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An Introduction to the Ancient and Modern Geometry of Conics, Being A CORNELL UNIVERSITY LIBRARIES Mathematict Library White Half CORNELL UNIVERSITY LIBRARY 924 059 55 469 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059551469 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volvime was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1991. THE ANCIENT AND MODERN GEOMETRY OP CONICS AN INTRODUCTION TO THE ANCIENT AND MODERN GEOMETRY OF CONICS BEING A GEOMETRICAL TREATISE ON THE CONIC SECTIONS "WITH A GOLLEOTION OF PROBLEMS AND HISTORICAL NOTES AND PROLEGOMENA By CHARLES TAYLOR M.A. FELLOW OF ST JOHN'S OOLLEOB OAMBRIDGB iv rh y£<i>[UTpt<f naaiy iariv oSoc /Ji'a CAMBRIDGE DEIGHTON BELL AND GO LONDON GEOKGE BELL AND SONS 1881 Eh CAMBRIDGE: PBIKTED BT W. METCALFE AND BOK, TRIKITT BTREET. PKEFACE. Geometrj, at once ancient and modern, is the science of Euclid, Archimedes and ApoUonius, of Kepler, Desargues, Newton and Poncelet. Geome- trical processes have indeed been simplified and their applications greatly extended in recent times, but the modern methods may be traced to the ancient as their germ and source, and 1;hus it remains in a sense still true that there is but one road for all, ev t§ yecafierpia iraaiv iariv oSoi fiia. The modern infinitesimal calculus is an adaptation of the ancient method of exhaustions, the method of Descartes diflfers only in the manner of its appli- cation from that of ApoUonius, the idea of perspective was already formulated by Serenus, and the principle of anharmonic section with the leading properties of transversals are found in the lenimas of Pappus to the lost three books of Porisms of Euclid. Tliere is not however in the works of the Greek geometers VI PfiEPACE. any distinct foreshadowing of Kepler's doctrine of the infinite, of his principles of analogy and con- tinuity, or of the theory of ideal chords and points, at length completed by Poncelet's discovery of the so-called circular points at infinity in any plane. In the present as in a former work (1863) I have commenced with an elementary treatment of the general conic in piano, following out a suggestion made by Professor Adams in a course of lectures on the Lunar Theory delivered in 1861. This department of the subject has now been made more complete with the help of the Eccentric Circle, the characteristic feature of a masterly though neglected work of Boscovich. In the chapter on the Cone the focal Spheres are more fully discussed, and the angle-properties of the sections as well as their metric properties are deduced. The chapter on Orthogonal Projection contains proofs of Lambert's theorem* in elliptic motion. To the chapter on Conical Projection is appended some account of the homographic method of Rever- sion, which springs out of the above mentioned con- struction of Boscovich. * It has been remarked that, so far as relates to the parabola, Lambert's theorem is implicitly contained in Newton's Principia lib. HI. lemma 10. See Lagrange Mecanique Analytique tome II, p. 28, ed. 3 (1853—5) ; Brougham and Routh An analytical view of Sir Isaac Newton'i Principia p. 430 (Lond. 1855). FBEFACE. TU Abundant references will be found to the works of authors to whom I am indebted. Suffice it here to add that my warmest thanks are due to the Reverend Professor Richard Townsend, F.R.S., Fellow of Trinity College, Dublin, who has been at all times ready, in the midst of pressing engagements, to aid me with his criticism and advice, and has from first to last shewn as great an interest in the work now brought to a close as if it had been bis own. C. TAYLOR. St. John'i College, Secembtr 81, 1880. EBBATA. Page 42, Ex. 73 for ig nod varies OS. Page 82, line 26 „ 1657 ., 1710. Page 136, note* „ Le Sueuf » Lc Seuf. Page 194, Schoiinm „ Erastosthenes „ Eratosthenes. Page 206, Ex. 545 „ tWO Of more „ tWO. Page 257, line 29 ,, Dynamics „ Dynamic. CONTENTS. PEOLEGOMENA. FACB Section I. Geometry before Eaclid .... zvii Section II. Fiom Euclid to Serenus ..... xxxir Section m. Eepler. Desargues. Newton . , Ivi Section IV. Modem Qeometiy . btxiii DEFINITIONS. A conic defined by focns, directrix and eccentricity . 1 The eccentric circle ...... 3 CHAPTER I. DSBORIPTION OF THE CUIITK. The three conies traced ...... 5 The circle a limit of the ellipse ... .7 A conic is a curve of the second order ..... 8 To describe a conic by the eccentric circle .... 9 The intersections of any line with a conic determined . .10 A conic is a curre of the second class .... 11 Every conic is concave to its axis . .12 Examples 1—10. ...... „ CHAPTER II. THE SENEBAL CONIC. Tangent subtends right angle at focus ... .14 The directrix the polar of the focus .... 16 Geometrical equation to a tangent . , . „ The tangent as defined by Euclid ..... 16 for Construction tangents from given point . „ Two tangents from any point subtend equal angles at focus . 17 CONTENTS. FAOB Harmonic division of any chord by focal distance of its pole and diiectiiz 18 Oeometrical equation to any chord .... 19 Length of intercept by normal on the axis Projection of normal on focal distance constant 20 Properties of angles subtended at focns . ft Angle-properties of the circle deduced 22 Parallel chords bisected by a right line . 23 Ordinates to any diameter parallel to tangents at its extremities 24 The diameter of any chord passes through its pole 26 The second focus and directrix, and the centre M The centre of the parabola is at infinity . 2i Condition for conjugate diameters .... It Sum of reciprocals of segments of focal chord constant 27 Batio of products of segments of intersecting chords . 29 Intersection of a ciide trith a conic It Properties of polars ..... 30 The straight line at infinity .... 32 Envelope of polar of any point on a conic with respect to a conic having the same focus and directrix .... Polar of external point meets the conic in real points 84 Polar equations of tangent, normal, chord and polar . Examples 11—80 ..... 86 CHAPTEBIIL THE FABABOLA. The parabola regarded as a central conic .... Square of principal ordinate varies as abscissa Construction for two mean proportionals .... 45 Every diameter is parallel to the axis .... 46 Length of any focal chord determined .... 47 Square of any ordinate varies as abscissa 48 Two sides of any triangle ou base parallel to axis are as the parallel tangents 49 Batio of segments of any chord by a diameter It Products of segments of intersecting chords are as the parallel focal chords 60 The tangent equally inclined to axis focal and distance . 61 Tangents at right angles meet on directrix . 62 The principal snbtangent double of the abscissa . Normal bisects angle between focal distance and diameter The subnormal constant and equal to half latus rectum 63 The focal perpendicular on the tangent . , The angle between two tangents, and their inclinations to the axis . 64 tangents with focus determine similar triangles Two . 65 Envelope of line cut in constant ratios by three fixed lines . Lambert's circle through intersections of three tangents Steiner's property of directrix of parabola inscribed in triangle 66 The parabola touching four given lines determined Proportionality of segments of two tangents by a third 67 Steiner's theorem in relation to Pascal's 68 The subtangent on any diameter double of the abscissa . CONTENTS. XI PAOB Tangent and diameter at any point cut any chord proportionally 69 Area of triangle circumscribed to a parabola determined . Theorem of Archimedes on quadrature of parabola Examples 81—200 ..... 61 QHAPTEE IV. CBNTBAL CONIOS. The principal ordinate 76 Length of minor axis of hyperbola defined 76 Projective property of central conies 77 The axis or its complement a degenerate form of conic Square of any ordinate varies as product of abscisss Length of imaginary diameter defined . 78 Sum or difference of focal distances constant » Belations between segments of the axis 80 Lengths of latus rectum and any focal chord 81 The names parabola, ellipse, hyperbola explained 82 Tangent bisects angle between focal distances 83 Confocal conies intersect at right angles M The focal perpendiculars npou the tangent . Constant intercept on focal distajice by diameter parallel to tangent Any diameter cut harmonically by tangent and ordinate at any point 87 The director circle or orthocycle .... 165, 280 Locns of vertex of right angle ciictimacribed to two confocala , 89 Property of polars ...... 90 Normal bisects angle between focal distances 92 licngth of intercept on normal by either axis . 98 The subnormal varies as the abscissa 94 Supplemental chords parallel to conjugate diameters 96 Conjugate diameters correspond to diameters at right angles in auxiliary circle 96 One of every two conjugate diameters meets hyperbola Construction for conjugate diameters containing a given angle Sum or diSeience of squares of conjugate diameters constant . Conjugate diameter expressed in terms of normal, and of focal distances 98 Area of conjugate parallelogram constant 99 Product of segments of fixed tangent by conjugate diameters constant 100 Product of segments of parallel tangents by any third 101 The hyperbola identified with the ellipse >i The bifocal definition ..... 102 Magnitude of angle between two tangents to a conic .
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