APOLLONIUS OF PERGACONICS. BOOKS ONE - SEVEN INTRODUCTION A. Apollonius at Perga Apollonius was born at Perga (Περγα) on the Southern coast of Asia Mi- nor, near the modern Turkish city of Bursa. Little is known about his life before he arrived in Alexandria, where he studied. Certain information about Apollonius’ life in Asia Minor can be obtained from his preface to Book 2 of Conics. The name “Apollonius”(Apollonius) means “devoted to Apollo”, similarly to “Artemius” or “Demetrius” meaning “devoted to Artemis or Demeter”. In the mentioned preface Apollonius writes to Eudemus of Pergamum that he sends him one of the books of Conics via his son also named Apollonius. The coincidence shows that this name was traditional in the family, and in all prob- ability Apollonius’ ancestors were priests of Apollo. Asia Minor during many centuries was for Indo-European tribes a bridge to Europe from their pre-fatherland south of the Caspian Sea. The Indo-European nation living in Asia Minor in 2nd and the beginning of the 1st millennia B.C. was usually called Hittites. Hittites are mentioned in the Bible and in Egyptian papyri. A military leader serving under the Biblical king David was the Hittite Uriah. His wife Bath- sheba, after his death, became the wife of king David and the mother of king Solomon. Hittites had a cuneiform writing analogous to the Babylonian one and hi- eroglyphs analogous to Egyptian ones. The Czech historian Bedrich Hrozny (1879-1952) who has deciphered Hittite cuneiform writing had established that the Hittite language belonged to the Western group of Indo-European languages [Hro]. Hence it is clear that such nations of Europe as Greeks and Romans, Galls and Goths, Slavies and Lithuani- ans were descendants of Hittite tribes. As the masculine words in the most an- cient of these languages have the endings of -os, -us, -as, -es, -is, the Hittite masculine words had ending of -ash, -ush, -ish. The Hittite word “vadar” for water is near to the Russian and Czech “voda”, English “water”, German “Wasser”, and Greek “υδωρ “. The Hittite word “pahhur” for fire is near to the English word “fire”, German “Feuer”, and Greek “πυρ”. The Hittite word “gordion” for town is near to Russian “gorod” and “ograda”, Czech “hrad”, English “garden”, and German “Garten”. The Hittite 0 word “eshmi” for “I am” is near to Russian “yesm’ “, Czech “jsem”, Latin “sum”, Greek “ειµι “ and English “I am”. In the first millennium B.C., after migration of Hittite tribes from the East to the west of Asia Minor and to Europe, the Hittite Empire disintegrated and many separate Hittite kingdoms appeared. The most important of these king- doms were situated in the Western part of Asia Minor. The most famous cities of these Hittite kingdoms were Ilion in Troy, Pergamum in Moesia, Sardis in Lydia, Gordion in Phrygia, and Myres in Lycia. The king of Lydia Croeses was fa- mous for his richness; with the name of the king of Phrygia Gordias was con- nected the legend of “Gordias’ knot”. City of Pergamum was the first city where pergament was made. In the same millennium on the Jonian coast of Asia Minor the Greek cities Miletus, Ephesus and others appeared. During the Greek - Persian wars all of Asia Minor was occupied by the Per- sians. After the victory of Greeks all Hittite states of Asia Minor became Greek states. In this period Pergamum was the cultural and scientific center of Asia Minor. Later all these states were conquered by Romans and became provinces of the Roman Empire. After the division of this empire into Western and Eastern parts, Asia Minor entered into Byzantium. In 14-15th centuries Asia Minor was conquered by Turks and entered into Turkey. The Greek state where the city Perga was located had the name Pam- phylia. This name, as well as its Hittite prototype, meant “belonging to all tribes”. This name shows that Pamphylia played an exclusive role among Hittite states. It is explained by the fact that main shrines common for all Hittite tribes were situated there. B.Hrozny proved that Greeks borrowed from Hittites the cults of the god of thunder, Zavaya, the god of Sun, Apulunash, and his sister- twin goddess of Moon, Artimu, whom they called Zeus, Apollo and Artemis [Hro, p.147]. The Hittite name “Perga” is near to Greek “πυργος” and German “Burg” and means “tower, castle”; in the original sense of the word “perga”, “rock”, is near to German “Berg” - “mountain”. This word was connected with the words “perunash” and “perginash” meaning “god of thunder, destroyer of rocks”. The word “perga” enters in the name of the city Pergamum. Hittite Perga was the center of the cults of Zavaya, Apulunash, and Ar- timu. When Perga became a Greek town, the main shrines of Zeus and Apollo were moved to Olympia and Delphi, and the main shrine of Artemis was left in 1 Perga. The other shrine of Artemis, one of the “Seven Wonders of the World”, was also situated in Asia Minor at Ephesus. Herodotus in his History wrote that kings of some Hittite states sent rich gifts to the Apollo’s shrine in Delphi, where the shrine was situated in his time. No doubt that they in fact sent their gifts into Perga. It is very probable that Apollonius’ kin comes from priests of Apulunash. B. Apollonius at Ephesus In the preface to Book 2 of Conics, Apollonius writes to Eudemus of Per- gamum that he sends him his son Apollonius bringing the second book of Con- ics. He asks Eudemus to acquaint with this book Philonides, the geometer, whom Apollonius introduced to Eudemus in Ephesus, if ever he happens to be about Pergamum. German historian Cronert [Cro] reports that Philonides was a student of Eudemus, mathematician and philosopher - Epicurean, who later worked at the court of Seleucid kings Antioch IV Epiphanus (183-175 B.C.) and Demetrius I Soter (162-150 B.C.). Eudemus was the first teacher of Philonides. No doubt that Eudemus was also the teacher of Apollonius at Ephesus, and it is natural that Apollonius sent him his main work. When Apollonius finished his study at Ephesus, Eudemus recommended that he continue his study at Alexandria. C. Apollonius at Alexandria Apollonius’ teachers at Alexandria were pupils of Euclid. In the preface to Book 1 of Conics, Apollonius writes that he composed this work at Alexandria. Apollonius’ nickname in this scientific capital of the Hellenistic world was “Epsilon”. Since the nickname of Eratosthenes was “Beta”, it is clear that the most great Alexandria mathematicians had as nicknames the first letters of the Greek alphabet: Euclid - “Alpha”, Archimedes - “Gamma”, and Conon of Samos - “Delta” Apollonius’ first works were on astronomy. Claudius Ptolemy quotes in Chapter 1 of Book 12 of Almagest Apollonius’ non-extant work on equivalence of epicyclic and eccentric hypotheses of motion of planets. This quotation shows that Apollonius was one of the initiators of the theory of motion of plan- ets by means of deferents and epicycles presented in Almagest. 2 Further works of Apollonius were devoted to mathematics. Since his main work Conics and many treatises were on geometry, Apollonius was called at Al- exandria “Great Geometer”. D. Conic sections before Apollonius The appearance of conic sections was also connected with the cult of Apollo. There sections were used for solving the so-called Delic problem of du- plication of cube. This problem was connected with following legend: on the island Delos, believed to be the place of birth of Apollo and Artemis, a plague epidemic broke out. The inhabitants of the island appealed to the shrine of Apollo at Delphi for aid. The priests of the shrine told them that they must duplicate the cubic altar of the shrine. The Delians made the second cube equal to the first one and stood over it, but the plague did not cease. Then the priests told that the dou- ble altar must be cubic like the old one. If the edge of the old altar was equal to a, the edge of the new altar must be equal to the root of the equation x3 = 2a3 . (0.1) It is possibly that the legend on the duplication of Apollo’s cubic altar ap- peared earlier when the main shrine of Apollo was at Perga. The problem of duplication of a cube was solved by some Greek mathe- maticians of the 4th c. B.C. Menaechmus found that this problem can be re- duced to the finding two mean proportionals between a and b, that is a : x = x : y = y : b (0.2) for b = 2a. Menaechmus found that the solution x of equation (0.1) is equal to the abscissa of the point of intersection of two parabolas x2 = ay and y2 = 2ax or of one of these parabolas with the hyperbola xy = 2a2. Menaechmus defined a parabola as the section of the surface of a right circular cone with right angle at its vertex by a plane orthogonal to a rectilinear generator of the cone, and a hyperbola as the analogous section of the surface of a right circular cone with obtuse angle at its vertex. The equations of these conic sections are determined by equalities (0.2). The works of Menaechmus are lost. The first known titles of works on conic sections are On Solid Loci (Περι στερεοι τοποι) by Aristaeus and Elements 3 of Conics (Κωνικων στοιξεια) by Euclid. Both of these works are also non-extant, but it is known that Aristaeus’ work consisted of 5 books and Euclid’s work con- sisted of 4 books. Ancient mathematicians used the word “locus” for lines and surfaces. Modern mathematicians regard lines and surfaces as sets of points, but this viewpoint was impossible for ancient scientists because they could not conceive that a set of points having no sizes has a non-zero length or a non-zero area. Aristotle wrote in his Physics: “Nothing that is continuous can be composed of indivisible parts: e.g., a line cannot be composed of points, the line being con- tinuous and the point indivisible [Ar, p. 231a]. Therefore ancient mathemati- cians regarded lines and surfaces only as “loci” (τοποι), that is places for points. Greek mathematicians called straight lines and circumferences of circles that can be drawn by a ruler and compass “plane loci” and conic sections they called “solid loci”. Conic sections are considered in many works of Archimedes who called a parabola a “section of right-angled cone”, single branch of a hyperbola - a “sec- tion of obtuse-angled cone”, and an ellipse - a “section of acute-angled cone”. Archimedes called a paraboloid of revolution a “right-angled conoid” and a single sheet of a hyperboloid of revolution of two sheets an “obtuse-angled conoid”. No doubt that Menaechmus, Aristaeus, and Euclid used the same names of conic sections. The equations of parabolas used by Menaechmus for solving the Delic problem are particular cases of the equation y2 = 2px (0.3) in the system of rectangular coordinates whose axis 0x is the axis of symmetry of this parabola and whose axis 0y is the tangent to this parabola at its vertex. The magnitude p is now called the parameter of the parabola. Euclid in Prop. II.14 of Elements proves that if Β is an arbitrary point of the circumference of a circle with the diameter ΑΧ, and Δ is the basis of the perpendicular dropped from Β onto ΑΧ, the line ΒΔ is mean proportional be- tween ΑΔ and ΔΧ, that is ΑΔ:ΒΔ = ΒΔ:ΔΧ. If we denote ΑΔ = x, ΔΧ = x’, ΒΔ = y, we obtain the equation y2 = xx’ (0.4) 4 of the circumference with “two abscissas” in the system of rectangular coordinates whose axis 0x = ΑΧ and axes 0y and 0y’ are tangents to the cir- cumference at the points Α and Χ. Archimedes in Prop. I.4 of his treatise On Conoids and Spheroids proves that an ellipse can be obtained from a circumference of a circle by the contrac- tion to its diameter in the direction perpendicular to this diameter x’ = x , y’ = ky (0.5) where k < 1. Therefore the equation with two abscissas of an ellipse in the system of rectangular coordinates whose axis 0x is the major axis of the el- lipse and axes of ordinates are tangents to the ellipse at the ends of its major axis has the form y2 = k2xx’ . (0.6) The branch of a hyperbola used by Menaechmus in the system of rectan- gular coordinates whose axes are asymptotes of the hyperbola is determined by the equation xy = const. In another system of rectangular coordinates, whose axis 0x is the axis of symmetry of the hyperbola, and axes of ordinates are tan- gents to both branches of the hyperbola at their vertices, this hyperbola is de- termined by equation (0.4). An arbitrary hyperbola can be obtained from the equilateral hyperbola used by Menaechmus by transformation (0.5), which is a contraction to the axis of symmetry of this hyperbola for k <1 and a dilatation from this axis for k >1. Therefore the equation with two abscissas of an arbitrary hyperbola in the sys- tem of rectangular coordinates whose axis 0x is the axis of symmetry of the hyperbola and the axes of ordinates are tangents to both branches of the hy- perbola at their vertices has form (0.6). Archimedes determined ellipses and hyperbolas by equations (0.6). If the major axis of an ellipse and the real axis of a hyperbola are equal to 2a and the minor axis of an ellipse and the imaginary axis of a hyperbola are equal to 2b, the coefficient k in equations (0.6) is equal to b/a. in the case of the ellipse x’= 2a - x and in the case of the hyperbola x’ = 2a + x. Therefore these equations have the form y2 = (b2/a2)x(2a - x) (0.7) for the ellipse and 5 y2 = (b2/a2)x(2a + x) . (0.8) for the hyperbola. If we denote b2/a = p, equations (0.7) of an ellipse can be rewritten as y2 = 2px - (p/a)x2 , (0.9) equations (0.8) of a hyperbola can be rewritten as y2 = 2px + (p/a)x2 . (0.10) Equations (0.9) and (0.10) are given in the systems of the rectangular coordinates whose axis 0x is the major axis of the ellipse and the real axis of the hyperbola, and whose axis 0y is tangent to the ellipse at the left end of its major axis and tangent to the hyperbola at the right end of its real axis. Magnitudes p in these equations are called parameters of the ellipse and hyper- bola. E. Structure of Conics Apollonius’ Conics consisted of 8 books. Books 1-4 are extant in Greek original, Books 5-7 are extant only in medieval Arabic translations by Thabit ibn Qurra edited by his teachers Ahmad and al-Hasan banu Musa ibn Shakir, Book 8 is lost. The books of Conics consist of prefaces addressed to Eudemus or Attalus of Pergamum, definitions, and propositions. Apollonius’ propositions, like propositions of Euclid’s Elements, are theo- rems or problems. In the beginning of every proposition, its general statement in italic and its formulation with notations of points and lines are given. The formulations of propositions Apollonius begins with the words Λεγω - “I say”. After that, the proof of a theorem or the solution of a problem follows. In beginning of the solution of every problem its analysis is given, where known points and lines are indicated; next, the synthesis, that is the required construc- tion, is described. Apollonius’ style is very concise, therefore the translators insert in the text explanatory words in brackets and references to Euclid and Apollonius’ propositions in parentheses. 6 F. Editions of Conics The most important editions of Apollonius’ Conics are: [Ap1] - the first Latin translation of Books 1-4 published by Federigo Commandino (1509-1575). [Ap 2] - the Greek text of Books 1-4 and the Latin translation of all 7 books published by Edmund Halley (1656-1742). [Ap 3] - the critical Greek text of Books 1-4 established by Johan Ludvig Heiberg (1854-1928) and published by him with the Latin translation. [Ap 4] - the English translation of Books 1-3 by Robert Catesby Taliaferro (1907-1987) published by Encyclopedia Britannica in the Great Books of the Western World series. The translation of Book 1 was first published in 1939 by St. John’s College at Annapolis in The Classics of the St. John’s Program series. [Ap 5] - the revised edition of the translation [Ap4] published by Dana Densmore and William H. Donahue. [Ap 6] - the English translation of Book 4 by Michael N. Fried (b. 1960). This translation was first published as Appendix to the book [FU](pp.416 -485). [Ap 7] - the critical Arabic text of Books 5-7 established by Gerald James Toomer (b. 1934) and published by him with the English translation and com- mentary Critical Arabic text is based on 3 manuscripts: Oxford one, translated by Halley; Istanbul one, published in [Ap12]; and Teheran one. [Ap 8] - the detailed English exposition of all 7 books on the basis of the editions [Ap 2] and [Ap3] published by Thomas Little Heath (18611940). [Ap9] - commented French translation of all 7 books published by Paul Ver Eecke. [Ap10] - German translation of Books 1- 4 published by Arthur Czwalina. [Ap11] - the Greek text of Heiberg reproduced and published with the Modern Greek translation of all 7 books by Euangelos Stamatis (1898-1990). [Ap12] - facsimile edition of the Istanbul manuscript of the medieval Ara- bic translation of all 7 books by Hilal al-Himsi and Thabit ibn Qurra copied by the famous mathematician and physicist al-Hasan Ibn al-Haytham (965-ca.1050) prepared by Nazim Terzioglu (1912- 1976). [Ap13] - commented Russian translation of 20 propositions by I. Yagodinsky (1928). [Ap14] - commented Russian translation of all 7 books published by B. A. Rosenfeld - in press. 7 Many mathematicians undertook attempts of restoration of Book 8. Let us mention the attempt by Ibn al-Haytham [IH] published with the English trans- lation by Jan Pieter Hogendijk (b.1955) and the attempt by Halley added to his translation [Ap2]. Let us mention the excellent exposition of Apollonius’ Conics: [Ze] - The Theory of Conic Sections in Antiquity by Hieronymus Georg Zeuthen (1839-1920). [Hea, pp.126-196] - in the book A History of Greek Mathematics by T.L. Heath. [VdW, pp.241-261] - in the book The Science Awakening by Bartel Leendert Van der Waerden (1903-1996). [VZ, pp.97-108] - in the book History of Mathematics by Michail E. Vash- chenko-Zakharchenko (1825-1912). [IM, pp.129-139] - in the book History of Mathematics from most ancient times to beginning of 19th century, vol.1 by Adolf P. Yushkevich (1906-1993). [Too] - the article Apollonius of Perga by G. J. Toomer. See also Introduction to his edition [Ap7], [FU] Apollonius of Perga’s Conica. Text, Context, Subtext by M.N.Fried and Sabetai Unguru. [Rho] - Apollonius of Perga, Doctoral Thesis by Diana L. Rodes (2005) [Ro3] - Apollonius of Perga (in Russian by B.A.Rosenfeld 2003). See also his article [Ro4]. G. Other mathematical works of Apollonius Besides Conics Apollonius was the author of following mathematical works: 1) Cutting off of a ratio (Λογου αποτοµα) in two books. 2) Cutting off of an area (Χωριου αποτοµα) in two books. 3) Determinate section (Διωρισµενα τοµα) ιν two books. 4) Inclinations (Νευσεις) in two books. 5) Tangencies (Επαφαι) in two books 6) Plane loci (Τοποι επιπεδοι) in two books. 7) Comparison of dodecahedron and isocahedron (Συγκρισις δωδεκαεδρου και εικοσαεδρου). 8) On non-ordered irrationals (Περι των ατακτων αλογων). 9) Rapid obtaining of a result (Ωκυτοκιον). 10) Screw lines (Κοξλιας). 11) Treatise on great numbers. 8 12) General treatise (Καθολου πραγµατεια). From these works only treatise (1) is extant in medieval Arabic transla- tion. There are the Latin translation [Ap15] by E. Halley and the English transla- tion [Ap16] by E.M.Macierowski of this treatise. The short expositions of treatises (1) - (6) are given by Pappus of Alex- andria (3rd c. A.D.) in Book 7 of Mathematical Collection [Pa, pp. 510 -546; Ap11, vol.1, pp.100 - 120]. The fragments of medieval Arabic translations of these treatises and Eng- lish translations of these fragments are published by J.P.Hogendijk [Ho]. In works (1) and (2) the following problems are solved: given two straight lines ΑΒ and ΧΔ with fixed points Α and Χ, to find two points Β and Δ, such that, in the case of treatise (1), the ratio ΑΒ/ΧΔ would be equal to the given ratio, and, in the case of treatise (2), the product ΑΒ.ΧΔ would be equal to the given area. In treatise (3) the problems of the following type are solved: given four points Α, Β, Χ, Δ on a straight line, to find a point Π such that ratio ΑΠ.ΧΠ/ΒΠ.ΔΠ would have the given or an extremal value. The last problem is equivalent to the problem of determining an extremum of a function that is a ratio of two quadratic polynomials. In work (4) the problems equivalent to quadratic and cubic equations are solved by geometrical means called “inclinations”. In treatise (5) the problem of construction of a circle tangent to given objects of three kinds, which can be circles, straight lines, and points, is solved. In treatise (6) theorems on plane loci, which is on circles and straight lines, are proven. In this treatise, homotheties, inversions with respect to cir- cles, and other transformations mapping plane loci to plane loci are considered. There is only the commentary on work (7) by Hypsicles (2nd -1st c. B.C.) added to Euclid’s Elements as Book 14 [Ap11, vol.1, pp.60-66]. In this work, Aristaeus’ treatise Comparison of five solids is mentioned, where the theorem, that if a cube and a regular octahedron are inscribed in the same sphere, then as their volumes are one to the other, so their surfaces are one to the other, is proven. Apollonius proves analogous theorem on regular dodecahedron and icosahedron inscribed in the same sphere. The commentary by Pappus on the work (8) is extant only in the medieval Arabic translation [Ap11, vol.1, pp. 134-144]. This commentary shows that in this treatise, besides quadratic irrationals considered in Book 10 of Euclid’s Ele- ments, cubic and higher irrationals are also considered. Work (9) is mentioned by Eutocius (6th c. A.D.) on Archimedes Measuring a circle [Ap11, vol.1, p. 48]. This information shows that in the treatise, the 9