), after giving Averroes’ account of why pyramids fill a space, remarks in his typical carping manner that he spent almost twenty years with experts on this material without finding anyone who even understood the terminology and that, even when he taught the truth to students and explained the terminology, they were unable to discuss it. He also relates that a prominent Parisian philosopher proclaimed that twenty pyramids would fill a space, a view which he rejects on the Averroean ground that 20(3^{.}60) ≠ 2160. Roger goes beyond Averroes by pointing out that by Averroes’ criteria nine octahedra also fill a space since 9(4^{.}60) = 2160 = 8(3^{.}90). He concludes that one cannot obtain certainty on this subject unless one has bodies, that is, I assume, physical models, which satisfy the descriptions in Elements 13, but he is satisfied that at least dodecahedra and icosahedra do not fill a space.
26 Neither Ibn alSalah nor Averroes nor Albert nor Roger had access to Simplicius’ commentary on De Caelo, but it became available in the Latin West in a translation made at the request of Thomas Aquinas by William of Moerbeke. Thomas’ commentary on the De Caelo was broken off at the end of book 3, chapter 5 by his death in 1273. Peter of Auvergne, writing after Thomas’ death, wrote a commentary on books 3 and 4, the part picking up where Thomas leaves off being printed most recently in the Parma edition of Thomas’ complete works. His discussion of plane figures [28] is clearly based on Simplicius, but he relies more on Averroes in the case of solids. There Peter begins his discussion by invoking the way eight cubes fill a space without mentioning Averroes’ three straight lines. When he turns to the pyramid, he first brings in the triangular prism of Potamon’s argument as a kind of (quaedam) pyramid, contrasting it with the regular tetrahedron and showing that such a figure fills the space around a point:
But one kind of pyramid is that which has one solid right angle; different are those which are equilateral and in which all four angles are equal, and each is less than a right angle. [29] It is evident that the pyramid which has one solid right angle fills a space just as the cube also does. For if eight rectangular pyramids are applied with their right angles around one point they will fill the whole space around it, since eight angles of this kind are equal to eight cubic angles; and in this way it is true to say that a pyramid is not different from a cube. [30]
28 He then gives a version of the dimensional analogy: this kind of pyramid has the same ratio < in solids > as the rightangled triangle has in planes. And like Potamon he says that it is evident to perception that eight of these figures fill the space around a point.
29 For the regular pyramid, Peter cites Averroes for the view that twelve fill a space, and explains his reasons for saying this.
Moreover, Averroes says that the equiangular pyramid fills a space. For if twelve pyramids having equal angles are taken and applied at one point they will fill the entire space around it. He posits this for two reasons. For he posits that the solid angle of a pyramid is composed of two right angles which consist of three plane angles, < together > equal to two right angles; and the angle of a cube consists of three right angles. Because of this three angles of a pyramid will be equal to two cubic angles since they are equal to six plane right angles. As a result six angles of a pyramid will be equal to four cubic angles and twelve will be equal to eight. [31]
31 He continues in a difficult passage, which I interpret as follows:
Moreover, the ratio of an angle of a triangle to four plane right angles is the same as that of the pyramidal angle to some number of right cubic angles. So if some number of angles of a triangle are equal to four plane right angles and as a result fill a < plane > space, it would seem that some number of angles of a pyramid fill a plane space (? replent locum superficialiter) if, indeed, some number <of pyramidal angles> is equal to four cubic angles. [32]
33 However, Peter says, this view seems to contradict both perception and reason. He himself has done experiments (hoc ad sensum expertus sum), and seen that twelve regular pyramids do not fill a space. As for reason,
If twelve equilateral pyramids are applied at one point in any way, they do not occupy the whole space around the point in length, breadth, and depth. For they only touch one another at the extremities of their angles and not orthogonally along a straight line at the given point, a line through which another dimension is indicated. Therefore they do not fill a whole space in one < of the three > dimensions, and therefore do not fill a space corporeally. [33]
35 Moreover, Averroes’ claim that a solid pyramidal angle is equal to two plane right angles is incomprehensible (non est intelligibile):
For it is necessary that equal magnitudes be of the same ratio; and so a line is not equal to a surface, nor is any of those bodies. And, in fact, the angle of a pyramid and a plane angle are not of the same ratio, because the one is solid, the other plane. And so they are not equal to one another; nor is a solid angle composed of plane angles, since a body is not composed of planes. [34]
37 Peter continues by invoking Aristotle’s criticism [35] of Plato for attempting to construct solids out of plane surfaces:
Moreover, if a pyramidal angle were two plane right angles it would be composed from them, so that a pyramid would also be composed from plane triangles; for the ratio is the same. But this is false and against the thinking of Aristotle, who does not want bodies to be composed from planes, and — therefore and first — contrary to the commentator himself. But what Averroes says — that as the angle of an equilateral triangle is to a right angle among planes so is the angle of a pyramid to the angle of a cube — should be said not to be true in the case of filling a space and perhaps not true at all. [36]
39 Peter adds that if Averroes were right there should also be a spacefilling solid with hexagonal faces, something which neither Aristotle nor Averroes thinks is true, and concludes his discussion by referring to the other regular solids and saying that Averroes seems to explain why they do not fill a threedimensional space, but that he will leave the evaluation of his explanation to “the industrious investigator” (diligenti inquisitori).
40 Before turning to Thomas Bradwardine (d. 1349) it is necessary to say one thing about the manuscripts of his Theoretical Geometry [37]: they vary considerably in that passages included in some are not included in others. The editor of the work, George Molland, takes these to be interpolations, and in his edition relies on three texts which he calls “unelaborated” and two whose additions he inserts in brackets. I shall make clear when I am citing these additions, but remark here that they were almost certainly written before the authors I introduce in the remainder of this paper other than Witelo. [38]
41 In his presentation of the theory of regular solids Bradwardine first proves [39] the remark made after 13.18 in the Elements, according to which there are only five regular solids. Taking up the issue of spacefilling solids, [40] he suggests that Averroes thought that only cubes and pyramids fill a space because of their correspondence (correspondet) with square and triangle respectively, there being no regular solid with hexagonal faces. This, he says, is only a case of conviction (persuasio). In truth the cube fills a space, but cube and pyramid fill a space only according to Averroes’ opinion. To validate the claim about the cube he cites sense experience, and as a secondary, “cogent enough” (satis cogens) confirmation from reason a pointless arithmetic argument that, since the product of two cubic numbers is a cube and eight is the first cubic number eight cubes will fill a space. [41] Turning to the pyramid he first explains Averroes’ view that twelve fill a space and then notes that others, citing experience, say that twenty do so, a view which Bradwardine commends (in an elaboration) on the basis of a “subtle imagining of straight lines being drawn from the vertices of an icosahedron to the centre of the body”. Bradwardine objects to Averroes’ treatment of solid angles in terms of their containing plane angles on the grounds that greater plane angles might contain an angle of less “corpulence” (corpulentia) than that contained by smaller angles, [42] just as greater lines can contain lesser areas than shorter ones. He also makes Roger Bacon’s point that the view entails that nine octahedra will fill a space, something which is not asserted by either Averroes or Aristotle. In an elaboration Bradwardine goes on to argue that if twelve pyramids filled a space they would produce a regular dodecahedron with triangular faces, which he has shown to be impossible. He continues:
Concerning whether twenty pyramids fill a space, although this seems plausible (probabile), it is in no way certain, because anyone who would say that eight pyramids fill a space would say similarly that from them a body of eight faces results … and again he would similarly resolve that octahedron into eight pyramids by the subtlety of imagination. [43]
43 Now, of course, Elements 13.13 and 14 show that neither the twenty nor the eight pyramids are regular, but Bradwardine is unaware of this, [44] since he says, “If … it were established that the pyramids into which the icosahedron [45] was resolved … were regular, the matter would no longer appear to be in doubt”. In all mss. Bradwardine concludes by saying that the matter must be left in doubt.
44 I now jump ahead to Giuseppe Biancani (15661624), apparently the first person in Europe to publish the proof that Aristotle’s assertion about the pyramid is wrong. In his Mathematical Passages in Aristotle, [46] Biancani, obviously unaware of Ibn alSalah, indicates that everyone, Greek, Arab, or Latin, has been mistaken about Aristotle’s assertion. After giving essentially the same proof for the pyramid as Ibn alSalah, Biancani expresses his sense of uncertainty in the light of the fact that none of his predecessors had questioned Aristotle’s authority. He communicated his ideas and uncertainty to his teacher, Christopher Clavius (d. 1612), who, instituting a tradition of explaining away Aristotle’s mistake, took the option of saying that Aristotle only intended to say that six pyramids lying on a hexagon fill a twodimensional space around a point, just as four cubes lying on top of a square do. Biancani, professing his desire to follow truth more than Aristotle, rejects this reading of the text. He also rejects the claim that twenty pyramids fill a threedimensional space and constitute a regular icosahedron, pointing out that the pyramids into which an icosahedron is divisible are not regular and referring his reader to Elements 13.
45 Biancani completes his discussion by mentioning to two earlier authors Francesco Maurolico (14941575) and Giovanni Battista Benedetti (15301590), whose work, he says, he came across after writing up his analysis of Aristotle’s claim and who recognize the falsehood of the claim. One of the six sections of Benedetti’s Book of Diverse Mathematical and Physical Speculations [47] is a set of 39 ‘disputations’ on the opinions of Aristotle, in the nexttolast of which Benedetti refers to the question of spacefilling pyramids. His doing so comes in oddly since his real concern is to reject the idea that the pyramid is more mobile than the cube. [48] However, he takes the trouble to point out Aristotle’s error and says that he made it because he thought that six pyramids constructed on either side of a hexagon fill a space, although, in fact these pyramids leave more empty than filled space on either side of the hexagon. His opinion here might well be based only on experiments, like those of Peter of Auvergne. Biancani rejects Benedetti’s explanation as childish and unworthy of Aristotle’s genius. Benedetti, after pointing out what he takes to be Aristotle’s error, says that Aristotle would have done better to have asserted that the pyramid and not the cube should be associated with earth on the grounds of being stable, and gives one of his “reasons”.
46 With Maurolico I believe we leave the domain of Elements 13 and enter the domain of trigonometry. Biancani cites Maurolico’s Cosmographia [49], at the beginning of which Maurolico gives an unpaginated descriptive catalogue of his “works”, which includes the following:
Our little book on regular plane and solid figures which fill a space. Although it is certain that John Regiomontanus has written very precisely about this matter, as far as I know, the book has not yet been published. However, in < our > book we demonstrate that of the regular solids cubes fill a space by themselves but pyramids do so only when conjoined with octahedra [50], from which it will be manifest that Averroes has made a childish mistake on this subject.
48 The text which Maurolico describes here does not appear to have been pblished, and I have not seen it. But the manuscript of the text (some twenty folios) is preserved in Rome, [51] dated 1529, with the title “On the five solids, which are commonly called regular, which of them clearly do fill a space, and which do not, contra Averroes, the commentator of Aristotle”. The text was described by de Marchi in “Di tre manoscritti” as complete and ready for publication. De Marchi gives a brief summary of the contents with no indication of the proofs, but it seems clear that Maurolico gave proofs based on a proper notion of a solid angle. [52]
49 Maurolico’s reference to Regiomontanus is almost certainly based on a list, [53] printed in 1473 or 1474, of works (by himself and others) which he intended to publish in Nürnberg, to which he had moved in 1471, some five years before his death. Most of the works were never published, including the one to which Maurolico refers, which in Regiomontanus’ list bears the title ‘On the five equilateral bodies which are commonly designated regular, which of them clearly do fill a natural space and which do not, contra Averroes, the commentator on Aristotle’. There is no way to know whether or not Regiomontanus completed the work which he announced for publication, but a list of problems he sent to Christian Roder after his arrival in Nürnberg includes three which make clear that Regiomontanus’ approach to the question of spacefilling polyhedra was like Maurolico’s rather than iancani’s. One problem is to determine the ratio of the angle of an icosahedron to that of the dodecahedron, another to find the area of a spherical triangle with sides of a specific given size. The third runs:
If some number of tetrahedral angles fills a solid space and also some number of octahedral angles fills such a space, the ratio of the latter number of angles to the former is to be sought. But if neither fills a solid space, it is to be asked of each whether the maximum number of angles is less than the eight solid right angles which fill a solid space. [54]
51 The first published application of trigonometry to the problem of spacefilling regular solids was made by Jan Brozek (15851652) in his defense of Aristotle and Euclid against Petrus Ramus (15151572) and others. [55] Ramus, in his Two Books of Arithmetic and Twenty Seven of Geometry (1569), had defended on Averroean grounds the position that both twelve pyramids and also nine octahedra fill a space. [56] After stating the rule for calculating the size of a solid angle, [57] Brozek gives arguments based on the assertion that the angle of the pyramid is 34 degrees (gradus), 42 minutes (scrupulae), [58] so that twelve such angles fall well short of the 720 degrees of the full sphere [59]; moreover, the angle of the octahedron is 73 degrees, 44 minutes so that eight pyramids plus six octahedra fill a space (6(73^{o} 44’) + 8(34^{o} 42’) = 444^{o} 24’ + 277^{o} 36’ = 720^{o}) (p. 8485). [60] But four pyramidal angles are less than six octahedral ones, so how can twelve pyramids fill a space? (And one can also argue that nine octahedral angles don’t do so either.) Brozek is only interested in refutation and so does not give arguments about the icosahedron and dodecahedron. He also tries to defend Aristotle by saying that he was not talking about regular pyramids. [61] And defending Aristotle is his only concern in his discussions of Biancani, Benedetti, and Maurolico. [62] He does not show any sense of the legitimacy of Biancani’s demonstration that regular pyramids do not fill a space.
52 Brozek cites [63] as inspirations for his thinking about solid angles Witelo (d. ca. 1280?), who in book 1 of his Perspectiva [64] gives an unsatisfactory proof of the claim that “the ratio of part of a spherical surface to the whole surface of its sphere must necessarily be equal to < the ratio > of the solid angles falling in the same < part of the spherical surface > from the center of the sphere to eight right angles”, [65] and Johannes Werner (14681528), author of a lengthy treatise on spherical triangles. [66] Brozek also says (p. 79) that Henry Briggs (15611630) ascribes to Thomas Harriot (ca. 15601621) the “demonstration” of the size of a solid angle and calls Harriot the first person to accomplish this. It is not clear where Brozek got his information, but, in fact, Briggs did write a letter dated March 10, 1625 [67] to Johannes Kepler (15711630) in which he ascribes the discovery to Harriot. Briggs gives a very accurate 350958/1000000 (90) (approximately 31^{o} 35’) as the size of the angle of a pyramid and rejects the view that twelve such angles fill a space. The evidence that Harriot did make the discovery is, I think, conclusive, [68] but the first published proofs of it are by Girard [69] (1629) and Cavalieri [70] (1632) (the latter of which I have not seen).
53 Finally, I move back in time to Paul of Middelburg (14451534), who is best known as a proponent of the calendrical reform which ultimately produced the Gregorian calendar, adopted in 1582. One of the services of Dirk Struik’s foundational paper on my subject was to call attention to passages in two texts of prognostication (for 1480 and 1481) in which Paul touches on the question of spacefilling polyhedra. The texts are available only as manuscripts. [71] In the first, which is generally contemptuous of contemporary scientists, Paul proposes 100 problems for them to solve, the 49th of which (unfortunately I haven’t seen the text) concerns spacefilling solids. After stating it he says that the solution will make clear that Aristotle and Averroes have gone astray in asserting that twelve angles of a regular pyramid could fill a space and, that, indeed, neither are twenty sufficient … and, furthermore, that Thomas Bradwardine also fell short in his considerations of the subject. After giving his solution, which again I have not seen, Paul says that the mistake of Aristotle and Averroes is clear from Elements 13.15 because the pyramids from which an icosahedron is constructed are not regular. Paul presumably means the proposition on the icosahedron which we call 13.16, although no text of Euclid I know of calls that proposition 15. Paul concludes that, therefore, regular pyramids cannot fill a space even if there are twenty of them; indeed, no other number of them will produce a regular body — that is impossible. I am inclined to think that Paul understood a version of the proof given by Ibn alSalah and Biancani rather than a trigonometric one, although Struik seems to have thought otherwise. He sees Paul and Maurolico as continuators of the program announced by Regiomontanus. [72]
54 Mathematically the question of spacefilling pyramids was resolved first in the twelfth century in Baghdad and then in the sixteenth in the West. But the resolution has for the most part been forgotten in nineteenth and twentiethcentury Aristotelian studies. So far as I know, Aristotle’s mistake was first recognized in the twentieth century by Thomas Heath, who says that Aristotle was wrong and refers to Biancani as “taking credit for having been the first to point this out, when all commentators before him […] accepted Aristotle’s statement as correct”. [73] It seems likely to me that writing his book on mathematics in Aristotle led Heath to the work of his Renaissance predecessor and to the truth on the matter. Heath’s remarks are picked up on by Leo Elders in his commentary on the De Caelo [74] and recently by Dalimier and Pellegrin [75] in their translation of the work. But, as far as I know, Aristotle’s claim has been passed over in silence by all the other nineteenth – and twentiethcentury Aristotelian scholars whom one would have expected to remark on Aristotle’s mistake if they were aware of it. To be sure a number of Islamicists, starting with Muhabat Türker, have become aware of it, and we can hope that by the end of the present century Aristotelians in general will have caught up with their Islamicist colleagues.
Notes

[1]
Aristote, Du ciel, ed. and trans. by P. Moraux, Paris, Les Belles Lettres, 1965, 3.8.306b39 [thereafter quoted as De Caelo].

[2]
The standard modern treatment of spacefilling figures in this sense is H. S. M. Coxeter, Regular Polytopes, 2^{nd} ed., New York, Macmillan Co., 1963, where it is shown (pp. 6869) that of the five regular solids only cubes fill space. The first author I know of to bring in this kind of consideration is Giuseppe Biancani, who on p. 121 of his Mathematical Passages in Aristotle (Aristotelis loca mathematica ex universis ipsius operibus collecta et explicata …, Bologna, 1615) says that plane figures which fill a space in the first sense can be used to make a pavement and solid figures which do so are suitable for constructing walls (a pagebypage transcription is available at: http://archimedes2.mpiwgberlin.mpg.de/archimedes_templates/biography.html?table=archimedes_authors&author=Biancani,%20Giuseppe&find)

[3]
Simplicii in Aristotelis De Caelo Commentaria, ed. J. L. Heiberg, Berlin, G. Reimer, 1894, p. 651,2652,8; a digitalized version is available at http://gallica.bnf.fr/ark:/12148/bpt6k950617.

[4]
Simplicius argues for this result after citing what is Elements 1.13 in Euclidis Elementa, ed. J. L. Heiberg, Leipzig, Teubner 1883, vol. 1 (“In whatever way a straight line which is set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles”), which Simplicius calls “the thirteenth theorem of the first book of the Elements”. Among other authors, Themistius in his paraphrase of De Caelo (Themistii in Libros Aristotelis De Caelo Paraphrasis, Hebraice et Latine, ed. S. Landauer, Berlin, G. Reimer, 1912, pp. 197,34198,28; digitalized version available at http://gallica.bnf.fr/ark:/12148/bpt6k255019), writing in the fourth century, and Averroes in his long commentary on it (Averrois Cordubensis commentum magnum super libro de celo et mundo Aristotelis, ed. F. J. Carmody and R. Arnzen, pref. by G. Endress, Leuven, Peeters, 2003, v. 2, p. 629; cp. Averroes’ paraphrase of De Caelo in Aristoteles omnia quae extant Opera … . Averrois Cordubensis in ea opera omnes, qui ad haec usque tempore pervenere, commentarii, Venice, 1562, 5 (reproduced as volume V of Aristotelis Opera cum Averrois Commentariis, Frankfurt am Main, Minerva, 1962, 325v326r), writing in the later twelfth, take the result for granted. Ibn alSalah (see M. Türker, “Ibnu’sSalah’in De Coelo ve onun serhleri hakkindaki tenkitleri,” Arastirma, 1964, 2, pp. 179), writing in the earlier twelfth century, does not really discuss the plane case. In his commentary on De Caelo Albert the Great (Alberti Magni De Caelo et Mundo, ed. P. Hossfeld, Münster, Aschendorff, 1971, p. 238, 2022), writing in the thirteenth, cites a confused version of 1.13 (Duae enim lineae se secantes in superficie corporali faciunt duos angulos rectos), but he calls it the fifteenth theorem of Elements 1. In fact, medieval versions of Euclid often contain the result as part of 1.15 (if two straight lines intersect, they make the opposite angles equal), and sometimes it is treated as a corollary to 15. And in his commentary on the last part of De Caelo Peter of Auvergne (Sancti Thomae Aquinatis … Opera Omnia, Parma, P. Fiaccadori, 1865, vol. 19, p. 179a; reprinted in New York, Musurgia Publishers, 1949; unpaged transcription available at http://www.corpusthomisticum.org/xcm.html), writing after Albert, follows Simplicius. Biancani (Aristotelis loca mathematica, op. cit., p. 84) relies on Clavius’ version of Euclid (Christophori Clavii Bambergensis E Societate Iesu Opera Mathematica, Mainz, 1611, vol. 1; digitalized versions of all five volumes of the Opera mathematica are available at http://mathematics.library.nd.edu/clavius/), where the result is stated as a second scholium to 1.15.

[5]
For the regular pentagon, each angle is 108^{o}, and 3 ^{.} 108 < 360 < 4 ^{.} 108; and for any n > 6, three times the angle of a regular nagon is greater than 360^{o}.

[6]
Simplicius’ claim (in De Caelo, op. cit., p. 650, 27) is made all the more perplexing by the fact that he later reproduces, without comment, Potamon’s assertion that eight ‘pyramids’ fill a space (see below).

[7]
I assume this Potamon is the eclectic philosopher, probably of early imperial times mentioned by Diogenes Laertius (Diogenes Laertii Vitae Philosophorum, ed. M. Marcovich, Stuttgart and Leipzig, De Gruyter, 1999, 1.21); cf. the Suda (Suidae Lexicon, ed. A. Adler, Leipzig, Teubner, 1930, pt. 4, s.v. Potamôn [2126]). For a thorough discussion of the evidence concerning Potamon, which argues that Simplicius’ Potamon is to be identified with the person described by the Suda rather than the one mentioned by Diogenes (the two have usually been assumed to be the same person), see A. Rescigno, “Potamone, interprete del De Caelo di Aristotele,” Lexis, 2001, 19, pp. 267282.

[8]
Simplicius, in De Caelo, op. cit., p. 655, 918.

[9]
See L. de Marchi, “Di tre manoscritti del Maurolico che si trovano nella Bibliotheca Vittorio Emmanuele di Roma (Continuazione e fine),” Bibliotheca Mathematica, 1885, pp. 193195 (on p. 194).

[10]
It appears that Alexander had some doubts about the principle, since he says, « < Potamon is saying > nothing else than that what happens in the case of plane figures also happens in the case of solid ones and happens in three dimensions in the case of solid figures in the same way as it happens in one dimension in the case of plane figures. » (Simplicius, in De Caelo, op. cit., p. 655, 2931).Unfortunately, Simplicius reports no more than this and turns instead to query Alexander’s perplexing reference to “one dimension”. Simplicius himself characterizes the principle as “elegant” (philokalos), and goes on to raise the question whether space might be filled with solids in a variety of sizes. (I note in passing that in his discussion of the difficulties raised by Aristotle, Simplicius also brings in the responses of Proclus: he gives no indication that Proclus had any difficulty with spacefilling pyramids.)

[11]
Simplicius, in De Caelo, op. cit., p. 655, 1827.

[12]
This is the way he was understood by Peter of Auvergne (see below). [nde – Ian Mueller is here relying on a passage of Peter of Auvergne which he quotes below, but I think that he misunderstood Peter, and that Peter in fact agrees with Dirk Struik’s interpretation, which Mueller goes on to discuss. – (SPM)] D. J. Struik (“Het Probleem ‘de impletione loci’,” Nieuw Archief voor Wiskunde, 1925, 2:15, pp. 121137; on p. 123) takes Potamon to have construed the pyramid as the solid angle at the vertex of a cube, essentially identifying the pyramid with a solid like AΒΔE in the figure and assuming falsely that such figures bisect a cube and that they are regular. [nde – ABΔE is a pyramid on an equilateral triangular base but BE = EΔ = ΔB = √2 ^{.} AB; so it is not regular even if eight of them clearly fill the space around A. There is no need to attribute to Potamon the strange mistake of thinking that these pyramids are regular; at least Peter is clear that they are not. But Struik may well be right to attribute to Potamon the easy mistake of thinking that two of these pyramids in opposite corners of a cube jointly exhaust the cube. That is what the text that Mueller cites from Simplicius seems to say. The alternative is, with Mueller, to say that Potamon is here talking about prisms based on isosceles right triangles, two of which do indeed jointly exhaust a cube. But it is hard to see why Potamon would call a prism a pyramid. (SPM)] I wish to record here my great debt to Struik’s paper and also to Marjorie Senechal, who gave me a copy of her unpublished English translation of it, and to Stephen Menn for enlightening remarks about spacefilling figures.

[13]
On Landauer’s edition and its problems, see M. Zonta, “Hebraica veritas: Temistio, Parafrasi del De coelo. Tradizione e critica del testo”, Athenaeum, 1994, 82, pp. 403428; the relevant passage can be found in Themistius, Themistii in Libros Aristotelis, op. cit., pp. 197,34199,34

[14]
[nde – That is, suppose a solid angle is formed by three planes, bounded by three lines and all meeting at a vertex. Themistius, and many of the other authors here discussed, assume wrongly that the size of the solid angle is the sum of the three angles between any two of the three lines meeting at the vertex. The correct definition of the size of the solid angle is that it is the sum of the three angles between any two of the three planes meeting at the vertex, minus two right angles. As Ian Mueller shows further down, Jan Brozek, and some earlier authors that Brozek cites, were aware of the correct definition. (SPM)]

[15]
My discussion of Ibn alSalah is based entirely on the French essay “Les critiques d’Ibn aṣṢalah sur le De Caelo d’Aristote et sur ses commentaires” (pp. 1930) in M. Türker, “Ibnu’sSalah’in De Coelo”, art. cit., which includes the Arabic text of Ibn alSalah’s work (pp. 5379), a Turkish translation of it (pp. 3152), and a somewhat longer Turkish version of the French essay (pp. 118) ; the French essay is also available in La Filosofia della natura nel medioevo (Atti del 3^{o} congesso internazionale di filosofia mediovale, Passo della Mendola; Trento, 31 agosto5 settembre 1964), Milan, Società editrice Vita e pensiero, 1966, pp. 242252.

[16]
Ibn alSalah says (M. Türker, “Les critiques d’Ibn aṣṢalah,” art.cit., p. 24) that Themistius made the dimensional analogy the principle of his demonstration. As what I have said in the previous paragraph indicates, this interpretation is not true of our text of Themistius. And there are other discrepancies, of which I mention three:
(i) In Ibn alSalah’s version of Themistius argument, the twelve pyramids are said to be supplemented by twelve other pyramids to fill out the space; Ibn alSalah’s response to his version of Themistius’ argument is a restricted version of his argument that no number of pyramids can fill a space, which I give below (Ibid., pp. 24 & 26);
(ii) Ibn alSalah also ascribes to Themistius an argument involving the fact that (by Elements 12.7) the eight triangular prisms of Potamon’s argument can each be divided into three equal pyramids, to which Ibn alSalah correctly objects that the pyramids in question are not regular (Id.);
(iii) he commends (overgenerously, it would seem) Themistius for invoking something like the remark after Elements 13.18 as an answer to the question why, given the dimensional analogy, there is no spacefilling solid corresponding to the hexagon (Ibid., p. 27). 
[17]
Ibid., pp. 2728.

[18]
[nde – “I say next that it is not possible to construct beside the said five figures – (regular) pyramid, octahedron, cube, icosahedron, dodecahedron – another figure contained by equilateral and equiangular (plane figures) equal to one another.” (BV)] This ‘remark’, which is not in the standard style of the Elements, is found in all the [preserved] Greek manuscripts [for 13.18. Furthermore,] the result is ascribed to Euclid “and others” in our text of the Collection of Pappus, who is usually dated to the earlier fourth century. However, [it does not exist in Arabic and AraboLatin translations and] we know that Thabit ibn Qurra (d. 901) found the proposition in some Greek texts of the Elements and added it to Ishaq ibn Hunain’s translation. See Euclide d’Alexandrie, Les Éléments, trans. B. Vitrac, Paris, Presses universitaires de France, 2001, vol. 4, pp. 471475. [So, it was apparently present in some Greek manuscripts, but not in all of them. This ‘remark’ is probably an old addition inserted in Euclid’s text before Pappus’ time. (BV)]

[19]
13.13 – “… the diameter of the (circumscribing) sphere is one and a half times in square the side of the pyramid” [in modern terms: D^{2} = (3/2) ^{ .} (a_{4})^{2}, so D = √(3/2) ^{.} a_{4 }(BV)]
13.14: “… the diameter of the sphere is double in square the side of the octahedron” [in modern terms: D^{2} = 2(a_{8})^{2}, so D = √2 ^{.} a_{8} (BV)]
13.15: “… the diameter of the sphere is triple in square the side of the cube” [in modern terms: D^{2} = 3(a_{6})^{2}, so D = √3 ^{.} a_{6} (BV)]
13.16: “… the side of the icosahedron is an irrational straight line, the one called minor” [the diameter of the sphere is supposed to be a rational straight line; in modern terms: D^{2} = {(10 + 2√5)/4} ^{.} (a_{20})^{2}(BV)]
13.17: “… the side of the dodecahedron is an irrational straight line, the one called apotome” [nde – The diameter of the sphere is supposed to be a rational straight line; in modern terms: D^{2} = (3/2) ^{ .} (3 + √5) ^{ .} (a_{12})^{2}. From these results, it is clear that the ratio D/a_{n} (n = 4, 6, 8, 20, 12) is irrational for the five regular figures; so, it cannot be 1, the condition for the n pyramids (n = 4, 6, 8, 20, 12) which compose each of them to be regular. (BV)] 
[20]
M. Türker, “Les critiques d’Ibn aṣṢalah,” art.cit., pp. 2830.

[21]
[nde – Combining 13.8: “if straight lines subtend two successive angles of an equilateral and equiangular pentagon, they cut one another in extreme and mean ratio and their greater segments are equal to the side of the pentagon”; and 13.6: “if a straight line is cut in extreme and mean ratio, each of the segments is an irrational straight line, the one called apotome.” (BV)]

[22]
I give a quasialgebraic proof of this claim in an appendix.

[23]
The whole commentary is preserved only in Latin and available in Carmody’s edition, Averrois Cordubensis commentum magnum, op. cit., the text I have used; the relevant material is in vol. 2, pp. 629631. A facsimile of the only known Arabic ms. of parts of the commentary on Books 1 and 2 is found in Ibn Rushd, Commentary on Aristotle’s Book on the Heaven and the Universe, ed. G. Endress, Frankfurt am Main, Publications of the Institute for the History of ArabicIslamic Sciences, 1994. Averroes says the same sort of thing in his paraphrase of De Caelo at 325v326v.

[24]
Averroes, Commentum magnum, op. cit., vol. 2, p. 630.

[25]
In our text Averroes actually says “for no n does n(5 ^{.} 60) = 4(3 ^{.} 90)”.

[26]
Albert the Great, De Caelo et Mundo, op. cit., pp. 237239.

[27]
Fr. Rogeri Bacon Opera Quaedam Hactenus Inedita, ed. J. S. Brewer, London, Longman, Green, Longman & Roberts, 1859, vol. 1, pp. 136140 (digitalized version available at http://books.google.com/books?id=wMUKAAAAYAAJ).

[28]
Peter of Auvergne, Commentary on De Caelo, op. cit., pp. 178b179b.

[29]
[nde – The “kind of pyramid … which has one solid right angle” seems to be Struik’s nonregular pyramid ABΔE, described and depicted in note 12 above, and not as Mueller says a prism on an isosceles right triangle, which would have two solid right angles. (SPM)]

[30]
Peter of Auvergne, Commentary on De Caelo, op. cit., op. cit., p. 179b.

[31]
Ibid.

[32]
Ibid.

[33]
Ibid., p. 179b180a.

[34]
Ibid. p. 180a.

[35]
See Aristotle, De Caelo, op. cit., 3.1.299a1300a19.

[36]
Peter of Auvergne, Commentary on De Caelo, op. cit., p. 180a.

[37]
Thomas Bradwardine, Geometria Speculativa, ed. and trans. by G. Molland, Stuttgart, Franz Steiner Verlag, 1989.

[38]
Of the two manuscripts which Molland relies on for the interpolations one is dated to the 14th century (see M. Clagett, Archimedes in the Middle Ages, Madison, University of Wisconsin Press, 1964, pp. 370 & 372), the other to the 14th or 15th (see W. Wislocki, Katalog rekopisow Biblijoteki Uniwersytetu Jagiellonskiego, Krakow, Kraków Nakładem Akad. Umiejetności w Krakowie, 18771881, vol. 2, no 1919).

[39]
Bradwardine, Speculative Geometry, op. cit., pp. 127134.

[40]
Ibid., p. 135.

[41]
Bradwardine goes on to “explain” why, e.g. 27 cubes don’t fill a space.

[42]
Compare, e.g., the solid angle contained by three right angles with the smaller angle contained by two angles of 170o and one angle of 1o.

[43]
Bradwardine, Speculative Geometry, op. cit., p. 139.

[44]
Perhaps surprisingly, since [the propositions on which these results are based] are included in Campanus’ version of Euclid (Campanus of Novara and Euclid’s Elements, ed. H. L. L. Busard, Stuttgart, Franz Steiner Verlag, 2005, [vol. 1, pp. 480484; nde – Note that Euclid’s 13.14 and 16 = Campanus 13.1516) (BV)], which, according to Molland (“An examination of Bradwardine’s geometry,” Archive for History of Exact Sciences, 1978, 19, pp. 113175, on p. 120; reprinted in G. Molland, Mathematics and the Medieval Ancestry of Physics, Aldershot, Variorum, 1995), was Bradwardine’s principal source for his Speculative Geometry.

[45]
Why is there no mention of octahedra here? The authority of Aristotle and Averroes? The prominence of the view about the icosahedron? The fact that the ratio of edge to radius of circumscribing sphere is much closer to unity in the case of the icosahedron [(a20/r = 1/5() ≈ 21/20 = 1.05)] than in the case of the octahedron [(a8/r = √2 ≈ 7/5 = 1.4)]?

[46]
Biancani, Aristotelis loca mathematica, op. cit., pp. 8487.

[47]
Io. Baptistae Benedicti … Diversarum Speculationum Mathematicarum, & Physicarum Liber, Turin, 1585 (digitalized version at http://echo.mpiwgberlin.mpg.de/ECHOdocuView/ECHOzogiLib?mode=imagepath&url=/mpiwg/online/permanent/library/S46RNB52/pageimg); printed again as Speculationum Mathematicarum, et Physicarum, Fertilissimus, Pariterque Utilissimus Tractatus, Venice, 1586; and again as Speculationum Liber, Venice, 1599; the relevant text is on pp. 196197.

[48]
Cp. Aristotle, De Caelo, op. cit., 3.8.306b29307a13.

[49]
Cosmographia Francisci Maurolyci, Venice, 1543 (digitalized version at http://gallica.bnf.fr/ark:/12148/bpt6k830579); another edition: Paris, 1558.

[50]
For a trigonometrical argument for this second claim, see the discussion of Jan Brozek below.

[51]
At the Biblioteca Nazionale Centrale Vittorio Emanuele II as San Pantaleo 117/33; see the Maurolico Project website at http://www.dm.unipi.it/pages/maurolic/instrume/catalogi/topograf/autog/17sp117.htm.

[52]
I summarize De Marchi’s summary of the contents of the essay:
chap. II. defines filling a space for plane figures and for solid figures distinguishing for solid figures between filling a space along a line (angulariter) and filling a space at a point (verticaliter).
chap. III. only regular figures to be discussed.
chap. IV. plane figures fill a space if their angles add up to 360o; the same is true of solids and filling space angulariter; for filling verticaliter the angles meeting at a point must make their angles equal to eight solid right angles.
chap. V. three plane figures fill a space.
chap. VI. similarly for right prisms.
chap. VII. describes the five regular solids, giving the number of faces and vertices in each case.
chap. VIII. (on solids) any number of cubes fills a space angulariter and verticaliter; six octahedra and eight pyramids together do so as well; one octahedron, one pyramid and two cubes together fill a space angulariter; no other combination of the five regular solids fills a space.
chap. IX, XIXVII proof that the three combinations of chap. VIII fill a space in the sense of chapter IV.
chap. X. proof that Averroes’ claim that twelve pyramids fill a space is absurd.
chap. XVIIILXVII. proof that no other combination of the regular solids fill a space, followed by a numerical demonstration of the preceding propositions. 
[53]
A smallscale reproduction of the list is on p. 533 of Joannis Regiomontani Opera Collectanea, ed. F. Schmeidler, Osnabrück, Otto Zeller Verlag, 1949. An even smaller one is plate 26 in E. Zinner, Leben und Wirken des Joh. Müller von Königsberg gennant Regiomontanus, 2nd ed. Osnabrück, Otto Zeller Verlag, 1968. The English translation of this work (E. Zinner, Regiomontanus; His Life and Work, vol. 1, trans. by E. Brown, Amsterdam, NorthHolland, 1990) does not include the plates.

[54]
M. Curtze (ed.), Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance, Leipzig, Teubner, p. 332. The other two problems are printed on the same page.

[55]
Apologia pro Aristotele & Euclide, contra Petrum Ramum, & alios … Dantzig, 1652 (digital version at at http://imgbasescdulp.ustrasbg.fr/displayimage.php?album=585&pos=0). Brozek’s discussion is very rhetorical and very long, covering pages 69109.

[56]
P. Rami Arithmeticae Libri Duo, Geometriae septem et viginti, Basel, 1569; other printings in 1580, 1599, 1604, 1612, and 1627; I consulted the 1599 edition, revised and enlarged by Lazarus Schonerus, Frankfurt, in which the two claims are made on p. 147 and p. 162.

[57]
J. Brozek, Apologia, op. cit., pp. 7477.

[58]
Brozek’s numbers are incorrect. [nde – The solid angle of a tetrahedron or regular pyramid is 90°3 arcsin(1/3), which is about 31°35’. The solid angle of an octahedron is 4 arcsin(1/3), which is about 77°53’. Six octahedral angles and eight tetrahedral angles equal 720° or eight solid right angles, which is a necessary but not a sufficient condition for their filling the space around a point. But in fact they fill, not only the space around a point, but all of space. Imagine unit cubes filling space, with their vertices at the points (i, j, k) where i, j and k are integers. Then if we take only those points (i, j, k) where i + j + k is an even number, these points will be the vertices of octahedra [like the one whose vertices are the points (0, 0, 0), (2, 0, 0), (1, ± 1, 0) and (1, 0, ± 1)] and tetrahedra [like the one whose vertices are the points (0, 0, 0), (1, 1, 0), (1, 0, 1) and (0, 1, 1)] which together fill space. Six such octahedra and eight such tetrahedra will share the vertex (0, 0, 0). (SPM)]

[59]
J. Brozek, Apologia, op. cit., pp. 7980.

[60]
Because he had seen only Maurolico’s general announcement that pyramids and octahedra together fill a space, Brozek thinks (Apologia, op. cit., pp. 103104) he has made Maurolico’s result more precise, but Maurolico has the correct numbers in the manuscript.

[61]
J. Brozek, Apologia, op. cit., p. 90.

[62]
Ibid., pp. 92105.

[63]
Ibid., p. 84.

[64]
Witelonis Perspectivae Liber Primus, ed. S. Unguru, Wroclaw, Polish Academy of the Sciences, 1977.

[65]
Ibid., Proposition 87.

[66]
Ioannis Verneri De Triangulis Sphaericis Libri Quatuor, De Meteoroscopiis, Libri Sex …, ed. A. A. Björnbo, Leipzig, Teubner, 1907, vol. 1. This is the first publication of Werner’s book; for the curious history of the manuscript see Björnbo, pp. 150175.

[67]
See Joannis Kepleri Astronomi Opera Omnia, ed. C. Frisch, Frankfurt and Erlangen, Heyder & Zimmer, 18581871, vol. 4, pp. 661662.

[68]
For this and other evidence see G. Vacca, “Notizie storiche sulla misura degli anguli solidi e dei poligone sferici,” Bibliotheca Mathematica, 1902, 3:3, pp. 191197; see also J. Tropfke, Geschichte der ElementarMathematik, Berlin and Leipzig, Veit, 1923, vol. 5, pp. 128131.

[69]
A. Girard, Invention Nouvelle en l’Algèbre, Amsterdam, 1629; reprinted in Leiden, Muré Frères, 1884 (digitalized version of this second edition at http://books.google.com/books?vid=UOM39015034807001).

[70]
Bonaventura Cavalieri, Directorium Generale Uranometricum, Bologna, 1632.

[71]
Struik, “Het Probleem,” (p. 128) lists the texts as Incun. 277 and 220 of the Biblioteca Alessandrina in Rome.

[72]
Ibid., pp. 128130.

[73]
Thomas Heath, Mathematics in Aristotle, Oxford, Clarendon Press, 1949, p. 178.

[74]
Leo Elders, Aristotle’s Cosmology. A Commentary on the De Caelo, Assen, Van Gorcum, 1966, p. 322.

[75]
Aristote, Traité du Ciel, trans. by C. Dalimier and P. Pellegrin, Paris, Flammarion, 2004, p. 449.