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APOLLONIUS OF PERGA CONICS. BOOKS ONE - SEVEN English translation by Boris Rosenfeld The Pennsylvania State University Apollonius of Perga (ca 250 B.C. - ca 170 B.C.) was one of the greatest mathematicians of antiquity. During 1990 - 2002 first English translations of Apollonius’ main work Conics were published. These translations [Ap5](Books 1-3), [Ap6](Book 4), [Ap7] (Books5-7) are very different. The best of these editions is [Ap6]. The editions [Ap4] and [Ap5] are very careless and often are far from the Greek original. The editors of [Ap5] have corrected many defects of [Ap4], but not all; they did not compare this text with the Greek original. Some defects remain also in the edition [Ap6]. The translation [Ap7], being the first rate work, is not a translation of Greek text because this text is lost, and is the translation of Arabic exposition by Thabit ibn Qurra (826 - 901). Therefore we present the new English translation of this classic work writ- ten in one style more near to Greek text by Apollonius, in our translation some expressions of the translations [Ap5], [Ap6], and [Ap7] are used. The authors of the translations [Ap5], [Ap6], and [Ap7] are linguists and in their translations many discoveries of Apollonius in affine, projective, confor- mal, and differential geometries in Apollonius’ Conics being special cases of gen- eral theorems proved in Western Europe only in 17th -19th centuries were not mentioned. The commentary to our translation from the standpoint of modern mathematics uses books [Ro1] and [Ro2] by the translator. I am very grateful to my master student, now Ph.D. and the author of the thesis[Rho1] and [Rho2] Diana L. Raodes, possessing ancient Greek. This work could not be completed without the help of translator’s daughter, Professor of the Pennsylvania State University, Svetlana R. Katok, and also Ph.D. Daniel Genin and Nicholas Ahlbin. Diagrams to Books I-IV should be taken from editions [AP3] Heiberg or [AP12] of Stamatis, diagrams to Books V-VII should be taken from the edition [AP7] of Toomer. 1 BOOK ONE Preface Apollonius greets Eudemus1 If you are restored in body, and other things go with you to your mind, well; and we too fare pretty well. At the time I was with you in Pergamum, I ob- served you were quite eager to be kept informed of the work I was doing in con- ics. And so I am sending you this first book revised. I will send you other books when I will be satisfied with them. For I don’t believe you have forgotten hear- ing from me how I worked out the plan for these conics at the request of Nauc- rates2, the geometer, at the time he was with me in Alexandria lecturing, and how on arranging them in eight books I immediately communicated them in great haste because of his near departure, not revising them but putting down whatever came to me with the intention of a final going over. And so finding now the occasion of correcting them, one book after another, I will publish them. And since it happened that some others among those frequenting me got acquainted with the first and second books before the revision, don’t be sur- prised if you come upon them in a different for. Of the eight books the first four belong to a course in the elements 3. The first book contains the generation of the three sections and of the opposite [sections]4, and the principal properties in them worked out more fully and universally than in the writings of others. The second book contains the properties having to do with the diameters and axes and also the asymptotes, and other things of a general and necessary use for limits of possibility. And what I call diameters and what I call axes you will know from this book. The third book contains many unexpected theorems of use for the con- struction of solid loci and for limits of possibility of which the greatest part and the most beautiful are new. And when I had grasped these, I knew that the 2 three-line and four-line locus5 had not been constructed by Euclid, but only a chance part of it and that not very happily. For it was not possible for this con- struction to be completed without the additional things found by me. The fourth book shows in how many ways the sections of cone intersect with each other and with the circumference of a circle, and contains other things in addition none of which has been written up by my predecessors, that is in how many points the section of a cone or the circumference of a circle and the opposite sections meet the opposite sections. The last four books are fuller in treatment. For there is one [the fifth book] dealing more fully with maxima and minima, and one [the sixth book] with equal and similar sections of a cone, and one [the seventh book] with limiting theorems, and one [the eighth book] with determinate problems. And so indeed, with all of them published, those happening upon them can judge them as they see fit. Let the happiness will be to you. First definitions 1. If a point is joined by a straight line with a point in the circumference of a circle which is not in the same plane with the point, and the line is continued in both directions, and if, with the point remaining fixed, the straight line being ro- tated about the circumference of the circle returns to the same place from which it began, then the generated surface composed of the two surfaces lying vertically opposite one another, each of which increases indefinitely as the gen- erating straight line is continued indefinitely, I call a conic surface 6, and I call the fixed point the vertex, and the straight line drawn from the vertex to the center of the circle I call the axis. 2. And the figure contained by the circle and by the conic surface be- tween the vertex and the circumference of the circle I call a cone7, and the point which is also the vertex of the surface I call the vertex of the cone, and the straight line drawn from the vertex to the center of the circle I call the axis, and the circle I call the base of the cone. 3. I call right cones those having axes perpendicular to their bases, and I call oblique those not having axes perpendicular to their bases. 4. For any curved line that is in one plane, I call straight line drawn from the curved line that bisects all straight lines drawn to this curved line parallel to some straight line the diameter 8,9. And I call the end of the diameter situated on the curved line the vertex of the curved line, and I call these parallels the ordinates drawn to the diameter 10 . 3 5. Likewise, for any two curved lines lying in one plane, I call the straight line which cuts the two curved lines and bisects all straight lines drawn to either of the curved lines parallel to some straight line the transverse diameter. And I call the ends of the [transverse] diameter situated on the curved lines the ver- tices of the curved lines. And I call the straight line lying between the two curved lines, bisecting all straight lines intercepted between the curved lines and drawn parallel to some straight lines the upright diameter 11. And I call the parallels the ordinates drawn to the [transverse or upright] diameter. 6. The two straight lines, each of which, being a diameter, bisecting the straight lines parallel to the other, I call the conjugate diameters12 of a curved line and of two curved lines. 7. And I call that straight line which is a diameter of the curved line or lines cutting the parallel straight lines at right angles the axis of curved line and of two curved lines 13,14. 8. And I call those straight lines which are conjugate diameters cutting the straight lines parallel to each other at right angles the conjugate axes of a curved line and of two curved lines. [Proposition] 1 The straight lines drawn from the vertex of the conic surface to points on the surface are on that surface 15. Let there be a conic surface whose vertex is the point Α, and let there be taken some point Β on the conic surface, and let a straight line ΑΓΒ be joined. I say that the straight line ΑΓΒ is on the conic surface. [Proof]. For if possible, let it not be [and the straight line ΑΒ is not on the conic surface], and let the straight line ΔΕ be the line generating the surface, and ΕΖ be the circle along which ΕΔ is moved. Then if, the point Α remaining fixed, the straight line ΔΕ is moved along the circumference of the circle ΕΖ. This straight line [according Definition 1] will also go through the point Β, and two straight lines will have the same ends. And this is impossible. Therefore, the straight line joined from Α to Β cannot not be on the surface. Therefore, it is on the surface. Porism It is also evident that, if a straight line is joined from the vertex to some point among those within the surface, it will fall within the conic surface. And if it is joined to some point among those without, it will be outside the surface. 4 [Proposition] 2 If on either one of the two vertically opposite surfaces two points are taken, and the straight line joining the points, when continued, does not pass through the vertex, then it will fall within the surface, and continued it will fall outside 16. Let there be a conic surface whose vertex is the point Α, and a circle ΒΓ along whose circumference the generating straight line is moved, and let two points Δ and Ε be taken on either one of the two vertically opposite surfaces, and let the joining straight line ΔΕ, when continued not pass through the point Α. I say that ΔΕ will be within the surface, and continued will be without. [Proof]. Let ΑΕ and ΑΔ be joined and continued. Then [according to PropositionI.1] they will fall on the circumference of the circle. Let them fall to Β and Γ, and let ΒΓ be joined. Therefore the ΒΓ will be within the circle, and so too within the conic surface. Then let Ζ be taken at random on ΔΕ, and let ΑΖ be joined and continued. Then it will fall on ΒΓ, for the triangle ΒΓΑ is in one plane [according to Proposition XI.2 of Euclid]. Let it fall to Η. Since then H is within the conic surface, therefore [according to the porism to Proposition I.1] the straight line ΑΗ is also within the conic surface, and so too the point Ζ is within the conic surface. Then likewise it will be shown that all the points on the straight line ΔΕ are within the surface. Therefore the straight line ΔΕ is within the conic surface. Then let ΔΕ be continued to Θ. I say that it will fall outside the conic sur- face. For it possible, let there be some point Θ of it not outside the conic sur- face, and let ΑΘ be joined and continued. Then it will fall either on the circum- ference of the circle or within [according to Proposition I.1 and its porism]. And this is impossible, for it falls on ΒΓ continued; as for example to the point Κ. Therefore the straight line ΕΘ is outside the surface. Therefore the straight line ΔΕ is within the conic surface, and continued is outside. [Proposition] 3 If a cone is cut by a plane through the vertex, the section is a triangle 17. Let there be a cone whose vertex is the point Α and whose base is the circle ΒΓ, and let it be cut by some plane through the point Α, and let it make, as section, lines ΑΒ and ΑΓ on the surface, and the straight line ΒΓ in the base. 5 I say that ΑΒΓ is a triangle. [Proof]. For since the line joined from Α to Β is the common section of the cutting plane and of the surface of the cone, therefore ΑΒ is a straight line. And likewise also ΑΓ. And ΒΓ is also a straight line. Therefore ΑΒΓ is a triangle. If then a cone is cut by some plane through the vertex, the section is a triangle. [Proposition] 4 If either one of the vertically opposite surfaces is cut by some plane paral- lel to the circle along which the straight line generating the surface is moved, the plane cut off within the surface will be a circle having its center on the axis, and the figure contained by the circle and the conic surface intercepted by the cutting plane on the side of the vertex will be a cone 18. Let there be a conic surface whose vertex is the point Α and whose circle along which the straight line generating the surface is moved is ΒΓ, and let it be cut by some plane parallel to the circle ΒΓ, and let it make on the surface as a section the line ΔΕ. I say that the line ΔΕ is a circle having the center on the axis. [Proof]. For let Ζ be taken as the center of the circle ΒΓ, and let ΑΖ be joined. Therefore [according to Definition 1] ΑΖ is the axis and meets the cut- ting plane. Let it meet it at Η, and let some plane be drawn through ΑΖ. Then [according to Proposition I.3] the section will be the triangle ΑΒΓ. And since the points Δ, Η, Ε are points in the cutting plane, and are also in the plane of the tri- angle ΑΒΓ, [according to Proposition XI.3 of Euclid] ΔΗΕ is a straight line. Then let some point Θ be taken on the line ΔΕ, let ΑΘ be joined and con- tinued. Then [according to Proposition I.1] it falls on the circumference ΒΓ. Let it meet it at Κ, and let ΗΘ and ΖΚ be joined. And since two parallel planes, ΔΕ and ΒΓ, are cut by a plane ΑΒΓ, [according to Proposition XI.16 of Euclid] their common sections are parallel. Therefore ΔΕ is parallel to ΒΓ. Then for the same reason ΗΘ is also parallel to ΚΖ. Therefore [according to Proposition VI.4 of Euclid] as ΖΑ is to ΑΗ, so ΖΒ is to ΔΗ, and ΖΓ is to ΗΕ, and ΖΚ is to ΗΘ. Since ΒΖ is equal to ΚΖ and to ΖΓ [according to Proposition V.9 of Euclid] ΔΗ is equal to ΗΘ and to ΗΕ. Then likewise we could show also that all the straight lines falling from the point Η on the line ΔΕ are equal to each other. Therefore the line ΔΕ is a circle having its center on the axis. And it is evident that the figure contained by the circle ΔΕ and the conic surface cut off by it on the side of the point Α is a cone. 6 And it is there with proved that the common section of the cutting plane and of the axial triangle [that is triangle through the axis] is a diameter of the circle. [Proposition] 5 If an oblique cone is cut by a plane through the axis at right angles to the base, and is also cut by another plane on the one hand at right angles to the ax- ial triangle, and on the other hand cutting off on the side of the vertex a trian- gle similar to the axial triangle and situated antiparallel, then the section is a cir- cle, and let such a section be called antiparallel 19. Let there be an oblique cone whose vertex is the point Α and whose base is the circle ΒΓ, and let it be cut through the axis by a plane perpendicular to the circle ΒΓ, and let it make as a section the triangle ΑΒΓ. Then let it also be cut by another plane perpendicular to the triangle ΑΒΓ and cutting off on the side of Α the triangle ΑΚΗ similar to the triangle ΑΒΓ and situated antiparallel, that is so that the angle ΑΚΗ is equal to the angle ΑΒΓ. And let it make as a section on the surface [of the cone] the line ΗΘΚ. I say that the line ΗΘΚ is a circle. [Proof]. For let any points Θ and Λ be taken on the lines ΗΘΚ and ΒΓ, and from Θ and ΛΛ let perpendiculars be dropped to the plane of the triangle ΑΒΓ. Then [according to Definition XI.4 of Euclid] they will fall to the common sections of the planes. Let them fall for example as ΖΘ and ΛΜ. Therefore [according to Proposition XI.6 of Euclid] ΖΘ is parallel to ΛΜ. Then ΔΖΕ be drawn through Ζ parallel to ΒΓ, and ΖΘ is parallel to ΛΜ. Therefore [according to Proposition XI.15 of Euclid] the plane through ΖΘ and ΔΕ is parallel to the base of the cone. Therefore [according to Proposition I.4] it is a circle whose diameter is ΔΕ. Therefore [according to Proposition II.14 of Euclid] 20 pl. ΔΖΕ is equal to sq. ΖΘ. And since ΕΔ is parallel to ΒΓ, the angle ΑΔΕ is equal to the angle ΑΒΓ. And the angle ΑΚΗ is supposed equal to the angle ΑΒΓ. Therefore the angle ΑΚΗ is equal to the angle ΑΔΕ. And the vertical angles at Ζ are also equal. Therefore the triangle ΔΖΗ is similar to the triangle ΚΖΕ, and therefore [accord- ing to Proposition VI.4 of Euclid] as ΕΖ is to ΖΚ, so ΗΖ is to ΖΔ. Therefore [according to Proposition VI.16 of Euclid] pl. ΕΖΔ is equal to pl.ΚΖΗ. But it has been shown that sq.ΖΘ is equal to pl.ΕΖΔ. Therefore pl.ΚΖΗ is equal to sq.ΖΘ. 7 Likewise then all the perpendiculars drawn from the line ΗΘΚ to ΗΚ could also be shown to be equal in square to the rectangular plane, in each case under the segments of ΗΚ. Therefore the section is a circle21 whose diameter is ΗΚ. [Proposition] 6 If a cone is cut by a plane through the axis, and if on the surface of the cone some point is taken which is not on a side of the axial triangle, and if from this point is drawn a straight line parallel to some straight line which is a per- pendicular from the circumference of the circle to the base of the triangle, then that drawn straight line meets the axial triangle, and on being continued to the other side of the surface the drawn straight line will be bisected by the trian- gle22. Let there be a cone whose vertex is the point Α and whose base is the circle ΒΓ, and let the cone be cut by a plane through the axis, and let it make as a common section the triangle ΑΒΓ, and from some point Μ on the circumfer- ence let ΜΝ be drawn perpendicular to [the straight line] ΕΒΓ. Then let some point Δ be taken on the surface of the cone, and through Δ let ΔΕ be drawn par- allel to ΜΝ. I say that the continued ΔΕ will meet the plane of the triangle ΑΒΓ, and if further continued toward the other side of the cone until it meet its surface, will be bisected by the triangle ΑΒΓ. [Proof]. Let ΑΔ be joined and continued. Therefore it will meet the cir- cumference of the circle ΒΓ. Let it meet it at Κ and from Κ let ΚΘΛ be drawn perpendicular to ΒΓ.Therefore ΚΘ is parallel to ΜΝ, and therefore [according to Proposition XI.9 of Euclid] also to ΔΕ. Let ΑΘ be joined. Since then in the triangle ΑΘΚ [the straight line] ΔΕ is parallel to ΘΚ, therefore ΔΕ continued will meet ΑΘ. But ΑΘ is in the plane of the triangle ΑΒΓ; therefore ΔΕ will meet this plane. For the same reasons it also meets ΑΘ, let it meet it at Ζ, and let ΔΖ be continued in a straight line until it meet the surface of the cone. Let it meet it at Η. I say that ΔΖ is equal to ΖΗ. For since Α, Η, Λ are points on the surface of the cone, but also in the plane drawn through ΑΘ, ΑΚ, ΔΗ, ΚΛ, which is a triangle through the vertex of the cone, therefore Α, Η, Λ are points of the common section of the cone’s sur- face and of the triangle. Therefore the line through Α, Η, and Λ is a straight line. Since then in the triangle ΑΛΚ [the straight line] ΔΗ has been drawn parallel in the base ΚΘΛ, and some straight line ΑΖΘ has been drawn across them from Α, 8 therefore [according to Proposition VI.4 of Euclid] as ΚΘ is to ΘΛ, so ΔΖ is to ΖΗ. But ΚΘ [according to Proposition III.3 of Euclid] is equal to ΘΛ since ΚΛ is a chord in the circle ΒΓ perpendicular to the diameter. Therefore ΔΖ is equal to ΖΗ. [Proposition] 7 If a cone is cut by a plane through the axis, and if the cone is also cut by another plane, so that the plane of the base of the cone is cut in a straight line perpendicular either to the base of the axial triangle or to it continued, and if from the cutting plane’s resulting section on the cone’s surface, straight lines are drawn parallel to the straight line perpendicular to the base of the triangle, then these straight lines will fall on the common section of the cutting plane and of the axial triangle, and further continued to the other side of the section, these straight lines will be bisected by the common section, and if the cone is right, then the straight line in the base will be perpendicular to the common section of the cutting plane and of the axial triangle, but if the cone is oblique, then the straight line in the base will be perpendicular to that common section only whenever the plane through the axis is perpendicular to the base of the cone23,24. Let there be a cone whose vertex is the point Α and whose base is the circle ΒΓ, and let it be cut by a plane through the axis, and let it make as a common section the triangle ΑΒΓ. And let it also be cut by another plane cut- ting the plane of the circle ΒΓ in ΔΕ perpendicular either to ΒΓ or to it contin- ued, and let it make as a section on the surface of the cone the line ΔΖΕ. Then ΖΗ is the common section of the cutting plane and of the triangle ΑΒΓ. And let some point Θ be taken on the section ΔΖΕ, and let ΘK be drawn through Θ parallel to ΔΕ. I say that ΘΚ meets ΖΗ, and if continued to the other side of the section ΔΖΕ will be bisected by ΖΗ. [Proof]. For since a cone whose vertex is the point Α and whose base is the circle ΒΓ has been cut by a plane through its axis, and makes as a section the triangle ΑΒΓ, and since some point Θ on the surface, not on a side of the triangle ΑΒΓ, has been taken, and since ΔΗ is perpendicular to [the straight line] ΒΓ, therefore the straight line drawn through Θ parallel to ΔΗ, that is ΘΚ, meets the triangle ΑΒΓ, and [according to Proposition I.6] if further continued to the other side of the surface, will be bisected by the triangle. Then since the straight line drawn through Θ parallel to ΔΕ meets the tri- angle ΑΒΓ and is in the planes of the section ΔΖΕ, therefore it will fall on the 9 common section of the cutting plane and of the triangle ΑΒΓ. But ΖΗ is the common section of the planes. Therefore the straight line drawn through Θ par- allel to ΔΕ will fall on ΖΗ, and, if further continued to the other side of the sec- tion ΔΖΕ, will be bisected by ΖΗ. Then either the cone is right, or the axial triangle ΑΒΓ is perpendicular to the circle ΒΓ, or neither. First let the cone be right. Then [according to Definition 3 and according to Proposition XI.18 of Euclid] the triangle ΑΒΓ would be perpendicular to the circle ΒΓ. Since then the plane ΑΒΓ is perpendicular to the plane [of the circle] ΒΓ, and ΔΕ has been drawn in one of these two planes, [the plane of the circle] ΒΓ, perpendicular to their common section, [the straight line] ΒΓ, therefore [ac- cording to Definition XI.4 of Euclid] ΔΕ is perpendicular to the triangle ΑΒΓ, and therefore to all straight lines touching it and situated in the triangle ΑΒΓ.And so ΔΕ is also perpendicular to ΖΗ. Then let the cone not be right. If now the axial triangle is perpendicular to the circle ΒΓ, we could likewise show that ΔΕ is perpendicular to ΖΗ. Then let the axial triangle ΑΒΓ not be perpendicular to the circle ΒΓ. I say that ΔΕ is not perpendicular to ΖΗ. For, if possible, let it be so. And it is also perpendicular to [the straight line] ΒΓ.Therefore ΔΕ is perpendicular to both ΒΓ and ΖΗ, and therefore it will be perpendicular to the plane through ΒΓ and ΖΗ. But the plane of through ΒΓ and HZ is the [plane of the] triangle ΑΒΓ, and therefore ΔΕ is perpendicular to the triangle ΑΒΓ. And therefore all planes through it are perpendicular to the triangle ΑΒΓ. But one of the planes through ΔΕ is the [plane of the] circle ΒΓ. Therefore the circle ΒΓ is perpendicular to the triangle ΑΒΓ. And so the triangle ΑΒΓ will also be perpendicular to the circle ΒΓ. And this is not supposed. Therefore ΔΕ is not perpendicular to ΖΗ. Porism Then from this it is evident that ΖΗ is the diameter of the section ΔΖΕ, since it bisects the straight lines drawn parallel to some straight line ΔΕ, and it is evident that it is possible for some parallels to be bisected by the diameter ΖΗ and not be perpendicular to ΖΗ. [Proposition] 8 If a cone is cut by a plane through its axis, and if the cone is cut by an- other plane cutting the base of the cone in a straight line perpendicular to the 10

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