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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 04:50:08 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 04:50:08 -0700 |
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diff --git a/17001-h/17001-h.htm b/17001-h/17001-h.htm new file mode 100644 index 0000000..e0a4cfd --- /dev/null +++ b/17001-h/17001-h.htm @@ -0,0 +1,5914 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> +<head> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta http-equiv="Content-Style-Type" content="text/css" /> +<title>The Project Gutenberg eBook of An Elementary Course in Synthetic Projective Geometry by, Derrick Norman Lehmer</title> + +<style type="text/css"> + +.tei { margin: 0; padding: 0; + font-size: 100%; font-weight: normal; font-style: normal } + +.block { display: block; } +.inline { display: inline; } +.floatleft { float: left; margin: 1em 2em 1em 0; } +.floatright { float: right; margin: 1em 0 1em 2em; } +.shaded { margin-top: 1em; margin-bottom: 1em; + padding: 1em; background-color: #eee; } +.boxed { margin-top: 1em; margin-bottom: 1em; + padding: 1em; border: 1px solid black; } + +body.tei { margin: 4ex 10%; text-align: justify } +div.tei { margin: 2em 0em } +p.tei { margin: 0em 0em 1em 0em; text-indent: 0em; } +blockquote.tei { margin: 2em 4em } + +div.tei-lg { margin: 1em 0em; } +div.tei-l { margin: 0em; text-align: left; } +div.tei-tb { text-align: center; } +div.tei-epigraph { margin: 0em 0em 1em 10em; } +div.tei-dateline { margin: 1ex 0em; text-align: right } +div.tei-salute { margin: 1ex 0em; } +div.tei-signed { margin: 1ex 0em; text-align: right } +div.tei-byline { margin: 1ex 0em; } + + /* calculate from size of body = 80% */ +div.tei-marginnote { margin: 0em 0em 0em -12%; width: 11%; float: left; } + +div.tei-sp { margin: 1em 0em 1em 2em } +div.tei-speaker { margin: 0em 0em 1em -2em; + font-weight: bold; text-indent: 0em } +div.tei-stage { margin: 1em 0em; font-weight: normal; font-style: italic } +span.tei-stage { font-weight: normal; font-style: italic } + +div.tei-eg { padding: 1em; + color: black; background-color: #eee } + +hr.doublepage { margin: 4em 0em; height: 5px; } +hr.page { margin: 4em 0em; height: 2px; } + +ul.tei-index { list-style-type: none } + +dl.tei { margin: 1em 0em } + +dt.tei-notelabel { font-weight: normal; text-align: right; + float: left; width: 3em } +dd.tei-notetext { margin: 0em 0em 1ex 4em } + +span.tei-pb { position: absolute; left: 1%; width: 8%; + font-style: normal; } + +span.code { font-family: monospace; font-size: 110%; } + +ul.tei-castlist { margin: 0em; list-style-type: none } +li.tei-castitem { margin: 0em; } +table.tei-castgroup { margin: 0em; } +ul.tei-castgroup { margin: 0em; list-style-type: none; + padding-right: 2em; border-right: solid black 2px; } +*.tei-roledesc { font-style: italic } +*.tei-set { font-style: italic } + +table.rules { border-collapse: collapse; } +table.rules th, +table.rules td { border: 1px solid black; } + +table.tei { border-collapse: collapse; } +table.tei-list { width: 100% } + +th.tei-head-table { padding: 0.5ex 1em } + +th.tei-cell { padding: 0em 1em } +td.tei-cell { padding: 0em 1em } + +td.tei-item { padding: 0; font-weight: normal; + vertical-align: top; text-align: left; } +th.tei-label, +td.tei-label { width: 3em; padding: 0; font-weight: normal; + vertical-align: top; text-align: right; } + +th.tei-label-gloss, +td.tei-label-gloss { text-align: left } + +td.tei-item-gloss, +th.tei-headItem-gloss { padding-left: 4em; } + +img.tei-formula { vertical-align: middle; } + +</style> +</head> +<body class="tei"> + +<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of An Elementary Course in Synthetic Projective Geometry by, Derrick Norman Lehmer</div> +<div style='display:block; margin:1em 0'> +This eBook is for the use of anyone anywhere in the United States and +most other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms +of the Project Gutenberg License included with this eBook or online +at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you +are not located in the United States, you will have to check the laws of the +country where you are located before using this eBook. +</div> +<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: An Elementary Course in Synthetic Projective Geometry</div> +<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: Derrick Norman Lehmer</div> +<div style='display:block; margin:1em 0'>Release Date: November 4, 2005 [eBook #17001]<br /> +[Most recently updated: June 22, 2021]</div> +<div style='display:block; margin:1em 0'>Language: English</div> +<div style='display:block; margin:1em 0'>Character set encoding: UTF-8</div> +<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK AN ELEMENTARY COURSE IN SYNTHETIC PROJECTIVE GEOMETRY ***</div> + +<div class="tei tei-text" style="margin-bottom: 2.00em; margin-top: 2.00em"> +<div class="tei tei-front" style="margin-bottom: 6.00em; margin-top: 2.00em"> + +<div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<div class="block tei tei-docTitle"> +<div class="block tei tei-titlePart" style="text-align: left; margin-bottom: 3.46em"> +<span style="font-size: 173%">An Elementary Course in Synthetic Projective Geometry</span> +</div> +</div> +<div class="block tei tei-byline" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"> +<span style="font-size: 173%">by </span> +<span class="inline tei tei-docAuthor" style="text-align: left"> +<span style="font-size: 173%">Lehmer, Derrick Norman</span></span> +</div> +<div class="tei tei-div" style="text-align: left; margin-bottom: 5.76em; margin-top: 5.76em"> +<span class="tei tei-docEdition" style="text-align: left"> +<span class="tei tei-edition" style="text-align: left"> +<span style="font-size: 144%">Edition 1</span></span> +</span><span style="font-size: 144%">, (</span><span class="tei tei-docDate" style="text-align: left"> +<span class="tei tei-date" style="text-align: left"> +<span style="font-size: 144%">November 4, 2005</span></span></span> +<span style="font-size: 144%">)</span> +</div> +</div> + +<hr class="doublepage" /> +<div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc1" id="toc1"></a><a name="pdf2" id="pdf2"></a> +<span class="tei tei-pb" id="pageiii">[pg iii]</span><a name="Pgiii" id="Pgiii" class="tei tei-anchor"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">Preface</span></h1> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The following course is intended to give, in as simple +a way as possible, the essentials of synthetic projective +geometry. While, in the main, the theory is developed +along the well-beaten track laid out by the great masters +of the subject, it is believed that there has been a slight +smoothing of the road in some places. Especially will +this be observed in the chapter on Involution. The +author has never felt satisfied with the usual treatment +of that subject by means of circles and anharmonic +ratios. A purely projective notion ought not to be based +on metrical foundations. Metrical developments should +be made there, as elsewhere in the theory, by the +introduction of infinitely distant elements.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The author has departed from the century-old custom +of writing in parallel columns each theorem and its +dual. He has not found that it conduces to sharpness +of vision to try to focus his eyes on two things at once. +Those who prefer the usual method of procedure can, +of course, develop the two sets of theorems side by side; +the author has not found this the better plan in actual +teaching.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">As regards nomenclature, the author has followed +the lead of the earlier writers in English, and has called +the system of lines in a plane which all pass through a +point a <span class="tei tei-hi"><span style="font-style: italic">pencil of rays</span></span> instead of a <span class="tei tei-hi"><span style="font-style: italic">bundle of rays</span></span>, as later +writers seem inclined to do. For a point considered +<span class="tei tei-pb" id="pageiv">[pg iv]</span><a name="Pgiv" id="Pgiv" class="tei tei-anchor"></a> +as made up of all the lines and planes through it he +has ventured to use the term <span class="tei tei-hi"><span style="font-style: italic">point system</span></span>, as being +the natural dualization of the usual term <span class="tei tei-hi"><span style="font-style: italic">plane system</span></span>. +He has also rejected the term <span class="tei tei-hi"><span style="font-style: italic">foci of an involution</span></span>, and +has not used the customary terms for classifying involutions—<span class="tei tei-hi"><span style="font-style: italic">hyperbolic +involution</span></span>, <span class="tei tei-hi"><span style="font-style: italic">elliptic involution</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">parabolic involution</span></span>. He has found that all these terms +are very confusing to the student, who inevitably tries +to connect them in some way with the conic sections.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Enough examples have been provided to give the +student a clear grasp of the theory. Many are of sufficient +generality to serve as a basis for individual investigation +on the part of the student. Thus, the third +example at the end of the first chapter will be found +to be very fruitful in interesting results. A correspondence +is there indicated between lines in space and +circles through a fixed point in space. If the student +will trace a few of the consequences of that correspondence, +and determine what configurations of circles +correspond to intersecting lines, to lines in a plane, to +lines of a plane pencil, to lines cutting three skew lines, +etc., he will have acquired no little practice in picturing +to himself figures in space.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The writer has not followed the usual practice of +inserting historical notes at the foot of the page, and +has tried instead, in the last chapter, to give a consecutive +account of the history of pure geometry, or, at +least, of as much of it as the student will be able to +appreciate who has mastered the course as given in the +preceding chapters. One is not apt to get a very wide +view of the history of a subject by reading a hundred +<span class="tei tei-pb" id="pagev">[pg v]</span><a name="Pgv" id="Pgv" class="tei tei-anchor"></a> +biographical footnotes, arranged in no sort of sequence. +The writer, moreover, feels that the proper time to +learn the history of a subject is after the student has +some general ideas of the subject itself.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The course is not intended to furnish an illustration +of how a subject may be developed, from the smallest +possible number of fundamental assumptions. The +author is aware of the importance of work of this sort, +but he does not believe it is possible at the present +time to write a book along such lines which shall be of +much use for elementary students. For the purposes of +this course the student should have a thorough grounding +in ordinary elementary geometry so far as to include +the study of the circle and of similar triangles. No solid +geometry is needed beyond the little used in the proof +of Desargues' theorem (25), and, except in certain +metrical developments of the general theory, there will +be no call for a knowledge of trigonometry or analytical +geometry. Naturally the student who is equipped with +these subjects as well as with the calculus will be a +little more mature, and may be expected to follow the +course all the more easily. The author has had no +difficulty, however, in presenting it to students in the +freshman class at the University of California.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The subject of synthetic projective geometry is, in +the opinion of the writer, destined shortly to force its +way down into the secondary schools; and if this little +book helps to accelerate the movement, he will feel +amply repaid for the task of working the materials into +a form available for such schools as well as for the +lower classes in the university.</p> + +<span class="tei tei-pb" id="pagevi">[pg vi]</span><a name="Pgvi" id="Pgvi" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The material for the course has been drawn from +many sources. The author is chiefly indebted to the +classical works of Reye, Cremona, Steiner, Poncelet, and +Von Staudt. Acknowledgments and thanks are also +due to Professor Walter C. Eells, of the U.S. Naval +Academy at Annapolis, for his searching examination +and keen criticism of the manuscript; also to Professor +Herbert Ellsworth Slaught, of The University of Chicago, +for his many valuable suggestions, and to Professor +B. M. Woods and Dr. H. N. Wright, of the University +of California, who have tried out the methods of +presentation, in their own classes.</p> + +<p class="tei tei-p" style="text-align: right; margin-bottom: 1.00em">D. N. LEHMER</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-variant: small-caps">Berkeley, California</span></span></p> + +</div> + +<hr class="doublepage" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> + <a name="toc3" id="toc3"></a><a name="pdf4" id="pdf4"></a> + <h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">Contents</span></h1> + <ul class="tei tei-index tei-index-toc"><li><a href="#toc1">Preface</a></li><li><a href="#toc3">Contents</a></li><li><a href="#toc5">CHAPTER I - ONE-TO-ONE CORRESPONDENCE</a></li><li style="margin-left: 2em"><a href="#toc7">1. Definition of one-to-one correspondence</a></li><li style="margin-left: 2em"><a href="#toc9">2. Consequences of one-to-one correspondence</a></li><li style="margin-left: 2em"><a href="#toc11">3. Applications in mathematics</a></li><li style="margin-left: 2em"><a href="#toc13">4. One-to-one correspondence and enumeration</a></li><li style="margin-left: 2em"><a href="#toc15">5. Correspondence between a part and the whole</a></li><li style="margin-left: 2em"><a href="#toc17">6. Infinitely distant point</a></li><li style="margin-left: 2em"><a href="#toc19">7. Axial pencil; fundamental forms</a></li><li style="margin-left: 2em"><a href="#toc21">8. Perspective position</a></li><li style="margin-left: 2em"><a href="#toc23">9. Projective relation</a></li><li style="margin-left: 2em"><a href="#toc25">10. Infinity-to-one correspondence</a></li><li style="margin-left: 2em"><a href="#toc27">11. Infinitudes of different orders</a></li><li style="margin-left: 2em"><a href="#toc29">12. Points in a plane</a></li><li style="margin-left: 2em"><a href="#toc31">13. Lines through a point</a></li><li style="margin-left: 2em"><a href="#toc33">14. Planes through a point</a></li><li style="margin-left: 2em"><a href="#toc35">15. Lines in a plane</a></li><li style="margin-left: 2em"><a href="#toc37">16. Plane system and point system</a></li><li style="margin-left: 2em"><a href="#toc39">17. Planes in space</a></li><li style="margin-left: 2em"><a href="#toc41">18. Points of space</a></li><li style="margin-left: 2em"><a href="#toc43">19. Space system</a></li><li style="margin-left: 2em"><a href="#toc45">20. Lines in space</a></li><li style="margin-left: 2em"><a href="#toc47">21. Correspondence between points and numbers</a></li><li style="margin-left: 2em"><a href="#toc49">22. Elements at infinity</a></li><li style="margin-left: 2em"><a href="#toc51">PROBLEMS</a></li><li><a href="#toc53">CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE +CORRESPONDENCE WITH EACH OTHER</a></li><li style="margin-left: 2em"><a href="#toc55">23. Seven fundamental forms</a></li><li style="margin-left: 2em"><a href="#toc57">24. Projective properties</a></li><li style="margin-left: 2em"><a href="#toc59">25. Desargues's theorem</a></li><li style="margin-left: 2em"><a href="#toc61">26. Fundamental theorem concerning two complete quadrangles</a></li><li style="margin-left: 2em"><a href="#toc63">27. Importance of the theorem</a></li><li style="margin-left: 2em"><a href="#toc65">28. Restatement of the theorem</a></li><li style="margin-left: 2em"><a href="#toc67">29. Four harmonic points</a></li><li style="margin-left: 2em"><a href="#toc69">30. Harmonic conjugates</a></li><li style="margin-left: 2em"><a href="#toc71">31. Importance of the notion of four harmonic points</a></li><li style="margin-left: 2em"><a href="#toc73">32. Projective invariance of four harmonic points</a></li><li style="margin-left: 2em"><a href="#toc75">33. Four harmonic lines</a></li><li style="margin-left: 2em"><a href="#toc77">34. Four harmonic planes</a></li><li style="margin-left: 2em"><a href="#toc79">35. Summary of results</a></li><li style="margin-left: 2em"><a href="#toc81">36. Definition of projectivity</a></li><li style="margin-left: 2em"><a href="#toc83">37. Correspondence between harmonic conjugates</a></li><li style="margin-left: 2em"><a href="#toc85">38. Separation of harmonic conjugates</a></li><li style="margin-left: 2em"><a href="#toc87">39. Harmonic conjugate of the point at infinity</a></li><li style="margin-left: 2em"><a href="#toc89">40. Projective theorems and metrical theorems. Linear construction</a></li><li style="margin-left: 2em"><a href="#toc91">41. Parallels and mid-points</a></li><li style="margin-left: 2em"><a href="#toc93">42. Division of segment into equal parts</a></li><li style="margin-left: 2em"><a href="#toc95">43. Numerical relations</a></li><li style="margin-left: 2em"><a href="#toc97">44. Algebraic formula connecting four harmonic points</a></li><li style="margin-left: 2em"><a href="#toc99">45. Further formulae</a></li><li style="margin-left: 2em"><a href="#toc101">46. Anharmonic ratio</a></li><li style="margin-left: 2em"><a href="#toc103">PROBLEMS</a></li><li><a href="#toc105">CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED +FUNDAMENTAL FORMS</a></li><li style="margin-left: 2em"><a href="#toc107">47. Superposed fundamental forms. Self-corresponding elements</a></li><li style="margin-left: 2em"><a href="#toc109">48. Special case</a></li><li style="margin-left: 2em"><a href="#toc111">49. Fundamental theorem. Postulate of continuity</a></li><li style="margin-left: 2em"><a href="#toc113">50. Extension of theorem to pencils of rays and planes</a></li><li style="margin-left: 2em"><a href="#toc115">51. Projective point-rows having a self-corresponding point in common</a></li><li style="margin-left: 2em"><a href="#toc117">52. Point-rows in perspective position</a></li><li style="margin-left: 2em"><a href="#toc119">53. Pencils in perspective position</a></li><li style="margin-left: 2em"><a href="#toc121">54. Axial pencils in perspective position</a></li><li style="margin-left: 2em"><a href="#toc123">55. Point-row of the second order</a></li><li style="margin-left: 2em"><a href="#toc125">56. Degeneration of locus</a></li><li style="margin-left: 2em"><a href="#toc127">57. Pencils of rays of the second order</a></li><li style="margin-left: 2em"><a href="#toc129">58. Degenerate case</a></li><li style="margin-left: 2em"><a href="#toc131">59. Cone of the second order</a></li><li style="margin-left: 2em"><a href="#toc133">PROBLEMS</a></li><li><a href="#toc135">CHAPTER IV - POINT-ROWS OF THE SECOND ORDER</a></li><li style="margin-left: 2em"><a href="#toc137">60. Point-row of the second order defined</a></li><li style="margin-left: 2em"><a href="#toc139">61. Tangent line</a></li><li style="margin-left: 2em"><a href="#toc141">62. Determination of the locus</a></li><li style="margin-left: 2em"><a href="#toc143">63. Restatement of the problem</a></li><li style="margin-left: 2em"><a href="#toc145">64. Solution of the fundamental problem</a></li><li style="margin-left: 2em"><a href="#toc147">65. Different constructions for the figure</a></li><li style="margin-left: 2em"><a href="#toc149">66. Lines joining four points of the locus to a fifth</a></li><li style="margin-left: 2em"><a href="#toc151">67. Restatement of the theorem</a></li><li style="margin-left: 2em"><a href="#toc153">68. Further important theorem</a></li><li style="margin-left: 2em"><a href="#toc155">69. Pascal's theorem</a></li><li style="margin-left: 2em"><a href="#toc157">70. Permutation of points in Pascal's theorem</a></li><li style="margin-left: 2em"><a href="#toc159">71. Harmonic points on a point-row of the second order</a></li><li style="margin-left: 2em"><a href="#toc161">72. Determination of the locus</a></li><li style="margin-left: 2em"><a href="#toc163">73. Circles and conics as point-rows of the second order</a></li><li style="margin-left: 2em"><a href="#toc165">74. Conic through five points</a></li><li style="margin-left: 2em"><a href="#toc167">75. Tangent to a conic</a></li><li style="margin-left: 2em"><a href="#toc169">76. Inscribed quadrangle</a></li><li style="margin-left: 2em"><a href="#toc171">77. Inscribed triangle</a></li><li style="margin-left: 2em"><a href="#toc173">78. Degenerate conic</a></li><li style="margin-left: 2em"><a href="#toc175">PROBLEMS</a></li><li><a href="#toc177">CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER</a></li><li style="margin-left: 2em"><a href="#toc179">79. Pencil of rays of the second order defined</a></li><li style="margin-left: 2em"><a href="#toc181">80. Tangents to a circle</a></li><li style="margin-left: 2em"><a href="#toc183">81. Tangents to a conic</a></li><li style="margin-left: 2em"><a href="#toc185">82. Generating point-rows lines of the system</a></li><li style="margin-left: 2em"><a href="#toc187">83. Determination of the pencil</a></li><li style="margin-left: 2em"><a href="#toc189">84. Brianchon's theorem</a></li><li style="margin-left: 2em"><a href="#toc191">85. Permutations of lines in Brianchon's theorem</a></li><li style="margin-left: 2em"><a href="#toc193">86. Construction of the penvil by Brianchon's theorem</a></li><li style="margin-left: 2em"><a href="#toc195">87. Point of contact of a tangent to a conic</a></li><li style="margin-left: 2em"><a href="#toc197">88. Circumscribed quadrilateral</a></li><li style="margin-left: 2em"><a href="#toc199">89. Circumscribed triangle</a></li><li style="margin-left: 2em"><a href="#toc201">90. Use of Brianchon's theorem</a></li><li style="margin-left: 2em"><a href="#toc203">91. Harmonic tangents</a></li><li style="margin-left: 2em"><a href="#toc205">92. Projectivity and perspectivity</a></li><li style="margin-left: 2em"><a href="#toc207">93. Degenerate case</a></li><li style="margin-left: 2em"><a href="#toc209">94. Law of duality</a></li><li style="margin-left: 2em"><a href="#toc211">PROBLEMS</a></li><li><a href="#toc213">CHAPTER VI - POLES AND POLARS</a></li><li style="margin-left: 2em"><a href="#toc215">95. Inscribed and circumscribed quadrilaterals</a></li><li style="margin-left: 2em"><a href="#toc217">96. Definition of the polar line of a point</a></li><li style="margin-left: 2em"><a href="#toc219">97. Further defining properties</a></li><li style="margin-left: 2em"><a href="#toc221">98. Definition of the pole of a line</a></li><li style="margin-left: 2em"><a href="#toc223">99. Fundamental theorem of poles and polars</a></li><li style="margin-left: 2em"><a href="#toc225">100. Conjugate points and lines</a></li><li style="margin-left: 2em"><a href="#toc227">101. Construction of the polar line of a given point</a></li><li style="margin-left: 2em"><a href="#toc229">102. Self-polar triangle</a></li><li style="margin-left: 2em"><a href="#toc231">103. Pole and polar projectively related</a></li><li style="margin-left: 2em"><a href="#toc233">104. Duality</a></li><li style="margin-left: 2em"><a href="#toc235">105. Self-dual theorems</a></li><li style="margin-left: 2em"><a href="#toc237">106. Other correspondences</a></li><li style="margin-left: 2em"><a href="#toc239">PROBLEMS</a></li><li><a href="#toc241">CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS</a></li><li style="margin-left: 2em"><a href="#toc243">107. Diameters. Center</a></li><li style="margin-left: 2em"><a href="#toc245">108. Various theorems</a></li><li style="margin-left: 2em"><a href="#toc247">109. Conjugate diameters</a></li><li style="margin-left: 2em"><a href="#toc249">110. Classification of conics</a></li><li style="margin-left: 2em"><a href="#toc251">111. Asymptotes</a></li><li style="margin-left: 2em"><a href="#toc253">112. Various theorems</a></li><li style="margin-left: 2em"><a href="#toc255">113. Theorems concerning asymptotes</a></li><li style="margin-left: 2em"><a href="#toc257">114. Asymptotes and conjugate diameters</a></li><li style="margin-left: 2em"><a href="#toc259">115. Segments cut off on a chord by hyperbola and its asymptotes</a></li><li style="margin-left: 2em"><a href="#toc261">116. Application of the theorem</a></li><li style="margin-left: 2em"><a href="#toc263">117. Triangle formed by the two asymptotes and a tangent</a></li><li style="margin-left: 2em"><a href="#toc265">118. Equation of hyperbola referred to the asymptotes</a></li><li style="margin-left: 2em"><a href="#toc267">119. Equation of parabola</a></li><li style="margin-left: 2em"><a href="#toc269">120. Equation of central conics referred to conjugate diameters</a></li><li style="margin-left: 2em"><a href="#toc271">PROBLEMS</a></li><li><a href="#toc273">CHAPTER VIII - INVOLUTION</a></li><li style="margin-left: 2em"><a href="#toc275">121. Fundamental theorem</a></li><li style="margin-left: 2em"><a href="#toc277">122. Linear construction</a></li><li style="margin-left: 2em"><a href="#toc279">123. Definition of involution of points on a line</a></li><li style="margin-left: 2em"><a href="#toc281">124. Double-points in an involution</a></li><li style="margin-left: 2em"><a href="#toc283">125. Desargues's theorem concerning conics through four points</a></li><li style="margin-left: 2em"><a href="#toc285">126. Degenerate conics of the system</a></li><li style="margin-left: 2em"><a href="#toc287">127. Conics through four points touching a given line</a></li><li style="margin-left: 2em"><a href="#toc289">128. Double correspondence</a></li><li style="margin-left: 2em"><a href="#toc291">129. Steiner's construction</a></li><li style="margin-left: 2em"><a href="#toc293">130. Application of Steiner's construction to double correspondence</a></li><li style="margin-left: 2em"><a href="#toc295">131. Involution of points on a point-row of the second order.</a></li><li style="margin-left: 2em"><a href="#toc297">132. Involution of rays</a></li><li style="margin-left: 2em"><a href="#toc299">133. Double rays</a></li><li style="margin-left: 2em"><a href="#toc301">134. Conic through a fixed point touching four lines</a></li><li style="margin-left: 2em"><a href="#toc303">135. Double correspondence</a></li><li style="margin-left: 2em"><a href="#toc305">136. Pencils of rays of the second order in involution</a></li><li style="margin-left: 2em"><a href="#toc307">137. Theorem concerning pencils of the second order in involution</a></li><li style="margin-left: 2em"><a href="#toc309">138. Involution of rays determined by a conic</a></li><li style="margin-left: 2em"><a href="#toc311">139. Statement of theorem</a></li><li style="margin-left: 2em"><a href="#toc313">140. Dual of the theorem</a></li><li style="margin-left: 2em"><a href="#toc315">PROBLEMS</a></li><li><a href="#toc317">CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS</a></li><li style="margin-left: 2em"><a href="#toc319">141. Introduction of infinite point; center of involution</a></li><li style="margin-left: 2em"><a href="#toc321">142. Fundamental metrical theorem</a></li><li style="margin-left: 2em"><a href="#toc323">143. Existence of double points</a></li><li style="margin-left: 2em"><a href="#toc325">144. Existence of double rays</a></li><li style="margin-left: 2em"><a href="#toc327">145. Construction of an involution by means of circles</a></li><li style="margin-left: 2em"><a href="#toc329">146. Circular points</a></li><li style="margin-left: 2em"><a href="#toc331">147. Pairs in an involution of rays which are at right angles. Circular involution</a></li><li style="margin-left: 2em"><a href="#toc333">148. Axes of conics</a></li><li style="margin-left: 2em"><a href="#toc335">149. Points at which the involution determined by a conic is circular</a></li><li style="margin-left: 2em"><a href="#toc337">150. Properties of such a point</a></li><li style="margin-left: 2em"><a href="#toc339">151. Position of such a point</a></li><li style="margin-left: 2em"><a href="#toc341">152. Discovery of the foci of the conic</a></li><li style="margin-left: 2em"><a href="#toc343">153. The circle and the parabola</a></li><li style="margin-left: 2em"><a href="#toc345">154. Focal properties of conics</a></li><li style="margin-left: 2em"><a href="#toc347">155. Case of the parabola</a></li><li style="margin-left: 2em"><a href="#toc349">156. Parabolic reflector</a></li><li style="margin-left: 2em"><a href="#toc351">157. Directrix. Principal axis. Vertex</a></li><li style="margin-left: 2em"><a href="#toc353">158. Another definition of a conic</a></li><li style="margin-left: 2em"><a href="#toc355">159. Eccentricity</a></li><li style="margin-left: 2em"><a href="#toc357">160. Sum or difference of focal distances</a></li><li style="margin-left: 2em"><a href="#toc359">PROBLEMS</a></li><li><a href="#toc361">CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY</a></li><li style="margin-left: 2em"><a href="#toc363">161. Ancient results</a></li><li style="margin-left: 2em"><a href="#toc365">162. Unifying principles</a></li><li style="margin-left: 2em"><a href="#toc367">163. Desargues</a></li><li style="margin-left: 2em"><a href="#toc369">164. Poles and polars</a></li><li style="margin-left: 2em"><a href="#toc371">165. Desargues's theorem concerning conics through four points</a></li><li style="margin-left: 2em"><a href="#toc373">166. Extension of the theory of poles and polars to space</a></li><li style="margin-left: 2em"><a href="#toc375">167. Desargues's method of describing a conic</a></li><li style="margin-left: 2em"><a href="#toc377">168. Reception of Desargues's work</a></li><li style="margin-left: 2em"><a href="#toc379">169. Conservatism in Desargues's time</a></li><li style="margin-left: 2em"><a href="#toc381">170. Desargues's style of writing</a></li><li style="margin-left: 2em"><a href="#toc383">171. Lack of appreciation of Desargues</a></li><li style="margin-left: 2em"><a href="#toc385">172. Pascal and his theorem</a></li><li style="margin-left: 2em"><a href="#toc387">173. Pascal's essay</a></li><li style="margin-left: 2em"><a href="#toc389">174. Pascal's originality</a></li><li style="margin-left: 2em"><a href="#toc391">175. De la Hire and his work</a></li><li style="margin-left: 2em"><a href="#toc393">176. Descartes and his influence</a></li><li style="margin-left: 2em"><a href="#toc395">177. Newton and Maclaurin</a></li><li style="margin-left: 2em"><a href="#toc397">178. Maclaurin's construction</a></li><li style="margin-left: 2em"><a href="#toc399">179. Descriptive geometry and the second revival</a></li><li style="margin-left: 2em"><a href="#toc401">180. Duality, homology, continuity, contingent relations</a></li><li style="margin-left: 2em"><a href="#toc403">181. Poncelet and Cauchy</a></li><li style="margin-left: 2em"><a href="#toc405">182. The work of Poncelet</a></li><li style="margin-left: 2em"><a href="#toc407">183. The debt which analytic geometry owes to synthetic geometry</a></li><li style="margin-left: 2em"><a href="#toc409">184. Steiner and his work</a></li><li style="margin-left: 2em"><a href="#toc411">185. Von Staudt and his work</a></li><li style="margin-left: 2em"><a href="#toc413">186. Recent developments</a></li><li><a href="#toc415">INDEX</a></li></ul> +</div> + +</div> + +<div class="tei tei-body" style="margin-bottom: 6.00em; margin-top: 6.00em"> +<hr class="doublepage" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<span class="tei tei-pb" id="page1">[pg 1]</span><a name="Pg1" id="Pg1" class="tei tei-anchor"></a> +<a name="toc5" id="toc5"></a><a name="pdf6" id="pdf6"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER I - ONE-TO-ONE CORRESPONDENCE</span></h1> +<p class="tei tei-p" style="margin-bottom: 1.00em"></p> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc7" id="toc7"></a><a name="pdf8" id="pdf8"></a> + +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p1" id="p1" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">1. Definition of one-to-one correspondence.</span></span> +Given any two sets of individuals, if it is possible to set up such +a correspondence between the two sets that to any +individual in one set corresponds one and only one +individual in the other, then the two sets are said to +be in <span class="tei tei-hi"><span style="font-style: italic">one-to-one correspondence</span></span> with each other. This +notion, simple as it is, is of fundamental importance +in all branches of science. The process of counting is +nothing but a setting up of a one-to-one correspondence +between the objects to be counted and certain +words, 'one,' 'two,' 'three,' etc., in the mind. Many +savage peoples have discovered no better method of +counting than by setting up a one-to-one correspondence +between the objects to be counted and their fingers. +The scientist who busies himself with naming and +classifying the objects of nature is only setting up a +one-to-one correspondence between the objects and certain +words which serve, not as a means of counting the +<span class="tei tei-pb" id="page2">[pg 2]</span><a name="Pg2" id="Pg2" class="tei tei-anchor"></a> +objects, but of listing them in a convenient way. Thus +he may be able to marshal and array his material in +such a way as to bring to light relations that may +exist between the objects themselves. Indeed, the whole +notion of language springs from this idea of one-to-one +correspondence.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc9" id="toc9"></a><a name="pdf10" id="pdf10"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p2" id="p2" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">2. Consequences of one-to-one correspondence.</span></span> +The most useful and interesting problem that may arise in +connection with any one-to-one correspondence is to +determine just what relations existing between the +individuals of one assemblage may be carried over to +another assemblage in one-to-one correspondence with +it. It is a favorite error to assume that whatever holds +for one set must also hold for the other. Magicians are +apt to assign magic properties to many of the words +and symbols which they are in the habit of using, and +scientists are constantly confusing objective things with +the subjective formulas for them. After the physicist +has set up correspondences between physical facts and +mathematical formulas, the "interpretation" of these +formulas is his most important and difficult task.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc11" id="toc11"></a><a name="pdf12" id="pdf12"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p3" id="p3" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">3.</span></span> In mathematics, effort is constantly being made +to set up one-to-one correspondences between simple +notions and more complicated ones, or between the well-explored +fields of research and fields less known. Thus, +by means of the mechanism employed in analytic geometry, +algebraic theorems are made to yield geometric +ones, and vice versa. In geometry we get at the properties +of the conic sections by means of the properties +of the straight line, and cubic surfaces are studied by +means of the plane.</p> + +<span class="tei tei-pb" id="page3">[pg 3]</span><a name="Pg3" id="Pg3" class="tei tei-anchor"></a> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc13" id="toc13"></a><a name="pdf14" id="pdf14"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image01.png" width="404" height="288" alt="Figure 1" title="Fig. 1" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 1</div></div> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image02.png" width="444" height="360" alt="Figure 2" title="Fig. 2" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 2</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p4" id="p4" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">4. One-to-one correspondence and enumeration.</span></span> If a +one-to-one correspondence has been set up between the +objects of one set and the objects of another set, then +the inference may usually be drawn that they have the +same number of elements. If, however, there is an +infinite number of individuals in +each of the two sets, the notion +of counting is necessarily ruled +out. It may be possible, nevertheless, +to set up a one-to-one +correspondence between the elements +of two sets even when the +number is infinite. Thus, it is easy to set up such a +correspondence between the points of a line an inch +long and the points of a line two inches long. For let +the lines (Fig. 1) be <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span>. Join <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span>, +and let these joining lines meet in <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. For every point <span class="tei tei-hi"><span style="font-style: italic">C</span></span> +on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> a point <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> may be found +on <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span> by joining <span class="tei tei-hi"><span style="font-style: italic">C</span></span> to <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and +noting the point <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> where <span class="tei tei-hi"><span style="font-style: italic">CS</span></span> +meets <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span>. Similarly, a point <span class="tei tei-hi"><span style="font-style: italic">C</span></span> +may be found on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> for any +point <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> on <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span>. The correspondence +is clearly one-to-one, +but it would be absurd to infer +from this that there were just +as many points on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> as on <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span>. In fact, it would +be just as reasonable to infer that there were twice as +many points on <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span> as on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>. For if we bend <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span> +into a circle with center at <span class="tei tei-hi"><span style="font-style: italic">S</span></span> (Fig. 2), we see that for +every point <span class="tei tei-hi"><span style="font-style: italic">C</span></span> on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> there are two points on <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span>. Thus +<span class="tei tei-pb" id="page4">[pg 4]</span><a name="Pg4" id="Pg4" class="tei tei-anchor"></a> +it is seen that the notion of one-to-one correspondence +is more extensive than the notion of counting, and +includes the notion of counting only when applied to +finite assemblages.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc15" id="toc15"></a><a name="pdf16" id="pdf16"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p5" id="p5" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">5. Correspondence between a part and the whole of an +infinite assemblage.</span></span> In the discussion of the last paragraph +the remarkable fact was brought to light that it +is sometimes possible to set the elements of an assemblage +into one-to-one correspondence with a part of +those elements. A moment's reflection will convince +one that this is never possible when there is a finite +number of elements in the assemblage.—Indeed, we +may take this property as our definition of an infinite +assemblage, and say that an infinite assemblage is one +that may be put into one-to-one correspondence with +part of itself. This has the advantage of being a positive +definition, as opposed to the usual negative definition of +an infinite assemblage as one that cannot be counted.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc17" id="toc17"></a><a name="pdf18" id="pdf18"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p6" id="p6" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">6. Infinitely distant point.</span></span> We have illustrated above +a simple method of setting the points of two lines into +one-to-one correspondence. The same illustration will +serve also to show how it is possible to set the points +on a line into one-to-one correspondence with the lines +through a point. Thus, for any point <span class="tei tei-hi"><span style="font-style: italic">C</span></span> on the line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> +there is a line <span class="tei tei-hi"><span style="font-style: italic">SC</span></span> through <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. We must assume the line +<span class="tei tei-hi"><span style="font-style: italic">AB</span></span> extended indefinitely in both directions, however, if +we are to have a point on it for every line through <span class="tei tei-hi"><span style="font-style: italic">S</span></span>; +and even with this extension there is one line through +<span class="tei tei-hi"><span style="font-style: italic">S</span></span>, according to Euclid's postulate, which does not meet +the line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and which therefore has no point on +<span class="tei tei-hi"><span style="font-style: italic">AB</span></span> to correspond to it. In order to smooth out this +<span class="tei tei-pb" id="page5">[pg 5]</span><a name="Pg5" id="Pg5" class="tei tei-anchor"></a>discrepancy we are accustomed to assume the existence +of an <span class="tei tei-hi"><span style="font-style: italic">infinitely distant</span></span> point on the line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and to assign +this point as the corresponding point of the exceptional +line of <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. With this understanding, then, we may say +that we have set the lines through a point and the +points on a line into one-to-one correspondence. This +correspondence is of such fundamental importance in +the study of projective geometry that a special name is +given to it. Calling the totality of points on a line a +<span class="tei tei-hi"><span style="font-style: italic">point-row</span></span>, and the totality of lines through a point a +<span class="tei tei-hi"><span style="font-style: italic">pencil of rays</span></span>, we say that the point-row and the pencil +related as above are in <span class="tei tei-hi"><span style="font-style: italic">perspective position</span></span>, or that they +are <span class="tei tei-hi"><span style="font-style: italic">perspectively related</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc19" id="toc19"></a><a name="pdf20" id="pdf20"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p7" id="p7" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">7. Axial pencil; fundamental forms.</span></span> A similar correspondence +may be set up between the points on a +line and the planes through another line which does not +meet the first. Such a system of planes is called an +<span class="tei tei-hi"><span style="font-style: italic">axial pencil</span></span>, and the three assemblages—the point-row, +the pencil of rays, and the axial pencil—are called +<span class="tei tei-hi"><span style="font-style: italic">fundamental forms</span></span>. The fact that they may all be set +into one-to-one correspondence with each other is expressed +by saying that they are of the same order. It is +usual also to speak of them as of the first order. We +shall see presently that there are other assemblages +which cannot be put into this sort of one-to-one correspondence +with the points on a line, and that they +will very reasonably be said to be of a higher order.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc21" id="toc21"></a><a name="pdf22" id="pdf22"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p8" id="p8" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">8. Perspective position.</span></span> We have said that a point-row +and a pencil of rays are in perspective position if +each ray of the pencil goes through the point of the +point-row which corresponds to it. Two pencils of rays +<span class="tei tei-pb" id="page6">[pg 6]</span><a name="Pg6" id="Pg6" class="tei tei-anchor"></a> +are also said to be in perspective position if corresponding +rays meet on a straight line which is called the +axis of perspectivity. Also, two point-rows are said to +be in perspective position if corresponding points lie on +straight lines through a point which is called the center +of perspectivity. A point-row and an axial pencil are +in perspective position if each plane of the pencil goes +through the point on the point-row which corresponds +to it, and an axial pencil and a pencil of rays are in +perspective position if each ray lies in the plane which +corresponds to it; and, finally, two axial pencils are +perspectively related if corresponding planes meet in +a plane.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc23" id="toc23"></a><a name="pdf24" id="pdf24"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p9" id="p9" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">9. Projective relation.</span></span> It is easy to imagine a more +general correspondence between the points of two point-rows +than the one just described. If we take two +perspective pencils, <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, then a point-row <span class="tei tei-hi"><span style="font-style: italic">a</span></span> perspective +to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> will be in one-to-one correspondence with +a point-row <span class="tei tei-hi"><span style="font-style: italic">b</span></span> perspective to <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, but corresponding points +will not, in general, lie on lines which all pass through +a point. Two such point-rows are said to be <span class="tei tei-hi"><span style="font-style: italic">projectively +related</span></span>, or simply projective to each other. Similarly, +two pencils of rays, or of planes, are projectively related +to each other if they are perspective to two perspective +point-rows. This idea will be generalized later on. It is +important to note that between the elements of two +projective fundamental forms there is a one-to-one correspondence, +and also that this correspondence is in +general <span class="tei tei-hi"><span style="font-style: italic">continuous</span></span>; that is, by taking two elements of +one form sufficiently close to each other, the two corresponding +elements in the other form may be made to +<span class="tei tei-pb" id="page7">[pg 7]</span><a name="Pg7" id="Pg7" class="tei tei-anchor"></a> +approach each other arbitrarily close. In the case of +point-rows this continuity is subject to exception in the +neighborhood of the point "at infinity."</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc25" id="toc25"></a><a name="pdf26" id="pdf26"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p10" id="p10" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">10. Infinity-to-one correspondence.</span></span> It might be inferred +that any infinite assemblage could be put into one-to-one +correspondence with any other. Such is not the case, +however, if the correspondence is to be continuous, +between the points on a line and the points on a plane. +Consider two lines which lie in different planes, and +take <span class="tei tei-hi"><span style="font-style: italic">m</span></span> points on one and <span class="tei tei-hi"><span style="font-style: italic">n</span></span> points on the other. The +number of lines joining the <span class="tei tei-hi"><span style="font-style: italic">m</span></span> points of one to the +<span class="tei tei-hi"><span style="font-style: italic">n</span></span> points jof the other is clearly <span class="tei tei-hi"><span style="font-style: italic">mn</span></span>. If we symbolize +the totality of points on a line by [infinity], then a reasonable +symbol for the totality of lines drawn to cut two lines +would be [infinity]<span class="tei tei-hi"><span style="vertical-align: super">2</span></span>. Clearly, for every point on one line there +are [infinity] lines cutting across the other, so that the correspondence +might be called [infinity]-to-one. Thus the assemblage +of lines cutting across two lines is of higher +order than the assemblage of points on a line; and as +we have called the point-row an assemblage of the first +order, the system of lines cutting across two lines ought +to be called of the second order.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc27" id="toc27"></a><a name="pdf28" id="pdf28"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p11" id="p11" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">11. Infinitudes of different orders.</span></span> Now it is easy to +set up a one-to-one correspondence between the points +in a plane and the system of lines cutting across two +lines which lie in different planes. In fact, each line of +the system of lines meets the plane in one point, and +each point in the plane determines one and only one line +cutting across the two given lines—namely, the line of +intersection of the two planes determined by the given +point with each of the given lines. The assemblage +<span class="tei tei-pb" id="page8">[pg 8]</span><a name="Pg8" id="Pg8" class="tei tei-anchor"></a> +of points in the plane is thus of the same order as +that of the lines cutting across two lines which lie in +different planes, and ought therefore to be spoken of +as of the second order. We express all these results +as follows:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc29" id="toc29"></a><a name="pdf30" id="pdf30"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p12" id="p12" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">12.</span></span> If the infinitude of points on a line is taken as +the infinitude of the first order, then the infinitude of +lines in a pencil of rays and the infinitude of planes in +an axial pencil are also of the first order, while the +infinitude of lines cutting across two "skew" lines, as +well as the infinitude of points in a plane, are of the +second order.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc31" id="toc31"></a><a name="pdf32" id="pdf32"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p13" id="p13" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">13.</span></span> If we join each of the points of a plane to a point +not in that plane, we set up a one-to-one correspondence +between the points in a plane and the lines through +a point in space. <span class="tei tei-hi"><span style="font-style: italic">Thus the infinitude of lines through a +point in space is of the second order.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc33" id="toc33"></a><a name="pdf34" id="pdf34"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p14" id="p14" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">14.</span></span> If to each line through a point in space we make +correspond that plane at right angles to it and passing +through the same point, we see that <span class="tei tei-hi"><span style="font-style: italic">the infinitude of +planes through a point in space is of the second order.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc35" id="toc35"></a><a name="pdf36" id="pdf36"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p15" id="p15" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">15.</span></span> If to each plane through a point in space we +make correspond the line in which it intersects a given +plane, we see that <span class="tei tei-hi"><span style="font-style: italic">the infinitude of lines in a plane is of +the second order.</span></span> This may also be seen by setting up +a one-to-one correspondence between the points on a +plane and the lines of that plane. Thus, take a point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> +not in the plane. Join any point <span class="tei tei-hi"><span style="font-style: italic">M</span></span> of the plane to <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. +Through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> draw a plane at right angles to <span class="tei tei-hi"><span style="font-style: italic">MS</span></span>. This +meets the given plane in a line <span class="tei tei-hi"><span style="font-style: italic">m</span></span> which may be taken as +corresponding to the point <span class="tei tei-hi"><span style="font-style: italic">M</span></span>. Another very important +<span class="tei tei-pb" id="page9">[pg 9]</span><a name="Pg9" id="Pg9" class="tei tei-anchor"></a> +method of setting up a one-to-one correspondence between +lines and points in a plane will be given later, and +many weighty consequences will be derived from it.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc37" id="toc37"></a><a name="pdf38" id="pdf38"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p16" id="p16" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">16. Plane system and point system.</span></span> The plane, considered +as made up of the points and lines in it, is called +a <span class="tei tei-hi"><span style="font-style: italic">plane system</span></span> and is a fundamental form of the second +order. The point, considered as made up of all the lines +and planes passing through it, is called a <span class="tei tei-hi"><span style="font-style: italic">point system</span></span> +and is also a fundamental form of the second order.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc39" id="toc39"></a><a name="pdf40" id="pdf40"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p17" id="p17" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">17.</span></span> If now we take three lines in space all lying in +different planes, and select <span class="tei tei-hi"><span style="font-style: italic">l</span></span> points on the first, <span class="tei tei-hi"><span style="font-style: italic">m</span></span> points +on the second, and <span class="tei tei-hi"><span style="font-style: italic">n</span></span> points on the third, then the total +number of planes passing through one of the selected +points on each line will be <span class="tei tei-hi"><span style="font-style: italic">lmn</span></span>. It is reasonable, therefore, +to symbolize the totality of planes that are determined +by the [infinity] points on each of the three lines by +[infinity]<span class="tei tei-hi"><span style="vertical-align: super">3</span></span>, and to call it an infinitude of the <span class="tei tei-hi"><span style="font-style: italic">third</span></span> order. But +it is easily seen that every plane in space is included in +this totality, so that <span class="tei tei-hi"><span style="font-style: italic">the totality of planes in space is an +infinitude of the third order.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc41" id="toc41"></a><a name="pdf42" id="pdf42"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p18" id="p18" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">18.</span></span> Consider now the planes perpendicular to these +three lines. Every set of three planes so drawn will +determine a point in space, and, conversely, through +every point in space may be drawn one and only one +set of three planes at right angles to the three given +lines. It follows, therefore, that <span class="tei tei-hi"><span style="font-style: italic">the totality of points +in space is an infinitude of the third order.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc43" id="toc43"></a><a name="pdf44" id="pdf44"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p19" id="p19" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">19. Space system.</span></span> Space of three dimensions, considered +as made up of all its planes and points, is then +a fundamental form of the <span class="tei tei-hi"><span style="font-style: italic">third</span></span> order, which we shall +call a <span class="tei tei-hi"><span style="font-style: italic">space system.</span></span></p> + +<span class="tei tei-pb" id="page10">[pg 10]</span><a name="Pg10" id="Pg10" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc45" id="toc45"></a><a name="pdf46" id="pdf46"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p20" id="p20" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">20. Lines in space.</span></span> If we join the twofold infinity +of points in one plane with the twofold infinity of +points in another plane, we get a totality of lines of +space which is of the fourth order of infinity. <span class="tei tei-hi"><span style="font-style: italic">The +totality of lines in space gives, then, a fundamental form +of the fourth order.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc47" id="toc47"></a><a name="pdf48" id="pdf48"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p21" id="p21" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">21. Correspondence between points and numbers.</span></span> In +the theory of analytic geometry a one-to-one correspondence +is assumed to exist between points on a +line and numbers. In order to justify this assumption +a very extended definition of number must be made +use of. A one-to-one correspondence is then set up between +points in the plane and pairs of numbers, and +also between points in space and sets of three numbers. +A single constant will serve to define the position of +a point on a line; two, a point in the plane; three, a +point in space; etc. In the same theory a one-to-one +correspondence is set up between loci in the plane and +equations in two variables; between surfaces in space +and equations in three variables; etc. The equation of +a line in a plane involves two constants, either of which +may take an infinite number of values. From this it +follows that there is an infinity of lines in the plane +which is of the second order if the infinity of points on +a line is assumed to be of the first. In the same way +a circle is determined by three conditions; a sphere by +four; etc. We might then expect to be able to set up +a one-to-one correspondence between circles in a plane +and points, or planes in space, or between spheres and +lines in space. Such, indeed, is the case, and it is +often possible to infer theorems concerning spheres +<span class="tei tei-pb" id="page11">[pg 11]</span><a name="Pg11" id="Pg11" class="tei tei-anchor"></a> +from theorems concerning lines, and vice versa. It is +possibilities such as these that, give to the theory of +one-to-one correspondence its great importance for the +mathematician. It must not be forgotten, however, that +we are considering only <span class="tei tei-hi"><span style="font-style: italic">continuous</span></span> correspondences. It +is perfectly possible to set, up a one-to-one correspondence +between the points of a line and the points of a +plane, or, indeed, between the points of a line and the +points of a space of any finite number of dimensions, if +the correspondence is not restricted to be continuous.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc49" id="toc49"></a><a name="pdf50" id="pdf50"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p22" id="p22" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">22. Elements at infinity.</span></span> A final word is necessary +in order to explain a phrase which is in constant use in +the study of projective geometry. We have spoken of +the "point at infinity" on a straight line—a fictitious +point only used to bridge over the exceptional case +when we are setting up a one-to-one correspondence +between the points of a line and the lines through a +point. We speak of it as "a point" and not as "points," +because in the geometry studied by Euclid we assume +only one line through a point parallel to a given line. +In the same sense we speak of all the points at infinity +in a plane as lying on a line, "the line at infinity," +because the straight line is the simplest locus we can +imagine which has only one point in common with any +line in the plane. Likewise we speak of the "plane at +infinity," because that seems the most convenient way +of imagining the points at infinity in space. It must not +be inferred that these conceptions have any essential +connection with physical facts, or that other means of +picturing to ourselves the infinitely distant configurations +are not possible. In other branches of mathematics, +<span class="tei tei-pb" id="page12">[pg 12]</span><a name="Pg12" id="Pg12" class="tei tei-anchor"></a> +notably in the theory of functions of a complex variable, +quite different assumptions are made and quite +different conceptions of the elements at infinity are used. +As we can know nothing experimentally about such +things, we are at liberty to make any assumptions we +please, so long as they are consistent and serve some +useful purpose.</p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc51" id="toc51"></a><a name="pdf52" id="pdf52"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. Since there is a threefold infinity of points in space, +there must be a sixfold infinity of pairs of points in space. +Each pair of points determines a line. Why, then, is there +not a sixfold infinity of lines in space?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. If there is a fourfold infinity of lines in space, why +is it that there is not a fourfold infinity of planes through +a point, seeing that each line in space determines a plane +through that point?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Show that there is a fourfold infinity of circles in +space that pass through a fixed point. (Set up a one-to-one +correspondence between the axes of the circles and lines +in space.)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Find the order of infinity of all the lines of space +that cut across a given line; across two given lines; across +three given lines; across four given lines.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. Find the order of infinity of all the spheres in space +that pass through a given point; through two given points; +through three given points; through four given points.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Find the order of infinity of all the circles on a +sphere; of all the circles on a sphere that pass through a +fixed point; through two fixed points; through three fixed +points; of all the circles in space; of all the circles that +cut across a given line.</p> + +<span class="tei tei-pb" id="page13">[pg 13]</span><a name="Pg13" id="Pg13" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. Find the order of infinity of all lines tangent to a +sphere; of all planes tangent to a sphere; of lines and +planes tangent to a sphere and passing through a fixed point.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Set up a one-to-one correspondence between the series +of numbers <span class="tei tei-hi"><span style="font-style: italic">1</span></span>, <span class="tei tei-hi"><span style="font-style: italic">2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">3</span></span>, <span class="tei tei-hi"><span style="font-style: italic">4</span></span>, ... and the series of even numbers +<span class="tei tei-hi"><span style="font-style: italic">2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">4</span></span>, <span class="tei tei-hi"><span style="font-style: italic">6</span></span>, <span class="tei tei-hi"><span style="font-style: italic">8</span></span> .... Are we justified in saying that there are just +as many even numbers as there are numbers altogether?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">9. Is the axiom "The whole is greater than one of its +parts" applicable to infinite assemblages?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">10. Make out a classified list of all the infinitudes of the +first, second, third, and fourth orders mentioned in this +chapter.</p> + +</div> +</div> + +<span class="tei tei-pb" id="page14">[pg 14]</span><a name="Pg14" id="Pg14" class="tei tei-anchor"></a> +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc53" id="toc53"></a><a name="pdf54" id="pdf54"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE +CORRESPONDENCE WITH EACH OTHER</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc55" id="toc55"></a><a name="pdf56" id="pdf56"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p23" id="p23" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">23. Seven fundamental forms.</span></span> In the preceding chapter +we have called attention to seven fundamental forms: +the point-row, the pencil of rays, the axial pencil, the +plane system, the point system, the space system, and +the system of lines in space. These fundamental forms +are the material which we intend to use in building up +a general theory which will be found to include ordinary +geometry as a special case. We shall be concerned, not +with measurement of angles and areas or line segments +as in the study of Euclid, but in combining and +comparing these fundamental forms and in "generating" +new forms by means of them. In problems of construction +we shall make no use of measurement, either +of angles or of segments, and except in certain special +applications of the general theory we shall not find it +necessary to require more of ourselves than the ability +to draw the line joining two points, or to find the point +of intersections of two lines, or the line of intersection +of two planes, or, in general, the common elements of +two fundamental forms.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc57" id="toc57"></a><a name="pdf58" id="pdf58"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p24" id="p24" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">24. Projective properties.</span></span> Our chief interest in this +chapter will be the discovery of relations between +the elements of one form which hold between the +<span class="tei tei-pb" id="page15">[pg 15]</span><a name="Pg15" id="Pg15" class="tei tei-anchor"></a> +corresponding elements of any other form in one-to-one +correspondence with it. We have already called attention +to the danger of assuming that whatever relations +hold between the elements of one assemblage must also +hold between the corresponding elements of any assemblage +in one-to-one correspondence with it. This false +assumption is the basis of the so-called "proof by +analogy" so much in vogue among speculative theorists. +When it appears that certain relations existing between +the points of a given point-row do not necessitate the +same relations between the corresponding elements of +another in one-to-one correspondence with it, we should +view with suspicion any application of the "proof by +analogy" in realms of thought where accurate judgments +are not so easily made. For example, if in a +given point-row <span class="tei tei-hi"><span style="font-style: italic">u</span></span> three points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, are taken +such that <span class="tei tei-hi"><span style="font-style: italic">B</span></span> is the middle point of the segment <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, +it does not follow that the three points <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> +in a point-row perspective to <span class="tei tei-hi"><span style="font-style: italic">u</span></span> will be so related. +Relations between the elements of any form which do +go over unaltered to the corresponding elements of +a form projectively related to it are called <span class="tei tei-hi"><span style="font-style: italic">projective +relations.</span></span> Relations involving measurement of lines or +of angles are not projective.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc59" id="toc59"></a><a name="pdf60" id="pdf60"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p25" id="p25" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">25. Desargues's theorem.</span></span> We consider first the following +beautiful theorem, due to Desargues and called +by his name.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If two triangles, </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">A'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">C'</span></span><span style="font-style: italic">, are so situated +that the lines </span><span class="tei tei-hi"><span style="font-style: italic">AA'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">BB'</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">CC'</span></span><span style="font-style: italic"> all meet in a point, then +the pairs of sides </span><span class="tei tei-hi"><span style="font-style: italic">AB</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">BC</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">B'C'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">CA</span></span><span style="font-style: italic"> and +</span><span class="tei tei-hi"><span style="font-style: italic">C'A'</span></span><span style="font-style: italic"> all meet on a straight line, and conversely.</span></span></p> + +<span class="tei tei-pb" id="page16">[pg 16]</span><a name="Pg16" id="Pg16" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image03.png" width="480" height="399" alt="Figure 3" title="Fig. 3" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 3</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Let the lines <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span> meet in the point <span class="tei tei-hi"><span style="font-style: italic">M</span></span> +(Fig. 3). Conceive of the figure as in space, so that +<span class="tei tei-hi"><span style="font-style: italic">M</span></span> is the vertex of a trihedral angle of which the given +triangles are plane sections. The lines <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span> are +in the same plane and must meet when produced, their +point of intersection +being clearly a point +in the plane of each +triangle and therefore +in the line of +intersection of these +two planes. Call this +point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. By similar +reasoning the point +<span class="tei tei-hi"><span style="font-style: italic">Q</span></span> of intersection of +the lines <span class="tei tei-hi"><span style="font-style: italic">BC</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">B'C'</span></span> must lie on this same line as well as the point <span class="tei tei-hi"><span style="font-style: italic">R</span></span> +of intersection of <span class="tei tei-hi"><span style="font-style: italic">CA</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C'A'</span></span>. Therefore the points +<span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">R</span></span> all lie on the same line <span class="tei tei-hi"><span style="font-style: italic">m</span></span>. If now we consider +the figure a plane figure, the points <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">R</span></span> +still all lie on a straight line, which proves the theorem. +The converse is established in the same manner.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc61" id="toc61"></a><a name="pdf62" id="pdf62"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p26" id="p26" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">26. Fundamental theorem concerning two complete +quadrangles.</span></span> This theorem throws into our hands the +following fundamental theorem concerning two complete +quadrangles, a <span class="tei tei-hi"><span style="font-style: italic">complete quadrangle</span></span> being defined +as the figure obtained by joining any four given points +by straight lines in the six possible ways.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given two complete quadrangles, </span><span class="tei tei-hi"><span style="font-style: italic">K</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">L</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N</span></span><span style="font-style: italic"> and +</span><span class="tei tei-hi"><span style="font-style: italic">K'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">L'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N'</span></span><span style="font-style: italic">, so related that </span><span class="tei tei-hi"><span style="font-style: italic">KL</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">K'L'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">MN</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M'N'</span></span><span style="font-style: italic"> all +meet in a point </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">; </span><span class="tei tei-hi"><span style="font-style: italic">LM</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">L'M'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">NK</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N'K'</span></span><span style="font-style: italic"> all meet in a +</span><span class="tei tei-pb" id="page17">[pg 17]</span><a name="Pg17" id="Pg17" class="tei tei-anchor"></a><span style="font-style: italic"> +point </span><span class="tei tei-hi"><span style="font-style: italic">Q</span></span><span style="font-style: italic">; and </span><span class="tei tei-hi"><span style="font-style: italic">LN</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">L'N'</span></span><span style="font-style: italic"> meet in a point </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic"> on the line +</span><span class="tei tei-hi"><span style="font-style: italic">AC</span></span><span style="font-style: italic">; then the lines </span><span class="tei tei-hi"><span style="font-style: italic">KM</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">K'M'</span></span><span style="font-style: italic"> also meet in a point </span><span class="tei tei-hi"><span style="font-style: italic">D</span></span><span style="font-style: italic"> +on the line </span><span class="tei tei-hi"><span style="font-style: italic">AC</span></span><span style="font-style: italic">.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image04.png" width="804" height="646" alt="Figure 4" title="Fig. 4" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 4</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">For, by the converse of the last theorem, <span class="tei tei-hi"><span style="font-style: italic">KK'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">LL'</span></span>, +and <span class="tei tei-hi"><span style="font-style: italic">NN'</span></span> all meet in a point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> (Fig. 4). Also <span class="tei tei-hi"><span style="font-style: italic">LL'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">MM'</span></span>, +and <span class="tei tei-hi"><span style="font-style: italic">NN'</span></span> meet in a point, and therefore in the same +point <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. Thus <span class="tei tei-hi"><span style="font-style: italic">KK'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">LL'</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">MM'</span></span> meet in a point, +and so, by Desargues's theorem itself, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are +on a straight line.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc63" id="toc63"></a><a name="pdf64" id="pdf64"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p27" id="p27" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">27. Importance of the theorem.</span></span> The importance of +this theorem lies in the fact that, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> being +given, an indefinite number of quadrangles <span class="tei tei-hi"><span style="font-style: italic">K'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N'</span></span> +my be found such that <span class="tei tei-hi"><span style="font-style: italic">K'L'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M'N'</span></span> meet in <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">K'N'</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">L'M'</span></span> in <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, with <span class="tei tei-hi"><span style="font-style: italic">L'N'</span></span> passing through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. Indeed, +the lines <span class="tei tei-hi"><span style="font-style: italic">AK'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">AM'</span></span> may be drawn arbitrarily +through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, and any line through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> may be used to +determine <span class="tei tei-hi"><span style="font-style: italic">L'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N'</span></span>. By joining these two points to +<span class="tei tei-hi"><span style="font-style: italic">C</span></span> the points <span class="tei tei-hi"><span style="font-style: italic">K'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M'</span></span> are determined. Then the line +<span class="tei tei-pb" id="page18">[pg 18]</span><a name="Pg18" id="Pg18" class="tei tei-anchor"></a> +joining <span class="tei tei-hi"><span style="font-style: italic">K'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M'</span></span>, found in this way, must pass +through the point <span class="tei tei-hi"><span style="font-style: italic">D</span></span> already determined by the quadrangle +<span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. <span class="tei tei-hi"><span style="font-style: italic">The three points </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic">, given in +order, serve thus to determine a fourth point </span><span class="tei tei-hi"><span style="font-style: italic">D</span></span><span style="font-style: italic">.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc65" id="toc65"></a><a name="pdf66" id="pdf66"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p28" id="p28" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">28.</span></span> In a complete quadrangle the line joining any +two points is called the <span class="tei tei-hi"><span style="font-style: italic">opposite side</span></span> to the line joining +the other two points. The result of the preceding +paragraph may then be stated as follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Given three points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, in a straight line, if a +pair of opposite sides of a complete quadrangle pass +through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, and another pair through <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, and one of the +remaining two sides goes through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, then the other of +the remaining two sides will go through a fixed point +which does not depend on the quadrangle employed.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc67" id="toc67"></a><a name="pdf68" id="pdf68"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p29" id="p29" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">29. Four harmonic points.</span></span> Four points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, +related as in the preceding theorem are called <span class="tei tei-hi"><span style="font-style: italic">four +harmonic points</span></span>. The point <span class="tei tei-hi"><span style="font-style: italic">D</span></span> is called the <span class="tei tei-hi"><span style="font-style: italic">fourth harmonic +of </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic"> with respect to </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span></span>. Since <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> play +exactly the same rôle in the above construction, <span class="tei tei-hi"><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic"> is +also the fourth harmonic of </span><span class="tei tei-hi"><span style="font-style: italic">D</span></span><span style="font-style: italic"> with respect to </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span></span>. +<span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are called <span class="tei tei-hi"><span style="font-style: italic">harmonic conjugates with respect to +</span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span></span>. We proceed to show that <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> are also +harmonic conjugates with respect to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>—that is, +that it is possible to find a quadrangle of which two +opposite sides shall pass through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, two through <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, +and of the remaining pair, one through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and the other +through <span class="tei tei-hi"><span style="font-style: italic">C</span></span>.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image05.png" width="679" height="454" alt="Figure 5" title="Fig. 5" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 5</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Let <span class="tei tei-hi"><span style="font-style: italic">O</span></span> be the intersection of <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> and <span class="tei tei-hi"><span style="font-style: italic">LN</span></span> (Fig. 5). +Join <span class="tei tei-hi"><span style="font-style: italic">O</span></span> to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. The joining lines cut out on the +sides of the quadrangle four points, <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, <span class="tei tei-hi"><span style="font-style: italic">R</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. Consider +the quadrangle <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, <span class="tei tei-hi"><span style="font-style: italic">O</span></span>. One pair of opposite sides +<span class="tei tei-pb" id="page19">[pg 19]</span><a name="Pg19" id="Pg19" class="tei tei-anchor"></a> +passes through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, one through <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, and one remaining side +through <span class="tei tei-hi"><span style="font-style: italic">D</span></span>; therefore the other remaining side must +pass through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. Similarly, <span class="tei tei-hi"><span style="font-style: italic">RS</span></span> passes through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">PS</span></span> and <span class="tei tei-hi"><span style="font-style: italic">QR</span></span> pass +through <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. The +quadrangle <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">R</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S</span></span> therefore +has two opposite +sides through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, +two through <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, +and the remaining +pair through +<span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">C</span></span> are thus harmonic conjugates with respect to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. +We may sum up the discussion, therefore, as follows:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc69" id="toc69"></a><a name="pdf70" id="pdf70"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p30" id="p30" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">30.</span></span> If <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> are harmonic conjugates with respect +to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are harmonic conjugates with +respect to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc71" id="toc71"></a><a name="pdf72" id="pdf72"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p31" id="p31" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">31. Importance of the notion.</span></span> The importance of the +notion of four harmonic points lies in the fact that it +is a relation which is carried over from four points in +a point-row <span class="tei tei-hi"><span style="font-style: italic">u</span></span> to the four points that correspond to +them in any point-row <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> perspective to <span class="tei tei-hi"><span style="font-style: italic">u</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">To prove this statement we construct a quadrangle +<span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span> such that <span class="tei tei-hi"><span style="font-style: italic">KL</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MN</span></span> pass through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">KN</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">LM</span></span> through <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">LN</span></span> through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> through <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. +Take now any point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> not in the plane of the quadrangle +and construct the planes determined by <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and +all the seven lines of the figure. Cut across this set of +planes by another plane not passing through <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. This +plane cuts out on the set of seven planes another +<span class="tei tei-pb" id="page20">[pg 20]</span><a name="Pg20" id="Pg20" class="tei tei-anchor"></a> +quadrangle which determines four new harmonic points, +<span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D'</span></span>, on the lines joining <span class="tei tei-hi"><span style="font-style: italic">S</span></span> to <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. But +<span class="tei tei-hi"><span style="font-style: italic">S</span></span> may be taken as any point, since the original quadrangle +may be taken in any plane through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>; +and, further, the points <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> are the intersection +of <span class="tei tei-hi"><span style="font-style: italic">SA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SD</span></span> by any line. We have, then, the +remarkable theorem:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc73" id="toc73"></a><a name="pdf74" id="pdf74"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p32" id="p32" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">32.</span></span> <span class="tei tei-hi"><span style="font-style: italic">If any point is joined to four harmonic points, and +the four lines thus obtained are cut by any fifth, the four +points of intersection are again harmonic.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc75" id="toc75"></a><a name="pdf76" id="pdf76"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p33" id="p33" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">33. Four harmonic lines.</span></span> We are now able to extend +the notion of harmonic elements to pencils of rays, and +indeed to axial pencils. For if we define <span class="tei tei-hi"><span style="font-style: italic">four harmonic +rays</span></span> as four rays which pass through a point and which +pass one through each of four harmonic points, we have +the theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Four harmonic lines are cut by any transversal in four +harmonic points.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc77" id="toc77"></a><a name="pdf78" id="pdf78"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p34" id="p34" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">34. Four harmonic planes.</span></span> We also define <span class="tei tei-hi"><span style="font-style: italic">four harmonic +planes</span></span> as four planes through a line which pass +one through each of four harmonic points, and we may +show that</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Four harmonic planes are cut by any plane not passing +through their common line in four harmonic lines, and also +by any line in four harmonic points.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">For let the planes α, β, γ, δ, which all pass through +the line <span class="tei tei-hi"><span style="font-style: italic">g</span></span>, pass also through the four harmonic points +<span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, so that α passes through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, etc. Then it is +clear that any plane π through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> will cut out +four harmonic lines from the four planes, for they are +<span class="tei tei-pb" id="page21">[pg 21]</span><a name="Pg21" id="Pg21" class="tei tei-anchor"></a> +lines through the intersection <span class="tei tei-hi"><span style="font-style: italic">P</span></span> of <span class="tei tei-hi"><span style="font-style: italic">g</span></span> with the plane +π, and they pass through the given harmonic points +<span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. Any other plane σ cuts <span class="tei tei-hi"><span style="font-style: italic">g</span></span> in a point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and +cuts α, β, γ, δ in four lines that meet π +in four points <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> lying on <span class="tei tei-hi"><span style="font-style: italic">PA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">PB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">PC</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">PD</span></span> respectively, +and are thus four harmonic hues. Further, any +ray cuts α, β, γ, δ in four harmonic points, since any +plane through the ray gives four harmonic lines of +intersection.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc79" id="toc79"></a><a name="pdf80" id="pdf80"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p35" id="p35" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">35.</span></span> These results may be put together as follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given any two assemblages of points, rays, or planes, +perspectively related to each other, four harmonic elements +of one must correspond to four elements of the other which +are likewise harmonic.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If, now, two forms are perspectively related to a third, +any four harmonic elements of one must correspond to +four harmonic elements in the other. We take this as +our definition of projective correspondence, and say:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc81" id="toc81"></a><a name="pdf82" id="pdf82"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p36" id="p36" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">36. Definition of projectivity.</span></span> <span class="tei tei-hi"><span style="font-style: italic">Two fundamental forms +are protectively related to each other when a one-to-one correspondence +exists between the elements of the two and when +four harmonic elements of one correspond to four harmonic +elements of the other.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image06.png" width="678" height="426" alt="Figure 6" title="Fig. 6" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 6</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc83" id="toc83"></a><a name="pdf84" id="pdf84"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p37" id="p37" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">37. Correspondence between harmonic conjugates.</span></span> Given +four harmonic points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>; if we fix <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, +then <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> vary together in a way that should be +thoroughly understood. To get a clear conception of +their relative motion we may fix the points <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> of +the quadrangle <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span> (Fig. 6). Then, as <span class="tei tei-hi"><span style="font-style: italic">B</span></span> describes +the point-row <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, the point <span class="tei tei-hi"><span style="font-style: italic">N</span></span> describes the point-row +<span class="tei tei-pb" id="page22">[pg 22]</span><a name="Pg22" id="Pg22" class="tei tei-anchor"></a> +<span class="tei tei-hi"><span style="font-style: italic">AM</span></span> perspective to it. Projecting <span class="tei tei-hi"><span style="font-style: italic">N</span></span> again from <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, we +get a point-row <span class="tei tei-hi"><span style="font-style: italic">K</span></span> on <span class="tei tei-hi"><span style="font-style: italic">AL</span></span> perspective to the point-row +<span class="tei tei-hi"><span style="font-style: italic">N</span></span> and thus projective to the point-row <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. Project the +point-row <span class="tei tei-hi"><span style="font-style: italic">K</span></span> from <span class="tei tei-hi"><span style="font-style: italic">M</span></span> and we get a point-row <span class="tei tei-hi"><span style="font-style: italic">D</span></span> on +<span class="tei tei-hi"><span style="font-style: italic">AC</span></span> again, which is projective to the point-row <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. For +every point <span class="tei tei-hi"><span style="font-style: italic">B</span></span> we have thus one and only one point +<span class="tei tei-hi"><span style="font-style: italic">D</span></span>, and conversely. +In other words, we +have set up a one-to-one +correspondence +between the +points of a single +point-row, which is +also a projective +correspondence because +four harmonic +points <span class="tei tei-hi"><span style="font-style: italic">B</span></span> correspond to four harmonic points <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. +We may note also that the correspondence is here characterized +by a feature which does not always appear in +projective correspondences: namely, the same process +that carries one from <span class="tei tei-hi"><span style="font-style: italic">B</span></span> to <span class="tei tei-hi"><span style="font-style: italic">D</span></span> will carry one back from +<span class="tei tei-hi"><span style="font-style: italic">D</span></span> to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> again. This special property will receive further +study in the chapter on Involution.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc85" id="toc85"></a><a name="pdf86" id="pdf86"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p38" id="p38" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">38.</span></span> It is seen that as <span class="tei tei-hi"><span style="font-style: italic">B</span></span> approaches <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> also approaches +<span class="tei tei-hi"><span style="font-style: italic">A</span></span>. As <span class="tei tei-hi"><span style="font-style: italic">B</span></span> moves from <span class="tei tei-hi"><span style="font-style: italic">A</span></span> toward <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> moves +from <span class="tei tei-hi"><span style="font-style: italic">A</span></span> in the opposite direction, passing through the +point at infinity on the line <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, and returns on the +other side to meet <span class="tei tei-hi"><span style="font-style: italic">B</span></span> at <span class="tei tei-hi"><span style="font-style: italic">C</span></span> again. In other words, as <span class="tei tei-hi"><span style="font-style: italic">B</span></span> +traverses <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> traverses the rest of the line from <span class="tei tei-hi"><span style="font-style: italic">A</span></span> to +<span class="tei tei-hi"><span style="font-style: italic">C</span></span> through infinity. In all positions of <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, except at <span class="tei tei-hi"><span style="font-style: italic">A</span></span> or +<span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are separated from each other by <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>.</p> + +<span class="tei tei-pb" id="page23">[pg 23]</span><a name="Pg23" id="Pg23" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc87" id="toc87"></a><a name="pdf88" id="pdf88"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p39" id="p39" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">39. Harmonic conjugate of the point at infinity.</span></span> It is +natural to inquire what position of <span class="tei tei-hi"><span style="font-style: italic">B</span></span> corresponds to the +infinitely distant position of <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. We have proved (§ 27) +that the particular quadrangle <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span> employed is +of no consequence. We shall therefore avail ourselves of +one that lends itself most readily to +the solution of the problem. We +choose the point <span class="tei tei-hi"><span style="font-style: italic">L</span></span> so that the triangle +<span class="tei tei-hi"><span style="font-style: italic">ALC</span></span> is isosceles (Fig. 7). Since +<span class="tei tei-hi"><span style="font-style: italic">D</span></span> is supposed to be at infinity, the +line <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> is parallel to <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>. Therefore +the triangles <span class="tei tei-hi"><span style="font-style: italic">KAC</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MAC</span></span> +are equal, and the triangle <span class="tei tei-hi"><span style="font-style: italic">ANC</span></span> is also isosceles. The +triangles <span class="tei tei-hi"><span style="font-style: italic">CNL</span></span> and <span class="tei tei-hi"><span style="font-style: italic">ANL</span></span> are therefore equal, and the line +<span class="tei tei-hi"><span style="font-style: italic">LB</span></span> bisects the angle <span class="tei tei-hi"><span style="font-style: italic">ALC</span></span>. <span class="tei tei-hi"><span style="font-style: italic">B</span></span> is therefore the middle +point of <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, and we have the theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The harmonic conjugate of the middle point of </span><span class="tei tei-hi"><span style="font-style: italic">AC</span></span><span style="font-style: italic"> is at +infinity.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image07.png" width="357" height="289" alt="Figure 7" title="Fig. 7" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 7</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc89" id="toc89"></a><a name="pdf90" id="pdf90"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p40" id="p40" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">40. Projective theorems and metrical theorems. Linear +construction.</span></span> This theorem is the connecting link between +the general protective theorems which we have +been considering so far and the metrical theorems of +ordinary geometry. Up to this point we have said nothing +about measurements, either of line segments or of +angles. Desargues's theorem and the theory of harmonic +elements which depends on it have nothing to do with +magnitudes at all. Not until the notion of an infinitely +distant point is brought in is any mention made of +distances or directions. We have been able to make +all of our constructions up to this point by means of +the straightedge, or ungraduated ruler. A construction +<span class="tei tei-pb" id="page24">[pg 24]</span><a name="Pg24" id="Pg24" class="tei tei-anchor"></a> +made with such an instrument we shall call a <span class="tei tei-hi"><span style="font-style: italic">linear</span></span> +construction. It requires merely that we be able to +draw the line joining two points or find the point of +intersection of two lines.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc91" id="toc91"></a><a name="pdf92" id="pdf92"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p41" id="p41" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">41. Parallels and mid-points.</span></span> It might be thought +that drawing a line through a given point parallel to +a given line was only a special case of drawing a line +joining two points. Indeed, it consists only in drawing +a line through the given point and through the +"infinitely distant point" on the given line. It must +be remembered, however, that the expression "infinitely +distant point" must not be taken literally. When we +say that two parallel lines meet "at infinity," we really +mean that they do not meet at all, and the only reason +for using the expression is to avoid tedious statement +of exceptions and restrictions to our theorems. We +ought therefore to consider the drawing of a line parallel +to a given line as a different accomplishment from +the drawing of the line joining two given points. It is +a remarkable consequence of the last theorem that a +parallel to a given line and the mid-point of a given +segment are equivalent data. For the construction is +reversible, and if we are given the middle point of a +given segment, we can construct <span class="tei tei-hi"><span style="font-style: italic">linearly</span></span> a line parallel to +that segment. Thus, given that <span class="tei tei-hi"><span style="font-style: italic">B</span></span> is the middle point of +<span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, we may draw any two lines through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, and any line +through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> cutting them in points <span class="tei tei-hi"><span style="font-style: italic">N</span></span> and <span class="tei tei-hi"><span style="font-style: italic">L</span></span>. Join <span class="tei tei-hi"><span style="font-style: italic">N</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">L</span></span> to <span class="tei tei-hi"><span style="font-style: italic">C</span></span> and get the points <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> on the two lines +through <span class="tei tei-hi"><span style="font-style: italic">A</span></span>. Then <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> is parallel to <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>. <span class="tei tei-hi"><span style="font-style: italic">The bisection of +a given segment and the drawing of a line parallel to the +segment are equivalent data when linear construction is used.</span></span></p> + +<span class="tei tei-pb" id="page25">[pg 25]</span><a name="Pg25" id="Pg25" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc93" id="toc93"></a><a name="pdf94" id="pdf94"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p42" id="p42" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">42.</span></span> It is not difficult to give a linear construction +for the problem to divide a given segment into <span class="tei tei-hi"><span style="font-style: italic">n</span></span> equal +parts, given only a parallel to the segment. This is +simple enough when <span class="tei tei-hi"><span style="font-style: italic">n</span></span> is a power of <span class="tei tei-hi"><span style="font-style: italic">2</span></span>. For any other +number, such as <span class="tei tei-hi"><span style="font-style: italic">29</span></span>, divide any segment on the line +parallel to <span class="tei tei-hi"><span style="font-style: italic">AC</span></span> into <span class="tei tei-hi"><span style="font-style: italic">32</span></span> equal parts, by a repetition of +the process just described. Take <span class="tei tei-hi"><span style="font-style: italic">29</span></span> of these, and join +the first to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and the last to <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. Let these joining lines +meet in <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. Join <span class="tei tei-hi"><span style="font-style: italic">S</span></span> to all the other points. Other +problems, of a similar sort, are given at the end of +the chapter.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc95" id="toc95"></a><a name="pdf96" id="pdf96"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p43" id="p43" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">43. Numerical relations.</span></span> Since three points, given in +order, are sufficient to determine a fourth, as explained +above, it ought to be possible to reproduce the process +numerically in view of the one-to-one correspondence +which exists between points on a line and numbers; a +correspondence which, to be sure, we have not established +here, but which is discussed in any treatise +on the theory of point sets. We proceed to discover +what relation between four numbers corresponds to the +harmonic relation between +four points.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image08.png" width="489" height="371" alt="Figure 8" title="Fig. 8" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 8</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc97" id="toc97"></a><a name="pdf98" id="pdf98"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p44" id="p44" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">44.</span></span> Let <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> be four +harmonic points (Fig. 8), and +let <span class="tei tei-hi"><span style="font-style: italic">SA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SD</span></span> be four +harmonic lines. Assume a +line drawn through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> parallel +to <span class="tei tei-hi"><span style="font-style: italic">SD</span></span>, meeting <span class="tei tei-hi"><span style="font-style: italic">SA</span></span> in <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">SC</span></span> in <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>. Then <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, and the infinitely distant +point on <span class="tei tei-hi"><span style="font-style: italic">A'C'</span></span> are four harmonic points, and therefore +<span class="tei tei-hi"><span style="font-style: italic">B</span></span> is the middle point of the segment <span class="tei tei-hi"><span style="font-style: italic">A'C'</span></span>. Then, since +<span class="tei tei-pb" id="page26">[pg 26]</span><a name="Pg26" id="Pg26" class="tei tei-anchor"></a> +the triangle <span class="tei tei-hi"><span style="font-style: italic">DAS</span></span> is similar to the triangle <span class="tei tei-hi"><span style="font-style: italic">BAA'</span></span>, we +may write the proportion</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AB : AD = BA' : SD.</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Also, from the similar triangles <span class="tei tei-hi"><span style="font-style: italic">DSC</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BCC'</span></span>, we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">CD : CB = SD : B'C.</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">From these two proportions we have, remembering that +<span class="tei tei-hi"><span style="font-style: italic">BA' = BC'</span></span>,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/1.png" alt="[formula]" width="100" height="31" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">the minus sign being given to the ratio on account of the +fact that <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> are always separated from <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, +so that one or three of the segments <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">CD</span></span>, <span class="tei tei-hi"><span style="font-style: italic">AD</span></span>, <span class="tei tei-hi"><span style="font-style: italic">CB</span></span> +must be negative.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc99" id="toc99"></a><a name="pdf100" id="pdf100"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p45" id="p45" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">45.</span></span> Writing the last equation in the form</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">CB : AB = -CD : AD,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and using the fundamental relation connecting three +points on a line,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">PR + RQ = PQ,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">which holds for all positions of the three points if +account be taken of the sign of the segments, the last +proportion may be written</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">(CB - BA) : AB = -(CA - DA) : AD,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">(AB - AC) : AB = (AC - AD) : AD;</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">so that <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> are three quantities in hamonic +progression, since the difference between the first +and second is to the first as the difference between the +second and third is to the third. Also, from this last +proportion comes the familiar relation</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/2.png" alt="[formula]" width="116" height="30" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">which is convenient for the computation of the distance +<span class="tei tei-hi"><span style="font-style: italic">AD</span></span> when <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">AC</span></span> are given numerically.</p> + +<span class="tei tei-pb" id="page27">[pg 27]</span><a name="Pg27" id="Pg27" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc101" id="toc101"></a><a name="pdf102" id="pdf102"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p46" id="p46" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">46. Anharmonic ratio.</span></span> The corresponding relations +between the trigonometric functions of the angles determined +by four harmonic lines are not difficult to obtain, +but as we shall not need them in building up the +theory of projective geometry, we will not discuss them +here. Students who have a slight acquaintance with +trigonometry may read in a later chapter (§ 161) a +development of the theory of a more general relation, +called the <span class="tei tei-hi"><span style="font-style: italic">anharmonic ratio</span></span>, or <span class="tei tei-hi"><span style="font-style: italic">cross ratio</span></span>, which connects +any four points on a line.</p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc103" id="toc103"></a><a name="pdf104" id="pdf104"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">1</span></span>. Draw through a given point a line which shall pass +through the inaccessible point of intersection of two given +lines. The following construction may be made to depend +upon Desargues's theorem: Through the given point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> draw +any two rays cutting the two lines in the points <span class="tei tei-hi"><span style="font-style: italic">AB'</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">A'B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, lying on one of the given lines and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, on the +other. Join <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span>, and find their point of intersection +<span class="tei tei-hi"><span style="font-style: italic">S</span></span>. Through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> draw any other ray, cutting the given +lines in <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span>. Join <span class="tei tei-hi"><span style="font-style: italic">BC'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'C</span></span>, and obtain their point +of intersection <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>. <span class="tei tei-hi"><span style="font-style: italic">PQ</span></span> is the desired line. Justify this +construction.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">2.</span></span> To draw through a given point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> a line which shall +meet two given lines in points <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, equally distant from +<span class="tei tei-hi"><span style="font-style: italic">P</span></span>. Justify the following construction: Join <span class="tei tei-hi"><span style="font-style: italic">P</span></span> to the point +<span class="tei tei-hi"><span style="font-style: italic">S</span></span> of intersection of the two given lines. Construct the +fourth harmonic of <span class="tei tei-hi"><span style="font-style: italic">PS</span></span> with respect to the two given lines. +Draw through <span class="tei tei-hi"><span style="font-style: italic">P</span></span> a line parallel to this line. This is the +required line.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">3.</span></span> Given a parallelogram in the same plane with a given +segment <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, to construct linearly the middle point of <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>.</p> + +<span class="tei tei-pb" id="page28">[pg 28]</span><a name="Pg28" id="Pg28" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">4.</span></span> Given four harmonic lines, of which one pair are at +right angles to each other, show that the other pair make +equal angles with them. This is a theorem of which frequent +use will be made.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">5.</span></span> Given the middle point of a line segment, to draw a +line parallel to the segment and passing through a given +point.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">6.</span></span> A line is drawn cutting the sides of a triangle <span class="tei tei-hi"><span style="font-style: italic">ABC</span></span> in +the points <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> the point <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> lying on the side <span class="tei tei-hi"><span style="font-style: italic">BC</span></span>, etc. +The harmonic conjugate of <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> with respect to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> is +then constructed and called <span class="tei tei-hi"><span style="font-style: italic">A"</span></span>. Similarly, <span class="tei tei-hi"><span style="font-style: italic">B"</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C"</span></span> are +constructed. Show that <span class="tei tei-hi"><span style="font-style: italic">A"B"C"</span></span> lie on a straight line. Find +other sets of three points on a line in the figure. Find also +sets of three lines through a point.</p> +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<span class="tei tei-pb" id="page29">[pg 29]</span><a name="Pg29" id="Pg29" class="tei tei-anchor"></a> +<a name="toc105" id="toc105"></a><a name="pdf106" id="pdf106"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED +FUNDAMENTAL FORMS</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc107" id="toc107"></a><a name="pdf108" id="pdf108"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image09.png" width="557" height="561" alt="Figure 9" title="Fig. 9" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 9</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p47" id="p47" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">47. Superposed fundamental forms. Self-corresponding +elements.</span></span> We have seen (§ 37) that two projective +point-rows may be superposed upon the same straight +line. This happens, for example, when two pencils +which are projective to each other are cut across by +a straight line. It is also possible for two projective +pencils to have the same center. This happens, for +example, when two projective point-rows are projected +to the same point. Similarly, two projective axial pencils +may have the same axis. We examine now the +possibility of two forms related in this way, having +an element or elements that correspond to themselves. +We have seen, indeed, that if <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are harmonic +conjugates with respect to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, then the point-row +described by <span class="tei tei-hi"><span style="font-style: italic">B</span></span> is projective to the point-row described +by <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, and that <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> are self-corresponding +points. Consider more generally the case of two pencils +perspective to each other with axis of perspectivity <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> +(Fig. 9). Cut across them by a line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. We get thus +two projective point-rows superposed on the same line +<span class="tei tei-hi"><span style="font-style: italic">u</span></span>, and a moment's reflection serves to show that the +point <span class="tei tei-hi"><span style="font-style: italic">N</span></span> of intersection <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> corresponds to itself +in the two point-rows. Also, the point <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, where <span class="tei tei-hi"><span style="font-style: italic">u</span></span> +<span class="tei tei-pb" id="page30">[pg 30]</span><a name="Pg30" id="Pg30" class="tei tei-anchor"></a> +intersects the line joining the centers of the two pencils, +is seen to correspond to itself. It is thus possible +for two projective point-rows, +superposed upon +the same line, to have two +self-corresponding points. +Clearly <span class="tei tei-hi"><span style="font-style: italic">M</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> may +fall together if the line +joining the centers of the +pencils happens to pass +through the point of intersection +of the lines <span class="tei tei-hi"><span style="font-style: italic">u</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image10.png" width="361" height="536" alt="Figure 10" title="Fig. 10" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 10</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc109" id="toc109"></a><a name="pdf110" id="pdf110"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p48" id="p48" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">48.</span></span> We may also give an illustration of a case +where two superposed projective point-rows have no +self-corresponding points at all. Thus we may take +two lines revolving about a fixed +point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and always making the +same angle a with each other +(Fig. 10). They will cut out on +any line <span class="tei tei-hi"><span style="font-style: italic">u</span></span> in the plane two point-rows +which are easily seen to be +projective. For, given any four +rays <span class="tei tei-hi"><span style="font-style: italic">SP</span></span> which are harmonic, the +four corresponding rays <span class="tei tei-hi"><span style="font-style: italic">SP'</span></span> must +also be harmonic, since they make +the same angles with each other. +Four harmonic points <span class="tei tei-hi"><span style="font-style: italic">P</span></span> correspond, +therefore, to four harmonic points <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>. It is clear, +however, that no point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> can coincide with its corresponding +point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, for in that case the lines <span class="tei tei-hi"><span style="font-style: italic">PS</span></span> and +<span class="tei tei-pb" id="page31">[pg 31]</span><a name="Pg31" id="Pg31" class="tei tei-anchor"></a> +<span class="tei tei-hi"><span style="font-style: italic">P'S</span></span> would coincide, which is impossible if the angle +between them is to be constant.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc111" id="toc111"></a><a name="pdf112" id="pdf112"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p49" id="p49" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">49. Fundamental theorem. Postulate of continuity.</span></span> +We have thus shown that two projective point-rows, +superposed one on the other, may have two points, one +point, or no point at all corresponding to themselves. +We proceed to show that</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If two projective point-rows, superposed upon the same +straight line, have more than two self-corresponding points, +they must have an infinite number, and every point corresponds +to itself; that is, the two point-rows are not +essentially distinct.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If three points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, are self-corresponding, +then the harmonic conjugate <span class="tei tei-hi"><span style="font-style: italic">D</span></span> of <span class="tei tei-hi"><span style="font-style: italic">B</span></span> with respect to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> must also correspond to itself. For four harmonic +points must always correspond to four harmonic points. +In the same way the harmonic conjugate of <span class="tei tei-hi"><span style="font-style: italic">D</span></span> with +respect to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> must correspond to itself. Combining +new points with old in this way, we may obtain as many +self-corresponding points as we wish. We show further +that every point on the line is the limiting point of a +finite or infinite sequence of self-corresponding points. +Thus, let a point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> lie between <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. Construct +now <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, the fourth harmonic of <span class="tei tei-hi"><span style="font-style: italic">C</span></span> with respect to <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">B</span></span>. <span class="tei tei-hi"><span style="font-style: italic">D</span></span> may coincide with <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, in which case the sequence +is closed; otherwise <span class="tei tei-hi"><span style="font-style: italic">P</span></span> lies in the stretch <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> or in the +stretch <span class="tei tei-hi"><span style="font-style: italic">DB</span></span>. If it lies in the stretch <span class="tei tei-hi"><span style="font-style: italic">DB</span></span>, construct the +fourth harmonic of <span class="tei tei-hi"><span style="font-style: italic">C</span></span> with respect to <span class="tei tei-hi"><span style="font-style: italic">D</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. This +point <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> may coincide with <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, in which case, as before, +the sequence is closed. If <span class="tei tei-hi"><span style="font-style: italic">P</span></span> lies in the stretch <span class="tei tei-hi"><span style="font-style: italic">DD'</span></span>, +we construct the fourth harmonic of <span class="tei tei-hi"><span style="font-style: italic">C</span></span> with respect +<span class="tei tei-pb" id="page32">[pg 32]</span><a name="Pg32" id="Pg32" class="tei tei-anchor"></a> +to <span class="tei tei-hi"><span style="font-style: italic">DD'</span></span>, etc. In each step the region in which <span class="tei tei-hi"><span style="font-style: italic">P</span></span> lies is +diminished, and the process may be continued until two +self-corresponding points are obtained on either side of +<span class="tei tei-hi"><span style="font-style: italic">P</span></span>, and at distances from it arbitrarily small.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">We now assume, explicitly, the fundamental postulate +that the correspondence is <span class="tei tei-hi"><span style="font-style: italic">continuous</span></span>, that is, that <span class="tei tei-hi"><span style="font-style: italic">the +distance between two points in one point-row may be made +arbitrarily small by sufficiently diminishing the distance +between the corresponding points in the other.</span></span> Suppose +now that <span class="tei tei-hi"><span style="font-style: italic">P</span></span> is not a self-corresponding point, but corresponds +to a point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> at a fixed distance <span class="tei tei-hi"><span style="font-style: italic">d</span></span> from <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. +As noted above, we can find self-corresponding points +arbitrarily close to <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, and it appears, then, that we can +take a point <span class="tei tei-hi"><span style="font-style: italic">D</span></span> as close to <span class="tei tei-hi"><span style="font-style: italic">P</span></span> as we wish, and yet the +distance between the corresponding points <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> +approaches <span class="tei tei-hi"><span style="font-style: italic">d</span></span> as a limit, and not zero, which contradicts +the postulate of continuity.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc113" id="toc113"></a><a name="pdf114" id="pdf114"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p50" id="p50" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">50.</span></span> It follows also that two projective pencils which +have the same center may have no more than two self-corresponding +rays, unless the pencils are identical. For +if we cut across them by a line, we obtain two projective +point-rows superposed on the same straight line, +which may have no more than two self-corresponding +points. The same considerations apply to two projective +axial pencils which have the same axis.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc115" id="toc115"></a><a name="pdf116" id="pdf116"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p51" id="p51" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">51. Projective point-rows having a self-corresponding +point in common.</span></span> Consider now two projective point-rows +lying on different lines in the same plane. Their +common point may or may not be a self-corresponding +point. If the two point-rows are perspectively related, +then their common point is evidently a self-corresponding +<span class="tei tei-pb" id="page33">[pg 33]</span><a name="Pg33" id="Pg33" class="tei tei-anchor"></a> +point. The converse is also true, and we have the very +important theorem:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc117" id="toc117"></a><a name="pdf118" id="pdf118"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p52" id="p52" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">52.</span></span> <span class="tei tei-hi"><span style="font-style: italic">If in two protective point-rows, the point of intersection +corresponds to itself, then the point-rows are in +perspective position.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image11.png" width="404" height="300" alt="Figure 11" title="Fig. 11" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 11</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Let the two point-rows be <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> (Fig. 11). Let +<span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, be corresponding points, and let +also the point <span class="tei tei-hi"><span style="font-style: italic">M</span></span> of intersection of <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> correspond +to itself. Let <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> meet in the point <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. Take +<span class="tei tei-hi"><span style="font-style: italic">S</span></span> as the center of two pencils, +one perspective to <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and the other +perspective to <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>. In these two +pencils <span class="tei tei-hi"><span style="font-style: italic">SA</span></span> coincides with its corresponding +ray <span class="tei tei-hi"><span style="font-style: italic">SA'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SB</span></span> with its +corresponding ray <span class="tei tei-hi"><span style="font-style: italic">SB'</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">SM</span></span> +with its corresponding ray <span class="tei tei-hi"><span style="font-style: italic">SM'</span></span>. +The two pencils are thus identical, by the preceding +theorem, and any ray <span class="tei tei-hi"><span style="font-style: italic">SD</span></span> must coincide with its corresponding +ray <span class="tei tei-hi"><span style="font-style: italic">SD'</span></span>. Corresponding points of <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, +therefore, all lie on lines through the point <span class="tei tei-hi"><span style="font-style: italic">S</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc119" id="toc119"></a><a name="pdf120" id="pdf120"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p53" id="p53" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">53.</span></span> An entirely similar discussion shows that</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If in two projective pencils the line joining their centers +is a self-corresponding ray, then the two pencils are +perspectively related.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc121" id="toc121"></a><a name="pdf122" id="pdf122"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p54" id="p54" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">54.</span></span> A similar theorem may be stated for two axial +pencils of which the axes intersect. Very frequent use +will be made of these fundamental theorems.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc123" id="toc123"></a><a name="pdf124" id="pdf124"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p55" id="p55" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">55. Point-row of the second order.</span></span> The question naturally +arises, What is the locus of points of intersection +of corresponding rays of two projective pencils +<span class="tei tei-pb" id="page34">[pg 34]</span><a name="Pg34" id="Pg34" class="tei tei-anchor"></a> +which are not in perspective position? This locus, +which will be discussed in detail in subsequent chapters, +is easily seen to have at most two points in common +with any line in the plane, and on account of this +fundamental property will be called a <span class="tei tei-hi"><span style="font-style: italic">point-row of the +second order</span></span>. For any line <span class="tei tei-hi"><span style="font-style: italic">u</span></span> in the plane of the two +pencils will be cut by them in two projective point-rows +which have at most two self-corresponding points. +Such a self-corresponding point is clearly a point of +intersection of corresponding rays of the two pencils.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc125" id="toc125"></a><a name="pdf126" id="pdf126"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p56" id="p56" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">56.</span></span> This locus degenerates in the case of two perspective +pencils to a pair of straight lines, one of which +is the axis of perspectivity and the other the common +ray, any point of which may be considered as the point +of intersection of corresponding rays of the two pencils.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc127" id="toc127"></a><a name="pdf128" id="pdf128"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p57" id="p57" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">57. Pencils of rays of the second order.</span></span> Similar investigations +may be made concerning the system of lines +joining corresponding points of two projective point-rows. +If we project the point-rows to any point in the +plane, we obtain two projective pencils having the same +center. At most two pairs of self-corresponding rays +may present themselves. Such a ray is clearly a line +joining two corresponding points in the two point-rows. +The result may be stated as follows: <span class="tei tei-hi"><span style="font-style: italic">The system of rays +joining corresponding points in two protective point-rows +has at most two rays in common with any pencil in the +plane.</span></span> For that reason the system of rays is called <span class="tei tei-hi"><span style="font-style: italic">a +pencil of rays of the second order.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc129" id="toc129"></a><a name="pdf130" id="pdf130"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p58" id="p58" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">58.</span></span> In the case of two perspective point-rows this +system of rays degenerates into two pencils of rays of +the first order, one of which has its center at the center +<span class="tei tei-pb" id="page35">[pg 35]</span><a name="Pg35" id="Pg35" class="tei tei-anchor"></a> +of perspectivity of the two point-rows, and the other at +the intersection of the two point-rows, any ray through +which may be considered as joining two corresponding +points of the two point-rows.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc131" id="toc131"></a><a name="pdf132" id="pdf132"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p59" id="p59" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">59. Cone of the second order.</span></span> The corresponding +theorems in space may easily be obtained by joining +the points and lines considered in the plane theorems +to a point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> in space. Two projective pencils give rise +to two projective axial pencils with axes intersecting. +Corresponding planes meet in lines which all pass +through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and through the points on a point-row of +the second order generated by the two pencils of rays. +They are thus generating lines of a <span class="tei tei-hi"><span style="font-style: italic">cone of the second +order</span></span>, or <span class="tei tei-hi"><span style="font-style: italic">quadric cone</span></span>, so called because every plane in +space not passing through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> cuts it in a point-row of +the second order, and every line also cuts it in at most +two points. If, again, we project two point-rows to a +point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> in space, we obtain two pencils of rays with a +common center but lying in different planes. Corresponding +lines of these pencils determine planes which +are the projections to <span class="tei tei-hi"><span style="font-style: italic">S</span></span> of the lines which join the corresponding +points of the two point-rows. At most two +such planes may pass through any ray through <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. It +is called <span class="tei tei-hi"><span style="font-style: italic">a pencil of planes of the second order</span></span>.</p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc133" id="toc133"></a><a name="pdf134" id="pdf134"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-weight: 700">1. </span></span> A man <span class="tei tei-hi"><span style="font-style: italic">A</span></span> moves along a straight road <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, and another +man <span class="tei tei-hi"><span style="font-style: italic">B</span></span> moves along the same road and walks so as always +to keep sight of <span class="tei tei-hi"><span style="font-style: italic">A</span></span> in a small mirror <span class="tei tei-hi"><span style="font-style: italic">M</span></span> at the side of the +road. How many times will they come together, <span class="tei tei-hi"><span style="font-style: italic">A</span></span> moving +always in the same direction along the road?</p> + +<span class="tei tei-pb" id="page36">[pg 36]</span><a name="Pg36" id="Pg36" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. How many times would the two men in the first problem +see each other in two mirrors <span class="tei tei-hi"><span style="font-style: italic">M</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> as they walk +along the road as before? (The planes of the two mirrors +are not necessarily parallel to <span class="tei tei-hi"><span style="font-style: italic">u</span></span>.)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. As A moves along <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, trace the path of B so that the +two men may always see each other in the two mirrors.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Two boys walk along two paths <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> each holding +a string which they keep stretched tightly between them. +They both move at constant but different rates of speed, +letting out the string or drawing it in as they walk. How +many times will the line of the string pass over any given +point in the plane of the paths?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. Trace the lines of the string when the two boys move +at the same rate of speed in the two paths but do not start +at the same time from the point where the two paths +intersect.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. A ship is sailing on a straight course and keeps a gun +trained on a point on the shore. Show that a line at right +angles to the direction of the gun at its muzzle will pass +through any point in the plane twice or not at all. (Consider +the point-row at infinity cut out by a line through the +point on the shore at right angles to the direction of +the gun.)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. Two lines <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> revolve about two points <span class="tei tei-hi"><span style="font-style: italic">U</span></span> and <span class="tei tei-hi"><span style="font-style: italic">U'</span></span> +respectively in the same plane. They go in the same direction +and at the same rate of speed, but one has an angle a +the start of the other. Show that they generate a point-row +of the second order.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Discuss the question given in the last problem when +the two lines revolve in opposite directions. Can you +recognize the locus?</p> +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<span class="tei tei-pb" id="page37">[pg 37]</span><a name="Pg37" id="Pg37" class="tei tei-anchor"></a> +<a name="toc135" id="toc135"></a><a name="pdf136" id="pdf136"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER IV - POINT-ROWS OF THE SECOND ORDER</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc137" id="toc137"></a><a name="pdf138" id="pdf138"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p60" id="p60" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">60. Point-row of the second order defined.</span></span> We have +seen that two fundamental forms in one-to-one correspondence +may sometimes generate a form of higher +order. Thus, two point-rows (§ 55) generate a system of +rays of the second order, and two pencils of rays (§ 57), +a system of points of the second order. As a system of +points is more familiar to most students of geometry +than a system of lines, we study first the point-row of +the second order.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc139" id="toc139"></a><a name="pdf140" id="pdf140"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p61" id="p61" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">61. Tangent line.</span></span> We have shown in the last chapter +(§ 55) that the locus of intersection of corresponding +rays of two projective pencils is a point-row of the +second order; that is, it has at most two points in common +with any line in the plane. It is clear, first of all, +that the centers of the pencils are points of the locus; +for to the line <span class="tei tei-hi"><span style="font-style: italic">SS'</span></span>, considered as a ray of <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, must +correspond some ray of <span class="tei tei-hi"><span style="font-style: italic">S'</span></span> which meets it in <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>. <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>, +and by the same argument <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, is then a point where +corresponding rays meet. Any ray through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> will meet +it in one point besides <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, namely, the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> where +it meets its corresponding ray. Now, by choosing the +ray through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> sufficiently close to the ray <span class="tei tei-hi"><span style="font-style: italic">SS'</span></span>, the point +<span class="tei tei-hi"><span style="font-style: italic">P</span></span> may be made to approach arbitrarily close to <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>, and +the ray <span class="tei tei-hi"><span style="font-style: italic">S'P</span></span> may be made to differ in position from the +<span class="tei tei-pb" id="page38">[pg 38]</span><a name="Pg38" id="Pg38" class="tei tei-anchor"></a> +tangent line at <span class="tei tei-hi"><span style="font-style: italic">S'</span></span> by as little as we please. We have, +then, the important theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The ray at </span><span class="tei tei-hi"><span style="font-style: italic">S'</span></span><span style="font-style: italic"> which corresponds to the common ray </span><span class="tei tei-hi"><span style="font-style: italic">SS'</span></span><span style="font-style: italic"> +is tangent to the locus at </span><span class="tei tei-hi"><span style="font-style: italic">S'</span></span><span style="font-style: italic">.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">In the same manner the tangent at <span class="tei tei-hi"><span style="font-style: italic">S</span></span> may be +constructed.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc141" id="toc141"></a><a name="pdf142" id="pdf142"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p62" id="p62" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">62. Determination of the locus.</span></span> We now show that +<span class="tei tei-hi"><span style="font-style: italic">it is possible to assign arbitrarily the position of three +points, </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic">, on the locus (besides the points </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic"> +and </span><span class="tei tei-hi"><span style="font-style: italic">S'</span></span><span style="font-style: italic">); but, these three points being chosen, the locus is +completely determined.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc143" id="toc143"></a><a name="pdf144" id="pdf144"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p63" id="p63" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">63.</span></span> This statement is equivalent to the following:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given three pairs of corresponding rays in two projective +pencils, it is possible to find a ray of one which corresponds +to any ray of the other.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc145" id="toc145"></a><a name="pdf146" id="pdf146"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p64" id="p64" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">64.</span></span> We proceed, then, to the solution of the fundamental</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-variant: small-caps">Problem</span></span>: <span class="tei tei-hi"><span style="font-style: italic">Given three pairs of rays, </span><span class="tei tei-hi"><span style="font-style: italic">aa'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">bb'</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">cc'</span></span><span style="font-style: italic">, +of two protective pencils, </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">S'</span></span><span style="font-style: italic">, to find the ray </span><span class="tei tei-hi"><span style="font-style: italic">d'</span></span><span style="font-style: italic"> of </span><span class="tei tei-hi"><span style="font-style: italic">S'</span></span><span style="font-style: italic"> +which corresponds to any ray </span><span class="tei tei-hi"><span style="font-style: italic">d</span></span><span style="font-style: italic"> of </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic">.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image12.png" width="811" height="439" alt="Figure 12" title="Fig. 12" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 12</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Call <span class="tei tei-hi"><span style="font-style: italic">A</span></span> the intersection of <span class="tei tei-hi"><span style="font-style: italic">aa'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span> the intersection of <span class="tei tei-hi"><span style="font-style: italic">bb'</span></span>, +and <span class="tei tei-hi"><span style="font-style: italic">C</span></span> the intersection of <span class="tei tei-hi"><span style="font-style: italic">cc'</span></span> (Fig. 12). Join <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> by the +line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">AC</span></span> by the line <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>. Consider <span class="tei tei-hi"><span style="font-style: italic">u</span></span> as a point-row +perspective to <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> as a point-row perspective +to <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>. <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> are projectively related to each other, +since <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and <span class="tei tei-hi"><span style="font-style: italic">S'</span></span> are, by hypothesis, so related. But their +point of intersection <span class="tei tei-hi"><span style="font-style: italic">A</span></span> is a self-corresponding point, since +<span class="tei tei-hi"><span style="font-style: italic">a</span></span> and <span class="tei tei-hi"><span style="font-style: italic">a'</span></span> were supposed to be corresponding rays. It follows +(§ 52) that <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> are in perspective position, +and that lines through corresponding points all pass +<span class="tei tei-pb" id="page39">[pg 39]</span><a name="Pg39" id="Pg39" class="tei tei-anchor"></a> +through a point <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, the center of perspectivity, the +position of which will be determined by any two such +lines. But the intersection of <span class="tei tei-hi"><span style="font-style: italic">a</span></span> with <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and the intersection +of <span class="tei tei-hi"><span style="font-style: italic">c'</span></span> with <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> are corresponding points on <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, +and the line joining them is clearly <span class="tei tei-hi"><span style="font-style: italic">c</span></span> itself. Similarly, +<span class="tei tei-hi"><span style="font-style: italic">b'</span></span> joins two corresponding points on <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, and so the +center <span class="tei tei-hi"><span style="font-style: italic">M</span></span> of perspectivity of <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> is the intersection +of <span class="tei tei-hi"><span style="font-style: italic">c</span></span> and <span class="tei tei-hi"><span style="font-style: italic">b'</span></span>. To find <span class="tei tei-hi"><span style="font-style: italic">d'</span></span> in <span class="tei tei-hi"><span style="font-style: italic">S'</span></span> corresponding to a given +line <span class="tei tei-hi"><span style="font-style: italic">d</span></span> of <span class="tei tei-hi"><span style="font-style: italic">S</span></span> we note the point <span class="tei tei-hi"><span style="font-style: italic">L</span></span> where <span class="tei tei-hi"><span style="font-style: italic">d</span></span> meets <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. Join +<span class="tei tei-hi"><span style="font-style: italic">L</span></span> to <span class="tei tei-hi"><span style="font-style: italic">M</span></span> and get the point <span class="tei tei-hi"><span style="font-style: italic">N</span></span> where this line meets <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>. +<span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> are corresponding points on <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">d'</span></span> +must therefore pass through <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. The intersection <span class="tei tei-hi"><span style="font-style: italic">P</span></span> of +<span class="tei tei-hi"><span style="font-style: italic">d</span></span> and <span class="tei tei-hi"><span style="font-style: italic">d'</span></span> is thus another point on the locus. In the same +manner any number of other points may be obtained.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc147" id="toc147"></a><a name="pdf148" id="pdf148"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p65" id="p65" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">65.</span></span> The lines <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> might have been drawn in +any direction through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> (avoiding, of course, the line +<span class="tei tei-hi"><span style="font-style: italic">a</span></span> for <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and the line <span class="tei tei-hi"><span style="font-style: italic">a'</span></span> for <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>), and the center of perspectivity +<span class="tei tei-hi"><span style="font-style: italic">M</span></span> would be easily obtainable; but the above +construction furnishes a simple and instructive figure. +An equally simple one is obtained by taking <span class="tei tei-hi"><span style="font-style: italic">a'</span></span> for <span class="tei tei-hi"><span style="font-style: italic">u</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">a</span></span> for <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>.</p> + +<span class="tei tei-pb" id="page40">[pg 40]</span><a name="Pg40" id="Pg40" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc149" id="toc149"></a><a name="pdf150" id="pdf150"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p66" id="p66" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">66. Lines joining four points of the locus to a fifth.</span></span> +Suppose that the points <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are fixed, +and that four points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">A</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">A</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, and <span class="tei tei-hi"><span style="font-style: italic">A</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>, are taken on the +locus at the intersection with it of any four harmonic +rays through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. These four harmonic rays give four +harmonic points, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span> etc., on the fixed ray <span class="tei tei-hi"><span style="font-style: italic">SD</span></span>. These, +in turn, project through the fixed point <span class="tei tei-hi"><span style="font-style: italic">M</span></span> into four +harmonic points, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span> etc., on the fixed line <span class="tei tei-hi"><span style="font-style: italic">DS'</span></span>. +These last four harmonic points give four harmonic +rays <span class="tei tei-hi"><span style="font-style: italic">CA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">CA</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">CA</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">CA</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>. Therefore the four points <span class="tei tei-hi"><span style="font-style: italic">A</span></span> +which project to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> in four harmonic rays also project +to <span class="tei tei-hi"><span style="font-style: italic">C</span></span> in four harmonic rays. But <span class="tei tei-hi"><span style="font-style: italic">C</span></span> may be any +point on the locus, and so we have the very important +theorem,</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Four points which are on the locus, and which project +to a fifth point of the locus in four harmonic rays, project +to any point of the locus in four harmonic rays.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc151" id="toc151"></a><a name="pdf152" id="pdf152"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p67" id="p67" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">67.</span></span> The theorem may also be stated thus:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The locus of points from which, four given points are +seen along four harmonic rays is a point-row of the second +order through them.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc153" id="toc153"></a><a name="pdf154" id="pdf154"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p68" id="p68" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">68.</span></span> A further theorem of prime importance also +follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Any two points on the locus may be taken as the centers +of two projective pencils which will generate the locus.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc155" id="toc155"></a><a name="pdf156" id="pdf156"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p69" id="p69" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">69. Pascal's theorem.</span></span> The points <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, and +<span class="tei tei-hi"><span style="font-style: italic">S'</span></span> may thus be considered as chosen arbitrarily on the +locus, and the following remarkable theorem follows +at once.</p> + +<span class="tei tei-pb" id="page41">[pg 41]</span><a name="Pg41" id="Pg41" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given six points, 1, 2, 3, 4, 5, 6, on the point-row of +the second order, if we call</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">L the intersection of 12 with 45,</span></span></p> +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">M the intersection of 23 with 56,</span></span></p> +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">N the intersection of 34 with 61,</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">then </span><span class="tei tei-hi"><span style="font-style: italic">L</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">N</span></span><span style="font-style: italic"> are on a straight line.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image13.png" width="671" height="596" alt="Figure 13" title="Fig. 13" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 13</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc157" id="toc157"></a><a name="pdf158" id="pdf158"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p70" id="p70" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">70.</span></span> To get the notation to correspond to the figure, we +may take (Fig. 13) <span class="tei tei-hi"><span style="font-style: italic">A = 1</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B = 2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S' = 3</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D = 4</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S = 5</span></span>, and +<span class="tei tei-hi"><span style="font-style: italic">C = 6</span></span>. If we make <span class="tei tei-hi"><span style="font-style: italic">A = 1</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C=2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S=3</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D = 4</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S'=5</span></span>, and. +<span class="tei tei-hi"><span style="font-style: italic">B = 6</span></span>, the points <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> are interchanged, but the line +is left unchanged. +It is clear that one +point may be named +arbitrarily and the +other five named in +<span class="tei tei-hi"><span style="font-style: italic">5! = 120</span></span> different +ways, but since, as +we have seen, two +different assignments +of names give the +same line, it follows +that there cannot be +more than 60 different +lines <span class="tei tei-hi"><span style="font-style: italic">LMN</span></span> obtained in this way from a given set of +six points. As a matter of fact, the number obtained in +this way is in general <span class="tei tei-hi"><span style="font-style: italic">60</span></span>. The above theorem, which +is of cardinal importance in the theory of the point-row +of the second order, is due to Pascal and was discovered +by him at the age of sixteen. It is, no doubt, the most +important contribution to the theory of these loci since +<span class="tei tei-pb" id="page42">[pg 42]</span><a name="Pg42" id="Pg42" class="tei tei-anchor"></a> +the days of Apollonius. If the six points be called the +vertices of a hexagon inscribed in the curve, then the +sides 12 and 45 may be appropriately called a pair of +opposite sides. Pascal's theorem, then, may be stated +as follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The three pairs of opposite sides of a hexagon inscribed in +a point-row of the second order meet in three points on a line.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc159" id="toc159"></a><a name="pdf160" id="pdf160"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p71" id="p71" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">71. Harmonic points on a point-row of the second order.</span></span> +Before proceeding to develop the consequences of this +theorem, we note another result of the utmost importance +for the higher developments of pure geometry, +which follows from the fact that if four points on the +locus project to a fifth in four harmonic rays, they will +project to any point of the locus in four harmonic rays. +It is natural to speak of four such points as four harmonic +points on the locus, and to use this notion to +define projective correspondence between point-rows of +the second order, or between a point-row of the second +order and any fundamental form of the first order. +Thus, in particular, the point-row of the second order, +σ, is said to be <span class="tei tei-hi"><span style="font-style: italic">perspectively related</span></span> to the pencil <span class="tei tei-hi"><span style="font-style: italic">S</span></span> when +every ray on <span class="tei tei-hi"><span style="font-style: italic">S</span></span> goes through the point on σ which +corresponds to it.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc161" id="toc161"></a><a name="pdf162" id="pdf162"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p72" id="p72" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">72. Determination of the locus.</span></span> It is now clear that +five points, arbitrarily chosen in the plane, are sufficient +to determine a point-row of the second order through +them. Two of the points may be taken as centers of +two projective pencils, and the three others will determine +three pairs of corresponding rays of the pencils, +and therefore all pairs. If four points of the locus are +<span class="tei tei-pb" id="page43">[pg 43]</span><a name="Pg43" id="Pg43" class="tei tei-anchor"></a> +given, together with the tangent at one of them, the +locus is likewise completely determined. For if the point +at which the tangent is given be taken as the center <span class="tei tei-hi"><span style="font-style: italic">S</span></span> +of one pencil, and any other of the points for <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>, then, +besides the two pairs of corresponding rays determined +by the remaining two points, we have one more pair, +consisting of the tangent at <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and the ray <span class="tei tei-hi"><span style="font-style: italic">SS'</span></span>. Similarly, +the curve is determined by three points and the +tangents at two of them.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc163" id="toc163"></a><a name="pdf164" id="pdf164"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p73" id="p73" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">73. Circles and conics as point-rows of the second order.</span></span> +It is not difficult to see that a circle is a point-row of +the second order. Indeed, take any point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> on the circle +and draw four harmonic rays through it. They will cut +the circle in four points, which will project to any other +point of the curve in four harmonic rays; for, by the +theorem concerning the angles inscribed in a circle, the +angles involved in the second set of four lines are +the same as those in the first set. If, moreover, we project +the figure to any point in space, we shall get a cone, +standing on a circular base, generated by two projective +axial pencils which are the projections of the pencils +at <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>. Cut across, now, by any plane, and we get +a conic section which is thus exhibited as the locus of +intersection of two projective pencils. It thus appears +that a conic section is a point-row of the second order. +It will later appear that a point-row of the second order +is a conic section. In the future, therefore, we shall +refer to a point-row of the second order as a conic.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image14.png" width="314" height="304" alt="Figure 14" title="Fig. 14" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 14</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc165" id="toc165"></a><a name="pdf166" id="pdf166"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p74" id="p74" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">74. Conic through five points.</span></span> Pascal's theorem furnishes +an elegant solution of the problem of drawing a +conic through five given points. To construct a sixth +<span class="tei tei-pb" id="page44">[pg 44]</span><a name="Pg44" id="Pg44" class="tei tei-anchor"></a> +point on the conic, draw through the point numbered 1 +an arbitrary line (Fig. 14), and let the desired point +6 be the second point of intersection +of this line with the conic. The point +<span class="tei tei-hi"><span style="font-style: italic">L = 12-45</span></span> is obtainable at once; also +the point <span class="tei tei-hi"><span style="font-style: italic">N = 34-61</span></span>. But <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> +determine Pascal's line, and the intersection +of 23 with 56 must be on +this line. Intersect, then, the line <span class="tei tei-hi"><span style="font-style: italic">LN</span></span> +with 23 and obtain the point <span class="tei tei-hi"><span style="font-style: italic">M</span></span>. Join +<span class="tei tei-hi"><span style="font-style: italic">M</span></span> to 5 and intersect with 61 for the desired point 6.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image15.png" width="371" height="439" alt="Figure 15" title="Fig. 15" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 15</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc167" id="toc167"></a><a name="pdf168" id="pdf168"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p75" id="p75" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">75. Tangent to a conic.</span></span> If two points of Pascal's hexagon +approach coincidence, then the line joining them +approaches as a limiting position the tangent line at that +point. Pascal's theorem thus affords a ready method of +drawing the tangent line to a conic +at a given point. If the conic is determined +by the points 1, 2, 3, 4, 5 +(Fig. 15), and it is desired to draw +the tangent at the point 1, we may +call that point 1, 6. The points +<span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> are obtained as usual, +and the intersection of 34 with <span class="tei tei-hi"><span style="font-style: italic">LM</span></span> +gives <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. Join <span class="tei tei-hi"><span style="font-style: italic">N</span></span> to the point 1 for +the desired tangent at that point.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc169" id="toc169"></a><a name="pdf170" id="pdf170"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p76" id="p76" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">76. Inscribed quadrangle.</span></span> Two pairs of vertices may +coalesce, giving an inscribed quadrangle. Pascal's theorem +gives for this case the very important theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Two pairs of opposite sides of any quadrangle inscribed +in a conic meet on a straight line, upon which line also +intersect the two pairs of tangents at the opposite vertices.</span></span></p> + +<span class="tei tei-pb" id="page45">[pg 45]</span><a name="Pg45" id="Pg45" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image16.png" width="789" height="457" alt="Figure 16" title="Fig. 16" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 16</div></div> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image17.png" width="432" height="546" alt="Figure 17" title="Fig. 17" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 17</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">For let the vertices be <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, and call the +vertex <span class="tei tei-hi"><span style="font-style: italic">A</span></span> the point 1, 6; <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, the point 2; <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, the point +3, 4; and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, the point 5 (Fig. 16). Pascal's theorem then +indicates that +<span class="tei tei-hi"><span style="font-style: italic">L = AB-CD</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">M = AD-BC</span></span>, +and <span class="tei tei-hi"><span style="font-style: italic">N</span></span>, which +is the intersection +of the +tangents at <span class="tei tei-hi"><span style="font-style: italic">A</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, are all +on a straight +line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. But +if we were to +call <span class="tei tei-hi"><span style="font-style: italic">A</span></span> the point 2, <span class="tei tei-hi"><span style="font-style: italic">B</span></span> the point 6, 1, <span class="tei tei-hi"><span style="font-style: italic">C</span></span> the point 5, and +<span class="tei tei-hi"><span style="font-style: italic">D</span></span> the point 4, 3, then the intersection <span class="tei tei-hi"><span style="font-style: italic">P</span></span> of the tangents +at <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are also on this same +line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. Thus <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">P</span></span> are +four points on a straight line. +The consequences of this theorem +are so numerous and important +that we shall devote a separate +chapter to them.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc171" id="toc171"></a><a name="pdf172" id="pdf172"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p77" id="p77" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">77. Inscribed triangle.</span></span> Finally, +three of the vertices of the hexagon +may coalesce, giving a triangle +inscribed in a conic. Pascal's +theorem then reads as follows (Fig. 17) for this case:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The three tangents at the vertices of a triangle inscribed +in a conic meet the opposite sides in three points on a +straight line.</span></span></p> + +<span class="tei tei-pb" id="page46">[pg 46]</span><a name="Pg46" id="Pg46" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image18.png" width="746" height="507" alt="Figure 18" title="Fig. 18" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 18</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc173" id="toc173"></a><a name="pdf174" id="pdf174"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p78" id="p78" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">78. Degenerate conic.</span></span> If we apply Pascal's theorem +to a degenerate conic made up of a pair of straight +lines, we get the +following theorem +(Fig. 18):</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If three points, +</span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic">, are +chosen on one +line, and three +points, </span><span class="tei tei-hi"><span style="font-style: italic">A'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B'</span></span><span style="font-style: italic">, +</span><span class="tei tei-hi"><span style="font-style: italic">C'</span></span><span style="font-style: italic">, are chosen on +another, then the +three points </span><span class="tei tei-hi"><span style="font-style: italic">L = AB'-A'B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M = BC'-B'C</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N = CA'-C'A</span></span><span style="font-style: italic"> +are all on a straight line.</span></span></p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc175" id="toc175"></a><a name="pdf176" id="pdf176"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. In Fig. 12, select different lines <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and trace the locus +of the center of perspectivity <span class="tei tei-hi"><span style="font-style: italic">M</span></span> of the lines <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. Given four points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, in the plane, construct +a fifth point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> such that the lines <span class="tei tei-hi"><span style="font-style: italic">PA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">PB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">PC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">PD</span></span> shall be +four harmonic lines.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Suggestion.</span></span> Draw a line <span class="tei tei-hi"><span style="font-style: italic">a</span></span> through the point <span class="tei tei-hi"><span style="font-style: italic">A</span></span> such that the four +lines <span class="tei tei-hi"><span style="font-style: italic">a</span></span>, <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> are harmonic. Construct now a conic through +<span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> having <span class="tei tei-hi"><span style="font-style: italic">a</span></span> for a tangent at <span class="tei tei-hi"><span style="font-style: italic">A</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Where are all the points <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, as determined in the +preceding question, to be found?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Select any five points in the plane and draw the tangent +to the conic through them at each of the five points.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. Given four points on the conic, and the tangent at one of +them, to construct the conic. ("To construct the conic" means +here to construct as many other points as may be desired.)</p> + +<span class="tei tei-pb" id="page47">[pg 47]</span><a name="Pg47" id="Pg47" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Given three points on the conic, and the tangent at +two of them, to construct the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. Given five points, two of which are at infinity in +different directions, to construct the conic. (In this, and +in the following examples, the student is supposed to be +able to draw a line parallel to a given line.)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Given four points on a conic (two of which are at infinity +and two in the finite part of the plane), together with +the tangent at one of the finite points, to construct the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">9. The tangents to a curve at its infinitely distant points +are called its <span class="tei tei-hi"><span style="font-style: italic">asymptotes</span></span> if they pass through a finite part +of the plane. Given the asymptotes and a finite point of a +conic, to construct the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">10. Given an asymptote and three finite points on the +conic, to determine the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">11. Given four points, one of which is at infinity, and +given also that the line at infinity is a tangent line, to +construct the conic.</p> +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc177" id="toc177"></a><a name="pdf178" id="pdf178"></a> +<span class="tei tei-pb" id="page48">[pg 48]</span><a name="Pg48" id="Pg48" class="tei tei-anchor"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc179" id="toc179"></a><a name="pdf180" id="pdf180"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p79" id="p79" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">79. Pencil of rays of the second order defined.</span></span> If the +corresponding points of two projective point-rows be +joined by straight lines, a system of lines is obtained +which is called a pencil of rays of the second order. +This name arises from the fact, easily shown (§ 57), that +at most two lines of the system may pass through any +arbitrary point in the plane. For if through any point +there should pass three lines of the system, then this +point might be taken as the center of two projective +pencils, one projecting one point-row and the other projecting +the other. Since, now, these pencils have three +rays of one coincident with the corresponding rays of +the other, the two are identical and the two point-rows +are in perspective position, which was not supposed.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image19.png" width="440" height="300" alt="Figure 19" title="Fig. 19" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 19</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc181" id="toc181"></a><a name="pdf182" id="pdf182"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p80" id="p80" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">80. Tangents to a circle.</span></span> To get a clear notion of this +system of lines, we may first show that the tangents +to a circle form a system of this kind. For take any +two tangents, <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, to a circle, and let <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> +be the points of contact (Fig. 19). Let now <span class="tei tei-hi"><span style="font-style: italic">t</span></span> be any +third tangent with point of contact at <span class="tei tei-hi"><span style="font-style: italic">C</span></span> and meeting <span class="tei tei-hi"><span style="font-style: italic">u</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> in <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> respectively. Join <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, and +<span class="tei tei-hi"><span style="font-style: italic">C</span></span> to <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, the center of the circle. Tangents from any +point to a circle are equal, and therefore the triangles +<span class="tei tei-hi"><span style="font-style: italic">POA</span></span> and <span class="tei tei-hi"><span style="font-style: italic">POC</span></span> are equal, as also are the triangles <span class="tei tei-hi"><span style="font-style: italic">P'OB</span></span> +<span class="tei tei-pb" id="page49">[pg 49]</span><a name="Pg49" id="Pg49" class="tei tei-anchor"></a> +and <span class="tei tei-hi"><span style="font-style: italic">P'OC</span></span>. Therefore the angle <span class="tei tei-hi"><span style="font-style: italic">POP'</span></span> is constant, being +equal to half the constant angle <span class="tei tei-hi"><span style="font-style: italic">AOC + COB</span></span>. This +being true, if we take any four harmonic points, <span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, +<span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">4</span></span></span>, on the line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, they will project to <span class="tei tei-hi"><span style="font-style: italic">O</span></span> in four +harmonic lines, and the tangents +to the circle from these four +points will meet <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> in four harmonic +points, <span class="tei tei-hi"><span style="font-style: italic">P'</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P'</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P'</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P'</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">4</span></span></span>, because +the lines from these points +to <span class="tei tei-hi"><span style="font-style: italic">O</span></span> inclose the same angles as +the lines from the points <span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, +<span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">P</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">4</span></span></span> on <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. The point-row on <span class="tei tei-hi"><span style="font-style: italic">u</span></span> is therefore projective +to the point-row on <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>. Thus the tangents to a circle +are seen to join corresponding points on two projective +point-rows, and so, according to the definition, form a +pencil of rays of the second order.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc183" id="toc183"></a><a name="pdf184" id="pdf184"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p81" id="p81" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">81. Tangents to a conic.</span></span> If now this figure be projected +to a point outside the plane of the circle, and +any section of the resulting cone be made by a plane, +we can easily see that the system of rays tangent to any +conic section is a pencil of rays of the second order. +The converse is also true, as we shall see later, and a +pencil of rays of the second order is also a set of lines +tangent to a conic section.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc185" id="toc185"></a><a name="pdf186" id="pdf186"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p82" id="p82" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">82.</span></span> The point-rows <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> are, themselves, lines of +the system, for to the common point of the two point-rows, +considered as a point of <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, must correspond some +point of <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, and the line joining these two corresponding +points is clearly <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> itself. Similarly for the line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc187" id="toc187"></a><a name="pdf188" id="pdf188"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p83" id="p83" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">83. Determination of the pencil.</span></span> We now show that +<span class="tei tei-hi"><span style="font-style: italic">it is possible to assign arbitrarily three lines, </span><span class="tei tei-hi"><span style="font-style: italic">a</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">b</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">c</span></span><span style="font-style: italic">, of +</span><span class="tei tei-pb" id="page50">[pg 50]</span><a name="Pg50" id="Pg50" class="tei tei-anchor"></a><span style="font-style: italic"> +the system (besides the lines </span><span class="tei tei-hi"><span style="font-style: italic">u</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">u'</span></span><span style="font-style: italic">); but if these three +lines are chosen, the system is completely determined.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">This statement is equivalent to the following:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given three pairs of corresponding points in two projective +point-rows, it is possible to find a point in one +which corresponds to any point of the other.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">We proceed, then, to the solution of the fundamental</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-variant: small-caps">Problem.</span></span> <span class="tei tei-hi"><span style="font-style: italic">Given three pairs of points, </span><span class="tei tei-hi"><span style="font-style: italic">AA'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">BB'</span></span><span style="font-style: italic">, and +</span><span class="tei tei-hi"><span style="font-style: italic">CC'</span></span><span style="font-style: italic">, of two projective point-rows </span><span class="tei tei-hi"><span style="font-style: italic">u</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">u'</span></span><span style="font-style: italic">, to find the point +</span><span class="tei tei-hi"><span style="font-style: italic">D'</span></span><span style="font-style: italic"> of </span><span class="tei tei-hi"><span style="font-style: italic">u'</span></span><span style="font-style: italic"> which corresponds to any given point </span><span class="tei tei-hi"><span style="font-style: italic">D</span></span><span style="font-style: italic"> of </span><span class="tei tei-hi"><span style="font-style: italic">u</span></span><span style="font-style: italic">.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image20.png" width="393" height="557" alt="Figure 20" title="Fig. 20" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 20</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">On the line <span class="tei tei-hi"><span style="font-style: italic">a</span></span>, joining <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, take two points, <span class="tei tei-hi"><span style="font-style: italic">S</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>, as centers of pencils perspective to <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> +respectively (Fig. 20). The figure +will be much simplified if we take +<span class="tei tei-hi"><span style="font-style: italic">S</span></span> on <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">S'</span></span> on <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span>. <span class="tei tei-hi"><span style="font-style: italic">SA</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">S'A'</span></span> are corresponding rays of <span class="tei tei-hi"><span style="font-style: italic">S</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>, and the two pencils are +therefore in perspective position. +It is not difficult to see that the +axis of perspectivity <span class="tei tei-hi"><span style="font-style: italic">m</span></span> is the line +joining <span class="tei tei-hi"><span style="font-style: italic">B'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. Given any point +<span class="tei tei-hi"><span style="font-style: italic">D</span></span> on <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, to find the corresponding +point <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> on <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> we proceed as +follows: Join <span class="tei tei-hi"><span style="font-style: italic">D</span></span> to <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and note +where the joining line meets <span class="tei tei-hi"><span style="font-style: italic">m</span></span>. Join this point to <span class="tei tei-hi"><span style="font-style: italic">S'</span></span>. +This last line meets <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> in the desired point <span class="tei tei-hi"><span style="font-style: italic">D'</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">We have now in this figure six lines of the system, +<span class="tei tei-hi"><span style="font-style: italic">a</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c</span></span>, <span class="tei tei-hi"><span style="font-style: italic">d</span></span>, <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>. Fix now the position of <span class="tei tei-hi"><span style="font-style: italic">u</span></span>, <span class="tei tei-hi"><span style="font-style: italic">u'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c</span></span>, and +<span class="tei tei-hi"><span style="font-style: italic">d</span></span>, and take four lines of the system, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">4</span></span></span>, which +meet <span class="tei tei-hi"><span style="font-style: italic">b</span></span> in four harmonic points. These points project to +<span class="tei tei-pb" id="page51">[pg 51]</span><a name="Pg51" id="Pg51" class="tei tei-anchor"></a> +<span class="tei tei-hi"><span style="font-style: italic">D</span></span>, giving four harmonic points on <span class="tei tei-hi"><span style="font-style: italic">m</span></span>. These again project +to <span class="tei tei-hi"><span style="font-style: italic">D'</span></span>, giving four harmonic points on <span class="tei tei-hi"><span style="font-style: italic">c</span></span>. It is thus clear +that the rays <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">1</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">2</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">3</span></span></span>, <span class="tei tei-hi"><span style="font-style: italic">a</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: sub">4</span></span></span> cut out two projective point-rows +on any two lines of the system. Thus <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> are +not special rays, and any two rays of the system will +serve as the point-rows to generate the system of lines.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc189" id="toc189"></a><a name="pdf190" id="pdf190"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p84" id="p84" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">84. Brianchon's theorem.</span></span> From the figure also appears +a fundamental theorem due to Brianchon:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If </span><span class="tei tei-hi"><span style="font-style: italic">1</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">2</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">3</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">4</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">5</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">6</span></span><span style="font-style: italic"> are any six rays of a pencil of the +second order, then the lines </span><span class="tei tei-hi"><span style="font-style: italic">l = (12, 45)</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">m = (23, 56)</span></span><span style="font-style: italic">, +</span><span class="tei tei-hi"><span style="font-style: italic">n = (34, 61)</span></span><span style="font-style: italic"> all pass through a point.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image21.png" width="618" height="518" alt="Figure 21" title="Fig. 21" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 21</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc191" id="toc191"></a><a name="pdf192" id="pdf192"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p85" id="p85" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">85.</span></span> To make the notation fit the figure (Fig. 21), make +<span class="tei tei-hi"><span style="font-style: italic">a=1</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b = 2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">u' = 3</span></span>, <span class="tei tei-hi"><span style="font-style: italic">d = 4</span></span>, <span class="tei tei-hi"><span style="font-style: italic">u = 5</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c = 6</span></span>; or, interchanging +two of the lines, <span class="tei tei-hi"><span style="font-style: italic">a = 1</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">c = 2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">u = 3</span></span>, <span class="tei tei-hi"><span style="font-style: italic">d = 4</span></span>, <span class="tei tei-hi"><span style="font-style: italic">u' = 5</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">b = 6</span></span>. Thus, by different +namings of the +lines, it appears that +not more than 60 different +<span class="tei tei-hi"><span style="font-style: italic">Brianchon points</span></span> +are possible. If we +call 12 and 45 opposite +vertices of a circumscribed +hexagon, +then Brianchon's theorem may be stated as follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The three lines joining the three pairs of opposite vertices +of a hexagon circumscribed about a conic meet in a point.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc193" id="toc193"></a><a name="pdf194" id="pdf194"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p86" id="p86" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">86. Construction of the pencil by Brianchon's theorem.</span></span> +Brianchon's theorem furnishes a ready method of determining +a sixth line of the pencil of rays of the second +<span class="tei tei-pb" id="page52">[pg 52]</span><a name="Pg52" id="Pg52" class="tei tei-anchor"></a> +order when five are given. Thus, select a point in line +1 and suppose that line 6 is to pass through it. Then +<span class="tei tei-hi"><span style="font-style: italic">l = (12, 45)</span></span>, <span class="tei tei-hi"><span style="font-style: italic">n = (34, 61)</span></span>, and the line <span class="tei tei-hi"><span style="font-style: italic">m = (23, 56)</span></span> must +pass through <span class="tei tei-hi"><span style="font-style: italic">(l, n)</span></span>. Then <span class="tei tei-hi"><span style="font-style: italic">(23, ln)</span></span> meets 5 in a point of +the required sixth line.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image22.png" width="596" height="386" alt="Figure 22" title="Fig. 22" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 22</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc195" id="toc195"></a><a name="pdf196" id="pdf196"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p87" id="p87" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">87. Point of contact +of a tangent to a conic.</span></span> +If the line 2 approach +as a limiting position the +line 1, then the intersection +<span class="tei tei-hi"><span style="font-style: italic">(1, 2)</span></span> approaches +as a limiting position +the point of contact of +1 with the conic. This suggests an easy way to construct +the point of contact of any tangent with the conic. +Thus (Fig. 22), given the lines 1, 2, 3, 4, 5 to construct +the point of contact of <span class="tei tei-hi"><span style="font-style: italic">1=6</span></span>. +Draw <span class="tei tei-hi"><span style="font-style: italic">l = (12,45)</span></span>, <span class="tei tei-hi"><span style="font-style: italic">m =(23,56)</span></span>; +then <span class="tei tei-hi"><span style="font-style: italic">(34, lm)</span></span> meets 1 in the +required point of contact <span class="tei tei-hi"><span style="font-style: italic">T</span></span>.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image23.png" width="468" height="554" alt="Figure 23" title="Fig. 23" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 23</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc197" id="toc197"></a><a name="pdf198" id="pdf198"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p88" id="p88" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">88. Circumscribed quadrilateral.</span></span> +If two pairs of lines in +Brianchon's hexagon coalesce, +we have a theorem concerning +a quadrilateral circumscribed +about a conic. It is +easily found to be (Fig. 23)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The four lines joining the two opposite pairs of vertices +and the two opposite points of contact of a quadrilateral +circumscribed about a conic all meet in a point.</span></span> The +consequences of this theorem will be deduced later.</p> + +<span class="tei tei-pb" id="page53">[pg 53]</span><a name="Pg53" id="Pg53" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image24.png" width="414" height="421" alt="Figure 24" title="Fig. 24" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 24</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc199" id="toc199"></a><a name="pdf200" id="pdf200"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p89" id="p89" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">89. Circumscribed triangle.</span></span> The hexagon may further +degenerate into a triangle, giving the theorem (Fig. 24) +<span class="tei tei-hi"><span style="font-style: italic">The lines joining the vertices to +the points of contact of the opposite +sides of a triangle circumscribed +about a conic all meet in a point.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc201" id="toc201"></a><a name="pdf202" id="pdf202"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p90" id="p90" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">90.</span></span> Brianchon's theorem may +also be used to solve the following +problems:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given four tangents and the point +of contact on any one of them, to construct other tangents to +a conic. Given three tangents and the points of contact of +any two of them, to construct other tangents to a conic.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc203" id="toc203"></a><a name="pdf204" id="pdf204"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p91" id="p91" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">91. Harmonic tangents.</span></span> We have seen that a variable +tangent cuts out on any two fixed tangents projective +point-rows. It follows that if four tangents cut a fifth +in four harmonic points, they must cut every tangent in +four harmonic points. It is possible, therefore, to make +the following definition:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Four tangents to a conic are said to be harmonic when +they meet every other tangent in four harmonic points.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc205" id="toc205"></a><a name="pdf206" id="pdf206"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p92" id="p92" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">92. Projectivity and perspectivity.</span></span> This definition suggests +the possibility of defining a projective correspondence +between the elements of a pencil of rays of the +second order and the elements of any form heretofore +discussed. In particular, the points on a tangent are +said to be <span class="tei tei-hi"><span style="font-style: italic">perspectively related</span></span> to the tangents of a conic +when each point lies on the tangent which corresponds +to it. These notions are of importance in the higher +developments of the subject.</p> + +<span class="tei tei-pb" id="page54">[pg 54]</span><a name="Pg54" id="Pg54" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image25.png" width="482" height="514" alt="Figure 25" title="Fig. 25" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 25</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc207" id="toc207"></a><a name="pdf208" id="pdf208"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p93" id="p93" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">93.</span></span> Brianchon's theorem may also be applied to a +degenerate conic made up of two points and the lines +through them. Thus(Fig. 25),</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If </span><span class="tei tei-hi"><span style="font-style: italic">a</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">b</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">c</span></span><span style="font-style: italic"> are three lines +through a point </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">a'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">b'</span></span><span style="font-style: italic">, +</span><span class="tei tei-hi"><span style="font-style: italic">c'</span></span><span style="font-style: italic"> are three lines through another +point </span><span class="tei tei-hi"><span style="font-style: italic">S'</span></span><span style="font-style: italic">, then the lines +</span><span class="tei tei-hi"><span style="font-style: italic">l = (ab', a'b)</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">m = (bc', b'c)</span></span><span style="font-style: italic">, +and </span><span class="tei tei-hi"><span style="font-style: italic">n = (ca', c'a)</span></span><span style="font-style: italic"> all meet in +a point.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc209" id="toc209"></a><a name="pdf210" id="pdf210"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p94" id="p94" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">94. Law of duality.</span></span> The +observant student will not +have failed to note the remarkable +similarity between the theorems of this chapter +and those of the preceding. He will have noted +that points have replaced lines and lines have replaced +points; that points on a curve have been replaced by +tangents to a curve; that pencils have been replaced +by point-rows, and that a conic considered as made up +of a succession of points has been replaced by a conic +considered as generated by a moving tangent line. The +theory upon which this wonderful <span class="tei tei-hi"><span style="font-style: italic">law of duality</span></span> is based +will be developed in the next chapter.</p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc211" id="toc211"></a><a name="pdf212" id="pdf212"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. Given four lines in the plane, to construct another +which shall meet them in four harmonic points.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. Where are all such lines found?</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Given any five lines in the plane, construct on each +the point of contact with the conic tangent to them all.</p> + +<span class="tei tei-pb" id="page55">[pg 55]</span><a name="Pg55" id="Pg55" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Given four lines and the point of contact on one, to +construct the conic. ("To construct the conic" means here +to draw as many other tangents as may be desired.)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. Given three lines and the point of contact on two of +them, to construct the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Given four lines and the line at infinity, to construct +the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. Given three lines and the line at infinity, together +with the point of contact at infinity, to construct the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Given three lines, two of which are asymptotes, to +construct the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">9. Given five tangents to a conic, to draw a tangent +which shall be parallel to any one of them.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">10. The lines <span class="tei tei-hi"><span style="font-style: italic">a</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c</span></span> are drawn parallel to each other. +The lines <span class="tei tei-hi"><span style="font-style: italic">a'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c'</span></span> are also drawn parallel to each other. +Show why the lines (<span class="tei tei-hi"><span style="font-style: italic">ab'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">a'b</span></span>), (<span class="tei tei-hi"><span style="font-style: italic">bc'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b'c</span></span>), (<span class="tei tei-hi"><span style="font-style: italic">ca'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c'a</span></span>) meet in a +point. (In problems 6 to 10 inclusive, parallel lines are to +be drawn.)</p> +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc213" id="toc213"></a><a name="pdf214" id="pdf214"></a> +<span class="tei tei-pb" id="page56">[pg 56]</span><a name="Pg56" id="Pg56" class="tei tei-anchor"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER VI - POLES AND POLARS</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc215" id="toc215"></a><a name="pdf216" id="pdf216"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p95" id="p95" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">95. Inscribed and circumscribed quadrilaterals.</span></span> The +following theorems have been noted as special cases of +Pascal's and Brianchon's theorems:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If a quadrilateral be inscribed in a conic, two pairs of +opposite sides and the tangents at opposite vertices intersect +in four points, all of which lie on a straight line.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If a quadrilateral be circumscribed about a conic, the +lines joining two pairs of opposite vertices and the lines +joining two opposite points of contact are four lines which +meet in a point.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image26.png" width="763" height="415" alt="Figure 26" title="Fig. 26" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 26</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc217" id="toc217"></a><a name="pdf218" id="pdf218"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p96" id="p96" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">96. Definition of the polar line of a point.</span></span> Consider +the quadrilateral <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span> inscribed in the conic +(Fig. 26). It +determines the +four harmonic +points <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">D</span></span> which project +from <span class="tei tei-hi"><span style="font-style: italic">N</span></span> in to +the four harmonic +points <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">O</span></span>. Now +the tangents at <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> meet in <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, a point on the +line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>. The line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> is thus determined entirely by +<span class="tei tei-pb" id="page57">[pg 57]</span><a name="Pg57" id="Pg57" class="tei tei-anchor"></a> +the point <span class="tei tei-hi"><span style="font-style: italic">O</span></span>. For if we draw any line through it, meeting +the conic in <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, and construct the harmonic +conjugate <span class="tei tei-hi"><span style="font-style: italic">B</span></span> of <span class="tei tei-hi"><span style="font-style: italic">O</span></span> with respect to <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, and also +the two tangents at <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> which meet in the point +<span class="tei tei-hi"><span style="font-style: italic">P</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">BP</span></span> is the line in question. It thus appears +that the line <span class="tei tei-hi"><span style="font-style: italic">LON</span></span> may be any line whatever through <span class="tei tei-hi"><span style="font-style: italic">O</span></span>; +and since <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span> are four harmonic points, we +may describe the line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> as the locus of points which +are harmonic conjugates of <span class="tei tei-hi"><span style="font-style: italic">O</span></span> with respect to the two +points where any line through <span class="tei tei-hi"><span style="font-style: italic">O</span></span> meets the curve.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc219" id="toc219"></a><a name="pdf220" id="pdf220"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p97" id="p97" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">97.</span></span> Furthermore, since the tangents at <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> meet +on this same line, it appears as the locus of intersections +of pairs of tangents drawn at the extremities of chords +through <span class="tei tei-hi"><span style="font-style: italic">O</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc221" id="toc221"></a><a name="pdf222" id="pdf222"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p98" id="p98" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">98.</span></span> This important line, which is completely determined +by the point <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, is called the <span class="tei tei-hi"><span style="font-style: italic">polar</span></span> of <span class="tei tei-hi"><span style="font-style: italic">O</span></span> with +respect to the conic; and the point <span class="tei tei-hi"><span style="font-style: italic">O</span></span> is called the <span class="tei tei-hi"><span style="font-style: italic">pole</span></span> +of the line with respect to the conic.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc223" id="toc223"></a><a name="pdf224" id="pdf224"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p99" id="p99" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">99.</span></span> If a point <span class="tei tei-hi"><span style="font-style: italic">B</span></span> is on the polar of <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, then it is harmonically +conjugate to <span class="tei tei-hi"><span style="font-style: italic">O</span></span> with respect to the two intersections +<span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> of the line <span class="tei tei-hi"><span style="font-style: italic">BC</span></span> with the conic. But +for the same reason <span class="tei tei-hi"><span style="font-style: italic">O</span></span> is on the polar of <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. We have, +then, the fundamental theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">If one point lies on the polar of a second, then the +second lies on the polar of the first.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc225" id="toc225"></a><a name="pdf226" id="pdf226"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p100" id="p100" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">100. Conjugate points and lines.</span></span> Such a pair of points +are said to be <span class="tei tei-hi"><span style="font-style: italic">conjugate</span></span> with respect to the conic. Similarly, +lines are said to be <span class="tei tei-hi"><span style="font-style: italic">conjugate</span></span> to each other with +respect to the conic if one, and consequently each, +passes through the pole of the other.</p> + +<span class="tei tei-pb" id="page58">[pg 58]</span><a name="Pg58" id="Pg58" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image27.png" width="314" height="221" alt="Figure 27" title="Fig. 27" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 27</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc227" id="toc227"></a><a name="pdf228" id="pdf228"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p101" id="p101" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">101. Construction of the polar line of a given point.</span></span> +Given a point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, if it is within the conic (that is, if no +tangents may be drawn from <span class="tei tei-hi"><span style="font-style: italic">P</span></span> to the +conic), we may construct its polar line +by drawing through it any two chords +and joining the two points of intersection +of the two pairs of tangents +at their extremities. If the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> is +outside the conic, we may draw the two tangents and +construct the chord of contact (Fig. 27).</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc229" id="toc229"></a><a name="pdf230" id="pdf230"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p102" id="p102" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">102. Self-polar triangle.</span></span> In Fig. 26 it is not difficult +to see that <span class="tei tei-hi"><span style="font-style: italic">AOC</span></span> is a <span class="tei tei-hi"><span style="font-style: italic">self-polar</span></span> triangle, that is, each +vertex is the pole of the opposite side. For <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, <span class="tei tei-hi"><span style="font-style: italic">K</span></span> +are four harmonic points, and they project to <span class="tei tei-hi"><span style="font-style: italic">C</span></span> in four +harmonic rays. The line <span class="tei tei-hi"><span style="font-style: italic">CO</span></span>, therefore, meets the line +<span class="tei tei-hi"><span style="font-style: italic">AMN</span></span> in a point on the polar of <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, being separated from +<span class="tei tei-hi"><span style="font-style: italic">A</span></span> harmonically by the points <span class="tei tei-hi"><span style="font-style: italic">M</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. Similarly, the +line <span class="tei tei-hi"><span style="font-style: italic">CO</span></span> meets <span class="tei tei-hi"><span style="font-style: italic">KL</span></span> in a point on the polar of <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, and +therefore <span class="tei tei-hi"><span style="font-style: italic">CO</span></span> is the polar of <span class="tei tei-hi"><span style="font-style: italic">A</span></span>. Similarly, <span class="tei tei-hi"><span style="font-style: italic">OA</span></span> is the +polar of <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, and therefore <span class="tei tei-hi"><span style="font-style: italic">O</span></span> is the pole of <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc231" id="toc231"></a><a name="pdf232" id="pdf232"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p103" id="p103" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">103. Pole and polar projectively related.</span></span> Another very +important theorem comes directly from Fig. 26.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">As a point </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> moves along a straight line its polar with +respect to a conic revolves about a fixed point and describes +a pencil projective to the point-row described by </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">For, fix the points <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span> and let the point <span class="tei tei-hi"><span style="font-style: italic">A</span></span> move +along the line <span class="tei tei-hi"><span style="font-style: italic">AQ</span></span>; then the point-row <span class="tei tei-hi"><span style="font-style: italic">A</span></span> is projective +to the pencil <span class="tei tei-hi"><span style="font-style: italic">LK</span></span>, and since <span class="tei tei-hi"><span style="font-style: italic">K</span></span> moves along the conic, +the pencil <span class="tei tei-hi"><span style="font-style: italic">LK</span></span> is projective to the pencil <span class="tei tei-hi"><span style="font-style: italic">NK</span></span>, which in +turn is projective to the point-row <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, which, finally, is +projective to the pencil <span class="tei tei-hi"><span style="font-style: italic">OC</span></span>, which is the polar of <span class="tei tei-hi"><span style="font-style: italic">A</span></span>.</p> + +<span class="tei tei-pb" id="page59">[pg 59]</span><a name="Pg59" id="Pg59" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc233" id="toc233"></a><a name="pdf234" id="pdf234"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p104" id="p104" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">104. Duality.</span></span> We have, then, in the pole and polar +relation a device for setting up a one-to-one correspondence +between the points and lines of the plane—a correspondence +which may be called projective, because to +four harmonic points or lines correspond always four +harmonic lines or points. To every figure made up of +points and lines will correspond a figure made up of +lines and points. To a point-row of the second order, +which is a conic considered as a point-locus, corresponds +a pencil of rays of the second order, which is a conic +considered as a line-locus. The name 'duality' is used +to describe this sort of correspondence. It is important +to note that the dual relation is subject to the same +exceptions as the one-to-one correspondence is, and +must not be appealed to in cases where the one-to-one +correspondence breaks down. We have seen that there +is in Euclidean geometry one and only one ray in a +pencil which has no point in a point-row perspective to +it for a corresponding point; namely, the line parallel +to the line of the point-row. Any theorem, therefore, +that involves explicitly the point at infinity is not to +be translated into a theorem concerning lines. Further, +in the pencil the angle between two lines has nothing +to correspond to it in a point-row perspective to the +pencil. Any theorem, therefore, that mentions angles is +not translatable into another theorem by means of the +law of duality. Now we have seen that the notion of +the infinitely distant point on a line involves the notion +of dividing a segment into any number of equal parts—in +other words, of <span class="tei tei-hi"><span style="font-style: italic">measuring</span></span>. If, therefore, we call any +theorem that has to do with the line at infinity or with +<span class="tei tei-pb" id="page60">[pg 60]</span><a name="Pg60" id="Pg60" class="tei tei-anchor"></a> +the measurement of angles a <span class="tei tei-hi"><span style="font-style: italic">metrical</span></span> theorem, and any +other kind a <span class="tei tei-hi"><span style="font-style: italic">projective</span></span> theorem, we may put the case +as follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Any projective theorem involves another theorem, dual to +it, obtainable by interchanging everywhere the words 'point' +and 'line.'</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc235" id="toc235"></a><a name="pdf236" id="pdf236"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p105" id="p105" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">105. Self-dual theorems.</span></span> The theorems of this chapter +will be found, upon examination, to be <span class="tei tei-hi"><span style="font-style: italic">self-dual</span></span>; +that is, no new theorem results from applying the +process indicated in the preceding paragraph. It is +therefore useless to look for new results from the theorem +on the circumscribed quadrilateral derived from +Brianchon's, which is itself clearly the dual of Pascal's +theorem, and in fact was first discovered by dualization +of Pascal's.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc237" id="toc237"></a><a name="pdf238" id="pdf238"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p106" id="p106" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">106.</span></span> It should not be inferred from the above discussion +that one-to-one correspondences may not be devised +that will control certain of the so-called metrical relations. +A very important one may be easily found that +leaves angles unaltered. The relation called <span class="tei tei-hi"><span style="font-style: italic">similarity</span></span> +leaves ratios between corresponding segments unaltered. +The above statements apply only to the particular one-to-one +correspondence considered.</p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc239" id="toc239"></a><a name="pdf240" id="pdf240"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. Given a quadrilateral, construct the quadrangle polar +to it with respect to a given conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. A point moves along a straight line. Show that its +polar lines with respect to two given conics generate a +point-row of the second order.</p> + +<span class="tei tei-pb" id="page61">[pg 61]</span><a name="Pg61" id="Pg61" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Given five points, draw the polar of a point with respect +to the conic passing through them, without drawing +the conic itself.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Given five lines, draw the polar of a point with respect +to the conic tangent to them, without drawing the +conic itself.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. Dualize problems 3 and 4.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Given four points on the conic, and the tangent at one +of them, draw the polar of a given point without drawing +the conic. Dualize.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. A point moves on a conic. Show that its polar line +with respect to another conic describes a pencil of rays of +the second order.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Suggestion.</span></span> Replace the given conic by a pair of protective pencils.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Show that the poles of the tangents of one conic with +respect to another lie on a conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">9. The polar of a point <span class="tei tei-hi"><span style="font-style: italic">A</span></span> with respect to one conic is <span class="tei tei-hi"><span style="font-style: italic">a</span></span>, +and the pole of <span class="tei tei-hi"><span style="font-style: italic">a</span></span> with respect to another conic is <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>. Show +that as <span class="tei tei-hi"><span style="font-style: italic">A</span></span> travels along a line, <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> also travels along another +line. In general, if <span class="tei tei-hi"><span style="font-style: italic">A</span></span> describes a curve of degree <span class="tei tei-hi"><span style="font-style: italic">n</span></span>, show +that <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> describes another curve of the same degree <span class="tei tei-hi"><span style="font-style: italic">n</span></span>. (The +degree of a curve is the greatest number of points that it +may have in common with any line in the plane.)</p> +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc241" id="toc241"></a><a name="pdf242" id="pdf242"></a> +<span class="tei tei-pb" id="page62">[pg 62]</span><a name="Pg62" id="Pg62" class="tei tei-anchor"></a> + +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc243" id="toc243"></a><a name="pdf244" id="pdf244"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p107" id="p107" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">107. Diameters. Center.</span></span> After what has been said in +the last chapter one would naturally expect to get at +the metrical properties of the conic sections by the +introduction of the infinite elements in the plane. Entering +into the theory of poles and polars with these +elements, we have the following definitions:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The polar line of an infinitely distant point is called +a <span class="tei tei-hi"><span style="font-style: italic">diameter</span></span>, and the pole of the infinitely distant line is +called the <span class="tei tei-hi"><span style="font-style: italic">center</span></span>, of the conic.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc245" id="toc245"></a><a name="pdf246" id="pdf246"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p108" id="p108" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">108.</span></span> From the harmonic properties of poles and polars,</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The center bisects all chords through it (§ 39).</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Every diameter passes through the center.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">All chords through the same point at infinity (that is, +each of a set of parallel chords) are bisected by the diameter +which is the polar of that infinitely distant point.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc247" id="toc247"></a><a name="pdf248" id="pdf248"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p109" id="p109" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">109. Conjugate diameters.</span></span> We have already defined +conjugate lines as lines which pass each through the +pole of the other (§ 100).</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Any diameter bisects all chords parallel to its conjugate.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The tangents at the extremities of any diameter are +parallel, and parallel to the conjugate diameter.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Diameters parallel to the sides of a circumscribed parallelogram +are conjugate.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">All these theorems are easy exercises for the student.</p> + +<span class="tei tei-pb" id="page63">[pg 63]</span><a name="Pg63" id="Pg63" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc249" id="toc249"></a><a name="pdf250" id="pdf250"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p110" id="p110" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">110. Classification of conics.</span></span> Conics are classified according +to their relation to the infinitely distant line. +If a conic has two points in common with the line at +infinity, it is called a <span class="tei tei-hi"><span style="font-style: italic">hyperbola</span></span>; if it has no point in +common with the infinitely distant line, it is called an +<span class="tei tei-hi"><span style="font-style: italic">ellipse</span></span>; if it is tangent to the line at infinity, it is called +a <span class="tei tei-hi"><span style="font-style: italic">parabola</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc251" id="toc251"></a><a name="pdf252" id="pdf252"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p111" id="p111" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">111.</span></span> <span class="tei tei-hi"><span style="font-style: italic">In a hyperbola the center is outside the curve</span></span> +(§ 101), since the two tangents to the curve at the points +where it meets the line at infinity determine by their +intersection the center. As previously noted, these two +tangents are called the <span class="tei tei-hi"><span style="font-style: italic">asymptotes</span></span> of the curve. The +ellipse and the parabola have no asymptotes.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc253" id="toc253"></a><a name="pdf254" id="pdf254"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p112" id="p112" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">112.</span></span> <span class="tei tei-hi"><span style="font-style: italic">The center of the parabola is at infinity, and therefore +all its diameters are parallel,</span></span> for the pole of a tangent +line is the point of contact.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The locus of the middle points of a series of parallel +chords in a parabola is a diameter, and the direction of +the line of centers is the same for all series of parallel +chords.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The center of an ellipse is within the curve.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image28.png" width="467" height="437" alt="Figure 28" title="Fig. 28" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 28</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc255" id="toc255"></a><a name="pdf256" id="pdf256"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p113" id="p113" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">113. Theorems concerning asymptotes.</span></span> We derived as +a consequence of the theorem of Brianchon (§ 89) the +proposition that if a triangle be circumscribed about +a conic, the lines joining the vertices to the points +of contact of the opposite sides all meet in a point. +Take, now, for two of the tangents the asymptotes of +a hyperbola, and let any third tangent cut them in <span class="tei tei-hi"><span style="font-style: italic">A</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> (Fig. 28). If, then, <span class="tei tei-hi"><span style="font-style: italic">O</span></span> is the intersection of the +asymptotes,—and therefore the center of the curve,— +<span class="tei tei-pb" id="page64">[pg 64]</span><a name="Pg64" id="Pg64" class="tei tei-anchor"></a> +then the triangle <span class="tei tei-hi"><span style="font-style: italic">OAB</span></span> is circumscribed about the curve. +By the theorem just quoted, the line through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> parallel +to <span class="tei tei-hi"><span style="font-style: italic">OB</span></span>, the line through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> parallel to <span class="tei tei-hi"><span style="font-style: italic">OA</span></span>, and the +line <span class="tei tei-hi"><span style="font-style: italic">OP</span></span> through the point of +contact of the tangent <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> +all meet in a point <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. But +<span class="tei tei-hi"><span style="font-style: italic">OACB</span></span> is a parallelogram, and +<span class="tei tei-hi"><span style="font-style: italic">PA = PB</span></span>. Therefore</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The asymptotes cut off on +each tangent a segment which is +bisected by the point of contact.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc257" id="toc257"></a><a name="pdf258" id="pdf258"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p114" id="p114" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">114.</span></span> If we draw a line <span class="tei tei-hi"><span style="font-style: italic">OQ</span></span> +parallel to <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">OP</span></span> and <span class="tei tei-hi"><span style="font-style: italic">OQ</span></span> are conjugate diameters, +since <span class="tei tei-hi"><span style="font-style: italic">OQ</span></span> is parallel to the tangent at the point +where <span class="tei tei-hi"><span style="font-style: italic">OP</span></span> meets the curve. Then, since <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and +the point at infinity on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> are four harmonic points, +we have the theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Conjugate diameters of the hyperbola are harmonic +conjugates with respect to the asymptotes.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc259" id="toc259"></a><a name="pdf260" id="pdf260"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p115" id="p115" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">115.</span></span> The chord <span class="tei tei-hi"><span style="font-style: italic">A"B"</span></span>, parallel to the diameter <span class="tei tei-hi"><span style="font-style: italic">OQ</span></span>, is +bisected at <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> by the conjugate diameter <span class="tei tei-hi"><span style="font-style: italic">OP</span></span>. If the +chord <span class="tei tei-hi"><span style="font-style: italic">A"B"</span></span> meet the asymptotes in <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, +and the point at infinity are four harmonic points, and +therefore <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> is the middle point of <span class="tei tei-hi"><span style="font-style: italic">A'B'</span></span>. Therefore +<span class="tei tei-hi"><span style="font-style: italic">A'A" = B'B"</span></span> and we have the theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The segments cut off on any chord between the hyperbola +and its asymptotes are equal.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc261" id="toc261"></a><a name="pdf262" id="pdf262"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p116" id="p116" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">116.</span></span> This theorem furnishes a ready means of constructing +the hyperbola by points when a point on the +curve and the two asymptotes are given.</p> + +<span class="tei tei-pb" id="page65">[pg 65]</span><a name="Pg65" id="Pg65" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image29.png" width="667" height="504" alt="Figure 29" title="Fig. 29" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 29</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc263" id="toc263"></a><a name="pdf264" id="pdf264"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p117" id="p117" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">117.</span></span> For the circumscribed quadrilateral, Brianchon's +theorem gave (§ 88) <span class="tei tei-hi"><span style="font-style: italic">The lines joining opposite vertices +and the lines joining opposite points of contact are four +lines meeting in a point.</span></span> Take now for two of the +tangents the asymptotes, and let <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">CD</span></span> be any +other two (Fig. 29). +If <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are opposite +vertices, and +also <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, then +<span class="tei tei-hi"><span style="font-style: italic">AC</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BD</span></span> are parallel, +and parallel to +<span class="tei tei-hi"><span style="font-style: italic">PQ</span></span>, the line joining +the points of contact +of <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">CD</span></span>, +for these are three of +the four lines of the +theorem just quoted. The fourth is the line at infinity +which joins the point of contact of the asymptotes. It +is thus seen that the triangles <span class="tei tei-hi"><span style="font-style: italic">ABC</span></span> and <span class="tei tei-hi"><span style="font-style: italic">ADC</span></span> are +equivalent, and therefore the triangles <span class="tei tei-hi"><span style="font-style: italic">AOB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">COD</span></span> +are also. The tangent AB may be fixed, and the tangent +<span class="tei tei-hi"><span style="font-style: italic">CD</span></span> chosen arbitrarily; therefore</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The triangle formed by any tangent to the hyperbola +and the two asymptotes is of constant area.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc265" id="toc265"></a><a name="pdf266" id="pdf266"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p118" id="p118" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">118. Equation of hyperbola referred to the asymptotes.</span></span> +Draw through the point of contact <span class="tei tei-hi"><span style="font-style: italic">P</span></span> of the tangent +<span class="tei tei-hi"><span style="font-style: italic">AB</span></span> two lines, one parallel to one asymptote and the +other parallel to the other. One of these lines meets +<span class="tei tei-hi"><span style="font-style: italic">OB</span></span> at a distance <span class="tei tei-hi"><span style="font-style: italic">y</span></span> from <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, and the other meets <span class="tei tei-hi"><span style="font-style: italic">OA</span></span> at +a distance <span class="tei tei-hi"><span style="font-style: italic">x</span></span> from <span class="tei tei-hi"><span style="font-style: italic">O</span></span>. Then, since <span class="tei tei-hi"><span style="font-style: italic">P</span></span> is the middle point +<span class="tei tei-pb" id="page66">[pg 66]</span><a name="Pg66" id="Pg66" class="tei tei-anchor"></a> +of <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">x</span></span> is one half of <span class="tei tei-hi"><span style="font-style: italic">OA</span></span> and <span class="tei tei-hi"><span style="font-style: italic">y</span></span> is one half of <span class="tei tei-hi"><span style="font-style: italic">OB</span></span>. +The area of the parallelogram whose adjacent sides are +<span class="tei tei-hi"><span style="font-style: italic">x</span></span> and <span class="tei tei-hi"><span style="font-style: italic">y</span></span> is one half the area of the triangle <span class="tei tei-hi"><span style="font-style: italic">AOB</span></span>, and +therefore, by the preceding paragraph, is constant. This +area is equal to <span class="tei tei-hi"><span style="font-style: italic">xy · </span><span class="tei tei-hi"><span style="font-style: normal">sin</span></span><span style="font-style: italic"> α</span></span>, where α is the constant angle +between the asymptotes. It follows that the product <span class="tei tei-hi"><span style="font-style: italic">xy</span></span> +is constant, and since <span class="tei tei-hi"><span style="font-style: italic">x</span></span> and <span class="tei tei-hi"><span style="font-style: italic">y</span></span> are the oblique coördinates +of the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, the asymptotes being the axes +of reference, we have</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The equation of the hyperbola, referred to the asymptotes +as axes, is </span><span class="tei tei-hi"><span style="font-style: italic">xy =</span></span><span style="font-style: italic"> constant.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">This identifies the curve with the hyperbola as defined +and discussed in works on analytic geometry.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc267" id="toc267"></a><a name="pdf268" id="pdf268"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image30.png" width="648" height="767" alt="Figure 30" title="Fig. 30" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 30</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p119" id="p119" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">119. Equation of +parabola.</span></span> We have +defined the parabola +as a conic which is +tangent to the line +at infinity (§ 110). +Draw now two tangents +to the curve +(Fig. 30), meeting in +<span class="tei tei-hi"><span style="font-style: italic">A</span></span>, the points of contact +being <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. +These two tangents, +together with the +line at infinity, form +a triangle circumscribed +about the +conic. Draw through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> a parallel to <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, and through +<span class="tei tei-hi"><span style="font-style: italic">C</span></span> a parallel to <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>. If these meet in <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> is a +<span class="tei tei-pb" id="page67">[pg 67]</span><a name="Pg67" id="Pg67" class="tei tei-anchor"></a> +diameter. Let <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> meet the curve in <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, and the chord +<span class="tei tei-hi"><span style="font-style: italic">BC</span></span> in <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>. <span class="tei tei-hi"><span style="font-style: italic">P</span></span> is then the middle point of <span class="tei tei-hi"><span style="font-style: italic">AQ</span></span>. Also, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span> +is the middle point of the chord <span class="tei tei-hi"><span style="font-style: italic">BC</span></span>, and therefore the +diameter <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> bisects all chords parallel to <span class="tei tei-hi"><span style="font-style: italic">BC</span></span>. In particular, +<span class="tei tei-hi"><span style="font-style: italic">AD</span></span> passes through <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, the point of contact of +the tangent drawn parallel to <span class="tei tei-hi"><span style="font-style: italic">BC</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Draw now another tangent, meeting <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> in <span class="tei tei-hi"><span style="font-style: italic">B'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">AC</span></span> +in <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>. Then these three, with the line at infinity, make +a circumscribed quadrilateral. But, by Brianchon's theorem +applied to a quadrilateral (§ 88), it appears that a +parallel to <span class="tei tei-hi"><span style="font-style: italic">AC</span></span> through <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, a parallel to <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> through <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, +and the line <span class="tei tei-hi"><span style="font-style: italic">BC</span></span> meet in a point <span class="tei tei-hi"><span style="font-style: italic">D'</span></span>. Also, from the similar +triangles <span class="tei tei-hi"><span style="font-style: italic">BB'D'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BAC</span></span> we have, for all positions of the +tangent line <span class="tei tei-hi"><span style="font-style: italic">B'C</span></span>,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">B'D' : BB' = AC : AB,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or, since <span class="tei tei-hi"><span style="font-style: italic">B'D' = AC'</span></span>,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AC': BB' = AC:AB =</span></span> constant. +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If another tangent meet <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> in <span class="tei tei-hi"><span style="font-style: italic">B"</span></span> and <span class="tei tei-hi"><span style="font-style: italic">AC</span></span> in <span class="tei tei-hi"><span style="font-style: italic">C"</span></span>, we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic"> +AC' : BB' = AC" : BB", +</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and by subtraction we get</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">C'C" : B'B" =</span></span> constant; +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">whence</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The segments cut off on any two tangents to a parabola +by a variable tangent are proportional.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If now we take the tangent <span class="tei tei-hi"><span style="font-style: italic">B'C'</span></span> as axis of ordinates, +and the diameter through the point of contact <span class="tei tei-hi"><span style="font-style: italic">O</span></span> as axis +of abscissas, calling the coordinates of <span class="tei tei-hi"><span style="font-style: italic">B(x, y)</span></span> and of +<span class="tei tei-hi"><span style="font-style: italic">C(x', y')</span></span>, then, from the similar triangles <span class="tei tei-hi"><span style="font-style: italic">BMD'</span></span> and +we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">y : y' = BD' : D'C = BB' : AB'.</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Also</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">y : y' = B'D' : C'C = AC' : C'C.</span></span> +</p> + +<span class="tei tei-pb" id="page68">[pg 68]</span><a name="Pg68" id="Pg68" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If now a line is drawn through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> parallel to a diameter, +meeting the axis of ordinates in <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AK : OQ' = AC' : CC' = y : y',</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OM : AK = BB' : AB' = y : y',</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and, by multiplication,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OM : OQ' = y</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic"> : y'</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic">,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">x : x' = y</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic"> : y'</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic">;</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">whence</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The abscissas of two points on a parabola are to each +other as the squares of the corresponding coördinates, a +diameter and the tangent to the curve at the extremity of +the diameter being the axes of reference.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The last equation may be written</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">y</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic"> = 2px,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">where <span class="tei tei-hi"><span style="font-style: italic">2p</span></span> stands for <span class="tei tei-hi"><span style="font-style: italic">y'</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic"> : x'</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The parabola is thus identified with the curve of the +same name studied in treatises on analytic geometry.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc269" id="toc269"></a><a name="pdf270" id="pdf270"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p120" id="p120" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">120. Equation of central conics referred to conjugate +diameters.</span></span> Consider now a <span class="tei tei-hi"><span style="font-style: italic">central conic</span></span>, that is, one +which is not a parabola and the center of which is +therefore at a finite distance. Draw any four tangents +to it, two of which are parallel (Fig. 31). Let the +parallel tangents meet one of the other tangents in <span class="tei tei-hi"><span style="font-style: italic">A</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and the other in <span class="tei tei-hi"><span style="font-style: italic">C</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, and let <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">Q</span></span> be +the points of contact of the parallel tangents <span class="tei tei-hi"><span style="font-style: italic">R</span></span> and <span class="tei tei-hi"><span style="font-style: italic">S</span></span> +of the others. Then <span class="tei tei-hi"><span style="font-style: italic">AC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">BD</span></span>, <span class="tei tei-hi"><span style="font-style: italic">PQ</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">RS</span></span> all meet in +a point <span class="tei tei-hi"><span style="font-style: italic">W</span></span> (§ 88). From the figure,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">PW : WQ = AP : QC = PD : BQ,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AP · BQ = PD · QC.</span></span> +</p> + +<span class="tei tei-pb" id="page69">[pg 69]</span><a name="Pg69" id="Pg69" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If now <span class="tei tei-hi"><span style="font-style: italic">DC</span></span> is a fixed tangent and <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> a variable one, +we have from this equation</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AP · BQ = </span><span class="tei tei-hi" style="text-align: center"><span style="font-style: normal">constant.</span></span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">This constant will be positive or negative according as +<span class="tei tei-hi"><span style="font-style: italic">PA</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BQ</span></span> are measured in the same or in opposite +directions. Accordingly we write</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AP · BQ = ± b</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic">.</span></span> +</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image31.png" width="500" height="568" alt="Figure 31" title="Fig. 31" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 31</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Since <span class="tei tei-hi"><span style="font-style: italic">AD</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BC</span></span> are parallel tangents, <span class="tei tei-hi"><span style="font-style: italic">PQ</span></span> is a diameter +and the conjugate diameter is parallel to <span class="tei tei-hi"><span style="font-style: italic">AD</span></span>. The +middle point of <span class="tei tei-hi"><span style="font-style: italic">PQ</span></span> is the +center of the conic. We take +now for the axis of abscissas +the diameter <span class="tei tei-hi"><span style="font-style: italic">PQ</span></span>, and the +conjugate diameter for the +axis of ordinates. Join <span class="tei tei-hi"><span style="font-style: italic">A</span></span> to +<span class="tei tei-hi"><span style="font-style: italic">Q</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> to <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and draw a +line through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> parallel to +the axis of ordinates. These +three lines all meet in a point +<span class="tei tei-hi"><span style="font-style: italic">N</span></span>, because <span class="tei tei-hi"><span style="font-style: italic">AP</span></span>, <span class="tei tei-hi"><span style="font-style: italic">BQ</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> +form a triangle circumscribed +to the conic. Let <span class="tei tei-hi"><span style="font-style: italic">NS</span></span> meet +<span class="tei tei-hi"><span style="font-style: italic">PQ</span></span> in <span class="tei tei-hi"><span style="font-style: italic">M</span></span>. Then, from the properties of the circumscribed +triangle (§ 89), <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, and the point at infinity +on <span class="tei tei-hi"><span style="font-style: italic">NS</span></span> are four harmonic points, and therefore <span class="tei tei-hi"><span style="font-style: italic">N</span></span> is the +middle point of <span class="tei tei-hi"><span style="font-style: italic">MS</span></span>. If the coördinates of <span class="tei tei-hi"><span style="font-style: italic">S</span></span> are <span class="tei tei-hi"><span style="font-style: italic">(x, y)</span></span>, +so that <span class="tei tei-hi"><span style="font-style: italic">OM</span></span> is <span class="tei tei-hi"><span style="font-style: italic">x</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MS</span></span> is <span class="tei tei-hi"><span style="font-style: italic">y</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">MN = y/2</span></span>. Now +from the similar triangles <span class="tei tei-hi"><span style="font-style: italic">PMN</span></span> and <span class="tei tei-hi"><span style="font-style: italic">PQB</span></span> we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">BQ : PQ = NM : PM,</span></span> +</p> + +<span class="tei tei-pb" id="page70">[pg 70]</span><a name="Pg70" id="Pg70" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and from the similar triangles <span class="tei tei-hi"><span style="font-style: italic">PQA</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MQN</span></span>,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">AP : PQ = MN : MQ,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">whence, multiplying, we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">±b</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic">/4 a</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic"> = y</span><span class="tei tei-hi" style="text-align: center"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic">/4 (a + x)(a - x),</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">where</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/3.png" alt="[formula]" width="55" height="30" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or, simplifying,</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/4.png" alt="[formula]" width="134" height="17" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">which is the equation of an ellipse when <span class="tei tei-hi"><span style="font-style: italic">b</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: super">2</span></span></span> has a positive +sign, and of a hyperbola when <span class="tei tei-hi"><span style="font-style: italic">b</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: super">2</span></span></span> has a negative +sign. We have thus identified point-rows of the second +order with the curves given by equations of the second +degree.</p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc271" id="toc271"></a><a name="pdf272" id="pdf272"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. Draw a chord of a given conic which shall be bisected +by a given point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. Show that all chords of a given conic that are bisected +by a given chord are tangent to a parabola.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Construct a parabola, given two tangents with their +points of contact.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Construct a parabola, given three points and the direction +of the diameters.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. A line <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> is drawn through the pole <span class="tei tei-hi"><span style="font-style: italic">U</span></span> of a line <span class="tei tei-hi"><span style="font-style: italic">u</span></span> and +at right angles to <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. The line <span class="tei tei-hi"><span style="font-style: italic">u</span></span> revolves about a point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. +Show that the line <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> is tangent to a parabola. (The lines <span class="tei tei-hi"><span style="font-style: italic">u</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">u'</span></span> are called normal conjugates.)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Given a circle and its center <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, to draw a line through +a given point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> parallel to a given line <span class="tei tei-hi"><span style="font-style: italic">q</span></span>. Prove the following +construction: Let <span class="tei tei-hi"><span style="font-style: italic">p</span></span> be the polar of <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span> the pole of +<span class="tei tei-hi"><span style="font-style: italic">q</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">A</span></span> the intersection of <span class="tei tei-hi"><span style="font-style: italic">p</span></span> with <span class="tei tei-hi"><span style="font-style: italic">OQ</span></span>. The polar of <span class="tei tei-hi"><span style="font-style: italic">A</span></span> is +the desired line.</p> +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<span class="tei tei-pb" id="page71">[pg 71]</span><a name="Pg71" id="Pg71" class="tei tei-anchor"></a> +<a name="toc273" id="toc273"></a><a name="pdf274" id="pdf274"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER VIII - INVOLUTION</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc275" id="toc275"></a><a name="pdf276" id="pdf276"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image32.png" width="756" height="504" alt="Figure 32" title="Fig. 32" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 32</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p121" id="p121" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">121. Fundamental theorem.</span></span> The important theorem +concerning two complete quadrangles (§ 26), upon which +the theory of four harmonic points was based, can easily +be extended to +the case where +the four lines +<span class="tei tei-hi"><span style="font-style: italic">KL</span></span>, <span class="tei tei-hi"><span style="font-style: italic">K'L'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">MN</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">M'N'</span></span> do not +all meet in the +same point <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, +and the more +general theorem +that results +may also +be made the basis of a theory no less important, which has +to do with six points on a line. The theorem is as follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given two complete quadrangles, </span><span class="tei tei-hi"><span style="font-style: italic">K</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">L</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N</span></span><span style="font-style: italic"> and +</span><span class="tei tei-hi"><span style="font-style: italic">K'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">L'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">M'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N'</span></span><span style="font-style: italic">, so related that </span><span class="tei tei-hi"><span style="font-style: italic">KL</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">K'L'</span></span><span style="font-style: italic"> meet in </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, +</span><span class="tei tei-hi"><span style="font-style: italic">MN</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">M'N'</span></span><span style="font-style: italic"> in </span><span class="tei tei-hi"><span style="font-style: italic">A'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">KN</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">K'N'</span></span><span style="font-style: italic"> in </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">LM</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">L'M'</span></span><span style="font-style: italic"> +in </span><span class="tei tei-hi"><span style="font-style: italic">B'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">LN</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">L'N'</span></span><span style="font-style: italic"> in </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">KM</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">K'M'</span></span><span style="font-style: italic"> in </span><span class="tei tei-hi"><span style="font-style: italic">C'</span></span><span style="font-style: italic">, then, +if </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">A'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B'</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic"> are in a straight line, the point </span><span class="tei tei-hi"><span style="font-style: italic">C'</span></span><span style="font-style: italic"> +also lies on that straight line.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The theorem follows from Desargues's theorem +(Fig. 32). It is seen that <span class="tei tei-hi"><span style="font-style: italic">KK'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">LL'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">MM'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">NN'</span></span> all +<span class="tei tei-pb" id="page72">[pg 72]</span><a name="Pg72" id="Pg72" class="tei tei-anchor"></a> +meet in a point, and thus, from the same theorem, applied +to the triangles <span class="tei tei-hi"><span style="font-style: italic">KLM</span></span> and <span class="tei tei-hi"><span style="font-style: italic">K'L'M'</span></span>, the point <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> is on +the same line with <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>. As in the simpler case, it +is seen that there is an indefinite number of quadrangles +which may be drawn, two sides of which go through +<span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, two through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, and one through <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. +The sixth side must then go through <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>. Therefore,</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc277" id="toc277"></a><a name="pdf278" id="pdf278"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p122" id="p122" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">122.</span></span> <span class="tei tei-hi"><span style="font-style: italic">Two pairs of points, </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">A'</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B'</span></span><span style="font-style: italic">, being given, +then the point </span><span class="tei tei-hi"><span style="font-style: italic">C'</span></span><span style="font-style: italic"> corresponding to any given point </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic"> is +uniquely determined.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The construction of this sixth point is easily accomplished. +Draw through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> any two lines, and +cut across them by any line through <span class="tei tei-hi"><span style="font-style: italic">C</span></span> in the points +<span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. Join <span class="tei tei-hi"><span style="font-style: italic">N</span></span> to <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">L</span></span> to <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, thus determining +the points <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> on the two lines through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, +The line <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> determines the desired point <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>. Manifestly, +starting from <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, we come in this way always to the +same point <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. The particular quadrangle employed is +of no consequence. Moreover, since one pair of opposite +sides in a complete quadrangle is not distinguishable +in any way from any other, the same set of six points +will be obtained by starting from the pairs <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">CC'</span></span>, or from the pairs <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc279" id="toc279"></a><a name="pdf280" id="pdf280"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p123" id="p123" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">123. Definition of involution of points on a line.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Three pairs of points on a line are said to be in involution +if through each pair may be drawn a pair of opposite +sides of a complete quadrangle. If two pairs are fixed and +one of the third pair describes the line, then the other also +describes the line, and the points of the line are said to be +paired in the involution determined by the two fixed pairs.</span></span></p> + +<span class="tei tei-pb" id="page73">[pg 73]</span><a name="Pg73" id="Pg73" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image33.png" width="739" height="511" alt="Figure 33" title="Fig. 33" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 33</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc281" id="toc281"></a><a name="pdf282" id="pdf282"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p124" id="p124" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">124. Double-points in an involution.</span></span> The points <span class="tei tei-hi"><span style="font-style: italic">C</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">C'</span></span> describe projective point-rows, as may be seen by fixing +the points <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span>. The self-corresponding points, of +which there are two or none, are called the <span class="tei tei-hi"><span style="font-style: italic">double-points</span></span> in +the involution. It is not difficult to see that the double-points +in the involution are harmonic conjugates with +respect to corresponding points in the involution. For, +fixing as before the points <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, let the intersection +of the lines <span class="tei tei-hi"><span style="font-style: italic">CL</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C'M</span></span> be <span class="tei tei-hi"><span style="font-style: italic">P</span></span> (Fig. 33). The locus of <span class="tei tei-hi"><span style="font-style: italic">P</span></span> is +a conic which goes through the double-points, because the +point-rows <span class="tei tei-hi"><span style="font-style: italic">C</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">C'</span></span> are projective, +and therefore so +are the pencils +<span class="tei tei-hi"><span style="font-style: italic">LC</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MC'</span></span> +which generate +the locus of <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. +Also, when <span class="tei tei-hi"><span style="font-style: italic">C</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> fall together, +the point +<span class="tei tei-hi"><span style="font-style: italic">P</span></span> coincides with +them. Further, the tangents at <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> to this conic +described by <span class="tei tei-hi"><span style="font-style: italic">P</span></span> are the lines <span class="tei tei-hi"><span style="font-style: italic">LB</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MB</span></span>. For in the +pencil at <span class="tei tei-hi"><span style="font-style: italic">L</span></span> the ray <span class="tei tei-hi"><span style="font-style: italic">LM</span></span> common to the two pencils which +generate the conic is the ray <span class="tei tei-hi"><span style="font-style: italic">LB'</span></span> and corresponds to the +ray <span class="tei tei-hi"><span style="font-style: italic">MB</span></span> of <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, which is therefore the tangent line to the +conic at <span class="tei tei-hi"><span style="font-style: italic">M</span></span>. Similarly for the tangent <span class="tei tei-hi"><span style="font-style: italic">LB</span></span> at <span class="tei tei-hi"><span style="font-style: italic">L</span></span>. <span class="tei tei-hi"><span style="font-style: italic">LM</span></span> is +therefore the polar of <span class="tei tei-hi"><span style="font-style: italic">B</span></span> with respect to this conic, and +<span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span> are therefore harmonic conjugates with respect +to the double-points. The same discussion applies to any +other pair of corresponding points in the involution.</p> + +<span class="tei tei-pb" id="page74">[pg 74]</span><a name="Pg74" id="Pg74" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image34.png" width="767" height="493" alt="Figure 34" title="Fig. 34" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 34</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc283" id="toc283"></a><a name="pdf284" id="pdf284"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p125" id="p125" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">125. Desargues's theorem concerning conics through +four points.</span></span> Let <span class="tei tei-hi"><span style="font-style: italic">DD'</span></span> be any pair of points in the involution +determined as above, and consider the conic +passing through the five points <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>. We +shall use Pascal's theorem to show that this conic also +passes through <span class="tei tei-hi"><span style="font-style: italic">D'</span></span>. The point <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> is determined as follows: +Fix <span class="tei tei-hi"><span style="font-style: italic">L</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span> as before (Fig. 34) and join <span class="tei tei-hi"><span style="font-style: italic">D</span></span> to <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, +giving on <span class="tei tei-hi"><span style="font-style: italic">MN</span></span> +the point <span class="tei tei-hi"><span style="font-style: italic">N'</span></span>. +Join <span class="tei tei-hi"><span style="font-style: italic">N'</span></span> to <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, +giving on <span class="tei tei-hi"><span style="font-style: italic">LK</span></span> +the point <span class="tei tei-hi"><span style="font-style: italic">K'</span></span>. +Then <span class="tei tei-hi"><span style="font-style: italic">MK'</span></span> determines +the +point <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> on +the line <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span>, +given by the +complete quadrangle +<span class="tei tei-hi"><span style="font-style: italic">K'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">L</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N'</span></span>. Consider the following six points, +numbering them in order: <span class="tei tei-hi"><span style="font-style: italic">D = 1</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D' = 2</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M = 3</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N = 4</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">K = 5</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">L = 6</span></span>. We have the following intersections: +<span class="tei tei-hi"><span style="font-style: italic">B = (12-45)</span></span>, <span class="tei tei-hi"><span style="font-style: italic">K' = (23-56)</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N' = (34-61)</span></span>; and since by +construction <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">K'</span></span> are on a straight line, it follows +from the converse of Pascal's theorem, which is +easily established, that the six points are on a conic. +We have, then, the beautiful theorem due to Desargues:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The system of conics through four points meets any line +in the plane in pairs of points in involution.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc285" id="toc285"></a><a name="pdf286" id="pdf286"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p126" id="p126" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">126.</span></span> It appears also that the six points in involution +determined by the quadrangle through the four fixed +<span class="tei tei-pb" id="page75">[pg 75]</span><a name="Pg75" id="Pg75" class="tei tei-anchor"></a> +points belong also to the same involution with the +points cut out by the system of conics, as indeed we +might infer from the fact that the three pairs of opposite +sides of the quadrangle may be considered as +degenerate conics of the system.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc287" id="toc287"></a><a name="pdf288" id="pdf288"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p127" id="p127" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">127. Conics through four points touching a given line.</span></span> +It is further evident that the involution determined on +a line by the system of conics will have a double-point +where a conic of the system is tangent to the line. We +may therefore infer the theorem</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Through four fixed points in the plane two conics or +none may be drawn tangent to any given line.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image35.png" width="404" height="522" alt="Figure 35" title="Fig. 35" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 35</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc289" id="toc289"></a><a name="pdf290" id="pdf290"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p128" id="p128" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">128. Double correspondence.</span></span> We have seen that corresponding +points in an involution form two projective +point-rows superposed on the same straight line. Two +projective point-rows superposed +on the same straight line are, however, +not necessarily in involution, +as a simple example will show. +Take two lines, <span class="tei tei-hi"><span style="font-style: italic">a</span></span> and <span class="tei tei-hi"><span style="font-style: italic">a'</span></span>, which +both revolve about a fixed point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> +and which always make the same +angle with each other (Fig. 35). +These lines cut out on any line +in the plane which does not pass +through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> two projective point-rows, +which are not, however, in +involution unless the angle between the lines is a right +angles. For a point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> may correspond to a point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, +which in turn will correspond to some other point +<span class="tei tei-pb" id="page76">[pg 76]</span><a name="Pg76" id="Pg76" class="tei tei-anchor"></a> +than <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. The peculiarity of point-rows in involution +is that any point will correspond to the same point, +in whichever point-row it is considered as belonging. +In this case, if a point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> corresponds to a point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, then +the point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> corresponds back again to the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. +The points <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> are then said to <span class="tei tei-hi"><span style="font-style: italic">correspond doubly</span></span>. +This notion is worthy of further study.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image36.png" width="868" height="425" alt="Figure 36" title="Fig. 36" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 36</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc291" id="toc291"></a><a name="pdf292" id="pdf292"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p129" id="p129" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">129. Steiner's construction.</span></span> It will be observed that +the solution of the fundamental problem given in § 83, +<span class="tei tei-hi"><span style="font-style: italic">Given three pairs of points of two protective point-rows, to +construct other pairs</span></span>, cannot be carried out if the two +point-rows lie on the same straight line. Of course the +method may be easily altered to cover that case also, +but it is worth while to give another solution of the +problem, due to Steiner, which will also give further +information regarding the theory of involution, and +which may, indeed, be used as a foundation for that +theory. Let the two point-rows <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, ... and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D'</span></span>, ... be superposed on the line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>. Project +them both to a point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> and pass any conic <span class="tei tei-hi"><span style="font-style: italic">κ</span></span> through <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. +We thus obtain two projective pencils, <span class="tei tei-hi"><span style="font-style: italic">a</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c</span></span>, <span class="tei tei-hi"><span style="font-style: italic">d</span></span>, ... and +<span class="tei tei-pb" id="page77">[pg 77]</span><a name="Pg77" id="Pg77" class="tei tei-anchor"></a> +<span class="tei tei-hi"><span style="font-style: italic">a'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">b'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">c'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">d'</span></span>, ... at <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, which meet the conic in the points +<span class="tei tei-hi"><span style="font-style: italic">α</span></span>, <span class="tei tei-hi"><span style="font-style: italic">β</span></span>, <span class="tei tei-hi"><span style="font-style: italic">γ</span></span>, <span class="tei tei-hi"><span style="font-style: italic">δ</span></span>, ... and +<span class="tei tei-hi"><span style="font-style: italic">α'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">β'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">γ'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">δ'</span></span>, ... (Fig. 36). Take now +<span class="tei tei-hi"><span style="font-style: italic">γ</span></span> as the center of a pencil projecting the points <span class="tei tei-hi"><span style="font-style: italic">α'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">β'</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">δ'</span></span>, ..., and take <span class="tei tei-hi"><span style="font-style: italic">γ'</span></span> as the center of a pencil projecting +the points <span class="tei tei-hi"><span style="font-style: italic">α</span></span>, <span class="tei tei-hi"><span style="font-style: italic">β</span></span>, <span class="tei tei-hi"><span style="font-style: italic">δ</span></span>, .... These two pencils are projective +to each other, and since they have a self-correspondin +ray in common, they are in perspective position and +corresponding rays meet on the line joining <span class="tei tei-hi"><span style="font-style: italic">(γα', γ'α)</span></span> +to <span class="tei tei-hi"><span style="font-style: italic">(γβ', γ'β)</span></span>. The correspondence between points in +the two point-rows on <span class="tei tei-hi"><span style="font-style: italic">u</span></span> is now easily traced.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc293" id="toc293"></a><a name="pdf294" id="pdf294"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p130" id="p130" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">130. Application of Steiner's construction to double +correspondence.</span></span> Steiner's construction throws into our +hands an important theorem concerning double correspondence: +<span class="tei tei-hi"><span style="font-style: italic">If two projective point-rows, superposed on +the same line, have one pair of points which correspond +to each other doubly, then all pairs correspond to each +other doubly, and the line is paired in involution.</span></span> To +make this appear, let us call the point <span class="tei tei-hi"><span style="font-style: italic">A</span></span> on <span class="tei tei-hi"><span style="font-style: italic">u</span></span> by two +names, <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, according as it is thought of as +belonging to the one or to the other of the two point-rows. +If this point is one of a pair which correspond to +each other doubly, then the points <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P</span></span> must coincide +(Fig. 37). Take now any point <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, which we will +also call <span class="tei tei-hi"><span style="font-style: italic">R'</span></span>. We must show that the corresponding +point <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> must also coincide with the point <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. Join all +the points to <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, as before, and it appears that the points +α and <span class="tei tei-hi"><span style="font-style: italic">π'</span></span> coincide, as also do the points <span class="tei tei-hi"><span style="font-style: italic">α'π</span></span> and <span class="tei tei-hi"><span style="font-style: italic">γρ'</span></span>. +By the above construction the line <span class="tei tei-hi"><span style="font-style: italic">γ'ρ</span></span> must meet <span class="tei tei-hi"><span style="font-style: italic">γρ'</span></span> +on the line joining <span class="tei tei-hi"><span style="font-style: italic">(γα', γ'α)</span></span> with +<span class="tei tei-hi"><span style="font-style: italic">(γπ', γ'π)</span></span>. But these +four points form a quadrangle inscribed in the conic, +and we know by § 95 that the tangents at the opposite +<span class="tei tei-pb" id="page78">[pg 78]</span><a name="Pg78" id="Pg78" class="tei tei-anchor"></a> +vertices <span class="tei tei-hi"><span style="font-style: italic">γ</span></span> and <span class="tei tei-hi"><span style="font-style: italic">γ'</span></span> meet on the line <span class="tei tei-hi"><span style="font-style: italic">v</span></span>. The line <span class="tei tei-hi"><span style="font-style: italic">γ'ρ</span></span> +is thus a tangent to the conic, and <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">R</span></span> are +the same point. That two projective point-rows superposed +on the same line are also in involution when +one pair, and therefore all pairs, correspond doubly +may be shown by taking <span class="tei tei-hi"><span style="font-style: italic">S</span></span> at one vertex of a complete +quadrangle which has two pairs of opposite sides going +through two pairs of points. The details we leave to +the student.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image37.png" width="968" height="607" alt="Figure 37" title="Fig. 37" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 37</div></div> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image38.png" width="379" height="332" alt="Figure 38" title="Fig. 38" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 38</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc295" id="toc295"></a><a name="pdf296" id="pdf296"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p131" id="p131" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">131. Involution of points on a point-row of the second +order.</span></span> It is important to note also, in Steiner's construction, +that we have obtained two point-rows of the +second order superposed on the same conic, and have +paired the points of one with the points of the other +in such a way that the correspondence is double. We +may then extend the notion of involution to point-rows +of the second order and say that <span class="tei tei-hi"><span style="font-style: italic">the points of a conic +are paired in involution when they are corresponding +</span><span class="tei tei-pb" id="page79">[pg 79]</span><a name="Pg79" id="Pg79" class="tei tei-anchor"></a><span style="font-style: italic"> +points of two projective point-rows superposed on the conic, +and when they correspond to each other doubly.</span></span> With this +definition we may prove the theorem: <span class="tei tei-hi"><span style="font-style: italic">The lines joining +corresponding points of a point-row of the second order in +involution all pass through a fixed point </span><span class="tei tei-hi"><span style="font-style: italic">U</span></span><span style="font-style: italic">, and the line +joining any two points </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic"> meets the line joining the +two corresponding points </span><span class="tei tei-hi"><span style="font-style: italic">A'</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B'</span></span><span style="font-style: italic"> in the +points of a line </span><span class="tei tei-hi"><span style="font-style: italic">u</span></span><span style="font-style: italic">, which is the polar +of </span><span class="tei tei-hi"><span style="font-style: italic">U</span></span><span style="font-style: italic"> with respect to the conic.</span></span> For +take <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> as the centers of two +pencils, the first perspective to the +point-row <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span> and the second +perspective to the point-row <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. +Then, since the common ray of the +two pencils corresponds to itself, they are in perspective +position, and their axis of perspectivity <span class="tei tei-hi"><span style="font-style: italic">u</span></span> (Fig. 38) +is the line which joins the point <span class="tei tei-hi"><span style="font-style: italic">(AB', A'B)</span></span> to the +point <span class="tei tei-hi"><span style="font-style: italic">(AC', A'C)</span></span>. It is then immediately clear, from +the theory of poles and polars, that <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span> pass +through the pole <span class="tei tei-hi"><span style="font-style: italic">U</span></span> of the line <span class="tei tei-hi"><span style="font-style: italic">u</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc297" id="toc297"></a><a name="pdf298" id="pdf298"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p132" id="p132" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">132. Involution of rays.</span></span> The whole theory thus far +developed may be dualized, and a theory of lines in +involution may be built up, starting with the complete +quadrilateral. Thus,</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The three pairs of rays which may be drawn from a +point through the three pairs of opposite vertices of a +complete quadrilateral are said to be in involution. If the +pairs </span><span class="tei tei-hi"><span style="font-style: italic">aa'</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">bb'</span></span><span style="font-style: italic"> are fixed, and the line </span><span class="tei tei-hi"><span style="font-style: italic">c</span></span><span style="font-style: italic"> describes a pencil, +the corresponding line </span><span class="tei tei-hi"><span style="font-style: italic">c'</span></span><span style="font-style: italic"> also describes a pencil, and the +rays of the pencil are said to be paired in the involution +determined by </span><span class="tei tei-hi"><span style="font-style: italic">aa'</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">bb'</span></span><span style="font-style: italic">.</span></span></p> + +<span class="tei tei-pb" id="page80">[pg 80]</span><a name="Pg80" id="Pg80" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc299" id="toc299"></a><a name="pdf300" id="pdf300"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p133" id="p133" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">133. Double rays.</span></span> The self-corresponding rays, of +which there are two or none, are called <span class="tei tei-hi"><span style="font-style: italic">double rays</span></span> of +the involution. Corresponding rays of the involution +are harmonic conjugates with respect to the double +rays. To the theorem of Desargues (§ 125) which has +to do with the system of conics through four points +we have the dual:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The tangents from a fixed point to a system of conics tangent +to four fixed lines form a pencil of rays in involution.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc301" id="toc301"></a><a name="pdf302" id="pdf302"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p134" id="p134" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">134.</span></span> If a conic of the system should go through the +fixed point, it is clear that the two tangents would coincide +and indicate a double ray of the involution. The +theorem, therefore, follows:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Two conics or none may be drawn through a fixed point +to be tangent to four fixed lines.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc303" id="toc303"></a><a name="pdf304" id="pdf304"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p135" id="p135" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">135. Double correspondence.</span></span> It further appears that +two projective pencils of rays which have the same +center are in involution if two pairs of rays correspond +to each other doubly. From this it is clear that we +might have deemed six rays in involution as six rays +which pass through a point and also through six points +in involution. While this would have been entirely in +accord with the treatment which was given the corresponding +problem in the theory of harmonic points and +lines, it is more satisfactory, from an aesthetic point of +view, to build the theory of lines in involution on its own +base. The student can show, by methods entirely analogous +to those used in the second chapter, that involution +is a projective property; that is, six rays in involution are +cut by any transversal in six points in involution.</p> + +<span class="tei tei-pb" id="page81">[pg 81]</span><a name="Pg81" id="Pg81" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc305" id="toc305"></a><a name="pdf306" id="pdf306"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p136" id="p136" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">136. Pencils of rays of the second order in involution.</span></span> +We may also extend the notion of involution to pencils +of rays of the second order. Thus, <span class="tei tei-hi"><span style="font-style: italic">the tangents to a +conic are in involution when they are corresponding rays +of two protective pencils of the second order superposed +upon the same conic, and when they correspond to each +other doubly.</span></span> We have then the theorem:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc307" id="toc307"></a><a name="pdf308" id="pdf308"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p137" id="p137" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">137.</span></span> <span class="tei tei-hi"><span style="font-style: italic">The intersections of corresponding rays of a pencil +of the second order in involution are all on a straight +line </span><span class="tei tei-hi"><span style="font-style: italic">u</span></span><span style="font-style: italic">, and the intersection of any two tangents </span><span class="tei tei-hi"><span style="font-style: italic">ab</span></span><span style="font-style: italic">, when +joined to the intersection of the corresponding tangents </span><span class="tei tei-hi"><span style="font-style: italic">a'b'</span></span><span style="font-style: italic">, +gives a line which passes through a fixed point </span><span class="tei tei-hi"><span style="font-style: italic">U</span></span><span style="font-style: italic">, the pole +of the line </span><span class="tei tei-hi"><span style="font-style: italic">u</span></span><span style="font-style: italic"> with respect to the conic.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc309" id="toc309"></a><a name="pdf310" id="pdf310"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p138" id="p138" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">138. Involution of rays determined by a conic.</span></span> We +have seen in the theory of poles and polars (§ 103) +that if a point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> moves along a line <span class="tei tei-hi"><span style="font-style: italic">m</span></span>, then the polar +of <span class="tei tei-hi"><span style="font-style: italic">P</span></span> revolves about a point. This pencil cuts out on <span class="tei tei-hi"><span style="font-style: italic">m</span></span> +another point-row <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, projective also to <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. Since the +polar of <span class="tei tei-hi"><span style="font-style: italic">P</span></span> passes through <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, the polar of <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> also passes +through <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, so that the correspondence between <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">P'</span></span> is double. The two point-rows are therefore in involution, +and the double points, if any exist, are the points +where the line <span class="tei tei-hi"><span style="font-style: italic">m</span></span> meets the conic. A similar involution +of rays may be found at any point in the plane, corresponding +rays passing each through the pole of the other. +We have called such points and rays <span class="tei tei-hi"><span style="font-style: italic">conjugate</span></span> with +respect to the conic (§ 100). We may then state the +following important theorem:</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc311" id="toc311"></a><a name="pdf312" id="pdf312"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p139" id="p139" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">139.</span></span> <span class="tei tei-hi"><span style="font-style: italic">A conic determines on every line in its plane an +involution of points, corresponding points in the involution +</span><span class="tei tei-pb" id="page82">[pg 82]</span><a name="Pg82" id="Pg82" class="tei tei-anchor"></a><span style="font-style: italic"> +being conjugate with respect to the conic. The double points, +if any exist, are the points where the line meets the conic.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc313" id="toc313"></a><a name="pdf314" id="pdf314"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p140" id="p140" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">140.</span></span> The dual theorem reads: <span class="tei tei-hi"><span style="font-style: italic">A conic determines at +every point in the plane an involution of rays, corresponding +rays being conjugate with respect to the conic. The +double rays, if any exist, are the tangents from the point +to the conic.</span></span></p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc315" id="toc315"></a><a name="pdf316" id="pdf316"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. Two lines are drawn through a point on a conic so +as always to make right angles with each other. Show that +the lines joining the points where they meet the conic again +all pass through a fixed point.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. Two lines are drawn through a fixed point on a conic +so as always to make equal angles with the tangent at that +point. Show that the lines joining the two points where the +lines meet the conic again all pass through a fixed point.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Four lines divide the plane into a certain number of +regions. Determine for each region whether two conics or +none may be drawn to pass through points of it and also +to be tangent to the four lines.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. If a variable quadrangle move in such a way as +always to remain inscribed in a fixed conic, while three of +its sides turn each around one of three fixed collinear points, +then the fourth will also turn around a fourth fixed point +collinear with the other three.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. State and prove the dual of problem 4.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Extend problem 4 as follows: If a variable polygon of +an even number of sides move in such a way as always to +remain inscribed in a fixed conic, while all its sides but one +pass through as many fixed collinear points, then the last side +will also pass through a fixed point collinear with the others.</p> + +<span class="tei tei-pb" id="page83">[pg 83]</span><a name="Pg83" id="Pg83" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. If a triangle <span class="tei tei-hi"><span style="font-style: italic">QRS</span></span> be inscribed in a conic, and if a +transversal <span class="tei tei-hi"><span style="font-style: italic">s</span></span> meet two of its sides in <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, the third +side and the tangent at the opposite vertex in <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, and +the conic itself in <span class="tei tei-hi"><span style="font-style: italic">C</span></span> and <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, then <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span> are three +pairs of points in an involution.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Use the last exercise to solve the problem: Given five +points, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, <span class="tei tei-hi"><span style="font-style: italic">R</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, on a conic, to draw the tangent at any +one of them.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">9. State and prove the dual of problem 7 and use it to +prove the dual of problem 8.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">10. If a transversal cut two tangents to a conic in <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, their chord of contact in <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, and the conic itself in <span class="tei tei-hi"><span style="font-style: italic">P</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, then the point <span class="tei tei-hi"><span style="font-style: italic">A</span></span> is a double point of the involution +determined by <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">PP'</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">11. State and prove the dual of problem 10.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">12. If a variable conic pass through two given points, +<span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, and if it be tangent to two given lines, the chord +of contact of these two tangents will always pass through +a fixed point on <span class="tei tei-hi"><span style="font-style: italic">PP'</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">13. Use the last theorem to solve the problem: Given +four points, <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, on a conic, and the tangent at one of +them, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, to draw the tangent at any one of the other points, <span class="tei tei-hi"><span style="font-style: italic">S</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">14. Apply the theorem of problem 9 to the case of a +hyperbola where the two tangents are the asymptotes. Show +in this way that if a hyperbola and its asymptotes be cut +by a transversal, the segments intercepted by the curve and +by the asymptotes respectively have the same middle point.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">15. In a triangle circumscribed about a conic, any side is +divided harmonically by its point of contact and the point +where it meets the chord joining the points of contact of the +other two sides.</p> +</div> +</div> + + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<span class="tei tei-pb" id="page84">[pg 84]</span><a name="Pg84" id="Pg84" class="tei tei-anchor"></a> +<a name="toc317" id="toc317"></a><a name="pdf318" id="pdf318"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc319" id="toc319"></a><a name="pdf320" id="pdf320"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image39.png" width="800" height="425" alt="Figure 39" title="Fig. 39" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 39</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p141" id="p141" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">141. Introduction of infinite point; center of involution.</span></span> +We connect the projective theory of involution with the +metrical, as usual, by the introduction of the elements at +infinity. In an involution of points on a line the point +which corresponds to the infinitely distant point is called +the <span class="tei tei-hi"><span style="font-style: italic">center</span></span> of the involution. Since corresponding points +in the involution have been shown to be harmonic conjugates +with respect to the double points, the center is +midway between the double points when they exist. To +construct the center (Fig. 39) we draw as usual through +<span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> any two rays and cut them by a line parallel +to <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> in the points <span class="tei tei-hi"><span style="font-style: italic">K</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M</span></span>. Join these points to +<span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, thus determining on <span class="tei tei-hi"><span style="font-style: italic">AK</span></span> and <span class="tei tei-hi"><span style="font-style: italic">AN</span></span> the points <span class="tei tei-hi"><span style="font-style: italic">L</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. <span class="tei tei-hi"><span style="font-style: italic">LN</span></span> meets <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> in the center <span class="tei tei-hi"><span style="font-style: italic">O</span></span> of the involution.</p> + +<span class="tei tei-pb" id="page85">[pg 85]</span><a name="Pg85" id="Pg85" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc321" id="toc321"></a><a name="pdf322" id="pdf322"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p142" id="p142" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">142. Fundamental metrical theorem.</span></span> From the figure +we see that the triangles <span class="tei tei-hi"><span style="font-style: italic">OLB'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">PLM</span></span> are similar, <span class="tei tei-hi"><span style="font-style: italic">P</span></span> +being the intersection of KM and LN. Also the triangles +<span class="tei tei-hi"><span style="font-style: italic">KPN</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BON</span></span> are similar. We thus have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OB : PK = ON : PN</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OB' : PM = OL : PL;</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">whence</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OB · OB' : PK · PM = ON · OL : PN · PL.</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">In the same way, from the similar triangles <span class="tei tei-hi"><span style="font-style: italic">OAL</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">PKL</span></span>, and also <span class="tei tei-hi"><span style="font-style: italic">OA'N</span></span> and <span class="tei tei-hi"><span style="font-style: italic">PMN</span></span>, we obtain</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OA · OA' : PK · PM = ON · OL : PN · PL,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">and this, with the preceding, gives at once the fundamental +theorem, which is sometimes taken also as the +definition of involution:</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">OA · OA' = OB · OB' = </span><span class="tei tei-hi" style="text-align: center"><span style="font-style: normal">constant</span></span><span style="font-style: italic">,</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or, in words,</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The product of the distances from the center to two corresponding +points in an involution of points is constant.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc323" id="toc323"></a><a name="pdf324" id="pdf324"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p143" id="p143" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">143. Existence of double points.</span></span> Clearly, according as +the constant is positive or negative the involution will +or will not have double points. The constant is the +square root of the distance from the center to the +double points. If <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> lie both on the same side +of the center, the product <span class="tei tei-hi"><span style="font-style: italic">OA · OA'</span></span> is positive; and if +they lie on opposite sides, it is negative. Take the case +where they both lie on the same side of the center, and +take also the pair of corresponding points <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span>. Then, +since <span class="tei tei-hi"><span style="font-style: italic">OA · OA' = OB · OB'</span></span>, it cannot happen that <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">B'</span></span> are separated from each other by <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>. This is +evident enough if the points are on opposite sides of +the center. If the pairs are on the same side of the +<span class="tei tei-pb" id="page86">[pg 86]</span><a name="Pg86" id="Pg86" class="tei tei-anchor"></a> +center, and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> lies between <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, so that <span class="tei tei-hi"><span style="font-style: italic">OB</span></span> is +greater, say, than <span class="tei tei-hi"><span style="font-style: italic">OA</span></span>, but less than <span class="tei tei-hi"><span style="font-style: italic">OA'</span></span>, then, by the +equation <span class="tei tei-hi"><span style="font-style: italic">OA · OA' = OB · OB'</span></span>, we must have <span class="tei tei-hi"><span style="font-style: italic">OB'</span></span> also +less than <span class="tei tei-hi"><span style="font-style: italic">OA'</span></span> and greater than <span class="tei tei-hi"><span style="font-style: italic">OA</span></span>. A similar discussion +may be made for the case where <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> lie on +opposite sides of <span class="tei tei-hi"><span style="font-style: italic">O</span></span>. The results may be stated as +follows, without any reference to the center:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">Given two pairs of points in an involution of points, if +the points of one pair are separated from each other by +the points of the other pair, then the involution has no +double points. If the points of one pair are not separated +from each other by the points of the other pair, then the +involution has two double points.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc325" id="toc325"></a><a name="pdf326" id="pdf326"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p144" id="p144" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">144.</span></span> An entirely similar criterion decides whether an +involution of rays has or has not double rays, or whether +an involution of planes has or has not double planes.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image40.png" width="461" height="289" alt="Figure 40" title="Fig. 40" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 40</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc327" id="toc327"></a><a name="pdf328" id="pdf328"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p145" id="p145" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">145. Construction of +an involution by means of circles.</span></span> The equation just derived, <span class="tei tei-hi"><span style="font-style: italic">OA · OA' = OB · OB'</span></span>, indicates another +simple way in which points of an involution of points may be +constructed. Through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">A'</span></span> draw any circle, and draw also any +circle through <span class="tei tei-hi"><span style="font-style: italic">B</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B'</span></span> to cut the first in the two points <span class="tei tei-hi"><span style="font-style: italic">G</span></span> and <span class="tei tei-hi"><span style="font-style: italic">G'</span></span> (Fig. 40). Then any circle through <span class="tei tei-hi"><span style="font-style: italic">G</span></span> and <span class="tei tei-hi"><span style="font-style: italic">G'</span></span> will meet the +line in pairs of points in the involution determined by <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span>. For if such a circle meets the line in the points <span class="tei tei-hi"><span style="font-style: italic">CC'</span></span>, then, by the theorem in the geometry +of the circle which says that <span class="tei tei-hi"><span style="font-style: italic">if any chord +is +</span><span class="tei tei-pb" id="page87">[pg 87]</span><a name="Pg87" id="Pg87" class="tei tei-anchor"></a><span style="font-style: italic"> +drawn through a fixed point within a circle, the product of its segments +is constant in whatever direction the chord is drawn, and if a secant +line be drawn from a fixed point without a circle, the product of the +secant and its external segment is constant in whatever direction the +secant line is drawn</span></span>, we have <span class="tei tei-hi"><span style="font-style: italic">OC · +OC' = OG · OG' =</span></span> constant. So that for all such points +<span class="tei tei-hi"><span style="font-style: italic">OA · OA' = OB · OB' = OC · +OC'</span></span>. Further, the line <span class="tei tei-hi"><span style="font-style: italic">GG'</span></span> +meets <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> in the center of the +involution. To find the double points, if they exist, we draw a tangent +from <span class="tei tei-hi"><span style="font-style: italic">O</span></span> to any of the circles through +<span class="tei tei-hi"><span style="font-style: italic">GG'</span></span>. Let <span class="tei tei-hi"><span style="font-style: italic">T</span></span> be the point of contact. Then lay off on the line <span class="tei tei-hi"><span style="font-style: italic">OA</span></span> a line <span class="tei tei-hi"><span style="font-style: italic">OF</span></span> equal to <span class="tei tei-hi"><span style="font-style: italic">OT</span></span>. Then, +since by the above theorem of elementary geometry +<span class="tei tei-hi"><span style="font-style: italic">OA · OA' = OT</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: super">2</span></span><span style="font-style: italic"> = OF</span><span class="tei tei-hi"><span style="font-style: italic; vertical-align: super">2</span></span></span>, we have one double +point <span class="tei tei-hi"><span style="font-style: italic">F</span></span>. The other is at an equal +distance on the other side of <span class="tei tei-hi"><span style="font-style: italic">O</span></span>. This +simple and effective method of constructing an involution of points is +often taken as the basis for the theory of involution. In projective +geometry, however, the circle, which is not a figure that remains +unaltered by projection, and is essentially a metrical notion, ought not +to be used to build up the purely projective part of the theory.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc329" id="toc329"></a><a name="pdf330" id="pdf330"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p146" id="p146" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">146.</span></span> It ought to be mentioned that the theory of +analytic geometry indicates that the circle is a special +conic section that happens to pass through two particular +imaginary points on the line at infinity, called the +<span class="tei tei-hi"><span style="font-style: italic">circular points</span></span> and usually denoted by <span class="tei tei-hi"><span style="font-style: italic">I</span></span> and <span class="tei tei-hi"><span style="font-style: italic">J</span></span>. The +above method of obtaining a point-row in involution is, +then, nothing but a special case of the general theorem +of the last chapter (§ 125), which asserted that a system +of conics through four points will cut any line in the +plane in a point-row in involution.</p> + +<span class="tei tei-pb" id="page88">[pg 88]</span><a name="Pg88" id="Pg88" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image41.png" width="468" height="221" alt="Figure 41" title="Fig. 41" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 41</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc331" id="toc331"></a><a name="pdf332" id="pdf332"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p147" id="p147" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">147. Pairs in an involution of rays which are at right +angles. Circular involution.</span></span> In an involution of rays +there is no one ray which may be distinguished from +all the others as the point at infinity is distinguished +from all other points on a line. There is one pair of +rays, however, which does differ from all the others in +that for this particular pair the angle is a right angle. +This is most easily shown by using the construction +that employs circles, as indicated above. The centers of +all the circles through <span class="tei tei-hi"><span style="font-style: italic">G</span></span> and <span class="tei tei-hi"><span style="font-style: italic">G'</span></span> lie on the perpendicular +bisector of the line <span class="tei tei-hi"><span style="font-style: italic">GG'</span></span>. Let +this line meet the line <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span> +in the point <span class="tei tei-hi"><span style="font-style: italic">C</span></span> (Fig. 41), and +draw the circle with center <span class="tei tei-hi"><span style="font-style: italic">C</span></span> +which goes through <span class="tei tei-hi"><span style="font-style: italic">G</span></span> and <span class="tei tei-hi"><span style="font-style: italic">G'</span></span>. +This circle cuts out two points +<span class="tei tei-hi"><span style="font-style: italic">M</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M'</span></span> in the involution. The rays <span class="tei tei-hi"><span style="font-style: italic">GM</span></span> and <span class="tei tei-hi"><span style="font-style: italic">GM'</span></span> are +clearly at right angles, being inscribed in a semicircle. +If, therefore, the involution of points is projected to +<span class="tei tei-hi"><span style="font-style: italic">G</span></span>, we have found two corresponding rays which are +at right angles to each other. Given now any involution +of rays with center <span class="tei tei-hi"><span style="font-style: italic">G</span></span>, we may cut across it +by a straight line and proceed to find the two points +<span class="tei tei-hi"><span style="font-style: italic">M</span></span> and <span class="tei tei-hi"><span style="font-style: italic">M'</span></span>. Clearly there will be only one such pair +unless the perpendicular bisector of <span class="tei tei-hi"><span style="font-style: italic">GG'</span></span> coincides with +the line <span class="tei tei-hi"><span style="font-style: italic">AA'</span></span>. In this case every ray is at right angles +to its corresponding ray, and the involution is called +<span class="tei tei-hi"><span style="font-style: italic">circular</span></span>.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc333" id="toc333"></a><a name="pdf334" id="pdf334"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p148" id="p148" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">148. Axes of conics.</span></span> At the close of the last chapter +(§ 140) we gave the theorem: <span class="tei tei-hi"><span style="font-style: italic">A conic determines at every +point in its plane an involution of rays, corresponding rays +</span><span class="tei tei-pb" id="page89">[pg 89]</span><a name="Pg89" id="Pg89" class="tei tei-anchor"></a><span style="font-style: italic"> +being conjugate with respect to the conic. The double rays, +if any exist, are the tangents from the point to the conic.</span></span> +In particular, taking the point as the center of the +conic, we find that conjugate diameters form a system +of rays in involution, of which the asymptotes, if there +are any, are the double rays. Also, conjugate diameters +are harmonic conjugates with respect to the asymptotes. +By the theorem of the last paragraph, there are two +conjugate diameters which are at right angles to each +other. These are called axes. In the case of the parabola, +where the center is at infinity, and on the curve, +there are, properly speaking, no conjugate diameters. +While the line at infinity might be considered as conjugate +to all the other diameters, it is not possible to +assign to it any particular direction, and so it cannot be +used for the purpose of defining an axis of a parabola. +There is one diameter, however, which is at right angles +to its conjugate system of chords, and this one is called +the <span class="tei tei-hi"><span style="font-style: italic">axis</span></span> of the parabola. The circle also furnishes an +exception in that every diameter is an axis. The involution +in this case is circular, every ray being at right +angles to its conjugate ray at the center.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc335" id="toc335"></a><a name="pdf336" id="pdf336"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p149" id="p149" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">149. Points at which the involution determined by +a conic is circular.</span></span> It is an important problem to discover +whether for any conic other than the circle it is +possible to find any point in the plane where the involution +determined as above by the conic is circular. +We shall proceed to the curious problem of proving the +existence of such points and of determining their number +and situation. We shall then develop the important +properties of such points.</p> + +<span class="tei tei-pb" id="page90">[pg 90]</span><a name="Pg90" id="Pg90" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc337" id="toc337"></a><a name="pdf338" id="pdf338"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p150" id="p150" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">150.</span></span> It is clear, in the first place, that such a point +cannot be on the outside of the conic, else the involution +would have double rays and such rays would have +to be at right angles to themselves. In the second +place, if two such points exist, the line joining them +must be a diameter and, indeed, an axis. For if <span class="tei tei-hi"><span style="font-style: italic">F</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">F'</span></span> were two such points, then, since the conjugate +ray at <span class="tei tei-hi"><span style="font-style: italic">F</span></span> to the line <span class="tei tei-hi"><span style="font-style: italic">FF'</span></span> must be at right angles to it, +and also since the conjugate ray at <span class="tei tei-hi"><span style="font-style: italic">F'</span></span> to the line <span class="tei tei-hi"><span style="font-style: italic">FF'</span></span> +must be at right angles to it, the pole of <span class="tei tei-hi"><span style="font-style: italic">FF'</span></span> must +be at infinity in a direction at right angles to <span class="tei tei-hi"><span style="font-style: italic">FF'</span></span>. +The line <span class="tei tei-hi"><span style="font-style: italic">FF'</span></span> is then a diameter, and since it is at +right angles to its conjugate diameter, it must be an +axis. From this it follows also that the points we are +seeking must all lie on one of the two axes, else we +should have a diameter which does not go through +the intersection of all axes—the center of the conic. +At least one axis, therefore, must be free from any +such points.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image42.png" width="512" height="469" alt="Figure 42" title="Fig. 42" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 42</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc339" id="toc339"></a><a name="pdf340" id="pdf340"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p151" id="p151" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">151.</span></span> Let now <span class="tei tei-hi"><span style="font-style: italic">P</span></span> be a point on one of the axes (Fig. 42), +and draw any ray through it, such as <span class="tei tei-hi"><span style="font-style: italic">q</span></span>. As <span class="tei tei-hi"><span style="font-style: italic">q</span></span> revolves +about <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, its pole <span class="tei tei-hi"><span style="font-style: italic">Q</span></span> moves along a line at right angles +to the axis on which <span class="tei tei-hi"><span style="font-style: italic">P</span></span> lies, describing a point-row <span class="tei tei-hi"><span style="font-style: italic">p</span></span> +projective to the pencil of rays <span class="tei tei-hi"><span style="font-style: italic">q</span></span>. The point at infinity +in a direction at right angles to <span class="tei tei-hi"><span style="font-style: italic">q</span></span> also describes a point-row +projective to <span class="tei tei-hi"><span style="font-style: italic">q</span></span>. The line joining corresponding +points of these two point-rows is always a conjugate +line to <span class="tei tei-hi"><span style="font-style: italic">q</span></span> and at right angles to <span class="tei tei-hi"><span style="font-style: italic">q</span></span>, or, as we may call it, +a <span class="tei tei-hi"><span style="font-style: italic">conjugate normal</span></span> to <span class="tei tei-hi"><span style="font-style: italic">q</span></span>. These conjugate normals to <span class="tei tei-hi"><span style="font-style: italic">q</span></span>, +joining as they do corresponding points in two projective +point-rows, form a pencil of rays of the second +<span class="tei tei-pb" id="page91">[pg 91]</span><a name="Pg91" id="Pg91" class="tei tei-anchor"></a> +order. But since the point at infinity on the point-row +<span class="tei tei-hi"><span style="font-style: italic">Q</span></span> corresponds to the point at infinity in a direction +at right angles to <span class="tei tei-hi"><span style="font-style: italic">q</span></span>, these point-rows are in perspective +position and the normal conjugates of all the lines +through <span class="tei tei-hi"><span style="font-style: italic">P</span></span> meet in a point. This point lies on the +same axis with <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, as is seen by taking <span class="tei tei-hi"><span style="font-style: italic">q</span></span> at right angles +to the axis on which <span class="tei tei-hi"><span style="font-style: italic">P</span></span> lies. The center of this pencil +may be called <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, and thus we have paired the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> +with the point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>. By moving the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> along the +axis, and by keeping the +ray <span class="tei tei-hi"><span style="font-style: italic">q</span></span> parallel to a fixed +direction, we may see that +the point-row <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and the +point-row <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> are projective. +Also the correspondence is +double, and by starting +from the point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> we arrive +at the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. Therefore +the point-rows <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> are +in involution, and if only +the involution has double points, we shall have found +in them the points we are seeking. For it is clear that +the rays through <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and the corresponding rays through +<span class="tei tei-hi"><span style="font-style: italic">P'</span></span> are conjugate normals; and if <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> coincide, we +shall have a point where all rays are at right angles +to their conjugates. We shall now show that the involution +thus obtained on one of the two axes must have +double points.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image43.png" width="364" height="382" alt="Figure 43" title="Fig. 43" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 43</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc341" id="toc341"></a><a name="pdf342" id="pdf342"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p152" id="p152" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">152. Discovery of the foci of the conic.</span></span> We know +that on one axis no such points as we are seeking can +lie (§ 150). The involution of points <span class="tei tei-hi"><span style="font-style: italic">PP'</span></span> on this axis +<span class="tei tei-pb" id="page92">[pg 92]</span><a name="Pg92" id="Pg92" class="tei tei-anchor"></a> +can therefore have no double points. Nevertheless, let +<span class="tei tei-hi"><span style="font-style: italic">PP'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">RR'</span></span> be two pairs of corresponding points on +this axis (Fig. 43). Then we know that <span class="tei tei-hi"><span style="font-style: italic">P</span></span> and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> are +separated from each other by <span class="tei tei-hi"><span style="font-style: italic">R</span></span> and <span class="tei tei-hi"><span style="font-style: italic">R'</span></span> (§ 143). Draw +a circle on <span class="tei tei-hi"><span style="font-style: italic">PP'</span></span> as a diameter, and one on <span class="tei tei-hi"><span style="font-style: italic">RR'</span></span> as a +diameter. These must intersect in +two points, <span class="tei tei-hi"><span style="font-style: italic">F</span></span> and <span class="tei tei-hi"><span style="font-style: italic">F'</span></span>, and since the +center of the conic is the center +of the involution <span class="tei tei-hi"><span style="font-style: italic">PP'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">RR'</span></span>, as is +easily seen, it follows that <span class="tei tei-hi"><span style="font-style: italic">F</span></span> and <span class="tei tei-hi"><span style="font-style: italic">F'</span></span> +are on the other axis of the conic. +Moreover, <span class="tei tei-hi"><span style="font-style: italic">FR</span></span> and <span class="tei tei-hi"><span style="font-style: italic">FR'</span></span> are conjugate +normal rays, since <span class="tei tei-hi"><span style="font-style: italic">RFR'</span></span> is +inscribed in a semicircle, and the +two rays go one through <span class="tei tei-hi"><span style="font-style: italic">R</span></span> and the other through <span class="tei tei-hi"><span style="font-style: italic">R'</span></span>. +The involution of points <span class="tei tei-hi"><span style="font-style: italic">PP'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">RR'</span></span> therefore projects +to the two points <span class="tei tei-hi"><span style="font-style: italic">F</span></span> and <span class="tei tei-hi"><span style="font-style: italic">F'</span></span> in two pencils of rays in +involution which have for corresponding rays conjugate +normals to the conic. We may, then, say:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">There are two and only two points of the plane where +the involution determined by the conic is circular. These +two points lie on one of the axes, at equal distances from +the center, on the inside of the conic. These points are +called the foci of the conic.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc343" id="toc343"></a><a name="pdf344" id="pdf344"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p153" id="p153" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">153. The circle and the parabola.</span></span> The above discussion +applies only to the central conics, apart from +the circle. In the circle the two foci fall together at the +center. In the case of the parabola, that part of the +investigation which proves the existence of two foci on +one of the axes will not hold, as we have but one +<span class="tei tei-pb" id="page93">[pg 93]</span><a name="Pg93" id="Pg93" class="tei tei-anchor"></a> +axis. It is seen, however, that as <span class="tei tei-hi"><span style="font-style: italic">P</span></span> moves to infinity, +carrying the line <span class="tei tei-hi"><span style="font-style: italic">q</span></span> with it, <span class="tei tei-hi"><span style="font-style: italic">q</span></span> becomes the line at infinity, +which for the parabola is a tangent line. Its pole +<span class="tei tei-hi"><span style="font-style: italic">Q</span></span> is thus at infinity and also the point <span class="tei tei-hi"><span style="font-style: italic">P'</span></span>, so that <span class="tei tei-hi"><span style="font-style: italic">P</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">P'</span></span> fall together at infinity, and therefore one focus +of the parabola is at infinity. There must therefore be +another, so that</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">A parabola has one and only one focus in the finite +part of the plane.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image44.png" width="375" height="271" alt="Figure 44" title="Fig. 44" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 44</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc345" id="toc345"></a><a name="pdf346" id="pdf346"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p154" id="p154" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">154. Focal properties of conics.</span></span> We proceed to develop +some theorems which will exhibit the importance +of these points in the theory of the conic section. +Draw a tangent to the conic, and also the normal +at the point of contact <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. These +two lines are clearly conjugate +normals. The two points <span class="tei tei-hi"><span style="font-style: italic">T</span></span> and +<span class="tei tei-hi"><span style="font-style: italic">N</span></span>, therefore, where they meet the +axis which contains the foci, are +corresponding points in the involution +considered above, and are +therefore harmonic conjugates with respect to the foci +(Fig. 44); and if we join them to the point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, we +shall obtain four harmonic lines. But two of them +are at right angles to each other, and so the others +make equal angles with them (Problem 4, Chapter II). +Therefore</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The lines joining a point on the conic to the foci make +equal angles with the tangent.</span></span></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">It follows that rays from a source of light at one +focus are reflected by an ellipse to the other.</p> + +<span class="tei tei-pb" id="page94">[pg 94]</span><a name="Pg94" id="Pg94" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc347" id="toc347"></a><a name="pdf348" id="pdf348"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p155" id="p155" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">155.</span></span> In the case of the parabola, where one of the +foci must be considered to be at infinity in the direction +of the diameter, we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image45.png" width="457" height="314" alt="Figure 45" title="Fig. 45" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 45</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">A diameter makes the same +angle with the tangent at its +extremity as that tangent does +with the line from its point of +contact to the focus (Fig. 45).</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc349" id="toc349"></a><a name="pdf350" id="pdf350"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p156" id="p156" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">156.</span></span> This last theorem is the basis for the construction +of the parabolic reflector. A ray of light from the +focus is reflected from such a reflector in a direction +parallel to the axis of the reflector.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc351" id="toc351"></a><a name="pdf352" id="pdf352"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p157" id="p157" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">157. Directrix. Principal axis. Vertex.</span></span> The polar of +the focus with respect to the conic is called the <span class="tei tei-hi"><span style="font-style: italic">directrix</span></span>. +The axis which contains the foci is called the <span class="tei tei-hi"><span style="font-style: italic">principal +axis</span></span>, and the intersection of the axis with the curve is +called the <span class="tei tei-hi"><span style="font-style: italic">vertex</span></span> of the curve. The directrix is at right +angles to the principal axis. In a parabola the vertex +is equally distant from the focus and the directrix, +these three points and the point at infinity on the axis +being four harmonic points. In the ellipse the vertex is +nearer to the focus than it is to the directrix, for the +same reason, and in the hyperbola it is farther from +the focus than it is from the directrix.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image46.png" width="479" height="536" alt="Figure 46" title="Fig. 46" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 46</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc353" id="toc353"></a><a name="pdf354" id="pdf354"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p158" id="p158" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">158. Another definition of a conic.</span></span> Let <span class="tei tei-hi"><span style="font-style: italic">P</span></span> be any point +on the directrix through which a line is drawn meeting +the conic in the points <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> (Fig. 46). Let the tangents +at <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span> meet in <span class="tei tei-hi"><span style="font-style: italic">T</span></span>, and call the focus <span class="tei tei-hi"><span style="font-style: italic">F</span></span>. Then +<span class="tei tei-hi"><span style="font-style: italic">TF</span></span> and <span class="tei tei-hi"><span style="font-style: italic">PF</span></span> are conjugate lines, and as they pass through +a focus they must be at right angles to each other. Let +<span class="tei tei-pb" id="page95">[pg 95]</span><a name="Pg95" id="Pg95" class="tei tei-anchor"></a> +<span class="tei tei-hi"><span style="font-style: italic">TF</span></span> meet <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> in <span class="tei tei-hi"><span style="font-style: italic">C</span></span>. Then <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span> are four harmonic +points. Project these four points parallel to <span class="tei tei-hi"><span style="font-style: italic">TF</span></span> upon +the directrix, and we then get +the four harmonic points <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">Q</span></span>, <span class="tei tei-hi"><span style="font-style: italic">N</span></span>. Since, now, <span class="tei tei-hi"><span style="font-style: italic">TFP</span></span> is +a right angle, the angles <span class="tei tei-hi"><span style="font-style: italic">MFQ</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">NFQ</span></span> are equal, as well +as the angles <span class="tei tei-hi"><span style="font-style: italic">AFC</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BFC</span></span>. +Therefore the triangles <span class="tei tei-hi"><span style="font-style: italic">MAF</span></span> +and <span class="tei tei-hi"><span style="font-style: italic">NFB</span></span> are similar, and +<span class="tei tei-hi"><span style="font-style: italic">FA : FM = FB : BN</span></span>. Dropping +perpendiculars <span class="tei tei-hi"><span style="font-style: italic">AA</span></span> and <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> +upon the directrix, this becomes +<span class="tei tei-hi"><span style="font-style: italic">FA : AA' = FB : BB'</span></span>. We +have thus the property often taken as the definition +of a conic:</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">The ratio of the distances from a point on the conic to +the focus and the directrix is constant.</span></span></p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image47.png" width="432" height="282" alt="Figure 47" title="Fig. 47" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 47</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc355" id="toc355"></a><a name="pdf356" id="pdf356"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p159" id="p159" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">159. Eccentricity.</span></span> By taking the point at the vertex +of the conic, we note that this ratio is less than unity +for the ellipse, greater than unity for the hyperbola, +and equal to unity for the parabola. This ratio is called the +<span class="tei tei-hi"><span style="font-style: italic">eccentricity</span></span>.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image48.png" width="512" height="488" alt="Figure 48" title="Fig. 48" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 48</div></div></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc357" id="toc357"></a><a name="pdf358" id="pdf358"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p160" id="p160" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">160. Sum or difference of focal +distances.</span></span> The ellipse and the +hyperbola have two foci and +two directrices. The eccentricity, of course, is the same +for one focus as for the other, since the curve is symmetrical +with respect to both. If the distances from +<span class="tei tei-pb" id="page96">[pg 96]</span><a name="Pg96" id="Pg96" class="tei tei-anchor"></a> +a point on a conic to the two foci are <span class="tei tei-hi"><span style="font-style: italic">r</span></span> and <span class="tei tei-hi"><span style="font-style: italic">r'</span></span>, and +the distances from the same point to the corresponding +directrices are <span class="tei tei-hi"><span style="font-style: italic">d</span></span> and <span class="tei tei-hi"><span style="font-style: italic">d'</span></span> +(Fig. 47), we have <span class="tei tei-hi"><span style="font-style: italic">r : d = r' : d'</span></span>; +<span class="tei tei-hi"><span style="font-style: italic">(r ± r') : (d ± d')</span></span>. In the +ellipse <span class="tei tei-hi"><span style="font-style: italic">(d + d')</span></span> is constant, +being the distance between +the directrices. In the hyperbola +this distance is <span class="tei tei-hi"><span style="font-style: italic">(d - d')</span></span>. +It follows (Fig. 48) that</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em"><span class="tei tei-hi"><span style="font-style: italic">In the ellipse the sum of the +focal distances of any point +on the curve is constant, and +in the hyperbola the difference between the focal distances +is constant.</span></span></p> +</div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc359" id="toc359"></a><a name="pdf360" id="pdf360"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"><span style="font-size: 144%">PROBLEMS</span></h2> + +<p class="tei tei-p" style="margin-bottom: 1.00em">1. Construct the axis of a parabola, given four tangents.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">2. Given two conjugate lines at right angles to each +other, and let them meet the axis which has no foci on it +in the points <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span>. The circle on <span class="tei tei-hi"><span style="font-style: italic">AB</span></span> as diameter will +pass through the foci of the conic.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">3. Given the axes of a conic in position, and also a +tangent with its point of contact, to construct the foci and +determine the length of the axes.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">4. Given the tangent at the vertex of a parabola, and +two other tangents, to find the focus.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">5. The locus of the center of a circle touching two given +circles is a conic with the centers of the given circles for +its foci.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">6. Given the axis of a parabola and a tangent, with its +point of contact, to find the focus.</p> + +<span class="tei tei-pb" id="page97">[pg 97]</span><a name="Pg97" id="Pg97" class="tei tei-anchor"></a> + +<p class="tei tei-p" style="margin-bottom: 1.00em">7. The locus of the center of a circle which touches a +given line and a given circle consists of two parabolas.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">8. Let <span class="tei tei-hi"><span style="font-style: italic">F</span></span> and <span class="tei tei-hi"><span style="font-style: italic">F'</span></span> be the foci of an ellipse, and <span class="tei tei-hi"><span style="font-style: italic">P</span></span> any +point on it. Produce <span class="tei tei-hi"><span style="font-style: italic">PF</span></span> to <span class="tei tei-hi"><span style="font-style: italic">G</span></span>, making <span class="tei tei-hi"><span style="font-style: italic">PG</span></span> equal to <span class="tei tei-hi"><span style="font-style: italic">PF'</span></span>. +Find the locus of <span class="tei tei-hi"><span style="font-style: italic">G</span></span>.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">9. If the points <span class="tei tei-hi"><span style="font-style: italic">G</span></span> of a circle be folded over upon a +point <span class="tei tei-hi"><span style="font-style: italic">F</span></span>, the creases will all be tangent to a conic. If <span class="tei tei-hi"><span style="font-style: italic">F</span></span> is +within the circle, the conic will be an ellipse; if <span class="tei tei-hi"><span style="font-style: italic">F</span></span> is without +the circle, the conic will be a hyperbola.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">10. If the points <span class="tei tei-hi"><span style="font-style: italic">G</span></span> in the last example be taken on a +straight line, the locus is a parabola.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">11. Find the foci and the length of the principal axis of +the conics in problems 9 and 10.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">12. In problem 10 a correspondence is set up between +straight lines and parabolas. As there is a fourfold infinity +of parabolas in the plane, and only a twofold infinity of +straight lines, there must be some restriction on the parabolas +obtained by this method. Find and explain this +restriction.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">13. State and explain the similar problem for problem 9.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">14. The last four problems are a study of the consequences +of the following transformation: A point <span class="tei tei-hi"><span style="font-style: italic">O</span></span> is fixed +in the plane. Then to any point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> is made to correspond +the line <span class="tei tei-hi"><span style="font-style: italic">p</span></span> at right angles to <span class="tei tei-hi"><span style="font-style: italic">OP</span></span> and bisecting it. In this +correspondence, what happens to <span class="tei tei-hi"><span style="font-style: italic">p</span></span> when <span class="tei tei-hi"><span style="font-style: italic">P</span></span> moves along a +straight line? What corresponds to the theorem that two +lines have only one point in common? What to the theorem +that the angle sum of a triangle is two right angles? Etc.</p> + +</div> +</div> + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc361" id="toc361"></a><a name="pdf362" id="pdf362"></a> +<span class="tei tei-pb" id="page98">[pg 98]</span><a name="Pg98" id="Pg98" class="tei tei-anchor"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY</span></h1> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc363" id="toc363"></a><a name="pdf364" id="pdf364"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p161" id="p161" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">161. Ancient results.</span></span> The theory of synthetic projective +geometry as we have built it up in this course is +less than a century old. This is not to say that many of +the theorems and principles involved were not discovered +much earlier, but isolated theorems do not make a +theory, any more than a pile of bricks makes a building. +The materials for our building have been contributed +by many different workmen from the days of Euclid +down to the present time. Thus, the notion of four +harmonic points was familiar to the ancients, who considered +it from the metrical point of view as the division +of a line internally and externally in the same ratio<a id="noteref_1" name="noteref_1" href="#note_1"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">1</span></span></a> +the involution of six points cut out by any transversal +which intersects the sides of a complete quadrilateral +<span class="tei tei-pb" id="page100">[pg 100]</span><a name="Pg100" id="Pg100" class="tei tei-anchor"></a> +as studied by Pappus<a id="noteref_2" name="noteref_2" href="#note_2"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">2</span></span></a>; +but these notions were not +made the foundation for any general theory. Taken by +themselves, they are of small consequence; it is their +relation to other theorems and sets of theorems that +gives them their importance. The ancients were doubtless +familiar with the theorem, <span class="tei tei-hi"><span style="font-style: italic">Two lines determine a +point, and two points determine a line</span></span>, but they had +no glimpse of the wonderful law of duality, of which +this theorem is a simple example. The principle of +projection, by which many properties of the conic sections +may be inferred from corresponding properties +of the circle which forms the base of the cone from +which they are cut—a principle so natural to modern +mathematicians—seems not to have occurred to the +Greeks. The ellipse, the hyperbola, and the parabola +<span class="tei tei-pb" id="page101">[pg 101]</span><a name="Pg101" id="Pg101" class="tei tei-anchor"></a> +were to them entirely different curves, to be treated +separately with methods appropriate to each. Thus the +focus of the ellipse was discovered some five hundred +years before the focus of the parabola! It was not till +1522 that Verner<a id="noteref_3" name="noteref_3" href="#note_3"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">3</span></span></a> +of Nürnberg undertook to demonstrate +the properties of the conic sections by means of +the circle.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc365" id="toc365"></a><a name="pdf366" id="pdf366"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p162" id="p162" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">162. Unifying principles.</span></span> In the early years of the +seventeenth century—that wonderful epoch in the +history of the world which produced a Galileo, a Kepler, +a Tycho Brahe, a Descartes, a Desargues, a Pascal, +a Cavalieri, a Wallis, a Fermat, a Huygens, a Bacon, +a Napier, and a goodly array of lesser lights, to say +nothing of a Rembrandt or of a Shakespeare—there +began to appear certain unifying principles connecting +the great mass of material dug out by the ancients. +Thus, in 1604 the great astronomer Kepler<a id="noteref_4" name="noteref_4" href="#note_4"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">4</span></span></a> introduced +the notion that parallel lines should be considered as +meeting at an infinite distance, and that a parabola is at +once the limiting case of an ellipse and of a hyperbola. +He also attributes to the parabola a "blind focus" +(<span class="tei tei-hi"><span style="font-style: italic">caecus focus</span></span>) at infinity on the axis.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc367" id="toc367"></a><a name="pdf368" id="pdf368"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p163" id="p163" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">163. Desargues.</span></span> In 1639 Desargues,<a id="noteref_5" name="noteref_5" href="#note_5"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">5</span></span></a> an architect of +Lyons, published a little treatise on the conic sections, +in which appears the theorem upon which we have +founded the theory of four harmonic points (§ 25). +<span class="tei tei-pb" id="page102">[pg 102]</span><a name="Pg102" id="Pg102" class="tei tei-anchor"></a> +Desargues, however, does not make use of it for that +purpose. Four harmonic points are for him a special +case of six points in involution when two of the three +pairs coincide giving double points. His development +of the theory of involution is also different from the +purely geometric one which we have adopted, and is +based on the theorem (§ 142) that the product of the +distances of two conjugate points from the center is +constant. He also proves the projective character of +an involution of points by showing that when six lines +pass through a point and through six points in involution, +then any transversal must meet them in six points +which are also in involution.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc369" id="toc369"></a><a name="pdf370" id="pdf370"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p164" id="p164" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">164. Poles and polars.</span></span> In this little treatise is also +contained the theory of poles and polars. The polar +line is called a <span class="tei tei-hi"><span style="font-style: italic">traversal</span></span>.<a id="noteref_6" name="noteref_6" href="#note_6"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">6</span></span></a> The harmonic properties of +poles and polars are given, but Desargues seems not +to have arrived at the metrical properties which result +when the infinite elements of the plane are introduced. +Thus he says, "When the <span class="tei tei-hi"><span style="font-style: italic">traversal</span></span> is at an infinite +distance, all is unimaginable."</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc371" id="toc371"></a><a name="pdf372" id="pdf372"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p165" id="p165" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">165. Desargues's theorem concerning conics through +four points.</span></span> We find in this little book the beautiful +theorem concerning a quadrilateral inscribed in a conic +section, which is given by his name in § 138. The +theorem is not given in terms of a system of conics +through four points, for Desargues had no conception of +<span class="tei tei-pb" id="page103">[pg 103]</span><a name="Pg103" id="Pg103" class="tei tei-anchor"></a> +any such system. He states the theorem, in effect, as +follows: <span class="tei tei-hi"><span style="font-style: italic">Given a simple quadrilateral inscribed in a conic +section, every transversal meets the conic and the four sides +of the quadrilateral in six points which are in involution.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc373" id="toc373"></a><a name="pdf374" id="pdf374"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p166" id="p166" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">166. Extension of the theory of poles and polars to +space.</span></span> As an illustration of his remarkable powers of +generalization, we may note that Desargues extended +the notion of poles and polars to space of three dimensions +for the sphere and for certain other surfaces of +the second degree. This is a matter which has not +been touched on in this book, but the notion is not +difficult to grasp. If we draw through any point <span class="tei tei-hi"><span style="font-style: italic">P</span></span> in +space a line to cut a sphere in two points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">S</span></span>, and +then construct the fourth harmonic of <span class="tei tei-hi"><span style="font-style: italic">P</span></span> with respect to +<span class="tei tei-hi"><span style="font-style: italic">A</span></span> and <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, the locus of this fourth harmonic, for various +lines through <span class="tei tei-hi"><span style="font-style: italic">P</span></span>, is a plane called the <span class="tei tei-hi"><span style="font-style: italic">polar plane</span></span> of <span class="tei tei-hi"><span style="font-style: italic">P</span></span> +with respect to the sphere. With this definition and theorem +one can easily find dual relations between points +and planes in space analogous to those between points and +lines in a plane. Desargues closes his discussion of this +matter with the remark, "Similar properties may be +found for those other solids which are related to the +sphere in the same way that the conic section is to the +circle." It should not be inferred from this remark, +however, that he was acquainted with all the different +varieties of surfaces of the second order. The ancients +were well acquainted with the surfaces obtained by +revolving an ellipse or a parabola about an axis. Even +the hyperboloid of two sheets, obtained by revolving the +hyperbola about its major axis, was known to them, +but probably not the hyperboloid of one sheet, which +<span class="tei tei-pb" id="page104">[pg 104]</span><a name="Pg104" id="Pg104" class="tei tei-anchor"></a> +results from revolving a hyperbola about the other +axis. All the other solids of the second degree were +probably unknown until their discovery by Euler.<a id="noteref_7" name="noteref_7" href="#note_7"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">7</span></span></a></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc375" id="toc375"></a><a name="pdf376" id="pdf376"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p167" id="p167" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">167.</span></span> Desargues had no conception of the conic section +of the locus of intersection of corresponding rays of two +projective pencils of rays. He seems to have tried to +describe the curve by means of a pair of compasses, +moving one leg back and forth along a straight line +instead of holding it fixed as in drawing a circle. He +does not attempt to define the law of the movement +necessary to obtain a conic by this means.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc377" id="toc377"></a><a name="pdf378" id="pdf378"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p168" id="p168" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">168. Reception of Desargues's work.</span></span> Strange to say, +Desargues's immortal work was heaped with the most violent +abuse and held up to ridicule and scorn! "Incredible +errors! Enormous mistakes and falsities! Really it +is impossible for anyone who is familiar with the science +concerning which he wishes to retail his thoughts, to +keep from laughing!" Such were the comments of reviewers +and critics. Nor were his detractors altogether +ignorant and uninstructed men. In spite of the devotion +of his pupils and in spite of the admiration and friendship +of men like Descartes, Fermat, Mersenne, and +Roberval, his book disappeared so completely that two +centuries after the date of its publication, when the +French geometer Chasles wrote his history of geometry, +there was no means of estimating the value of the work +done by Desargues. Six years later, however, in 1845, +Chasles found a manuscript copy of the "Bruillon-project," +made by Desargues's pupil, De la Hire.</p> + +<span class="tei tei-pb" id="page105">[pg 105]</span><a name="Pg105" id="Pg105" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc379" id="toc379"></a><a name="pdf380" id="pdf380"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p169" id="p169" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">169. Conservatism in Desargues's time.</span></span> It is not necessary +to suppose that this effacement of Desargues's work +for two centuries was due to the savage attacks of his +critics. All this was in accordance with the fashion of +the time, and no man escaped bitter denunciation who +attempted to improve on the methods of the ancients. +Those were days when men refused to believe that a +heavy body falls at the same rate as a lighter one, even +when Galileo made them see it with their own eyes +at the foot of the tower of Pisa. Could they not turn +to the exact page and line of Aristotle which declared +that the heavier body must fall the faster! "I have +read Aristotle's writings from end to end, many times," +wrote a Jesuit provincial to the mathematician and +astronomer, Christoph Scheiner, at Ingolstadt, whose +telescope seemed to reveal certain mysterious spots on +the sun, "and I can assure you I have nowhere found +anything similar to what you describe. Go, my son, and +tranquilize yourself; be assured that what you take for +spots on the sun are the faults of your glasses, or of +your eyes." The dead hand of Aristotle barred the +advance in every department of research. Physicians +would have nothing to do with Harvey's discoveries +about the circulation of the blood. "Nature is accused +of tolerating a vacuum!" exclaimed a priest when Pascal +began his experiments on the Puy-de-Dome to show +that the column of mercury in a glass tube varied in +height with the pressure of the atmosphere.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc381" id="toc381"></a><a name="pdf382" id="pdf382"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p170" id="p170" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">170. Desargues's style of writing.</span></span> Nevertheless, authority +counted for less at this time in Paris than it did in +Italy, and the tragedy enacted in Rome when Galileo +<span class="tei tei-pb" id="page106">[pg 106]</span><a name="Pg106" id="Pg106" class="tei tei-anchor"></a> +was forced to deny his inmost convictions at the bidding +of a brutal Inquisition could not have been staged +in France. Moreover, in the little company of scientists +of which Desargues was a member the utmost liberty +of thought and expression was maintained. One very +good reason for the disappearance of the work of Desargues +is to be found in his style of writing. He failed +to heed the very good advice given him in a letter from +his warm admirer Descartes.<a id="noteref_8" name="noteref_8" href="#note_8"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">8</span></span></a> "You may have two designs, +both very good and very laudable, but which do +not require the same method of procedure: The one is +to write for the learned, and show them some new properties +of the conic sections which they do not already +know; and the other is to write for the curious unlearned, +and to do it so that this matter which until +now has been understood by only a very few, and which +is nevertheless very useful for perspective, for painting, +architecture, etc., shall become common and easy to +all who wish to study them in your book. If you have +the first idea, then it seems to me that it is necessary +to avoid using new terms; for the learned are already +accustomed to using those of Apollonius, and will not +readily change them for others, though better, and thus +yours will serve only to render your demonstrations +more difficult, and to turn away your readers from your +book. If you have the second plan in mind, it is certain +that your terms, which are French, and conceived +with spirit and grace, will be better received by persons +not preoccupied with those of the ancients.... But, if +you have that intention, you should make of it a great +<span class="tei tei-pb" id="page107">[pg 107]</span><a name="Pg107" id="Pg107" class="tei tei-anchor"></a> +volume; explain it all so fully and so distinctly that +those gentlemen who cannot study without yawning; +who cannot distress their imaginations enough to grasp +a proposition in geometry, nor turn the leaves of a book +to look at the letters in a figure, shall find nothing in +your discourse more difficult to understand than the +description of an enchanted palace in a fairy story." +The point of these remarks is apparent when we note +that Desargues introduced some seventy new terms in +his little book, of which only one, <span class="tei tei-hi"><span style="font-style: italic">involution</span></span>, has survived. +Curiously enough, this is the one term singled +out for the sharpest criticism and ridicule by his reviewer, +De Beaugrand.<a id="noteref_9" name="noteref_9" href="#note_9"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">9</span></span></a> That Descartes knew the character +of Desargues's audience better than he did is also +evidenced by the fact that De Beaugrand exhausted his +patience in reading the first ten pages of the book.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc383" id="toc383"></a><a name="pdf384" id="pdf384"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p171" id="p171" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">171. Lack of appreciation of Desargues.</span></span> Desargues's +methods, entirely different from the analytic methods +just then being developed by Descartes and Fermat, +seem to have been little understood. "Between you +and me," wrote Descartes<a id="noteref_10" name="noteref_10" href="#note_10"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">10</span></span></a> to Mersenne, "I can hardly +form an idea of what he may have written concerning +conics." Desargues seems to have boasted that he owed +nothing to any man, and that all his results had come +from his own mind. His favorite pupil, De la Hire, did +not realize the extraordinary simplicity and generality +of his work. It is a remarkable fact that the only one +of all his associates to understand and appreciate the +methods of Desargues should be a lad of sixteen years!</p> + +<span class="tei tei-pb" id="page108">[pg 108]</span><a name="Pg108" id="Pg108" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc385" id="toc385"></a><a name="pdf386" id="pdf386"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p172" id="p172" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">172. Pascal and his theorem.</span></span> One does not have to +believe all the marvelous stories of Pascal's admiring +sisters to credit him with wonderful precocity. We have +the fact that in 1640, when he was sixteen years old, +he published a little placard, or poster, entitled "Essay +pour les conique,"<a id="noteref_11" name="noteref_11" href="#note_11"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">11</span></span></a> in which his great theorem appears +for the first time. His manner of putting it may be a +little puzzling to one who has only seen it in the form +given in this book, and it may be worth while for the +student to compare the two methods of stating it. It is +given as follows: <span class="tei tei-hi"><span style="font-style: italic">"If in the plane of </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">Q</span></span><span style="font-style: italic"> we draw +through </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic"> the two lines </span><span class="tei tei-hi"><span style="font-style: italic">MK</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">MV</span></span><span style="font-style: italic">, and through the +point </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic"> the two lines </span><span class="tei tei-hi"><span style="font-style: italic">SK</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">SV</span></span><span style="font-style: italic">, and let </span><span class="tei tei-hi"><span style="font-style: italic">K</span></span><span style="font-style: italic"> be the intersection +of </span><span class="tei tei-hi"><span style="font-style: italic">MK</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">SK</span></span><span style="font-style: italic">; </span><span class="tei tei-hi"><span style="font-style: italic">V</span></span><span style="font-style: italic"> the intersection of </span><span class="tei tei-hi"><span style="font-style: italic">MV</span></span><span style="font-style: italic"> and +</span><span class="tei tei-hi"><span style="font-style: italic">SV</span></span><span style="font-style: italic">; </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> the intersection of </span><span class="tei tei-hi"><span style="font-style: italic">MA</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">SA</span></span><span style="font-style: italic"> (</span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> is the intersection +of </span><span class="tei tei-hi"><span style="font-style: italic">SV</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">MK</span></span><span style="font-style: italic">), and </span><span class="tei tei-hi"><span style="font-style: italic">μ</span></span><span style="font-style: italic"> the intersection of </span><span class="tei tei-hi"><span style="font-style: italic">MV</span></span><span style="font-style: italic"> +and </span><span class="tei tei-hi"><span style="font-style: italic">SK</span></span><span style="font-style: italic">; and if through two of the four points </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">K</span></span><span style="font-style: italic">, +</span><span class="tei tei-hi"><span style="font-style: italic">μ</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">V</span></span><span style="font-style: italic">, which are not in the same straight line with </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic"> and +</span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic">, such as </span><span class="tei tei-hi"><span style="font-style: italic">K</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">V</span></span><span style="font-style: italic">, we pass the circumference of a circle +cutting the lines </span><span class="tei tei-hi"><span style="font-style: italic">MV</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">MP</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">SV</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">SK</span></span><span style="font-style: italic"> in the points </span><span class="tei tei-hi"><span style="font-style: italic">O</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">P</span></span><span style="font-style: italic">, +</span><span class="tei tei-hi"><span style="font-style: italic">Q</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">N</span></span><span style="font-style: italic">; I say that the lines </span><span class="tei tei-hi"><span style="font-style: italic">MS</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">NO</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">PQ</span></span><span style="font-style: italic"> are of the same +order."</span></span> (By "lines of the same order" Pascal means +lines which meet in the same point or are parallel.) By +projecting the figure thus described upon another plane +he is able to state his theorem for the case where the +circle is replaced by any conic section.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc387" id="toc387"></a><a name="pdf388" id="pdf388"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p173" id="p173" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">173.</span></span> It must be understood that the "Essay" was +only a résumé of a more extended treatise on conics +which, owing partly to Pascal's extreme youth, partly +to the difficulty of publishing scientific works in those +<span class="tei tei-pb" id="page109">[pg 109]</span><a name="Pg109" id="Pg109" class="tei tei-anchor"></a> +days, and also to his later morbid interest in religious +matters, was never published. Leibniz<a id="noteref_12" name="noteref_12" href="#note_12"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">12</span></span></a> examined a copy +of the complete work, and has reported that the great +theorem on the mystic hexagram was made the basis of +the whole theory, and that Pascal had deduced some four +hundred corollaries from it. This would indicate that +here was a man able to take the unconnected materials +of projective geometry and shape them into some such +symmetrical edifice as we have to-day. Unfortunately +for science, Pascal's early death prevented the further +development of the subject at his hands.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc389" id="toc389"></a><a name="pdf390" id="pdf390"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p174" id="p174" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">174.</span></span> In the "Essay" Pascal gives full credit to +Desargues, saying of one of the other propositions, +"We prove this property also, the original discoverer of +which is M. Desargues, of Lyons, one of the greatest +minds of this age ... and I wish to acknowledge that +I owe to him the little which I have discovered." This +acknowledgment led Descartes to believe that Pascal's +theorem should also be credited to Desargues. But in +the scientific club which the young Pascal attended +in company with his father, who was also a scientist +of some reputation, the theorem went by the name of +'la Pascalia,' and Descartes's remarks do not seem to +have been taken seriously, which indeed is not to be +wondered at, seeing that he was in the habit of giving +scant credit to the work of other scientific investigators +than himself.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc391" id="toc391"></a><a name="pdf392" id="pdf392"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p175" id="p175" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">175. De la Hire and his work.</span></span> De la Hire added +little to the development of the subject, but he did put +into print much of what Desargues had already worked +<span class="tei tei-pb" id="page110">[pg 110]</span><a name="Pg110" id="Pg110" class="tei tei-anchor"></a> +out, not fully realizing, perhaps, how much was his +own and how much he owed to his teacher. Writing in +1679, he says,<a id="noteref_13" name="noteref_13" href="#note_13"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">13</span></span></a> "I have just read for the first time +M. Desargues's little treatise, and have made a copy +of it in order to have a more perfect knowledge of it." +It was this copy that saved the work of his master +from oblivion. De la Hire should be credited, among +other things, with the invention of a method by which +figures in the plane may be transformed into others +of the same order. His method is extremely interesting, +and will serve as an exercise for the student in +synthetic projective geometry. It is as follows: <span class="tei tei-hi"><span style="font-style: italic">Draw +two parallel lines, </span><span class="tei tei-hi"><span style="font-style: italic">a</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">b</span></span><span style="font-style: italic">, and select a point </span><span class="tei tei-hi"><span style="font-style: italic">P</span></span><span style="font-style: italic"> in their +plane. Through any point </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic"> of the plane draw a line +meeting </span><span class="tei tei-hi"><span style="font-style: italic">a</span></span><span style="font-style: italic"> in </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">b</span></span><span style="font-style: italic"> in </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">. Draw a line through </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic"> +parallel to </span><span class="tei tei-hi"><span style="font-style: italic">AP</span></span><span style="font-style: italic">, and let it meet </span><span class="tei tei-hi"><span style="font-style: italic">MP</span></span><span style="font-style: italic"> in the point </span><span class="tei tei-hi"><span style="font-style: italic">M'</span></span><span style="font-style: italic">. It +may be shown that the point </span><span class="tei tei-hi"><span style="font-style: italic">M'</span></span><span style="font-style: italic"> thus obtained does not +depend at all on the particular ray </span><span class="tei tei-hi"><span style="font-style: italic">MAB</span></span><span style="font-style: italic"> used in determining +it, so that we have set up a one-to-one correspondence +between the points </span><span class="tei tei-hi"><span style="font-style: italic">M</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">M'</span></span><span style="font-style: italic"> in the plane.</span></span> The student +may show that as <span class="tei tei-hi"><span style="font-style: italic">M</span></span> describes a point-row, <span class="tei tei-hi"><span style="font-style: italic">M'</span></span> describes +a point-row projective to it. As <span class="tei tei-hi"><span style="font-style: italic">M</span></span> describes a conic, +<span class="tei tei-hi"><span style="font-style: italic">M'</span></span> describes another conic. This sort of correspondence +is called a <span class="tei tei-hi"><span style="font-style: italic">collineation</span></span>. It will be found that the +points on the line <span class="tei tei-hi"><span style="font-style: italic">b</span></span> transform into themselves, as does +also the single point <span class="tei tei-hi"><span style="font-style: italic">P</span></span>. Points on the line <span class="tei tei-hi"><span style="font-style: italic">a</span></span> transform +into points on the line at infinity. The student +should remove the metrical features of the construction +and take, instead of two parallel lines <span class="tei tei-hi"><span style="font-style: italic">a</span></span> and <span class="tei tei-hi"><span style="font-style: italic">b</span></span>, any +two lines which may meet in a finite part of the plane. +<span class="tei tei-pb" id="page111">[pg 111]</span><a name="Pg111" id="Pg111" class="tei tei-anchor"></a> +The collineation is a special one in that the general +one has an invariant triangle instead of an invariant +point and line.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc393" id="toc393"></a><a name="pdf394" id="pdf394"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p176" id="p176" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">176. Descartes and his influence.</span></span> The history of synthetic +projective geometry has little to do with the work +of the great philosopher Descartes, except in an indirect +way. The method of algebraic analysis invented by +him, and the differential and integral calculus which +developed from it, attracted all the interest of the +mathematical world for nearly two centuries after +Desargues, and synthetic geometry received scant attention +during the rest of the seventeenth century and for +the greater part of the eighteenth century. It is difficult +for moderns to conceive of the richness and variety of +the problems which confronted the first workers in the +calculus. To come into the possession of a method +which would solve almost automatically problems which +had baffled the keenest minds of antiquity; to be able +to derive in a few moments results which an Archimedes +had toiled long and patiently to reach or a Galileo had +determined experimentally; such was the happy experience +of mathematicians for a century and a half after +Descartes, and it is not to be wondered at that along +with this enthusiastic pursuit of new theorems in analysis +should come a species of contempt for the methods +of the ancients, so that in his preface to his "Méchanique +Analytique," published in 1788, Lagrange boasts, "One +will find no figures in this work." But at the close of +the eighteenth century the field opened up to research +by the invention of the calculus began to appear so +thoroughly explored that new methods and new objects +<span class="tei tei-pb" id="page112">[pg 112]</span><a name="Pg112" id="Pg112" class="tei tei-anchor"></a> +of investigation began to attract attention. Lagrange +himself, in his later years, turned in weariness from +analysis and mechanics, and applied himself to chemistry, +physics, and philosophical speculations. "This state of +mind," says Darboux,<a id="noteref_14" name="noteref_14" href="#note_14"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">14</span></span></a> "we find almost always at certain +moments in the lives of the greatest scholars." At any +rate, after lying fallow for almost two centuries, the +field of pure geometry was attacked with almost religious +enthusiasm.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc395" id="toc395"></a><a name="pdf396" id="pdf396"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p177" id="p177" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">177. Newton and Maclaurin.</span></span> But in hastening on +to the epoch of Poncelet and Steiner we should not +omit to mention the work of Newton and Maclaurin. +Although their results were obtained by analysis for the +most part, nevertheless they have given us theorems +which fall naturally into the domain of synthetic projective +geometry. Thus Newton's "organic method"<a id="noteref_15" name="noteref_15" href="#note_15"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">15</span></span></a> +of generating conic sections is closely related to the +method which we have made use of in Chapter III. +It is as follows: <span class="tei tei-hi"><span style="font-style: italic">If two angles, </span><span class="tei tei-hi"><span style="font-style: italic">AOS</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">AO'S</span></span><span style="font-style: italic">, of given +magnitudes turn about their respective vertices, </span><span class="tei tei-hi"><span style="font-style: italic">O</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">O'</span></span><span style="font-style: italic">, +in such a way that the point of intersection, </span><span class="tei tei-hi"><span style="font-style: italic">S</span></span><span style="font-style: italic">, of one pair +of arms always lies on a straight line, the point of intersection, +</span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, of the other pair of arms will describe a conic.</span></span> +The proof of this is left to the student.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc397" id="toc397"></a><a name="pdf398" id="pdf398"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p178" id="p178" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">178.</span></span> Another method of generating a conic is due to +Maclaurin.<a id="noteref_16" name="noteref_16" href="#note_16"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">16</span></span></a> The construction, which we also leave for +the student to justify, is as follows: <span class="tei tei-hi"><span style="font-style: italic">If a triangle </span><span class="tei tei-hi"><span style="font-style: italic">C'PQ</span></span><span style="font-style: italic"> +move in such a way that its sides, </span><span class="tei tei-hi"><span style="font-style: italic">PQ</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">QC'</span></span><span style="font-style: italic">, and </span><span class="tei tei-hi"><span style="font-style: italic">C'P</span></span><span style="font-style: italic">, turn +</span><span class="tei tei-pb" id="page113">[pg 113]</span><a name="Pg113" id="Pg113" class="tei tei-anchor"></a><span style="font-style: italic"> +around three fixed points, </span><span class="tei tei-hi"><span style="font-style: italic">R</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, respectively, while two of +its vertices, </span><span class="tei tei-hi"><span style="font-style: italic">P</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">Q</span></span><span style="font-style: italic">, slide along two fixed lines, </span><span class="tei tei-hi"><span style="font-style: italic">CB'</span></span><span style="font-style: italic"> and </span><span class="tei tei-hi"><span style="font-style: italic">CA'</span></span><span style="font-style: italic">, +respectively, then the remaining vertex will describe a conic.</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc399" id="toc399"></a><a name="pdf400" id="pdf400"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p179" id="p179" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">179. Descriptive geometry and the second revival.</span></span> +The second revival of pure geometry was again to take +place at a time of great intellectual activity. The period +at the close of the eighteenth and the beginning of +the nineteenth century is adorned with a glorious list +of mighty names, among which are Gauss, Lagrange, +Legendre, Laplace, Monge, Carnot, Poncelet, Cauchy, +Fourier, Steiner, Von Staudt, Möbius, Abel, and many +others. The renaissance may be said to date from the invention +by Monge<a id="noteref_17" name="noteref_17" href="#note_17"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">17</span></span></a> of the theory of <span class="tei tei-hi"><span style="font-style: italic">descriptive geometry</span></span>. +Descriptive geometry is concerned with the representation +of figures in space of three dimensions by means +of space of two dimensions. The method commonly +used consists in projecting the space figure on two +planes (a vertical and a horizontal plane being most +convenient), the projections being made most simply +for metrical purposes from infinity in directions perpendicular +to the two planes of projection. These two +planes are then made to coincide by revolving the horizontal +into the vertical about their common line. Such +is the method of descriptive geometry which in the +hands of Monge acquired wonderful generality and elegance. +Problems concerning fortifications were worked +so quickly by this method that the commandant at the +military school at Mézières, where Monge was a draftsman +and pupil, viewed the results with distrust. Monge +afterward became professor of mathematics at Mézières +<span class="tei tei-pb" id="page114">[pg 114]</span><a name="Pg114" id="Pg114" class="tei tei-anchor"></a> +and gathered around him a group of students destined +to have a share in the advancement of pure geometry. +Among these were Hachette, Brianchon, Dupin, Chasles, +Poncelet, and many others.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc401" id="toc401"></a><a name="pdf402" id="pdf402"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p180" id="p180" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">180. Duality, homology, continuity, contingent relations.</span></span> +Analytic geometry had left little to do in the +way of discovery of new material, and the mathematical +world was ready for the construction of the edifice. +The activities of the group of men that followed Monge +were directed toward this end, and we now begin to +hear of the great unifying notions of duality, homology, +continuity, contingent relations, and the like. The +devotees of pure geometry were beginning to feel the +need of a basis for their science which should be at +once as general and as rigorous as that of the analysts. +Their dream was the building up of a system of geometry +which should be independent of analysis. Monge, +and after him Poncelet, spent much thought on the so-called +"principle of continuity," afterwards discussed +by Chasles under the name of the "principle of contingent +relations." To get a clear idea of this principle, +consider a theorem in geometry in the proof of which +certain auxiliary elements are employed. These elements +do not appear in the statement of the theorem, +and the theorem might possibly be proved without them. +In drawing the figure for the proof of the theorem, +however, some of these elements may not appear, or, +as the analyst would say, they become imaginary. "No +matter," says the principle of contingent relations, "the +theorem is true, and the proof is valid whether the +elements used in the proof are real or imaginary."</p> + +<span class="tei tei-pb" id="page115">[pg 115]</span><a name="Pg115" id="Pg115" class="tei tei-anchor"></a></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc403" id="toc403"></a><a name="pdf404" id="pdf404"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p181" id="p181" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">181. Poncelet and Cauchy.</span></span> The efforts of Poncelet +to compel the acceptance of this principle independent +of analysis resulted in a bitter and perhaps fruitless +controversy between him and the great analyst Cauchy. +In his review of Poncelet's great work on the projective +properties of figures<a id="noteref_18" name="noteref_18" href="#note_18"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">18</span></span></a> +Cauchy says, "In his preliminary +discourse the author insists once more on the +necessity of admitting into geometry what he calls the +'principle of continuity.' We have already discussed +that principle ... and we have found that that principle +is, properly speaking, only a strong induction, +which cannot be indiscriminately applied to all sorts of +questions in geometry, nor even in analysis. The reasons +which we have given as the basis of our opinion +are not affected by the considerations which the author +has developed in his Traité des Propriétés Projectives +des Figures." Although this principle is constantly made +use of at the present day in all sorts of investigations, +careful geometricians are in agreement with Cauchy +in this matter, and use it only as a convenient working +tool for purposes of exploration. The one-to-one +correspondence between geometric forms and algebraic +analysis is subject to many and important exceptions. +The field of analysis is much more general than the +field of geometry, and while there may be a clear +notion in analysis to, correspond to every notion in +geometry, the opposite is not true. Thus, in analysis +we can deal with four coördinates as well as with +three, but the existence of a space of four dimensions +<span class="tei tei-pb" id="page116">[pg 116]</span><a name="Pg116" id="Pg116" class="tei tei-anchor"></a> +to correspond to it does not therefore follow. When +the geometer speaks of the two real or imaginary intersections +of a straight line with a conic, he is really +speaking the language of algebra. <span class="tei tei-hi"><span style="font-style: italic">Apart from the +algebra involved</span></span>, it is the height of absurdity to try to +distinguish between the two points in which a line +<span class="tei tei-hi"><span style="font-style: italic">fails to meet a conic!</span></span></p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc405" id="toc405"></a><a name="pdf406" id="pdf406"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p182" id="p182" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">182. The work of Poncelet.</span></span> But Poncelet's right to +the title "The Father of Modern Geometry" does not +stand or fall with the principle of contingent relations. +In spite of the fact that he considered this principle +the most important of all his discoveries, his reputation +rests on more solid foundations. He was the first to +study figures <span class="tei tei-hi"><span style="font-style: italic">in homology</span></span>, which is, in effect, the collineation +described in § 175, where corresponding points +lie on straight lines through a fixed point. He was the +first to give, by means of the theory of poles and polars, +a transformation by which an element is transformed +into another of a different sort. Point-to-point transformations +will sometimes generalize a theorem, but +the transformation discovered by Poncelet may throw a +theorem into one of an entirely different aspect. The +principle of duality, first stated in definite form by +Gergonne,<a id="noteref_19" name="noteref_19" href="#note_19"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">19</span></span></a> +the editor of the mathematical journal in +which Poncelet published his researches, was based by +Poncelet on his theory of poles and polars. He also put +into definite form the notions of the infinitely distant +elements in space as all lying on a plane at infinity.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc407" id="toc407"></a><a name="pdf408" id="pdf408"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p183" id="p183" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">183. The debt which analytic geometry owes to synthetic +geometry.</span></span> The reaction of pure geometry on +<span class="tei tei-pb" id="page117">[pg 117]</span><a name="Pg117" id="Pg117" class="tei tei-anchor"></a> +analytic geometry is clearly seen in the development of +the notion of the <span class="tei tei-hi"><span style="font-style: italic">class</span></span> of a curve, which is the number +of tangents that may be drawn from a point in a plane +to a given curve lying in that plane. If a point moves +along a conic, it is easy to show—and the student +is recommended to furnish the proof—that the polar +line with respect to a conic remains tangent to another +conic. This may be expressed by the statement that the +conic is of the second order and also of the second class. +It might be thought that if a point moved along a +cubic curve, its polar line with respect to a conic would +remain tangent to another cubic curve. This is not the +case, however, and the investigations of Poncelet and +others to determine the class of a given curve were +afterward completed by Plücker. The notion of geometrical +transformation led also to the very important +developments in the theory of invariants, which, geometrically, +are the elements and configurations which +are not affected by the transformation. The anharmonic +ratio of four points is such an invariant, since it remains +unaltered under all projective transformations.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc409" id="toc409"></a><a name="pdf410" id="pdf410"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p184" id="p184" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">184. Steiner and his work.</span></span> In the work of Poncelet +and his contemporaries, Chasles, Brianchon, Hachette, +Dupin, Gergonne, and others, the anharmonic ratio enjoyed +a fundamental rôle. It is made also the basis of +the great work of Steiner,<a id="noteref_20" name="noteref_20" href="#note_20"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">20</span></span></a> +who was the first to treat +of the conic, not as the projection of a circle, but as the +locus of intersection of corresponding rays of two projective +pencils. Steiner not only related to each other, +<span class="tei tei-pb" id="page118">[pg 118]</span><a name="Pg118" id="Pg118" class="tei tei-anchor"></a> +in one-to-one correspondence, point-rows and pencils +and all the other fundamental forms, but he set into +correspondence even curves and surfaces of higher degrees. +This new and fertile conception gave him an +easy and direct route into the most abstract and difficult +regions of pure geometry. Much of his work was +given without any indication of the methods by which +he had arrived at it, and many of his results have only +recently been verified.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc411" id="toc411"></a><a name="pdf412" id="pdf412"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p185" id="p185" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">185. Von Staudt and his work.</span></span> To complete the theory +of geometry as we have it to-day it only remained +to free it from its dependence on the semimetrical basis +of the anharmonic ratio. This work was accomplished by +Von Staudt,<a id="noteref_21" name="noteref_21" href="#note_21"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">21</span></span></a> +who applied himself to the restatement +of the theory of geometry in a form independent of +analytic and metrical notions. The method which has +been used in Chapter II to develop the notion of four +harmonic points by means of the complete quadrilateral +is due to Von Staudt. His work is characterized by a +most remarkable generality, in that he is able to discuss +real and imaginary forms with equal ease. Thus he +assumes a one-to-one correspondence between the points +and lines of a plane, and defines a conic as the locus +of points which lie on their corresponding lines, and a +pencil of rays of the second order as the system of lines +which pass through their corresponding points. The +point-row and pencil of the second order may be real +or imaginary, but his theorems still apply. An illustration +of a correspondence of this sort, where the conic +is imaginary, is given in § 15 of the first chapter. In +<span class="tei tei-pb" id="page119">[pg 119]</span><a name="Pg119" id="Pg119" class="tei tei-anchor"></a> +defining conjugate imaginary points on a line, Von +Staudt made use of an involution of points having no +double points. His methods, while elegant and powerful, +are hardly adapted to an elementary course, but +Reye<a id="noteref_22" name="noteref_22" href="#note_22"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">22</span></span></a> +and others have done much toward simplifying +his presentation.</p></div> + +<div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> +<a name="toc413" id="toc413"></a><a name="pdf414" id="pdf414"></a> +<h2 class="tei tei-head" style="text-align: left; margin-bottom: 2.88em; margin-top: 2.88em"></h2><p class="tei tei-p" style="margin-bottom: 1.00em"><a name="p186" id="p186" class="tei tei-anchor"></a><span class="tei tei-hi"><span style="font-weight: 700">186. Recent developments.</span></span> It would be only confusing +to the student to attempt to trace here the later +developments of the science of protective geometry. It +is concerned for the most part with curves and surfaces +of a higher degree than the second. Purely synthetic +methods have been used with marked success in the +study of the straight line in space. The struggle between +analysis and pure geometry has long since come +to an end. Each has its distinct advantages, and the +mathematician who cultivates one at the expense of the +other will never attain the results that he would attain +if both methods were equally ready to his hand. Pure +geometry has to its credit some of the finest discoveries +in mathematics, and need not apologize for having +been born. The day of its usefulness has not passed +with the invention of abridged notation and of short +methods in analysis. While we may be certain that any +geometrical problem may always be stated in analytic +form, it does not follow that that statement will be +simple or easily interpreted. For many mathematicians +the geometric intuitions are weak, and for such the +method will have little attraction. On the other hand, +there will always be those for whom the subject will +have a peculiar glamor—who will follow with delight +<span class="tei tei-pb" id="page120">[pg 120]</span><a name="Pg120" id="Pg120" class="tei tei-anchor"></a> +the curious and unexpected relations between the forms +of space. There is a corresponding pleasure, doubtless, +for the analyst in tracing the marvelous connections +between the various fields in which he wanders, and it +is as absurd to shut one's eyes to the beauties in one +as it is to ignore those in the other. "Let us cultivate +geometry, then," says Darboux,<a id="noteref_23" name="noteref_23" href="#note_23"><span class="tei tei-noteref"><span style="font-size: 60%; vertical-align: super">23</span></span></a> +"without wishing in +all points to equal it to its rival. Besides, if we were +tempted to neglect it, it would not be long in finding +in the applications of mathematics, as once it has already +done, the means of renewing its life and of +developing itself anew. It is like the Giant Antaeus, +who renewed, his strength by touching the earth."</p> +</div> +</div> + + +<hr class="page" /><div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> +<a name="toc415" id="toc415"></a><a name="pdf416" id="pdf416"></a> +<h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">INDEX</span></h1> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em">(The numbers refer to the paragraphs)</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Abel (1802-1829), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Analogy, <a href="#p24" class="tei tei-ref">24</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Analytic geometry, <a href="#p21" class="tei tei-ref">21</a>, <a href="#p118" class="tei tei-ref">118</a>, <a href="#p119" class="tei tei-ref">119</a>, +120, <a href="#p146" class="tei tei-ref">146</a>, <a href="#p176" class="tei tei-ref">176</a>, <a href="#p180" class="tei tei-ref">180</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Anharmonic ratio, <a href="#p46" class="tei tei-ref">46</a>, <a href="#p161" class="tei tei-ref">161</a>, <a href="#p184" class="tei tei-ref">184</a>, <a href="#p185" class="tei tei-ref">185</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Apollonius (second half of third +century B.C.), <a href="#p70" class="tei tei-ref">70</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Archimedes (287-212 B.C.), <a href="#p176" class="tei tei-ref">176</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Aristotle (384-322 B.C.), <a href="#p169" class="tei tei-ref">169</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Asymptotes, <a href="#p111" class="tei tei-ref">111</a>, <a href="#p113" class="tei tei-ref">113</a>, <a href="#p114" class="tei tei-ref">114</a>, <a href="#p115" class="tei tei-ref">115</a>, +116, <a href="#p117" class="tei tei-ref">117</a>, <a href="#p118" class="tei tei-ref">118</a>, <a href="#p148" class="tei tei-ref">148</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Axes of a conic, <a href="#p148" class="tei tei-ref">148</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Axial pencil, <a href="#p7" class="tei tei-ref">7</a>, <a href="#p8" class="tei tei-ref">8</a>, <a href="#p23" class="tei tei-ref">23</a>, <a href="#p50" class="tei tei-ref">50</a>, <a href="#p54" class="tei tei-ref">54</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Axis of perspectivity, <a href="#p8" class="tei tei-ref">8</a>, <a href="#p47" class="tei tei-ref">47</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Bacon (1561-1626), <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Bisection, <a href="#p41" class="tei tei-ref">41</a>, <a href="#p109" class="tei tei-ref">109</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Brianchon (1785-1864), <a href="#p84" class="tei tei-ref">84</a>, <a href="#p85" class="tei tei-ref">85</a>, <a href="#p86" class="tei tei-ref">86</a>, +88, <a href="#p89" class="tei tei-ref">89</a>, <a href="#p90" class="tei tei-ref">90</a>, <a href="#p95" class="tei tei-ref">95</a>, <a href="#p105" class="tei tei-ref">105</a>, <a href="#p113" class="tei tei-ref">113</a>, <a href="#p174" class="tei tei-ref">174</a>, <a href="#p184" class="tei tei-ref">184</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Calculus, <a href="#p176" class="tei tei-ref">176</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Carnot (1796-1832), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Cauchy (1789-1857), <a href="#p179" class="tei tei-ref">179</a>, <a href="#p181" class="tei tei-ref">181</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Cavalieri (1598-1647), <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Center of a conic, <a href="#p107" class="tei tei-ref">107</a>, <a href="#p112" class="tei tei-ref">112</a>, <a href="#p148" class="tei tei-ref">148</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Center of involution, <a href="#p141" class="tei tei-ref">141</a>, <a href="#p142" class="tei tei-ref">142</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Center of perspectivity, <a href="#p8" class="tei tei-ref">8</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Central conic, <a href="#p120" class="tei tei-ref">120</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Chasles (1793-1880), <a href="#p168" class="tei tei-ref">168</a>, <a href="#p179" class="tei tei-ref">179</a>, <a href="#p180" class="tei tei-ref">180</a>, +184<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Circle, <a href="#p21" class="tei tei-ref">21</a>, <a href="#p73" class="tei tei-ref">73</a>, <a href="#p80" class="tei tei-ref">80</a>, <a href="#p145" class="tei tei-ref">145</a>, <a href="#p146" class="tei tei-ref">146</a>, <a href="#p147" class="tei tei-ref">147</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Circular involution, <a href="#p147" class="tei tei-ref">147</a>, <a href="#p149" class="tei tei-ref">149</a>, <a href="#p150" class="tei tei-ref">150</a>, +151<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Circular points, <a href="#p146" class="tei tei-ref">146</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Class of a curve, <a href="#p183" class="tei tei-ref">183</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Classification of conics, <a href="#p110" class="tei tei-ref">110</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Collineation, <a href="#p175" class="tei tei-ref">175</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Concentric pencils, <a href="#p50" class="tei tei-ref">50</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Cone of the second order, <a href="#p59" class="tei tei-ref">59</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Conic, <a href="#p73" class="tei tei-ref">73</a>, <a href="#p81" class="tei tei-ref">81</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Conjugate diameters, <a href="#p114" class="tei tei-ref">114</a>, <a href="#p148" class="tei tei-ref">148</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Conjugate normal, <a href="#p151" class="tei tei-ref">151</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Conjugate points and lines, <a href="#p100" class="tei tei-ref">100</a>, +109, <a href="#p138" class="tei tei-ref">138</a>, <a href="#p139" class="tei tei-ref">139</a>, <a href="#p140" class="tei tei-ref">140</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Constants in an equation, <a href="#p21" class="tei tei-ref">21</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Contingent relations, <a href="#p180" class="tei tei-ref">180</a>, <a href="#p181" class="tei tei-ref">181</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Continuity, <a href="#p180" class="tei tei-ref">180</a>, <a href="#p181" class="tei tei-ref">181</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Continuous correspondence, <a href="#p9" class="tei tei-ref">9</a>, <a href="#p10" class="tei tei-ref">10</a>, +21, <a href="#p49" class="tei tei-ref">49</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Corresponding elements, <a href="#p64" class="tei tei-ref">64</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Counting, <a href="#p1" class="tei tei-ref">1</a>, <a href="#p4" class="tei tei-ref">4</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Cross ratio, <a href="#p46" class="tei tei-ref">46</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Darboux, <a href="#p176" class="tei tei-ref">176</a>, <a href="#p186" class="tei tei-ref">186</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">De Beaugrand, <a href="#p170" class="tei tei-ref">170</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Degenerate pencil of rays of the +second order, <a href="#p58" class="tei tei-ref">58</a>, <a href="#p93" class="tei tei-ref">93</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Degenerate point-row of the +second order, <a href="#p56" class="tei tei-ref">56</a>, <a href="#p78" class="tei tei-ref">78</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">De la Hire (1640-1718), <a href="#p168" class="tei tei-ref">168</a>, <a href="#p171" class="tei tei-ref">171</a>, +175<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Desargues (1593-1662), <a href="#p25" class="tei tei-ref">25</a>, <a href="#p26" class="tei tei-ref">26</a>, <a href="#p40" class="tei tei-ref">40</a>, +121, <a href="#p125" class="tei tei-ref">125</a>, <a href="#p162" class="tei tei-ref">162</a>, <a href="#p163" class="tei tei-ref">163</a>, <a href="#p164" class="tei tei-ref">164</a>, <a href="#p165" class="tei tei-ref">165</a>, <a href="#p166" class="tei tei-ref">166</a>, +167, <a href="#p168" class="tei tei-ref">168</a>, <a href="#p169" class="tei tei-ref">169</a>, <a href="#p170" class="tei tei-ref">170</a>, <a href="#p171" class="tei tei-ref">171</a>, <a href="#p174" class="tei tei-ref">174</a>, <a href="#p175" class="tei tei-ref">175</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Descartes (1596-1650), <a href="#p162" class="tei tei-ref">162</a>, <a href="#p170" class="tei tei-ref">170</a>, +171, <a href="#p174" class="tei tei-ref">174</a>, <a href="#p176" class="tei tei-ref">176</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Descriptive geometry, <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Diameter, <a href="#p107" class="tei tei-ref">107</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Directrix, <a href="#p157" class="tei tei-ref">157</a>, <a href="#p158" class="tei tei-ref">158</a>, <a href="#p159" class="tei tei-ref">159</a>, <a href="#p160" class="tei tei-ref">160</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Double correspondence, <a href="#p128" class="tei tei-ref">128</a>, <a href="#p130" class="tei tei-ref">130</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Double points of an involution, <a href="#p124" class="tei tei-ref">124</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Double rays of an involution, <a href="#p133" class="tei tei-ref">133</a>, +134<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Duality, <a href="#p94" class="tei tei-ref">94</a>, <a href="#p104" class="tei tei-ref">104</a>, <a href="#p161" class="tei tei-ref">161</a>, <a href="#p180" class="tei tei-ref">180</a>, <a href="#p182" class="tei tei-ref">182</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Dupin (1784-1873), <a href="#p174" class="tei tei-ref">174</a>, <a href="#p184" class="tei tei-ref">184</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Eccentricity of conic, <a href="#p159" class="tei tei-ref">159</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Ellipse, <a href="#p110" class="tei tei-ref">110</a>, <a href="#p111" class="tei tei-ref">111</a>, <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Equation of conic, <a href="#p118" class="tei tei-ref">118</a>, <a href="#p119" class="tei tei-ref">119</a>, <a href="#p120" class="tei tei-ref">120</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Euclid (ca. 300 B.C.), <a href="#p6" class="tei tei-ref">6</a>, <a href="#p22" class="tei tei-ref">22</a>, <a href="#p104" class="tei tei-ref">104</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Euler (1707-1783), <a href="#p166" class="tei tei-ref">166</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Fermat (1601-1665), <a href="#p162" class="tei tei-ref">162</a>, <a href="#p171" class="tei tei-ref">171</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Foci of a conic, <a href="#p152" class="tei tei-ref">152</a>, <a href="#p153" class="tei tei-ref">153</a>, <a href="#p154" class="tei tei-ref">154</a>, <a href="#p155" class="tei tei-ref">155</a>, +156, <a href="#p157" class="tei tei-ref">157</a>, <a href="#p158" class="tei tei-ref">158</a>, <a href="#p159" class="tei tei-ref">159</a>, <a href="#p160" class="tei tei-ref">160</a>, <a href="#p161" class="tei tei-ref">161</a>, <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Fourier (1768-1830), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Fourth harmonic, <a href="#p29" class="tei tei-ref">29</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Fundamental form, <a href="#p7" class="tei tei-ref">7</a>, <a href="#p16" class="tei tei-ref">16</a>, <a href="#p23" class="tei tei-ref">23</a>, <a href="#p36" class="tei tei-ref">36</a>, +47, <a href="#p60" class="tei tei-ref">60</a>, <a href="#p184" class="tei tei-ref">184</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Galileo (1564-1642), <a href="#p162" class="tei tei-ref">162</a>, <a href="#p169" class="tei tei-ref">169</a>, <a href="#p170" class="tei tei-ref">170</a>, +176<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Gauss (1777-1855), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Gergonne (1771-1859), <a href="#p182" class="tei tei-ref">182</a>, <a href="#p184" class="tei tei-ref">184</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Greek geometry, <a href="#p161" class="tei tei-ref">161</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Hachette (1769-1834), <a href="#p179" class="tei tei-ref">179</a>, <a href="#p184" class="tei tei-ref">184</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harmonic conjugates, <a href="#p29" class="tei tei-ref">29</a>, <a href="#p30" class="tei tei-ref">30</a>, <a href="#p39" class="tei tei-ref">39</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harmonic elements, <a href="#p86" class="tei tei-ref">86</a>, <a href="#p49" class="tei tei-ref">49</a>, <a href="#p91" class="tei tei-ref">91</a>, +163, <a href="#p185" class="tei tei-ref">185</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harmonic lines, <a href="#p33" class="tei tei-ref">33</a>, <a href="#p34" class="tei tei-ref">34</a>, <a href="#p35" class="tei tei-ref">35</a>, <a href="#p66" class="tei tei-ref">66</a>, <a href="#p67" class="tei tei-ref">67</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harmonic planes, <a href="#p34" class="tei tei-ref">34</a>, <a href="#p35" class="tei tei-ref">35</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harmonic points, <a href="#p29" class="tei tei-ref">29</a>, <a href="#p31" class="tei tei-ref">31</a>, <a href="#p32" class="tei tei-ref">32</a>, <a href="#p33" class="tei tei-ref">33</a>, +34, <a href="#p35" class="tei tei-ref">35</a>, <a href="#p36" class="tei tei-ref">36</a>, <a href="#p43" class="tei tei-ref">43</a>, <a href="#p71" class="tei tei-ref">71</a>, <a href="#p161" class="tei tei-ref">161</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harmonic tangents to a conic, +91, <a href="#p92" class="tei tei-ref">92</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Harvey (1578-1657), <a href="#p169" class="tei tei-ref">169</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Homology, <a href="#p180" class="tei tei-ref">180</a>, <a href="#p182" class="tei tei-ref">182</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Huygens (1629-1695), <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Hyperbola, <a href="#p110" class="tei tei-ref">110</a>, <a href="#p111" class="tei tei-ref">111</a>, <a href="#p113" class="tei tei-ref">113</a>, <a href="#p114" class="tei tei-ref">114</a>, <a href="#p115" class="tei tei-ref">115</a>, +116, <a href="#p117" class="tei tei-ref">117</a>, <a href="#p118" class="tei tei-ref">118</a>, <a href="#p162" class="tei tei-ref">162</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Imaginary elements, <a href="#p146" class="tei tei-ref">146</a>, <a href="#p180" class="tei tei-ref">180</a>, <a href="#p181" class="tei tei-ref">181</a>, +182, <a href="#p185" class="tei tei-ref">185</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Infinitely distant elements, <a href="#p6" class="tei tei-ref">6</a>, <a href="#p9" class="tei tei-ref">9</a>, +22, <a href="#p39" class="tei tei-ref">39</a>, <a href="#p40" class="tei tei-ref">40</a>, <a href="#p41" class="tei tei-ref">41</a>, <a href="#p104" class="tei tei-ref">104</a>, <a href="#p107" class="tei tei-ref">107</a>, <a href="#p110" class="tei tei-ref">110</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Infinity, <a href="#p4" class="tei tei-ref">4</a>, <a href="#p5" class="tei tei-ref">5</a>, <a href="#p10" class="tei tei-ref">10</a>, <a href="#p12" class="tei tei-ref">12</a>, <a href="#p13" class="tei tei-ref">13</a>, <a href="#p14" class="tei tei-ref">14</a>, <a href="#p15" class="tei tei-ref">15</a>, +17, <a href="#p18" class="tei tei-ref">18</a>, <a href="#p19" class="tei tei-ref">19</a>, <a href="#p20" class="tei tei-ref">20</a>, <a href="#p21" class="tei tei-ref">21</a>, <a href="#p22" class="tei tei-ref">22</a>, <a href="#p41" class="tei tei-ref">41</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Involution, <a href="#p37" class="tei tei-ref">37</a>, <a href="#p123" class="tei tei-ref">123</a>, <a href="#p124" class="tei tei-ref">124</a>, <a href="#p125" class="tei tei-ref">125</a>, <a href="#p126" class="tei tei-ref">126</a>, +127, <a href="#p128" class="tei tei-ref">128</a>, <a href="#p129" class="tei tei-ref">129</a>, <a href="#p130" class="tei tei-ref">130</a>, <a href="#p131" class="tei tei-ref">131</a>, <a href="#p132" class="tei tei-ref">132</a>, <a href="#p133" class="tei tei-ref">133</a>, +134, <a href="#p135" class="tei tei-ref">135</a>, <a href="#p136" class="tei tei-ref">136</a>, <a href="#p137" class="tei tei-ref">137</a>, <a href="#p138" class="tei tei-ref">138</a>, <a href="#p139" class="tei tei-ref">139</a>, <a href="#p140" class="tei tei-ref">140</a>, +161, <a href="#p163" class="tei tei-ref">163</a>, <a href="#p170" class="tei tei-ref">170</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Kepler (1571-1630), <a href="#p162" class="tei tei-ref">162</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Lagrange (1736-1813), <a href="#p176" class="tei tei-ref">176</a>, <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Laplace (1749-1827), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Legendre (1752-1833), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Leibniz (1646-1716), <a href="#p173" class="tei tei-ref">173</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Linear construction, <a href="#p40" class="tei tei-ref">40</a>, <a href="#p41" class="tei tei-ref">41</a>, <a href="#p42" class="tei tei-ref">42</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Maclaurin (1698-1746), <a href="#p177" class="tei tei-ref">177</a>, <a href="#p178" class="tei tei-ref">178</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Measurements, <a href="#p23" class="tei tei-ref">23</a>, <a href="#p40" class="tei tei-ref">40</a>, <a href="#p41" class="tei tei-ref">41</a>, <a href="#p104" class="tei tei-ref">104</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Mersenne (1588-1648), <a href="#p168" class="tei tei-ref">168</a>, <a href="#p171" class="tei tei-ref">171</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Metrical theorems, <a href="#p40" class="tei tei-ref">40</a>, <a href="#p104" class="tei tei-ref">104</a>, <a href="#p106" class="tei tei-ref">106</a>, +107, <a href="#p141" class="tei tei-ref">141</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Middle point, <a href="#p39" class="tei tei-ref">39</a>, <a href="#p41" class="tei tei-ref">41</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Möbius (1790-1868), <a href="#p179" class="tei tei-ref">179</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Monge (1746-1818), <a href="#p179" class="tei tei-ref">179</a>, <a href="#p180" class="tei tei-ref">180</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Napier (1550-1617), <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Newton (1642-1727), <a href="#p177" class="tei tei-ref">177</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Numbers, <a href="#p4" class="tei tei-ref">4</a>, <a href="#p21" class="tei tei-ref">21</a>, <a href="#p43" class="tei tei-ref">43</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Numerical computations, <a href="#p43" class="tei tei-ref">43</a>, <a href="#p44" class="tei tei-ref">44</a>, +46 <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">One-to-one correspondence, <a href="#p1" class="tei tei-ref">1</a>, <a href="#p2" class="tei tei-ref">2</a>, +3, <a href="#p4" class="tei tei-ref">4</a>, <a href="#p5" class="tei tei-ref">5</a>, <a href="#p6" class="tei tei-ref">6</a>, <a href="#p7" class="tei tei-ref">7</a>, <a href="#p9" class="tei tei-ref">9</a>, <a href="#p10" class="tei tei-ref">10</a>, <a href="#p11" class="tei tei-ref">11</a>, <a href="#p24" class="tei tei-ref">24</a>, <a href="#p36" class="tei tei-ref">36</a>, +87, <a href="#p43" class="tei tei-ref">43</a>, <a href="#p60" class="tei tei-ref">60</a>, <a href="#p104" class="tei tei-ref">104</a>, <a href="#p106" class="tei tei-ref">106</a>, <a href="#p184" class="tei tei-ref">184</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Opposite sides of a hexagon, <a href="#p70" class="tei tei-ref">70</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Opposite sides of a quadrilateral, +28, <a href="#p29" class="tei tei-ref">29</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Order of a form, <a href="#p7" class="tei tei-ref">7</a>, <a href="#p10" class="tei tei-ref">10</a>, <a href="#p11" class="tei tei-ref">11</a>, <a href="#p12" class="tei tei-ref">12</a>, <a href="#p13" class="tei tei-ref">13</a>, +14, <a href="#p15" class="tei tei-ref">15</a>, <a href="#p16" class="tei tei-ref">16</a>, <a href="#p17" class="tei tei-ref">17</a>, <a href="#p18" class="tei tei-ref">18</a>, <a href="#p19" class="tei tei-ref">19</a>, <a href="#p20" class="tei tei-ref">20</a>, <a href="#p21" class="tei tei-ref">21</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Pappus (fourth century A.D.), +161<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Parabola, <a href="#p110" class="tei tei-ref">110</a>, <a href="#p111" class="tei tei-ref">111</a>, <a href="#p112" class="tei tei-ref">112</a>, <a href="#p119" class="tei tei-ref">119</a>, <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Parallel lines, <a href="#p39" class="tei tei-ref">39</a>, <a href="#p41" class="tei tei-ref">41</a>, <a href="#p162" class="tei tei-ref">162</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Pascal (1623-1662), <a href="#p69" class="tei tei-ref">69</a>, <a href="#p70" class="tei tei-ref">70</a>, <a href="#p74" class="tei tei-ref">74</a>, <a href="#p75" class="tei tei-ref">75</a>, +76, <a href="#p77" class="tei tei-ref">77</a>, <a href="#p78" class="tei tei-ref">78</a>, <a href="#p95" class="tei tei-ref">95</a>, <a href="#p105" class="tei tei-ref">105</a>, <a href="#p125" class="tei tei-ref">125</a>, <a href="#p162" class="tei tei-ref">162</a>, +169, <a href="#p171" class="tei tei-ref">171</a>, <a href="#p172" class="tei tei-ref">172</a>, <a href="#p173" class="tei tei-ref">173</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Pencil of planes of the second +order, <a href="#p59" class="tei tei-ref">59</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Pencil of rays, <a href="#p6" class="tei tei-ref">6</a>, <a href="#p7" class="tei tei-ref">7</a>, <a href="#p8" class="tei tei-ref">8</a>, <a href="#p23" class="tei tei-ref">23</a>; + of the second order, <a href="#p57" class="tei tei-ref">57</a>, <a href="#p60" class="tei tei-ref">60</a>, <a href="#p79" class="tei tei-ref">79</a>, <a href="#p81" class="tei tei-ref">81</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Perspective position, <a href="#p6" class="tei tei-ref">6</a>, <a href="#p8" class="tei tei-ref">8</a>, <a href="#p35" class="tei tei-ref">35</a>, <a href="#p37" class="tei tei-ref">37</a>, +51, <a href="#p53" class="tei tei-ref">53</a>, <a href="#p71" class="tei tei-ref">71</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Plane system, <a href="#p16" class="tei tei-ref">16</a>, <a href="#p23" class="tei tei-ref">23</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Planes on space, <a href="#p17" class="tei tei-ref">17</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Point of contact, <a href="#p87" class="tei tei-ref">87</a>, <a href="#p88" class="tei tei-ref">88</a>, <a href="#p89" class="tei tei-ref">89</a>, <a href="#p90" class="tei tei-ref">90</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Point system, <a href="#p16" class="tei tei-ref">16</a>, <a href="#p23" class="tei tei-ref">23</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Point-row, <a href="#p6" class="tei tei-ref">6</a>, <a href="#p7" class="tei tei-ref">7</a>, <a href="#p8" class="tei tei-ref">8</a>, <a href="#p9" class="tei tei-ref">9</a>, <a href="#p23" class="tei tei-ref">23</a>; + of the second order, <a href="#p55" class="tei tei-ref">55</a>, <a href="#p60" class="tei tei-ref">60</a>, <a href="#p61" class="tei tei-ref">61</a>, <a href="#p66" class="tei tei-ref">66</a>, + <a href="#p67" class="tei tei-ref">67</a>, <a href="#p72" class="tei tei-ref">72</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Points in space, <a href="#p18" class="tei tei-ref">18</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Pole and polar, <a href="#p98" class="tei tei-ref">98</a>, <a href="#p99" class="tei tei-ref">99</a>, <a href="#p100" class="tei tei-ref">100</a>, <a href="#p101" class="tei tei-ref">101</a>, +138, <a href="#p164" class="tei tei-ref">164</a>, <a href="#p166" class="tei tei-ref">166</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Poncelet (1788-1867), <a href="#p177" class="tei tei-ref">177</a>, <a href="#p179" class="tei tei-ref">179</a>, +180, <a href="#p181" class="tei tei-ref">181</a>, <a href="#p182" class="tei tei-ref">182</a>, <a href="#p183" class="tei tei-ref">183</a>, <a href="#p184" class="tei tei-ref">184</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Principal axis of a conic, <a href="#p157" class="tei tei-ref">157</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Projection, <a href="#p161" class="tei tei-ref">161</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Protective axial pencils, <a href="#p59" class="tei tei-ref">59</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Projective correspondence, <a href="#p9" class="tei tei-ref">9</a>, <a href="#p35" class="tei tei-ref">35</a>, +36, <a href="#p37" class="tei tei-ref">37</a>, <a href="#p47" class="tei tei-ref">47</a>, <a href="#p71" class="tei tei-ref">71</a>, <a href="#p92" class="tei tei-ref">92</a>, <a href="#p104" class="tei tei-ref">104</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Projective pencils, <a href="#p53" class="tei tei-ref">53</a>, <a href="#p64" class="tei tei-ref">64</a>, <a href="#p68" class="tei tei-ref">68</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Projective point-rows, <a href="#p51" class="tei tei-ref">51</a>, <a href="#p79" class="tei tei-ref">79</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Projective properties, <a href="#p24" class="tei tei-ref">24</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Projective theorems, <a href="#p40" class="tei tei-ref">40</a>, <a href="#p104" class="tei tei-ref">104</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Quadrangle, <a href="#p26" class="tei tei-ref">26</a>, <a href="#p27" class="tei tei-ref">27</a>, <a href="#p28" class="tei tei-ref">28</a>, <a href="#p29" class="tei tei-ref">29</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Quadric cone, <a href="#p59" class="tei tei-ref">59</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Quadrilateral, <a href="#p88" class="tei tei-ref">88</a>, <a href="#p95" class="tei tei-ref">95</a>, <a href="#p96" class="tei tei-ref">96</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Roberval (1602-1675), <a href="#p168" class="tei tei-ref">168</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Ruler construction, <a href="#p40" class="tei tei-ref">40</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Scheiner, <a href="#p169" class="tei tei-ref">169</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Self-corresponding elements, <a href="#p47" class="tei tei-ref">47</a>, +48, <a href="#p49" class="tei tei-ref">49</a>, <a href="#p50" class="tei tei-ref">50</a>, <a href="#p51" class="tei tei-ref">51</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Self-dual, <a href="#p105" class="tei tei-ref">105</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Self-polar triangle, <a href="#p102" class="tei tei-ref">102</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Separation of elements in involution, +148<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Separation of harmonic conjugates, +38<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Sequence of points, <a href="#p49" class="tei tei-ref">49</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Sign of segment, <a href="#p44" class="tei tei-ref">44</a>, <a href="#p45" class="tei tei-ref">45</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Similarity, <a href="#p106" class="tei tei-ref">106</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Skew lines, <a href="#p12" class="tei tei-ref">12</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Space system, <a href="#p19" class="tei tei-ref">19</a>, <a href="#p23" class="tei tei-ref">23</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Sphere, <a href="#p21" class="tei tei-ref">21</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Steiner (1796-1863), <a href="#p129" class="tei tei-ref">129</a>, <a href="#p130" class="tei tei-ref">130</a>, <a href="#p131" class="tei tei-ref">131</a>, +177, <a href="#p179" class="tei tei-ref">179</a>, <a href="#p184" class="tei tei-ref">184</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Steiner's construction, <a href="#p129" class="tei tei-ref">129</a>, <a href="#p130" class="tei tei-ref">130</a>, +131<br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Superposed point-rows, <a href="#p47" class="tei tei-ref">47</a>, <a href="#p48" class="tei tei-ref">48</a>, <a href="#p49" class="tei tei-ref">49</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Surfaces of the second degree, <a href="#p166" class="tei tei-ref">166</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">System of lines in space, <a href="#p20" class="tei tei-ref">20</a>, <a href="#p23" class="tei tei-ref">23</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Systems of conics, <a href="#p125" class="tei tei-ref">125</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Tangent line, <a href="#p61" class="tei tei-ref">61</a>, <a href="#p80" class="tei tei-ref">80</a>, <a href="#p81" class="tei tei-ref">81</a>, <a href="#p87" class="tei tei-ref">87</a>, <a href="#p88" class="tei tei-ref">88</a>, +89, <a href="#p90" class="tei tei-ref">90</a>, <a href="#p91" class="tei tei-ref">91</a>, <a href="#p92" class="tei tei-ref">92</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Tycho Brahe (1546-1601), <a href="#p162" class="tei tei-ref">162</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Verner, <a href="#p161" class="tei tei-ref">161</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Vertex of conic, <a href="#p157" class="tei tei-ref">157</a>, <a href="#p159" class="tei tei-ref">159</a><br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Von Staudt (1798-1867), <a href="#p179" class="tei tei-ref">179</a>, <a href="#p185" class="tei tei-ref">185</a> <br /></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Wallis (1616-1703), <a href="#p162" class="tei tei-ref">162</a><br /></p> + +</div> + +</div> + +<hr class="doublepage" /><div class="tei tei-back" style="margin-bottom: 2.00em; margin-top: 6.00em"> +<div class="tei tei-div" style="margin-bottom: 5.00em; margin-top: 5.00em"> + + + + <div class="tei tei-div" style="margin-bottom: 4.00em; margin-top: 4.00em"> + <h1 class="tei tei-head" style="text-align: left; margin-bottom: 3.46em; margin-top: 3.46em"><span style="font-size: 173%">Footnotes</span></h1> + <dl class="tei tei-list-footnotes"><dt class="tei tei-notelabel"><a id="note_1" name="note_1" href="#noteref_1">1.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em">The +more general notion of <span class="tei tei-hi"><span style="font-style: italic">anharmonic ratio</span></span>, which includes +the harmonic ratio as a special case, was also known to the ancients. +While we have not found it necessary to make use of the anharmonic +ratio in building up our theory, it is so frequently met with in treatises +on geometry that some account of it should be given.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Consider any four points, <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, on a line, and join them to +any point <span class="tei tei-hi"><span style="font-style: italic">S</span></span> not on that line. Then the triangles <span class="tei tei-hi"><span style="font-style: italic">ASB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">GSD</span></span>, <span class="tei tei-hi"><span style="font-style: italic">ASD</span></span>, +<span class="tei tei-hi"><span style="font-style: italic">CSB</span></span>, having all the same altitude, are to each other as their bases. +Also, since the area of any triangle is one half the product of any two +of its sides by the sine of the angle included between them, we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"><img src="images/5.png" alt="[formula]" width="485" height="30" class="tei tei-formula tei-formula-tex" style="text-align: center"></img></p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Now the fraction on the right would be unchanged if instead of the +points <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> we should take any other four points <span class="tei tei-hi"><span style="font-style: italic">A'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C'</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> +lying on any other line cutting across <span class="tei tei-hi"><span style="font-style: italic">SA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SD</span></span>. In other +words, <span class="tei tei-hi"><span style="font-style: italic">the fraction on the left is unaltered in value if the points +</span><span class="tei tei-hi"><span style="font-style: italic">A</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">B</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">C</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">D</span></span><span style="font-style: italic"> are replaced by any other four points perspective to them.</span></span> +Again, the fraction on the left is unchanged if some other point were +taken instead of <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. In other words, <span class="tei tei-hi"><span style="font-style: italic">the fraction on the right is +unaltered if we replace the four lines </span><span class="tei tei-hi"><span style="font-style: italic">SA</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">SB</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">SC</span></span><span style="font-style: italic">, </span><span class="tei tei-hi"><span style="font-style: italic">SD</span></span><span style="font-style: italic"> by any other four +lines perspective to them.</span></span> The fraction on the left is called the <span class="tei tei-hi"><span style="font-style: italic">anharmonic +ratio</span></span> of the four points <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span>; the fraction on the right +is called the <span class="tei tei-hi"><span style="font-style: italic">anharmonic ratio</span></span> of the four lines <span class="tei tei-hi"><span style="font-style: italic">SA</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SB</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SC</span></span>, <span class="tei tei-hi"><span style="font-style: italic">SD</span></span>. The +anharmonic ratio of four points is sometimes written (<span class="tei tei-hi"><span style="font-style: italic">ABCD</span></span>), so that</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/6.png" alt="[formula]" width="146" height="31" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If we take the points in different order, the value of the anharmonic +ratio will not necessarily remain the same. The twenty-four different +ways of writing them will, however, give not more than six different +values for the anharmonic ratio, for by writing out the fractions +which define them we can find that <span class="tei tei-hi"><span style="font-style: italic">(ABCD) = (BADC) = (CDAB) = (DCBA)</span></span>. +If we write <span class="tei tei-hi"><span style="font-style: italic">(ABCD) = a</span></span>, it is not difficult to show that +the six values are</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/7.png" alt="[formula]" width="273" height="15" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">The proof of this we leave to the student.</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">If <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> are four harmonic points (see Fig. 6, p. *22), and a quadrilateral +<span class="tei tei-hi"><span style="font-style: italic">KLMN</span></span> is constructed such that <span class="tei tei-hi"><span style="font-style: italic">KL</span></span> and <span class="tei tei-hi"><span style="font-style: italic">MN</span></span> pass through +<span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">KN</span></span> and <span class="tei tei-hi"><span style="font-style: italic">LM</span></span> through <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">LN</span></span> through <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, and <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> through <span class="tei tei-hi"><span style="font-style: italic">D</span></span>, then, +projecting <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">C</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> from <span class="tei tei-hi"><span style="font-style: italic">L</span></span> upon <span class="tei tei-hi"><span style="font-style: italic">KM</span></span>, we have <span class="tei tei-hi"><span style="font-style: italic">(ABCD) = (KOMD)</span></span>, +where <span class="tei tei-hi"><span style="font-style: italic">O</span></span> is the intersection of <span class="tei tei-hi"><span style="font-style: italic">KM</span></span> with <span class="tei tei-hi"><span style="font-style: italic">LN</span></span>. But, projecting again +the points <span class="tei tei-hi"><span style="font-style: italic">K</span></span>, <span class="tei tei-hi"><span style="font-style: italic">O</span></span>, <span class="tei tei-hi"><span style="font-style: italic">M</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> from <span class="tei tei-hi"><span style="font-style: italic">N</span></span> back upon the line <span class="tei tei-hi"><span style="font-style: italic">AB</span></span>, we have +<span class="tei tei-hi"><span style="font-style: italic">(KOMD) = (CBAD)</span></span>. From this we have</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<span class="tei tei-hi" style="text-align: center"><span style="font-style: italic">(ABCD) = (CBAD),</span></span> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">or</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +<img src="images/8.png" alt="[formula]" width="89" height="15" class="tei tei-formula tei-formula-tex" style="text-align: center"></img> +</p> + +<p class="tei tei-p" style="margin-bottom: 1.00em">whence <span class="tei tei-hi"><span style="font-style: italic">a = 0</span></span> or <span class="tei tei-hi"><span style="font-style: italic">a = 2</span></span>. But it is easy to see that <span class="tei tei-hi"><span style="font-style: italic">a = 0</span></span> implies that two of the four points +coincide. For four harmonic points, therefore, the six values of the +anharmonic ratio reduce to three, namely, 2, <img src="images/9.png" alt="[formula]" width="6" height="18" class="tei tei-formula tei-formula-tex"></img>, and -1. Incidentally we see that +if an interchange of any two points in an anharmonic ratio does not +change its value, then the four points are harmonic.</p> + +<p class="tei tei-p" style="text-align: center; margin-bottom: 1.00em"> +</p><div class="tei tei-figure" style="width: 95%; text-align: center"><img src="images/image49.png" width="457" height="482" alt="Figure 49" title="Fig. 49" /><div class="tei tei-head" style="text-align: center; margin-bottom: 1.00em; margin-top: 1.00em"><span class="tei tei-hi" style="text-align: center"><span style="font-variant: small-caps">Fig.</span></span> 49</div></div> + +<p class="tei tei-p" style="margin-bottom: 1.00em">Many theorems of projective geometry are succinctly stated in +terms of anharmonic ratios. Thus, the <span class="tei tei-hi"><span style="font-style: italic">anharmonic ratio of any four +elements of a form is equal to the anharmonic ratio of the corresponding +four elements in any form projectively related to it. The anharmonic +ratio of the lines joining any four fixed points on a conic to a variable +fifthpoint on the conic is constant. The +locus of points from which four points +in a plane are seen along four rays of +constant anharmonic ratio is a conic +through the four points.</span></span> We leave these +theorems for the student, who may +also justify the following solution of +the problem: <span class="tei tei-hi"><span style="font-style: italic">Given three points and +a certain anharmonic ratio, to find a +fourth point which shall have with the +given three the given anharmonic ratio.</span></span> +Let <span class="tei tei-hi"><span style="font-style: italic">A</span></span>, <span class="tei tei-hi"><span style="font-style: italic">B</span></span>, <span class="tei tei-hi"><span style="font-style: italic">D</span></span> be the three given points +(Fig. 49). On any convenient line +through <span class="tei tei-hi"><span style="font-style: italic">A</span></span> take two points <span class="tei tei-hi"><span style="font-style: italic">B'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">D'</span></span> +such that <span class="tei tei-hi"><span style="font-style: italic">AB'/AD'</span></span> is equal to the +given anharmonic ratio. Join <span class="tei tei-hi"><span style="font-style: italic">BB'</span></span> and <span class="tei tei-hi"><span style="font-style: italic">DD'</span></span> and let the two lines +meet in <span class="tei tei-hi"><span style="font-style: italic">S</span></span>. Draw through <span class="tei tei-hi"><span style="font-style: italic">S</span></span> a parallel to <span class="tei tei-hi"><span style="font-style: italic">AB'</span></span>. This line will meet +<span class="tei tei-hi"><span style="font-style: italic">AB</span></span> in the required point <span class="tei tei-hi"><span style="font-style: italic">C</span></span>.</p></dd><dt class="tei tei-notelabel"><a id="note_2" name="note_2" href="#noteref_2">2.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Pappus, Mathematicae Collectiones, vii, 129.</p></dd><dt class="tei tei-notelabel"><a id="note_3" name="note_3" href="#noteref_3">3.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em">J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522.</p></dd><dt class="tei tei-notelabel"><a id="note_4" name="note_4" href="#noteref_4">4.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em">Kepler, Ad Vitellionem paralipomena quibus astronomiae pars +optica traditur. 1604.</p></dd><dt class="tei tei-notelabel"><a id="note_5" name="note_5" href="#noteref_5">5.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em">Desargues, Bruillon-project d'une atteinte aux événements des +rencontres d'un cône avec un plan. 1639. Edited and analyzed by +Poudra, 1864.</p></dd><dt class="tei tei-notelabel"><a id="note_6" name="note_6" href="#noteref_6">6.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em">The term 'pole' was first introduced, in the sense in which we +have used it, in 1810, by a French mathematician named Servois +(Gergonne, <span class="tei tei-hi"><span style="font-style: italic">Annales des Mathéématiques</span></span>, I, 337), and the corresponding +term 'polar' by the editor, Gergonne, of this same journal three years +later.</p></dd><dt class="tei tei-notelabel"><a id="note_7" name="note_7" href="#noteref_7">7.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Euler, Introductio in analysin infinitorum, Appendix, cap. V. +1748.</p></dd><dt class="tei tei-notelabel"><a id="note_8" name="note_8" href="#noteref_8">8.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Œuvres de Desargues, t. II, 132.</p></dd><dt class="tei tei-notelabel"><a id="note_9" name="note_9" href="#noteref_9">9.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Œuvres de Desargues, t. II, 370.</p></dd><dt class="tei tei-notelabel"><a id="note_10" name="note_10" href="#noteref_10">10.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Œuvres de Descartes, t. II, 499.</p></dd><dt class="tei tei-notelabel"><a id="note_11" name="note_11" href="#noteref_11">11.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Œuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252. +</p></dd><dt class="tei tei-notelabel"><a id="note_12" name="note_12" href="#noteref_12">12.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Chasles, Histoire de la Géométrie, 70.</p></dd><dt class="tei tei-notelabel"><a id="note_13" name="note_13" href="#noteref_13">13.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Œuvres de Desargues, t. I, 231.</p></dd><dt class="tei tei-notelabel"><a id="note_14" name="note_14" href="#noteref_14">14.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +See Ball, History of Mathematics, French edition, t. II, 233. +</p></dd><dt class="tei tei-notelabel"><a id="note_15" name="note_15" href="#noteref_15">15.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Newton, Principia, lib. i, lemma XXI.</p></dd><dt class="tei tei-notelabel"><a id="note_16" name="note_16" href="#noteref_16">16.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Maclaurin, Philosophical Transactions of the Royal Society of +London, 1735.</p></dd><dt class="tei tei-notelabel"><a id="note_17" name="note_17" href="#noteref_17">17.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Monge, Géométrie Descriptive. 1800.</p></dd><dt class="tei tei-notelabel"><a id="note_18" name="note_18" href="#noteref_18">18.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Poncelet, Traité des Propriétés Projectives des Figures. 1822. +(See p. 357, Vol. II, of the edition of 1866.)</p></dd><dt class="tei tei-notelabel"><a id="note_19" name="note_19" href="#noteref_19">19.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Gergonne, <span class="tei tei-hi"><span style="font-style: italic">Annales de Mathématiques, XVI, 209. 1826.</span></span></p></dd><dt class="tei tei-notelabel"><a id="note_20" name="note_20" href="#noteref_20">20.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Steiner, Systematische Ehtwickelung der Abhängigkeit geometrischer +Gestalten von einander. 1832.</p></dd><dt class="tei tei-notelabel"><a id="note_21" name="note_21" href="#noteref_21">21.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Von Staudt, Geometrie der Lage. 1847.</p></dd><dt class="tei tei-notelabel"><a id="note_22" name="note_22" href="#noteref_22">22.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Reye, Geometrie der Lage. Translated by Holgate, 1897.</p></dd><dt class="tei tei-notelabel"><a id="note_23" name="note_23" href="#noteref_23">23.</a></dt><dd class="tei tei-notetext"><p class="tei tei-p" style="margin-bottom: 1.00em"> +Ball, loc. cit. p. 261.</p></dd></dl> +</div> +</div> +</div> +</div> + +<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK AN ELEMENTARY COURSE IN SYNTHETIC PROJECTIVE GEOMETRY ***</div> +<div style='text-align:left'> + +<div style='display:block; margin:1em 0'> +Updated editions will replace the previous one—the old editions will +be renamed. +</div> + +<div style='display:block; margin:1em 0'> +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the United +States without permission and without paying copyright +royalties. 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