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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 http://www.gutenberg.org/license + + + +Title: An Elementary Course in Synthetic Projective Geometry + +Author: Lehmer, Derrick Norman + +Release Date: November 4, 2005 [Ebook #17001] + +Language: American + +Character set encoding: UTF-8 +--> + +<!DOCTYPE TEI.2 SYSTEM "http://www.gutenberg.org/tei/marcello/0.4/dtd/pgtei.dtd"> + +<TEI.2 lang="en-us"> +<teiHeader> + <fileDesc> + <titleStmt> + <title>An Elementary Course in Synthetic Projective Geometry</title> + <author>Lehmer, Derrick Norman</author> + </titleStmt> + <editionStmt> + <edition n="1">Edition 1</edition> + </editionStmt> + <publicationStmt> + <publisher>Project Gutenberg</publisher> + <date value="2005-11-4">November 4, 2005</date> + <idno type="etext-no">17001</idno> + <idno type="DPid">projectID3fa113afbb55e</idno> + <availability> + <p>This eBook is for the use of anyone anywhere 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 online at www.gutenberg.org/license</p> + </availability> + </publicationStmt> + + <sourceDesc> + <bibl> + <title>An Elementary Course in Synthetic Projective Geometry</title> + <author>Lehmer, Derrick Norman</author> + <imprint> + <pubPlace>Boston</pubPlace> + <publisher>Ginn and Company</publisher> + <date>1917</date> + </imprint> + </bibl> + </sourceDesc> + </fileDesc> + + <encodingDesc> + <classDecl> + <taxonomy id="lc"> + <bibl> + <title>Library of Congress Classification</title> + </bibl> + </taxonomy> + </classDecl> + </encodingDesc> + <profileDesc> + <langUsage> + <language id="en-us">United States English</language> + </langUsage> + <textClass> + <classCode scheme="lc"> + *** <!-- LoC Class (PR, PQ, ...) --> + </classCode> + <keywords> + <list> + <!-- <item></item> any keywords for PG search engine --> + </list> + </keywords> + </textClass> + </profileDesc> + <revisionDesc> + <change> + <date value="2005-11">November 2005</date> + <respStmt> + <name>Joshua Hutchinson, </name> + <name>Cornell University, </name> + <name>Online Distributed Proofreading Team</name> + </respStmt> + <item>Project Gutenberg Edition</item> + </change> + <change> + <date value="2006-6">June 2006</date> + <respStmt> + <name>Joshua Hutchinson</name> + </respStmt> + <item>Added PGHeader/PGFooter.</item> + </change> + </revisionDesc> +</teiHeader> + +<pgExtensions> + <pgStyleSheet> + figure { text-align: center; page-float: 'htb' } + .w95 { } + @media pdf { + .w95 { width: 95% } + } + </pgStyleSheet> +</pgExtensions> + +<text> +<front> + +<div> +<divGen type="pgheader" /> +</div> + +<div> +<divGen type="encodingDesc" /> +</div> + +<div rend="page-break-before: right"> +<divGen type="titlepage" /> +</div> + +<div rend="page-break-before: right"> +<index index="toc" /><index index="pdf" /> +<pb n="iii" /><anchor id="Pgiii" /> +<head>Preface</head> + +<p>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>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>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 <hi rend="font-style: italic">pencil of rays</hi> instead of a <hi rend="font-style: italic">bundle of rays</hi>, as later +writers seem inclined to do. For a point considered +<pb n="iv" /><anchor id="Pgiv" /> +as made up of all the lines and planes through it he +has ventured to use the term <hi rend="font-style: italic">point system</hi>, as being +the natural dualization of the usual term <hi rend="font-style: italic">plane system</hi>. +He has also rejected the term <hi rend="font-style: italic">foci of an involution</hi>, and +has not used the customary terms for classifying involutions—<hi rend="font-style: italic">hyperbolic +involution</hi>, <hi rend="font-style: italic">elliptic involution</hi> and +<hi rend="font-style: italic">parabolic involution</hi>. 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>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>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 +<pb n="v" /><anchor id="Pgv" /> +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>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>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> + +<pb n="vi" /><anchor id="Pgvi" /> + +<p>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 rend="text-align: right">D. N. LEHMER</p> + +<p><hi rend="font-variant: small-caps">Berkeley, California</hi></p> + +</div> + +<div rend="page-break-before: right"> + <index index="toc" /><index index="pdf" /> + <head>Contents</head> + <divGen type="toc" /> +</div> + +</front> + +<body> +<div rend="page-break-before: right"> +<pb n="1" /><anchor id="Pg1" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER I - ONE-TO-ONE CORRESPONDENCE</head> +<p></p> + +<div> +<index index="toc" level1="1. Definition of one-to-one correspondence" /><index index="pdf" /> + +<head></head><p><anchor id="p1" /><hi rend="font-weight: bold">1. Definition of one-to-one correspondence.</hi> +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 <hi rend="font-style: italic">one-to-one correspondence</hi> 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 +<pb n="2" /><anchor id="Pg2" /> +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> +<index index="toc" level1="2. Consequences of one-to-one correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p2" /><hi rend="font-weight: bold">2. Consequences of one-to-one correspondence.</hi> +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> +<index index="toc" level1="3. Applications in mathematics" /><index index="pdf" /> +<head></head><p><anchor id="p3" /><hi rend="font-weight: bold">3.</hi> 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> + +<pb n="3" /><anchor id="Pg3" /> +</div> + +<div> +<index index="toc" level1="4. One-to-one correspondence and enumeration" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image01.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 1</head> +<figDesc>Figure 1</figDesc> +</figure></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image02.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 2</head> +<figDesc>Figure 2</figDesc> +</figure></p> + +<p><anchor id="p4" /><hi rend="font-weight: bold">4. One-to-one correspondence and enumeration.</hi> 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 <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">A'B'</hi>. Join <hi rend="font-style: italic">AA'</hi> and <hi rend="font-style: italic">BB'</hi>, +and let these joining lines meet in <hi rend="font-style: italic">S</hi>. For every point <hi rend="font-style: italic">C</hi> +on <hi rend="font-style: italic">AB</hi> a point <hi rend="font-style: italic">C'</hi> may be found +on <hi rend="font-style: italic">A'B'</hi> by joining <hi rend="font-style: italic">C</hi> to <hi rend="font-style: italic">S</hi> and +noting the point <hi rend="font-style: italic">C'</hi> where <hi rend="font-style: italic">CS</hi> +meets <hi rend="font-style: italic">A'B'</hi>. Similarly, a point <hi rend="font-style: italic">C</hi> +may be found on <hi rend="font-style: italic">AB</hi> for any +point <hi rend="font-style: italic">C'</hi> on <hi rend="font-style: italic">A'B'</hi>. The correspondence +is clearly one-to-one, +but it would be absurd to infer +from this that there were just +as many points on <hi rend="font-style: italic">AB</hi> as on <hi rend="font-style: italic">A'B'</hi>. In fact, it would +be just as reasonable to infer that there were twice as +many points on <hi rend="font-style: italic">A'B'</hi> as on <hi rend="font-style: italic">AB</hi>. For if we bend <hi rend="font-style: italic">A'B'</hi> +into a circle with center at <hi rend="font-style: italic">S</hi> (Fig. 2), we see that for +every point <hi rend="font-style: italic">C</hi> on <hi rend="font-style: italic">AB</hi> there are two points on <hi rend="font-style: italic">A'B'</hi>. Thus +<pb n="4" /><anchor id="Pg4" /> +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> +<index index="toc" level1="5. Correspondence between a part and the whole" /><index index="pdf" /> +<head></head><p><anchor id="p5" /><hi rend="font-weight: bold">5. Correspondence between a part and the whole of an +infinite assemblage.</hi> 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> +<index index="toc" level1="6. Infinitely distant point" /><index index="pdf" /> +<head></head><p><anchor id="p6" /><hi rend="font-weight: bold">6. Infinitely distant point.</hi> 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 <hi rend="font-style: italic">C</hi> on the line <hi rend="font-style: italic">AB</hi> +there is a line <hi rend="font-style: italic">SC</hi> through <hi rend="font-style: italic">S</hi>. We must assume the line +<hi rend="font-style: italic">AB</hi> extended indefinitely in both directions, however, if +we are to have a point on it for every line through <hi rend="font-style: italic">S</hi>; +and even with this extension there is one line through +<hi rend="font-style: italic">S</hi>, according to Euclid's postulate, which does not meet +the line <hi rend="font-style: italic">AB</hi> and which therefore has no point on +<hi rend="font-style: italic">AB</hi> to correspond to it. In order to smooth out this +<pb n="5" /><anchor id="Pg5" />discrepancy we are accustomed to assume the existence +of an <hi rend="font-style: italic">infinitely distant</hi> point on the line <hi rend="font-style: italic">AB</hi> and to assign +this point as the corresponding point of the exceptional +line of <hi rend="font-style: italic">S</hi>. 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 +<hi rend="font-style: italic">point-row</hi>, and the totality of lines through a point a +<hi rend="font-style: italic">pencil of rays</hi>, we say that the point-row and the pencil +related as above are in <hi rend="font-style: italic">perspective position</hi>, or that they +are <hi rend="font-style: italic">perspectively related</hi>.</p></div> + +<div> +<index index="toc" level1="7. Axial pencil; fundamental forms" /><index index="pdf" /> +<head></head><p><anchor id="p7" /><hi rend="font-weight: bold">7. Axial pencil; fundamental forms.</hi> 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 +<hi rend="font-style: italic">axial pencil</hi>, and the three assemblages—the point-row, +the pencil of rays, and the axial pencil—are called +<hi rend="font-style: italic">fundamental forms</hi>. 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> +<index index="toc" level1="8. Perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p8" /><hi rend="font-weight: bold">8. Perspective position.</hi> 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 +<pb n="6" /><anchor id="Pg6" /> +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> +<index index="toc" level1="9. Projective relation" /><index index="pdf" /> +<head></head><p><anchor id="p9" /><hi rend="font-weight: bold">9. Projective relation.</hi> 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, <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">S</hi>, then a point-row <hi rend="font-style: italic">a</hi> perspective +to <hi rend="font-style: italic">A</hi> will be in one-to-one correspondence with +a point-row <hi rend="font-style: italic">b</hi> perspective to <hi rend="font-style: italic">B</hi>, but corresponding points +will not, in general, lie on lines which all pass through +a point. Two such point-rows are said to be <hi rend="font-style: italic">projectively +related</hi>, 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 <hi rend="font-style: italic">continuous</hi>; 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 +<pb n="7" /><anchor id="Pg7" /> +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> +<index index="toc" level1="10. Infinity-to-one correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p10" /><hi rend="font-weight: bold">10. Infinity-to-one correspondence.</hi> 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 <hi rend="font-style: italic">m</hi> points on one and <hi rend="font-style: italic">n</hi> points on the other. The +number of lines joining the <hi rend="font-style: italic">m</hi> points of one to the +<hi rend="font-style: italic">n</hi> points jof the other is clearly <hi rend="font-style: italic">mn</hi>. 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]<hi rend="vertical-align: super">2</hi>. 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> +<index index="toc" level1="11. Infinitudes of different orders" /><index index="pdf" /> +<head></head><p><anchor id="p11" /><hi rend="font-weight: bold">11. Infinitudes of different orders.</hi> 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 +<pb n="8" /><anchor id="Pg8" /> +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> +<index index="toc" level1="12. Points in a plane" /><index index="pdf" /> +<head></head><p><anchor id="p12" /><hi rend="font-weight: bold">12.</hi> 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> +<index index="toc" level1="13. Lines through a point" /><index index="pdf" /> +<head></head><p><anchor id="p13" /><hi rend="font-weight: bold">13.</hi> 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. <hi rend="font-style: italic">Thus the infinitude of lines through a +point in space is of the second order.</hi></p></div> + +<div> +<index index="toc" level1="14. Planes through a point" /><index index="pdf" /> +<head></head><p><anchor id="p14" /><hi rend="font-weight: bold">14.</hi> 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 <hi rend="font-style: italic">the infinitude of +planes through a point in space is of the second order.</hi></p></div> + +<div> +<index index="toc" level1="15. Lines in a plane" /><index index="pdf" /> +<head></head><p><anchor id="p15" /><hi rend="font-weight: bold">15.</hi> If to each plane through a point in space we +make correspond the line in which it intersects a given +plane, we see that <hi rend="font-style: italic">the infinitude of lines in a plane is of +the second order.</hi> 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 <hi rend="font-style: italic">S</hi> +not in the plane. Join any point <hi rend="font-style: italic">M</hi> of the plane to <hi rend="font-style: italic">S</hi>. +Through <hi rend="font-style: italic">S</hi> draw a plane at right angles to <hi rend="font-style: italic">MS</hi>. This +meets the given plane in a line <hi rend="font-style: italic">m</hi> which may be taken as +corresponding to the point <hi rend="font-style: italic">M</hi>. Another very important +<pb n="9" /><anchor id="Pg9" /> +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> +<index index="toc" level1="16. Plane system and point system" /><index index="pdf" /> +<head></head><p><anchor id="p16" /><hi rend="font-weight: bold">16. Plane system and point system.</hi> The plane, considered +as made up of the points and lines in it, is called +a <hi rend="font-style: italic">plane system</hi> 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 <hi rend="font-style: italic">point system</hi> +and is also a fundamental form of the second order.</p></div> + +<div> +<index index="toc" level1="17. Planes in space" /><index index="pdf" /> +<head></head><p><anchor id="p17" /><hi rend="font-weight: bold">17.</hi> If now we take three lines in space all lying in +different planes, and select <hi rend="font-style: italic">l</hi> points on the first, <hi rend="font-style: italic">m</hi> points +on the second, and <hi rend="font-style: italic">n</hi> points on the third, then the total +number of planes passing through one of the selected +points on each line will be <hi rend="font-style: italic">lmn</hi>. 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]<hi rend="vertical-align: super">3</hi>, and to call it an infinitude of the <hi rend="font-style: italic">third</hi> order. But +it is easily seen that every plane in space is included in +this totality, so that <hi rend="font-style: italic">the totality of planes in space is an +infinitude of the third order.</hi></p></div> + +<div> +<index index="toc" level1="18. Points of space" /><index index="pdf" /> +<head></head><p><anchor id="p18" /><hi rend="font-weight: bold">18.</hi> 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 <hi rend="font-style: italic">the totality of points +in space is an infinitude of the third order.</hi></p></div> + +<div> +<index index="toc" level1="19. Space system" /><index index="pdf" /> +<head></head><p><anchor id="p19" /><hi rend="font-weight: bold">19. Space system.</hi> Space of three dimensions, considered +as made up of all its planes and points, is then +a fundamental form of the <hi rend="font-style: italic">third</hi> order, which we shall +call a <hi rend="font-style: italic">space system.</hi></p> + +<pb n="10" /><anchor id="Pg10" /></div> + +<div> +<index index="toc" level1="20. Lines in space" /><index index="pdf" /> +<head></head><p><anchor id="p20" /><hi rend="font-weight: bold">20. Lines in space.</hi> 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. <hi rend="font-style: italic">The +totality of lines in space gives, then, a fundamental form +of the fourth order.</hi></p></div> + +<div> +<index index="toc" level1="21. Correspondence between points and numbers" /><index index="pdf" /> +<head></head><p><anchor id="p21" /><hi rend="font-weight: bold">21. Correspondence between points and numbers.</hi> 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 +<pb n="11" /><anchor id="Pg11" /> +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 <hi rend="font-style: italic">continuous</hi> 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> +<index index="toc" level1="22. Elements at infinity" /><index index="pdf" /> +<head></head><p><anchor id="p22" /><hi rend="font-weight: bold">22. Elements at infinity.</hi> 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, +<pb n="12" /><anchor id="Pg12" /> +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> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>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>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>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>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>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>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> + +<pb n="13" /><anchor id="Pg13" /> + +<p>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>8. Set up a one-to-one correspondence between the series +of numbers <hi rend="font-style: italic">1</hi>, <hi rend="font-style: italic">2</hi>, <hi rend="font-style: italic">3</hi>, <hi rend="font-style: italic">4</hi>, ... and the series of even numbers +<hi rend="font-style: italic">2</hi>, <hi rend="font-style: italic">4</hi>, <hi rend="font-style: italic">6</hi>, <hi rend="font-style: italic">8</hi> .... Are we justified in saying that there are just +as many even numbers as there are numbers altogether?</p> + +<p>9. Is the axiom "The whole is greater than one of its +parts" applicable to infinite assemblages?</p> + +<p>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> + +<pb n="14" /><anchor id="Pg14" /> +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE +CORRESPONDENCE WITH EACH OTHER</head> + +<div> +<index index="toc" level1="23. Seven fundamental forms" /><index index="pdf" /> +<head></head><p><anchor id="p23" /><hi rend="font-weight: bold">23. Seven fundamental forms.</hi> 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> +<index index="toc" level1="24. Projective properties" /><index index="pdf" /> +<head></head><p><anchor id="p24" /><hi rend="font-weight: bold">24. Projective properties.</hi> Our chief interest in this +chapter will be the discovery of relations between +the elements of one form which hold between the +<pb n="15" /><anchor id="Pg15" /> +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 <hi rend="font-style: italic">u</hi> three points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi>, are taken +such that <hi rend="font-style: italic">B</hi> is the middle point of the segment <hi rend="font-style: italic">AC</hi>, +it does not follow that the three points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi> +in a point-row perspective to <hi rend="font-style: italic">u</hi> 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 <hi rend="font-style: italic">projective +relations.</hi> Relations involving measurement of lines or +of angles are not projective.</p></div> + +<div> +<index index="toc" level1="25. Desargues's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p25" /><hi rend="font-weight: bold">25. Desargues's theorem.</hi> We consider first the following +beautiful theorem, due to Desargues and called +by his name.</p> + +<p><hi rend="font-style: italic">If two triangles, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi> and <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, are so situated +that the lines <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, and <hi rend="font-style: italic">CC'</hi> all meet in a point, then +the pairs of sides <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">A'B'</hi>, <hi rend="font-style: italic">BC</hi> and <hi rend="font-style: italic">B'C'</hi>, <hi rend="font-style: italic">CA</hi> and +<hi rend="font-style: italic">C'A'</hi> all meet on a straight line, and conversely.</hi></p> + +<pb n="16" /><anchor id="Pg16" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image03.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 3</head> +<figDesc>Figure 3</figDesc> +</figure></p> + +<p>Let the lines <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, and <hi rend="font-style: italic">CC'</hi> meet in the point <hi rend="font-style: italic">M</hi> +(Fig. 3). Conceive of the figure as in space, so that +<hi rend="font-style: italic">M</hi> is the vertex of a trihedral angle of which the given +triangles are plane sections. The lines <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">A'B'</hi> 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 <hi rend="font-style: italic">P</hi>. By similar +reasoning the point +<hi rend="font-style: italic">Q</hi> of intersection of +the lines <hi rend="font-style: italic">BC</hi> and +<hi rend="font-style: italic">B'C'</hi> must lie on this same line as well as the point <hi rend="font-style: italic">R</hi> +of intersection of <hi rend="font-style: italic">CA</hi> and <hi rend="font-style: italic">C'A'</hi>. Therefore the points +<hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, and <hi rend="font-style: italic">R</hi> all lie on the same line <hi rend="font-style: italic">m</hi>. If now we consider +the figure a plane figure, the points <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, and <hi rend="font-style: italic">R</hi> +still all lie on a straight line, which proves the theorem. +The converse is established in the same manner.</p></div> + +<div> +<index index="toc" level1="26. Fundamental theorem concerning two complete +quadrangles" /><index index="pdf" /> +<head></head><p><anchor id="p26" /><hi rend="font-weight: bold">26. Fundamental theorem concerning two complete +quadrangles.</hi> This theorem throws into our hands the +following fundamental theorem concerning two complete +quadrangles, a <hi rend="font-style: italic">complete quadrangle</hi> being defined +as the figure obtained by joining any four given points +by straight lines in the six possible ways.</p> + +<p><hi rend="font-style: italic">Given two complete quadrangles, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> and +<hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L'</hi>, <hi rend="font-style: italic">M'</hi>, <hi rend="font-style: italic">N'</hi>, so related that <hi rend="font-style: italic">KL</hi>, <hi rend="font-style: italic">K'L'</hi>, <hi rend="font-style: italic">MN</hi>, <hi rend="font-style: italic">M'N'</hi> all +meet in a point <hi rend="font-style: italic">A</hi>; <hi rend="font-style: italic">LM</hi>, <hi rend="font-style: italic">L'M'</hi>, <hi rend="font-style: italic">NK</hi>, <hi rend="font-style: italic">N'K'</hi> all meet in a +<pb n="17" /><anchor id="Pg17" /> +point <hi rend="font-style: italic">Q</hi>; and <hi rend="font-style: italic">LN</hi>, <hi rend="font-style: italic">L'N'</hi> meet in a point <hi rend="font-style: italic">B</hi> on the line +<hi rend="font-style: italic">AC</hi>; then the lines <hi rend="font-style: italic">KM</hi> and <hi rend="font-style: italic">K'M'</hi> also meet in a point <hi rend="font-style: italic">D</hi> +on the line <hi rend="font-style: italic">AC</hi>.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image04.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 4</head> +<figDesc>Figure 4</figDesc> +</figure></p> + +<p>For, by the converse of the last theorem, <hi rend="font-style: italic">KK'</hi>, <hi rend="font-style: italic">LL'</hi>, +and <hi rend="font-style: italic">NN'</hi> all meet in a point <hi rend="font-style: italic">S</hi> (Fig. 4). Also <hi rend="font-style: italic">LL'</hi>, <hi rend="font-style: italic">MM'</hi>, +and <hi rend="font-style: italic">NN'</hi> meet in a point, and therefore in the same +point <hi rend="font-style: italic">S</hi>. Thus <hi rend="font-style: italic">KK'</hi>, <hi rend="font-style: italic">LL'</hi>, and <hi rend="font-style: italic">MM'</hi> meet in a point, +and so, by Desargues's theorem itself, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">D</hi> are +on a straight line.</p></div> + +<div> +<index index="toc" level1="27. Importance of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p27" /><hi rend="font-weight: bold">27. Importance of the theorem.</hi> The importance of +this theorem lies in the fact that, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi> being +given, an indefinite number of quadrangles <hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L'</hi>, <hi rend="font-style: italic">M'</hi>, <hi rend="font-style: italic">N'</hi> +my be found such that <hi rend="font-style: italic">K'L'</hi> and <hi rend="font-style: italic">M'N'</hi> meet in <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">K'N'</hi> +and <hi rend="font-style: italic">L'M'</hi> in <hi rend="font-style: italic">C</hi>, with <hi rend="font-style: italic">L'N'</hi> passing through <hi rend="font-style: italic">B</hi>. Indeed, +the lines <hi rend="font-style: italic">AK'</hi> and <hi rend="font-style: italic">AM'</hi> may be drawn arbitrarily +through <hi rend="font-style: italic">A</hi>, and any line through <hi rend="font-style: italic">B</hi> may be used to +determine <hi rend="font-style: italic">L'</hi> and <hi rend="font-style: italic">N'</hi>. By joining these two points to +<hi rend="font-style: italic">C</hi> the points <hi rend="font-style: italic">K'</hi> and <hi rend="font-style: italic">M'</hi> are determined. Then the line +<pb n="18" /><anchor id="Pg18" /> +joining <hi rend="font-style: italic">K'</hi> and <hi rend="font-style: italic">M'</hi>, found in this way, must pass +through the point <hi rend="font-style: italic">D</hi> already determined by the quadrangle +<hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>. <hi rend="font-style: italic">The three points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, given in +order, serve thus to determine a fourth point <hi rend="font-style: italic">D</hi>.</hi></p></div> + +<div> +<index index="toc" level1="28. Restatement of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p28" /><hi rend="font-weight: bold">28.</hi> In a complete quadrangle the line joining any +two points is called the <hi rend="font-style: italic">opposite side</hi> to the line joining +the other two points. The result of the preceding +paragraph may then be stated as follows:</p> + +<p>Given three points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, in a straight line, if a +pair of opposite sides of a complete quadrangle pass +through <hi rend="font-style: italic">A</hi>, and another pair through <hi rend="font-style: italic">C</hi>, and one of the +remaining two sides goes through <hi rend="font-style: italic">B</hi>, 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> +<index index="toc" level1="29. Four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p29" /><hi rend="font-weight: bold">29. Four harmonic points.</hi> Four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, +related as in the preceding theorem are called <hi rend="font-style: italic">four +harmonic points</hi>. The point <hi rend="font-style: italic">D</hi> is called the <hi rend="font-style: italic">fourth harmonic +of <hi rend="font-style: italic">B</hi> with respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi></hi>. Since <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> play +exactly the same rôle in the above construction, <hi rend="font-style: italic"><hi rend="font-style: italic">B</hi> is +also the fourth harmonic of <hi rend="font-style: italic">D</hi> with respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi></hi>. +<hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are called <hi rend="font-style: italic">harmonic conjugates with respect to +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi></hi>. We proceed to show that <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are also +harmonic conjugates with respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>—that is, +that it is possible to find a quadrangle of which two +opposite sides shall pass through <hi rend="font-style: italic">B</hi>, two through <hi rend="font-style: italic">D</hi>, +and of the remaining pair, one through <hi rend="font-style: italic">A</hi> and the other +through <hi rend="font-style: italic">C</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image05.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 5</head> +<figDesc>Figure 5</figDesc> +</figure></p> + +<p>Let <hi rend="font-style: italic">O</hi> be the intersection of <hi rend="font-style: italic">KM</hi> and <hi rend="font-style: italic">LN</hi> (Fig. 5). +Join <hi rend="font-style: italic">O</hi> to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>. The joining lines cut out on the +sides of the quadrangle four points, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">S</hi>. Consider +the quadrangle <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">O</hi>. One pair of opposite sides +<pb n="19" /><anchor id="Pg19" /> +passes through <hi rend="font-style: italic">A</hi>, one through <hi rend="font-style: italic">C</hi>, and one remaining side +through <hi rend="font-style: italic">D</hi>; therefore the other remaining side must +pass through <hi rend="font-style: italic">B</hi>. Similarly, <hi rend="font-style: italic">RS</hi> passes through <hi rend="font-style: italic">B</hi> and +<hi rend="font-style: italic">PS</hi> and <hi rend="font-style: italic">QR</hi> pass +through <hi rend="font-style: italic">D</hi>. The +quadrangle <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, +<hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">S</hi> therefore +has two opposite +sides through <hi rend="font-style: italic">B</hi>, +two through <hi rend="font-style: italic">D</hi>, +and the remaining +pair through +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>. <hi rend="font-style: italic">A</hi> and +<hi rend="font-style: italic">C</hi> are thus harmonic conjugates with respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>. +We may sum up the discussion, therefore, as follows:</p></div> + +<div> +<index index="toc" level1="30. Harmonic conjugates" /><index index="pdf" /> +<head></head><p><anchor id="p30" /><hi rend="font-weight: bold">30.</hi> If <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are harmonic conjugates with respect +to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>, then <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are harmonic conjugates with +respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>.</p></div> + +<div> +<index index="toc" level1="31. Importance of the notion of four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p31" /><hi rend="font-weight: bold">31. Importance of the notion.</hi> 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 <hi rend="font-style: italic">u</hi> to the four points that correspond to +them in any point-row <hi rend="font-style: italic">u'</hi> perspective to <hi rend="font-style: italic">u</hi>.</p> + +<p>To prove this statement we construct a quadrangle +<hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> such that <hi rend="font-style: italic">KL</hi> and <hi rend="font-style: italic">MN</hi> pass through <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">KN</hi> +and <hi rend="font-style: italic">LM</hi> through <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">LN</hi> through <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">KM</hi> through <hi rend="font-style: italic">D</hi>. +Take now any point <hi rend="font-style: italic">S</hi> not in the plane of the quadrangle +and construct the planes determined by <hi rend="font-style: italic">S</hi> and +all the seven lines of the figure. Cut across this set of +planes by another plane not passing through <hi rend="font-style: italic">S</hi>. This +plane cuts out on the set of seven planes another +<pb n="20" /><anchor id="Pg20" /> +quadrangle which determines four new harmonic points, +<hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi>, on the lines joining <hi rend="font-style: italic">S</hi> to <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>. But +<hi rend="font-style: italic">S</hi> may be taken as any point, since the original quadrangle +may be taken in any plane through <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>; +and, further, the points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi> are the intersection +of <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi> by any line. We have, then, the +remarkable theorem:</p></div> + +<div> +<index index="toc" level1="32. Projective invariance of four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p32" /><hi rend="font-weight: bold">32.</hi> <hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="33. Four harmonic lines" /><index index="pdf" /> +<head></head><p><anchor id="p33" /><hi rend="font-weight: bold">33. Four harmonic lines.</hi> We are now able to extend +the notion of harmonic elements to pencils of rays, and +indeed to axial pencils. For if we define <hi rend="font-style: italic">four harmonic +rays</hi> as four rays which pass through a point and which +pass one through each of four harmonic points, we have +the theorem</p> + +<p><hi rend="font-style: italic">Four harmonic lines are cut by any transversal in four +harmonic points.</hi></p></div> + +<div> +<index index="toc" level1="34. Four harmonic planes" /><index index="pdf" /> +<head></head><p><anchor id="p34" /><hi rend="font-weight: bold">34. Four harmonic planes.</hi> We also define <hi rend="font-style: italic">four harmonic +planes</hi> as four planes through a line which pass +one through each of four harmonic points, and we may +show that</p> + +<p><hi rend="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.</hi></p> + +<p>For let the planes α, β, γ, δ, which all pass through +the line <hi rend="font-style: italic">g</hi>, pass also through the four harmonic points +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, so that α passes through <hi rend="font-style: italic">A</hi>, etc. Then it is +clear that any plane π through <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> will cut out +four harmonic lines from the four planes, for they are +<pb n="21" /><anchor id="Pg21" /> +lines through the intersection <hi rend="font-style: italic">P</hi> of <hi rend="font-style: italic">g</hi> with the plane +π, and they pass through the given harmonic points +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>. Any other plane σ cuts <hi rend="font-style: italic">g</hi> in a point <hi rend="font-style: italic">S</hi> and +cuts α, β, γ, δ in four lines that meet π +in four points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi> lying on <hi rend="font-style: italic">PA</hi>, <hi rend="font-style: italic">PB</hi>, <hi rend="font-style: italic">PC</hi>, and <hi rend="font-style: italic">PD</hi> 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> +<index index="toc" level1="35. Summary of results" /><index index="pdf" /> +<head></head><p><anchor id="p35" /><hi rend="font-weight: bold">35.</hi> These results may be put together as follows:</p> + +<p><hi rend="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.</hi></p> + +<p>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> +<index index="toc" level1="36. Definition of projectivity" /><index index="pdf" /> +<head></head><p><anchor id="p36" /><hi rend="font-weight: bold">36. Definition of projectivity.</hi> <hi rend="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.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image06.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 6</head> +<figDesc>Figure 6</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="37. Correspondence between harmonic conjugates" /><index index="pdf" /> +<head></head><p><anchor id="p37" /><hi rend="font-weight: bold">37. Correspondence between harmonic conjugates.</hi> Given +four harmonic points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>; if we fix <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>, +then <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> vary together in a way that should be +thoroughly understood. To get a clear conception of +their relative motion we may fix the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> of +the quadrangle <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> (Fig. 6). Then, as <hi rend="font-style: italic">B</hi> describes +the point-row <hi rend="font-style: italic">AC</hi>, the point <hi rend="font-style: italic">N</hi> describes the point-row +<pb n="22" /><anchor id="Pg22" /> +<hi rend="font-style: italic">AM</hi> perspective to it. Projecting <hi rend="font-style: italic">N</hi> again from <hi rend="font-style: italic">C</hi>, we +get a point-row <hi rend="font-style: italic">K</hi> on <hi rend="font-style: italic">AL</hi> perspective to the point-row +<hi rend="font-style: italic">N</hi> and thus projective to the point-row <hi rend="font-style: italic">B</hi>. Project the +point-row <hi rend="font-style: italic">K</hi> from <hi rend="font-style: italic">M</hi> and we get a point-row <hi rend="font-style: italic">D</hi> on +<hi rend="font-style: italic">AC</hi> again, which is projective to the point-row <hi rend="font-style: italic">B</hi>. For +every point <hi rend="font-style: italic">B</hi> we have thus one and only one point +<hi rend="font-style: italic">D</hi>, 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 <hi rend="font-style: italic">B</hi> correspond to four harmonic points <hi rend="font-style: italic">D</hi>. +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 <hi rend="font-style: italic">B</hi> to <hi rend="font-style: italic">D</hi> will carry one back from +<hi rend="font-style: italic">D</hi> to <hi rend="font-style: italic">B</hi> again. This special property will receive further +study in the chapter on Involution.</p></div> + +<div> +<index index="toc" level1="38. Separation of harmonic conjugates" /><index index="pdf" /> +<head></head><p><anchor id="p38" /><hi rend="font-weight: bold">38.</hi> It is seen that as <hi rend="font-style: italic">B</hi> approaches <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">D</hi> also approaches +<hi rend="font-style: italic">A</hi>. As <hi rend="font-style: italic">B</hi> moves from <hi rend="font-style: italic">A</hi> toward <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> moves +from <hi rend="font-style: italic">A</hi> in the opposite direction, passing through the +point at infinity on the line <hi rend="font-style: italic">AC</hi>, and returns on the +other side to meet <hi rend="font-style: italic">B</hi> at <hi rend="font-style: italic">C</hi> again. In other words, as <hi rend="font-style: italic">B</hi> +traverses <hi rend="font-style: italic">AC</hi>, <hi rend="font-style: italic">D</hi> traverses the rest of the line from <hi rend="font-style: italic">A</hi> to +<hi rend="font-style: italic">C</hi> through infinity. In all positions of <hi rend="font-style: italic">B</hi>, except at <hi rend="font-style: italic">A</hi> or +<hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are separated from each other by <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>.</p> + +<pb n="23" /><anchor id="Pg23" /></div> + +<div> +<index index="toc" level1="39. Harmonic conjugate of the point at infinity" /><index index="pdf" /> +<head></head><p><anchor id="p39" /><hi rend="font-weight: bold">39. Harmonic conjugate of the point at infinity.</hi> It is +natural to inquire what position of <hi rend="font-style: italic">B</hi> corresponds to the +infinitely distant position of <hi rend="font-style: italic">D</hi>. We have proved (§ 27) +that the particular quadrangle <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> 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 <hi rend="font-style: italic">L</hi> so that the triangle +<hi rend="font-style: italic">ALC</hi> is isosceles (Fig. 7). Since +<hi rend="font-style: italic">D</hi> is supposed to be at infinity, the +line <hi rend="font-style: italic">KM</hi> is parallel to <hi rend="font-style: italic">AC</hi>. Therefore +the triangles <hi rend="font-style: italic">KAC</hi> and <hi rend="font-style: italic">MAC</hi> +are equal, and the triangle <hi rend="font-style: italic">ANC</hi> is also isosceles. The +triangles <hi rend="font-style: italic">CNL</hi> and <hi rend="font-style: italic">ANL</hi> are therefore equal, and the line +<hi rend="font-style: italic">LB</hi> bisects the angle <hi rend="font-style: italic">ALC</hi>. <hi rend="font-style: italic">B</hi> is therefore the middle +point of <hi rend="font-style: italic">AC</hi>, and we have the theorem</p> + +<p><hi rend="font-style: italic">The harmonic conjugate of the middle point of <hi rend="font-style: italic">AC</hi> is at +infinity.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image07.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 7</head> +<figDesc>Figure 7</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="40. Projective theorems and metrical theorems. Linear +construction" /><index index="pdf" /> +<head></head><p><anchor id="p40" /><hi rend="font-weight: bold">40. Projective theorems and metrical theorems. Linear +construction.</hi> 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 +<pb n="24" /><anchor id="Pg24" /> +made with such an instrument we shall call a <hi rend="font-style: italic">linear</hi> +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> +<index index="toc" level1="41. Parallels and mid-points" /><index index="pdf" /> +<head></head><p><anchor id="p41" /><hi rend="font-weight: bold">41. Parallels and mid-points.</hi> 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 <hi rend="font-style: italic">linearly</hi> a line parallel to +that segment. Thus, given that <hi rend="font-style: italic">B</hi> is the middle point of +<hi rend="font-style: italic">AC</hi>, we may draw any two lines through <hi rend="font-style: italic">A</hi>, and any line +through <hi rend="font-style: italic">B</hi> cutting them in points <hi rend="font-style: italic">N</hi> and <hi rend="font-style: italic">L</hi>. Join <hi rend="font-style: italic">N</hi> and +<hi rend="font-style: italic">L</hi> to <hi rend="font-style: italic">C</hi> and get the points <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> on the two lines +through <hi rend="font-style: italic">A</hi>. Then <hi rend="font-style: italic">KM</hi> is parallel to <hi rend="font-style: italic">AC</hi>. <hi rend="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.</hi></p> + +<pb n="25" /><anchor id="Pg25" /></div> + +<div> +<index index="toc" level1="42. Division of segment into equal parts" /><index index="pdf" /> +<head></head><p><anchor id="p42" /><hi rend="font-weight: bold">42.</hi> It is not difficult to give a linear construction +for the problem to divide a given segment into <hi rend="font-style: italic">n</hi> equal +parts, given only a parallel to the segment. This is +simple enough when <hi rend="font-style: italic">n</hi> is a power of <hi rend="font-style: italic">2</hi>. For any other +number, such as <hi rend="font-style: italic">29</hi>, divide any segment on the line +parallel to <hi rend="font-style: italic">AC</hi> into <hi rend="font-style: italic">32</hi> equal parts, by a repetition of +the process just described. Take <hi rend="font-style: italic">29</hi> of these, and join +the first to <hi rend="font-style: italic">A</hi> and the last to <hi rend="font-style: italic">C</hi>. Let these joining lines +meet in <hi rend="font-style: italic">S</hi>. Join <hi rend="font-style: italic">S</hi> to all the other points. Other +problems, of a similar sort, are given at the end of +the chapter.</p></div> + +<div> +<index index="toc" level1="43. Numerical relations" /><index index="pdf" /> +<head></head><p><anchor id="p43" /><hi rend="font-weight: bold">43. Numerical relations.</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image08.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 8</head> +<figDesc>Figure 8</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="44. Algebraic formula connecting four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p44" /><hi rend="font-weight: bold">44.</hi> Let <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> be four +harmonic points (Fig. 8), and +let <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi> be four +harmonic lines. Assume a +line drawn through <hi rend="font-style: italic">B</hi> parallel +to <hi rend="font-style: italic">SD</hi>, meeting <hi rend="font-style: italic">SA</hi> in <hi rend="font-style: italic">A'</hi> and +<hi rend="font-style: italic">SC</hi> in <hi rend="font-style: italic">C'</hi>. Then <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, and the infinitely distant +point on <hi rend="font-style: italic">A'C'</hi> are four harmonic points, and therefore +<hi rend="font-style: italic">B</hi> is the middle point of the segment <hi rend="font-style: italic">A'C'</hi>. Then, since +<pb n="26" /><anchor id="Pg26" /> +the triangle <hi rend="font-style: italic">DAS</hi> is similar to the triangle <hi rend="font-style: italic">BAA'</hi>, we +may write the proportion</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AB : AD = BA' : SD.</hi> +</p> + +<p>Also, from the similar triangles <hi rend="font-style: italic">DSC</hi> and <hi rend="font-style: italic">BCC'</hi>, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">CD : CB = SD : B'C.</hi> +</p> + +<p>From these two proportions we have, remembering that +<hi rend="font-style: italic">BA' = BC'</hi>,</p> + +<p rend="text-align: center"> +<formula notation="tex">\[ +\frac{AB \cdot CD}{AD \cdot CB} = -1, +\]</formula> +</p> + +<p>the minus sign being given to the ratio on account of the +fact that <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are always separated from <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>, +so that one or three of the segments <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">CD</hi>, <hi rend="font-style: italic">AD</hi>, <hi rend="font-style: italic">CB</hi> +must be negative.</p></div> + +<div> +<index index="toc" level1="45. Further formulae" /><index index="pdf" /> +<head></head><p><anchor id="p45" /><hi rend="font-weight: bold">45.</hi> Writing the last equation in the form</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">CB : AB = -CD : AD,</hi> +</p> + +<p>and using the fundamental relation connecting three +points on a line,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">PR + RQ = PQ,</hi> +</p> + +<p>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 rend="text-align: center"> +<hi rend="font-style: italic">(CB - BA) : AB = -(CA - DA) : AD,</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">(AB - AC) : AB = (AC - AD) : AD;</hi> +</p> + +<p>so that <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">AC</hi>, and <hi rend="font-style: italic">AD</hi> 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 rend="text-align: center"> +<formula notation="tex">\[ +\frac{2}{AC} = \frac{1}{AB} + \frac{1}{AD}, +\]</formula> +</p> + +<p>which is convenient for the computation of the distance +<hi rend="font-style: italic">AD</hi> when <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">AC</hi> are given numerically.</p> + +<pb n="27" /><anchor id="Pg27" /></div> + +<div> +<index index="toc" level1="46. Anharmonic ratio" /><index index="pdf" /> +<head></head><p><anchor id="p46" /><hi rend="font-weight: bold">46. Anharmonic ratio.</hi> 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 <hi rend="font-style: italic">anharmonic ratio</hi>, or <hi rend="font-style: italic">cross ratio</hi>, which connects +any four points on a line.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p><hi rend="font-weight: bold">1</hi>. 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 <hi rend="font-style: italic">P</hi> draw +any two rays cutting the two lines in the points <hi rend="font-style: italic">AB'</hi> and +<hi rend="font-style: italic">A'B</hi>, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, lying on one of the given lines and <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, on the +other. Join <hi rend="font-style: italic">AA'</hi> and <hi rend="font-style: italic">BB'</hi>, and find their point of intersection +<hi rend="font-style: italic">S</hi>. Through <hi rend="font-style: italic">S</hi> draw any other ray, cutting the given +lines in <hi rend="font-style: italic">CC'</hi>. Join <hi rend="font-style: italic">BC'</hi> and <hi rend="font-style: italic">B'C</hi>, and obtain their point +of intersection <hi rend="font-style: italic">Q</hi>. <hi rend="font-style: italic">PQ</hi> is the desired line. Justify this +construction.</p> + +<p><hi rend="font-weight: bold">2.</hi> To draw through a given point <hi rend="font-style: italic">P</hi> a line which shall +meet two given lines in points <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>, equally distant from +<hi rend="font-style: italic">P</hi>. Justify the following construction: Join <hi rend="font-style: italic">P</hi> to the point +<hi rend="font-style: italic">S</hi> of intersection of the two given lines. Construct the +fourth harmonic of <hi rend="font-style: italic">PS</hi> with respect to the two given lines. +Draw through <hi rend="font-style: italic">P</hi> a line parallel to this line. This is the +required line.</p> + +<p><hi rend="font-weight: bold">3.</hi> Given a parallelogram in the same plane with a given +segment <hi rend="font-style: italic">AC</hi>, to construct linearly the middle point of <hi rend="font-style: italic">AC</hi>.</p> + +<pb n="28" /><anchor id="Pg28" /> + +<p><hi rend="font-weight: bold">4.</hi> 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><hi rend="font-weight: bold">5.</hi> Given the middle point of a line segment, to draw a +line parallel to the segment and passing through a given +point.</p> + +<p><hi rend="font-weight: bold">6.</hi> A line is drawn cutting the sides of a triangle <hi rend="font-style: italic">ABC</hi> in +the points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi> the point <hi rend="font-style: italic">A'</hi> lying on the side <hi rend="font-style: italic">BC</hi>, etc. +The harmonic conjugate of <hi rend="font-style: italic">A'</hi> with respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">C</hi> is +then constructed and called <hi rend="font-style: italic">A"</hi>. Similarly, <hi rend="font-style: italic">B"</hi> and <hi rend="font-style: italic">C"</hi> are +constructed. Show that <hi rend="font-style: italic">A"B"C"</hi> 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> + +<div rend="page-break-before: always"> +<pb n="29" /><anchor id="Pg29" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED +FUNDAMENTAL FORMS</head> + +<div> +<index index="toc" level1="47. Superposed fundamental forms. Self-corresponding +elements" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image09.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 9</head> +<figDesc>Figure 9</figDesc> +</figure></p> + +<p><anchor id="p47" /><hi rend="font-weight: bold">47. Superposed fundamental forms. Self-corresponding +elements.</hi> 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 <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are harmonic +conjugates with respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>, then the point-row +described by <hi rend="font-style: italic">B</hi> is projective to the point-row described +by <hi rend="font-style: italic">D</hi>, and that <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are self-corresponding +points. Consider more generally the case of two pencils +perspective to each other with axis of perspectivity <hi rend="font-style: italic">u'</hi> +(Fig. 9). Cut across them by a line <hi rend="font-style: italic">u</hi>. We get thus +two projective point-rows superposed on the same line +<hi rend="font-style: italic">u</hi>, and a moment's reflection serves to show that the +point <hi rend="font-style: italic">N</hi> of intersection <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> corresponds to itself +in the two point-rows. Also, the point <hi rend="font-style: italic">M</hi>, where <hi rend="font-style: italic">u</hi> +<pb n="30" /><anchor id="Pg30" /> +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 <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">N</hi> may +fall together if the line +joining the centers of the +pencils happens to pass +through the point of intersection +of the lines <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">u'</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image10.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 10</head> +<figDesc>Figure 10</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="48. Special case" /><index index="pdf" /> +<head></head><p><anchor id="p48" /><hi rend="font-weight: bold">48.</hi> 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 <hi rend="font-style: italic">S</hi> and always making the +same angle a with each other +(Fig. 10). They will cut out on +any line <hi rend="font-style: italic">u</hi> in the plane two point-rows +which are easily seen to be +projective. For, given any four +rays <hi rend="font-style: italic">SP</hi> which are harmonic, the +four corresponding rays <hi rend="font-style: italic">SP'</hi> must +also be harmonic, since they make +the same angles with each other. +Four harmonic points <hi rend="font-style: italic">P</hi> correspond, +therefore, to four harmonic points <hi rend="font-style: italic">P'</hi>. It is clear, +however, that no point <hi rend="font-style: italic">P</hi> can coincide with its corresponding +point <hi rend="font-style: italic">P'</hi>, for in that case the lines <hi rend="font-style: italic">PS</hi> and +<pb n="31" /><anchor id="Pg31" /> +<hi rend="font-style: italic">P'S</hi> would coincide, which is impossible if the angle +between them is to be constant.</p></div> + +<div> +<index index="toc" level1="49. Fundamental theorem. Postulate of continuity" /><index index="pdf" /> +<head></head><p><anchor id="p49" /><hi rend="font-weight: bold">49. Fundamental theorem. Postulate of continuity.</hi> +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><hi rend="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.</hi></p> + +<p>If three points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi>, are self-corresponding, +then the harmonic conjugate <hi rend="font-style: italic">D</hi> of <hi rend="font-style: italic">B</hi> with respect to <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">C</hi> must also correspond to itself. For four harmonic +points must always correspond to four harmonic points. +In the same way the harmonic conjugate of <hi rend="font-style: italic">D</hi> with +respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">C</hi> 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 <hi rend="font-style: italic">P</hi> lie between <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>. Construct +now <hi rend="font-style: italic">D</hi>, the fourth harmonic of <hi rend="font-style: italic">C</hi> with respect to <hi rend="font-style: italic">A</hi> and +<hi rend="font-style: italic">B</hi>. <hi rend="font-style: italic">D</hi> may coincide with <hi rend="font-style: italic">P</hi>, in which case the sequence +is closed; otherwise <hi rend="font-style: italic">P</hi> lies in the stretch <hi rend="font-style: italic">AD</hi> or in the +stretch <hi rend="font-style: italic">DB</hi>. If it lies in the stretch <hi rend="font-style: italic">DB</hi>, construct the +fourth harmonic of <hi rend="font-style: italic">C</hi> with respect to <hi rend="font-style: italic">D</hi> and <hi rend="font-style: italic">B</hi>. This +point <hi rend="font-style: italic">D'</hi> may coincide with <hi rend="font-style: italic">P</hi>, in which case, as before, +the sequence is closed. If <hi rend="font-style: italic">P</hi> lies in the stretch <hi rend="font-style: italic">DD'</hi>, +we construct the fourth harmonic of <hi rend="font-style: italic">C</hi> with respect +<pb n="32" /><anchor id="Pg32" /> +to <hi rend="font-style: italic">DD'</hi>, etc. In each step the region in which <hi rend="font-style: italic">P</hi> lies is +diminished, and the process may be continued until two +self-corresponding points are obtained on either side of +<hi rend="font-style: italic">P</hi>, and at distances from it arbitrarily small.</p> + +<p>We now assume, explicitly, the fundamental postulate +that the correspondence is <hi rend="font-style: italic">continuous</hi>, that is, that <hi rend="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.</hi> Suppose +now that <hi rend="font-style: italic">P</hi> is not a self-corresponding point, but corresponds +to a point <hi rend="font-style: italic">P'</hi> at a fixed distance <hi rend="font-style: italic">d</hi> from <hi rend="font-style: italic">P</hi>. +As noted above, we can find self-corresponding points +arbitrarily close to <hi rend="font-style: italic">P</hi>, and it appears, then, that we can +take a point <hi rend="font-style: italic">D</hi> as close to <hi rend="font-style: italic">P</hi> as we wish, and yet the +distance between the corresponding points <hi rend="font-style: italic">D'</hi> and <hi rend="font-style: italic">P'</hi> +approaches <hi rend="font-style: italic">d</hi> as a limit, and not zero, which contradicts +the postulate of continuity.</p></div> + +<div> +<index index="toc" level1="50. Extension of theorem to pencils of rays and planes" /><index index="pdf" /> +<head></head><p><anchor id="p50" /><hi rend="font-weight: bold">50.</hi> 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> +<index index="toc" level1="51. Projective point-rows having a self-corresponding +point in common" /><index index="pdf" /> +<head></head><p><anchor id="p51" /><hi rend="font-weight: bold">51. Projective point-rows having a self-corresponding +point in common.</hi> 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 +<pb n="33" /><anchor id="Pg33" /> +point. The converse is also true, and we have the very +important theorem:</p></div> + +<div> +<index index="toc" level1="52. Point-rows in perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p52" /><hi rend="font-weight: bold">52.</hi> <hi rend="font-style: italic">If in two protective point-rows, the point of intersection +corresponds to itself, then the point-rows are in +perspective position.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image11.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 11</head> +<figDesc>Figure 11</figDesc> +</figure></p> + +<p>Let the two point-rows be <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> (Fig. 11). Let +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, be corresponding points, and let +also the point <hi rend="font-style: italic">M</hi> of intersection of <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> correspond +to itself. Let <hi rend="font-style: italic">AA'</hi> and <hi rend="font-style: italic">BB'</hi> meet in the point <hi rend="font-style: italic">S</hi>. Take +<hi rend="font-style: italic">S</hi> as the center of two pencils, +one perspective to <hi rend="font-style: italic">u</hi> and the other +perspective to <hi rend="font-style: italic">u'</hi>. In these two +pencils <hi rend="font-style: italic">SA</hi> coincides with its corresponding +ray <hi rend="font-style: italic">SA'</hi>, <hi rend="font-style: italic">SB</hi> with its +corresponding ray <hi rend="font-style: italic">SB'</hi>, and <hi rend="font-style: italic">SM</hi> +with its corresponding ray <hi rend="font-style: italic">SM'</hi>. +The two pencils are thus identical, by the preceding +theorem, and any ray <hi rend="font-style: italic">SD</hi> must coincide with its corresponding +ray <hi rend="font-style: italic">SD'</hi>. Corresponding points of <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, +therefore, all lie on lines through the point <hi rend="font-style: italic">S</hi>.</p></div> + +<div> +<index index="toc" level1="53. Pencils in perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p53" /><hi rend="font-weight: bold">53.</hi> An entirely similar discussion shows that</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="54. Axial pencils in perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p54" /><hi rend="font-weight: bold">54.</hi> 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> +<index index="toc" level1="55. Point-row of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p55" /><hi rend="font-weight: bold">55. Point-row of the second order.</hi> The question naturally +arises, What is the locus of points of intersection +of corresponding rays of two projective pencils +<pb n="34" /><anchor id="Pg34" /> +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 <hi rend="font-style: italic">point-row of the +second order</hi>. For any line <hi rend="font-style: italic">u</hi> 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> +<index index="toc" level1="56. Degeneration of locus" /><index index="pdf" /> +<head></head><p><anchor id="p56" /><hi rend="font-weight: bold">56.</hi> 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> +<index index="toc" level1="57. Pencils of rays of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p57" /><hi rend="font-weight: bold">57. Pencils of rays of the second order.</hi> 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: <hi rend="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.</hi> For that reason the system of rays is called <hi rend="font-style: italic">a +pencil of rays of the second order.</hi></p></div> + +<div> +<index index="toc" level1="58. Degenerate case" /><index index="pdf" /> +<head></head><p><anchor id="p58" /><hi rend="font-weight: bold">58.</hi> 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 +<pb n="35" /><anchor id="Pg35" /> +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> +<index index="toc" level1="59. Cone of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p59" /><hi rend="font-weight: bold">59. Cone of the second order.</hi> The corresponding +theorems in space may easily be obtained by joining +the points and lines considered in the plane theorems +to a point <hi rend="font-style: italic">S</hi> in space. Two projective pencils give rise +to two projective axial pencils with axes intersecting. +Corresponding planes meet in lines which all pass +through <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">cone of the second +order</hi>, or <hi rend="font-style: italic">quadric cone</hi>, so called because every plane in +space not passing through <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">S</hi> of the lines which join the corresponding +points of the two point-rows. At most two +such planes may pass through any ray through <hi rend="font-style: italic">S</hi>. It +is called <hi rend="font-style: italic">a pencil of planes of the second order</hi>.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p><hi rend="font-weight: bold">1. </hi> A man <hi rend="font-style: italic">A</hi> moves along a straight road <hi rend="font-style: italic">u</hi>, and another +man <hi rend="font-style: italic">B</hi> moves along the same road and walks so as always +to keep sight of <hi rend="font-style: italic">A</hi> in a small mirror <hi rend="font-style: italic">M</hi> at the side of the +road. How many times will they come together, <hi rend="font-style: italic">A</hi> moving +always in the same direction along the road?</p> + +<pb n="36" /><anchor id="Pg36" /> + +<p>2. How many times would the two men in the first problem +see each other in two mirrors <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">N</hi> as they walk +along the road as before? (The planes of the two mirrors +are not necessarily parallel to <hi rend="font-style: italic">u</hi>.)</p> + +<p>3. As A moves along <hi rend="font-style: italic">u</hi>, trace the path of B so that the +two men may always see each other in the two mirrors.</p> + +<p>4. Two boys walk along two paths <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> 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>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>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>7. Two lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> revolve about two points <hi rend="font-style: italic">U</hi> and <hi rend="font-style: italic">U'</hi> +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>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> + +<div rend="page-break-before: always"> +<pb n="37" /><anchor id="Pg37" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER IV - POINT-ROWS OF THE SECOND ORDER</head> + +<div> +<index index="toc" level1="60. Point-row of the second order defined" /><index index="pdf" /> +<head></head><p><anchor id="p60" /><hi rend="font-weight: bold">60. Point-row of the second order defined.</hi> 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> +<index index="toc" level1="61. Tangent line" /><index index="pdf" /> +<head></head><p><anchor id="p61" /><hi rend="font-weight: bold">61. Tangent line.</hi> 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 <hi rend="font-style: italic">SS'</hi>, considered as a ray of <hi rend="font-style: italic">S</hi>, must +correspond some ray of <hi rend="font-style: italic">S'</hi> which meets it in <hi rend="font-style: italic">S'</hi>. <hi rend="font-style: italic">S'</hi>, +and by the same argument <hi rend="font-style: italic">S</hi>, is then a point where +corresponding rays meet. Any ray through <hi rend="font-style: italic">S</hi> will meet +it in one point besides <hi rend="font-style: italic">S</hi>, namely, the point <hi rend="font-style: italic">P</hi> where +it meets its corresponding ray. Now, by choosing the +ray through <hi rend="font-style: italic">S</hi> sufficiently close to the ray <hi rend="font-style: italic">SS'</hi>, the point +<hi rend="font-style: italic">P</hi> may be made to approach arbitrarily close to <hi rend="font-style: italic">S'</hi>, and +the ray <hi rend="font-style: italic">S'P</hi> may be made to differ in position from the +<pb n="38" /><anchor id="Pg38" /> +tangent line at <hi rend="font-style: italic">S'</hi> by as little as we please. We have, +then, the important theorem</p> + +<p><hi rend="font-style: italic">The ray at <hi rend="font-style: italic">S'</hi> which corresponds to the common ray <hi rend="font-style: italic">SS'</hi> +is tangent to the locus at <hi rend="font-style: italic">S'</hi>.</hi></p> + +<p>In the same manner the tangent at <hi rend="font-style: italic">S</hi> may be +constructed.</p></div> + +<div> +<index index="toc" level1="62. Determination of the locus" /><index index="pdf" /> +<head></head><p><anchor id="p62" /><hi rend="font-weight: bold">62. Determination of the locus.</hi> We now show that +<hi rend="font-style: italic">it is possible to assign arbitrarily the position of three +points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi>, on the locus (besides the points <hi rend="font-style: italic">S</hi> +and <hi rend="font-style: italic">S'</hi>); but, these three points being chosen, the locus is +completely determined.</hi></p></div> + +<div> +<index index="toc" level1="63. Restatement of the problem" /><index index="pdf" /> +<head></head><p><anchor id="p63" /><hi rend="font-weight: bold">63.</hi> This statement is equivalent to the following:</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="64. Solution of the fundamental problem" /><index index="pdf" /> +<head></head><p><anchor id="p64" /><hi rend="font-weight: bold">64.</hi> We proceed, then, to the solution of the fundamental</p> + +<p><hi rend="font-variant: small-caps">Problem</hi>: <hi rend="font-style: italic">Given three pairs of rays, <hi rend="font-style: italic">aa'</hi>, <hi rend="font-style: italic">bb'</hi>, and <hi rend="font-style: italic">cc'</hi>, +of two protective pencils, <hi rend="font-style: italic">S</hi> and <hi rend="font-style: italic">S'</hi>, to find the ray <hi rend="font-style: italic">d'</hi> of <hi rend="font-style: italic">S'</hi> +which corresponds to any ray <hi rend="font-style: italic">d</hi> of <hi rend="font-style: italic">S</hi>.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image12.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 12</head> +<figDesc>Figure 12</figDesc> +</figure></p> + +<p>Call <hi rend="font-style: italic">A</hi> the intersection of <hi rend="font-style: italic">aa'</hi>, <hi rend="font-style: italic">B</hi> the intersection of <hi rend="font-style: italic">bb'</hi>, +and <hi rend="font-style: italic">C</hi> the intersection of <hi rend="font-style: italic">cc'</hi> (Fig. 12). Join <hi rend="font-style: italic">AB</hi> by the +line <hi rend="font-style: italic">u</hi>, and <hi rend="font-style: italic">AC</hi> by the line <hi rend="font-style: italic">u'</hi>. Consider <hi rend="font-style: italic">u</hi> as a point-row +perspective to <hi rend="font-style: italic">S</hi>, and <hi rend="font-style: italic">u'</hi> as a point-row perspective +to <hi rend="font-style: italic">S'</hi>. <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are projectively related to each other, +since <hi rend="font-style: italic">S</hi> and <hi rend="font-style: italic">S'</hi> are, by hypothesis, so related. But their +point of intersection <hi rend="font-style: italic">A</hi> is a self-corresponding point, since +<hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">a'</hi> were supposed to be corresponding rays. It follows +(§ 52) that <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are in perspective position, +and that lines through corresponding points all pass +<pb n="39" /><anchor id="Pg39" /> +through a point <hi rend="font-style: italic">M</hi>, the center of perspectivity, the +position of which will be determined by any two such +lines. But the intersection of <hi rend="font-style: italic">a</hi> with <hi rend="font-style: italic">u</hi> and the intersection +of <hi rend="font-style: italic">c'</hi> with <hi rend="font-style: italic">u'</hi> are corresponding points on <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, +and the line joining them is clearly <hi rend="font-style: italic">c</hi> itself. Similarly, +<hi rend="font-style: italic">b'</hi> joins two corresponding points on <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, and so the +center <hi rend="font-style: italic">M</hi> of perspectivity of <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> is the intersection +of <hi rend="font-style: italic">c</hi> and <hi rend="font-style: italic">b'</hi>. To find <hi rend="font-style: italic">d'</hi> in <hi rend="font-style: italic">S'</hi> corresponding to a given +line <hi rend="font-style: italic">d</hi> of <hi rend="font-style: italic">S</hi> we note the point <hi rend="font-style: italic">L</hi> where <hi rend="font-style: italic">d</hi> meets <hi rend="font-style: italic">u</hi>. Join +<hi rend="font-style: italic">L</hi> to <hi rend="font-style: italic">M</hi> and get the point <hi rend="font-style: italic">N</hi> where this line meets <hi rend="font-style: italic">u'</hi>. +<hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> are corresponding points on <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, and <hi rend="font-style: italic">d'</hi> +must therefore pass through <hi rend="font-style: italic">N</hi>. The intersection <hi rend="font-style: italic">P</hi> of +<hi rend="font-style: italic">d</hi> and <hi rend="font-style: italic">d'</hi> is thus another point on the locus. In the same +manner any number of other points may be obtained.</p></div> + +<div> +<index index="toc" level1="65. Different constructions for the figure" /><index index="pdf" /> +<head></head><p><anchor id="p65" /><hi rend="font-weight: bold">65.</hi> The lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> might have been drawn in +any direction through <hi rend="font-style: italic">A</hi> (avoiding, of course, the line +<hi rend="font-style: italic">a</hi> for <hi rend="font-style: italic">u</hi> and the line <hi rend="font-style: italic">a'</hi> for <hi rend="font-style: italic">u'</hi>), and the center of perspectivity +<hi rend="font-style: italic">M</hi> would be easily obtainable; but the above +construction furnishes a simple and instructive figure. +An equally simple one is obtained by taking <hi rend="font-style: italic">a'</hi> for <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">a</hi> for <hi rend="font-style: italic">u'</hi>.</p> + +<pb n="40" /><anchor id="Pg40" /></div> + +<div> +<index index="toc" level1="66. Lines joining four points of the locus to a fifth" /><index index="pdf" /> +<head></head><p><anchor id="p66" /><hi rend="font-weight: bold">66. Lines joining four points of the locus to a fifth.</hi> +Suppose that the points <hi rend="font-style: italic">S</hi>, <hi rend="font-style: italic">S'</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">D</hi> are fixed, +and that four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">A<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">A<hi rend="vertical-align: sub">2</hi></hi>, and <hi rend="font-style: italic">A<hi rend="vertical-align: sub">3</hi></hi>, are taken on the +locus at the intersection with it of any four harmonic +rays through <hi rend="font-style: italic">B</hi>. These four harmonic rays give four +harmonic points, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">L<hi rend="vertical-align: sub">1</hi></hi> etc., on the fixed ray <hi rend="font-style: italic">SD</hi>. These, +in turn, project through the fixed point <hi rend="font-style: italic">M</hi> into four +harmonic points, <hi rend="font-style: italic">N</hi>, <hi rend="font-style: italic">N<hi rend="vertical-align: sub">1</hi></hi> etc., on the fixed line <hi rend="font-style: italic">DS'</hi>. +These last four harmonic points give four harmonic +rays <hi rend="font-style: italic">CA</hi>, <hi rend="font-style: italic">CA<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">CA<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">CA<hi rend="vertical-align: sub">3</hi></hi>. Therefore the four points <hi rend="font-style: italic">A</hi> +which project to <hi rend="font-style: italic">B</hi> in four harmonic rays also project +to <hi rend="font-style: italic">C</hi> in four harmonic rays. But <hi rend="font-style: italic">C</hi> may be any +point on the locus, and so we have the very important +theorem,</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="67. Restatement of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p67" /><hi rend="font-weight: bold">67.</hi> The theorem may also be stated thus:</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="68. Further important theorem" /><index index="pdf" /> +<head></head><p><anchor id="p68" /><hi rend="font-weight: bold">68.</hi> A further theorem of prime importance also +follows:</p> + +<p><hi rend="font-style: italic">Any two points on the locus may be taken as the centers +of two projective pencils which will generate the locus.</hi></p></div> + +<div> +<index index="toc" level1="69. Pascal's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p69" /><hi rend="font-weight: bold">69. Pascal's theorem.</hi> The points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, <hi rend="font-style: italic">S</hi>, and +<hi rend="font-style: italic">S'</hi> may thus be considered as chosen arbitrarily on the +locus, and the following remarkable theorem follows +at once.</p> + +<pb n="41" /><anchor id="Pg41" /> + +<p><hi rend="font-style: italic">Given six points, 1, 2, 3, 4, 5, 6, on the point-row of +the second order, if we call</hi></p> + +<p rend="text-align: center"><hi rend="font-style: italic">L the intersection of 12 with 45,</hi></p> +<p rend="text-align: center"><hi rend="font-style: italic">M the intersection of 23 with 56,</hi></p> +<p rend="text-align: center"><hi rend="font-style: italic">N the intersection of 34 with 61,</hi></p> + +<p><hi rend="font-style: italic">then <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, and <hi rend="font-style: italic">N</hi> are on a straight line.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image13.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 13</head> +<figDesc>Figure 13</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="70. Permutation of points in Pascal's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p70" /><hi rend="font-weight: bold">70.</hi> To get the notation to correspond to the figure, we +may take (Fig. 13) <hi rend="font-style: italic">A = 1</hi>, <hi rend="font-style: italic">B = 2</hi>, <hi rend="font-style: italic">S' = 3</hi>, <hi rend="font-style: italic">D = 4</hi>, <hi rend="font-style: italic">S = 5</hi>, and +<hi rend="font-style: italic">C = 6</hi>. If we make <hi rend="font-style: italic">A = 1</hi>, <hi rend="font-style: italic">C=2</hi>, <hi rend="font-style: italic">S=3</hi>, <hi rend="font-style: italic">D = 4</hi>, <hi rend="font-style: italic">S'=5</hi>, and. +<hi rend="font-style: italic">B = 6</hi>, the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> are interchanged, but the line +is left unchanged. +It is clear that one +point may be named +arbitrarily and the +other five named in +<hi rend="font-style: italic">5! = 120</hi> 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 <hi rend="font-style: italic">LMN</hi> 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 <hi rend="font-style: italic">60</hi>. 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 +<pb n="42" /><anchor id="Pg42" /> +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><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="71. Harmonic points on a point-row of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p71" /><hi rend="font-weight: bold">71. Harmonic points on a point-row of the second order.</hi> +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 <hi rend="font-style: italic">perspectively related</hi> to the pencil <hi rend="font-style: italic">S</hi> when +every ray on <hi rend="font-style: italic">S</hi> goes through the point on σ which +corresponds to it.</p></div> + +<div> +<index index="toc" level1="72. Determination of the locus" /><index index="pdf" /> +<head></head><p><anchor id="p72" /><hi rend="font-weight: bold">72. Determination of the locus.</hi> 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 +<pb n="43" /><anchor id="Pg43" /> +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 <hi rend="font-style: italic">S</hi> +of one pencil, and any other of the points for <hi rend="font-style: italic">S'</hi>, then, +besides the two pairs of corresponding rays determined +by the remaining two points, we have one more pair, +consisting of the tangent at <hi rend="font-style: italic">S</hi> and the ray <hi rend="font-style: italic">SS'</hi>. Similarly, +the curve is determined by three points and the +tangents at two of them.</p></div> + +<div> +<index index="toc" level1="73. Circles and conics as point-rows of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p73" /><hi rend="font-weight: bold">73. Circles and conics as point-rows of the second order.</hi> +It is not difficult to see that a circle is a point-row of +the second order. Indeed, take any point <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">S</hi> and <hi rend="font-style: italic">S'</hi>. 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 rend="text-align: center"> +<figure rend="w95" url="images/image14.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 14</head> +<figDesc>Figure 14</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="74. Conic through five points" /><index index="pdf" /> +<head></head><p><anchor id="p74" /><hi rend="font-weight: bold">74. Conic through five points.</hi> Pascal's theorem furnishes +an elegant solution of the problem of drawing a +conic through five given points. To construct a sixth +<pb n="44" /><anchor id="Pg44" /> +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 +<hi rend="font-style: italic">L = 12-45</hi> is obtainable at once; also +the point <hi rend="font-style: italic">N = 34-61</hi>. But <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> +determine Pascal's line, and the intersection +of 23 with 56 must be on +this line. Intersect, then, the line <hi rend="font-style: italic">LN</hi> +with 23 and obtain the point <hi rend="font-style: italic">M</hi>. Join +<hi rend="font-style: italic">M</hi> to 5 and intersect with 61 for the desired point 6.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image15.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 15</head> +<figDesc>Figure 15</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="75. Tangent to a conic" /><index index="pdf" /> +<head></head><p><anchor id="p75" /><hi rend="font-weight: bold">75. Tangent to a conic.</hi> 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 +<hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> are obtained as usual, +and the intersection of 34 with <hi rend="font-style: italic">LM</hi> +gives <hi rend="font-style: italic">N</hi>. Join <hi rend="font-style: italic">N</hi> to the point 1 for +the desired tangent at that point.</p></div> + +<div> +<index index="toc" level1="76. Inscribed quadrangle" /><index index="pdf" /> +<head></head><p><anchor id="p76" /><hi rend="font-weight: bold">76. Inscribed quadrangle.</hi> Two pairs of vertices may +coalesce, giving an inscribed quadrangle. Pascal's theorem +gives for this case the very important theorem</p> + +<p><hi rend="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.</hi></p> + +<pb n="45" /><anchor id="Pg45" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image16.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 16</head> +<figDesc>Figure 16</figDesc> +</figure></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image17.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 17</head> +<figDesc>Figure 17</figDesc> +</figure></p> + +<p>For let the vertices be <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">D</hi>, and call the +vertex <hi rend="font-style: italic">A</hi> the point 1, 6; <hi rend="font-style: italic">B</hi>, the point 2; <hi rend="font-style: italic">C</hi>, the point +3, 4; and <hi rend="font-style: italic">D</hi>, the point 5 (Fig. 16). Pascal's theorem then +indicates that +<hi rend="font-style: italic">L = AB-CD</hi>, +<hi rend="font-style: italic">M = AD-BC</hi>, +and <hi rend="font-style: italic">N</hi>, which +is the intersection +of the +tangents at <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">C</hi>, are all +on a straight +line <hi rend="font-style: italic">u</hi>. But +if we were to +call <hi rend="font-style: italic">A</hi> the point 2, <hi rend="font-style: italic">B</hi> the point 6, 1, <hi rend="font-style: italic">C</hi> the point 5, and +<hi rend="font-style: italic">D</hi> the point 4, 3, then the intersection <hi rend="font-style: italic">P</hi> of the tangents +at <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are also on this same +line <hi rend="font-style: italic">u</hi>. Thus <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>, and <hi rend="font-style: italic">P</hi> 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> +<index index="toc" level1="77. Inscribed triangle" /><index index="pdf" /> +<head></head><p><anchor id="p77" /><hi rend="font-weight: bold">77. Inscribed triangle.</hi> 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><hi rend="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.</hi></p> + +<pb n="46" /><anchor id="Pg46" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image18.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 18</head> +<figDesc>Figure 18</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="78. Degenerate conic" /><index index="pdf" /> +<head></head><p><anchor id="p78" /><hi rend="font-weight: bold">78. Degenerate conic.</hi> 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><hi rend="font-style: italic">If three points, +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, are +chosen on one +line, and three +points, <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, +<hi rend="font-style: italic">C'</hi>, are chosen on +another, then the +three points <hi rend="font-style: italic">L = AB'-A'B</hi>, <hi rend="font-style: italic">M = BC'-B'C</hi>, <hi rend="font-style: italic">N = CA'-C'A</hi> +are all on a straight line.</hi></p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. In Fig. 12, select different lines <hi rend="font-style: italic">u</hi> and trace the locus +of the center of perspectivity <hi rend="font-style: italic">M</hi> of the lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>.</p> + +<p>2. Given four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, in the plane, construct +a fifth point <hi rend="font-style: italic">P</hi> such that the lines <hi rend="font-style: italic">PA</hi>, <hi rend="font-style: italic">PB</hi>, <hi rend="font-style: italic">PC</hi>, <hi rend="font-style: italic">PD</hi> shall be +four harmonic lines.</p> + +<p><hi rend="font-style: italic">Suggestion.</hi> Draw a line <hi rend="font-style: italic">a</hi> through the point <hi rend="font-style: italic">A</hi> such that the four +lines <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">AC</hi>, <hi rend="font-style: italic">AD</hi> are harmonic. Construct now a conic through +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">D</hi> having <hi rend="font-style: italic">a</hi> for a tangent at <hi rend="font-style: italic">A</hi>.</p> + +<p>3. Where are all the points <hi rend="font-style: italic">P</hi>, as determined in the +preceding question, to be found?</p> + +<p>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>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> + +<pb n="47" /><anchor id="Pg47" /> + +<p>6. Given three points on the conic, and the tangent at +two of them, to construct the conic.</p> + +<p>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>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>9. The tangents to a curve at its infinitely distant points +are called its <hi rend="font-style: italic">asymptotes</hi> 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>10. Given an asymptote and three finite points on the +conic, to determine the conic.</p> + +<p>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> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="48" /><anchor id="Pg48" /> +<head>CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER</head> + +<div> +<index index="toc" level1="79. Pencil of rays of the second order defined" /><index index="pdf" /> +<head></head><p><anchor id="p79" /><hi rend="font-weight: bold">79. Pencil of rays of the second order defined.</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image19.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 19</head> +<figDesc>Figure 19</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="80. Tangents to a circle" /><index index="pdf" /> +<head></head><p><anchor id="p80" /><hi rend="font-weight: bold">80. Tangents to a circle.</hi> 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, <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, to a circle, and let <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi> +be the points of contact (Fig. 19). Let now <hi rend="font-style: italic">t</hi> be any +third tangent with point of contact at <hi rend="font-style: italic">C</hi> and meeting <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">u'</hi> in <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> respectively. Join <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">P'</hi>, and +<hi rend="font-style: italic">C</hi> to <hi rend="font-style: italic">O</hi>, the center of the circle. Tangents from any +point to a circle are equal, and therefore the triangles +<hi rend="font-style: italic">POA</hi> and <hi rend="font-style: italic">POC</hi> are equal, as also are the triangles <hi rend="font-style: italic">P'OB</hi> +<pb n="49" /><anchor id="Pg49" /> +and <hi rend="font-style: italic">P'OC</hi>. Therefore the angle <hi rend="font-style: italic">POP'</hi> is constant, being +equal to half the constant angle <hi rend="font-style: italic">AOC + COB</hi>. This +being true, if we take any four harmonic points, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">2</hi></hi>, +<hi rend="font-style: italic">P<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">4</hi></hi>, on the line <hi rend="font-style: italic">u</hi>, they will project to <hi rend="font-style: italic">O</hi> in four +harmonic lines, and the tangents +to the circle from these four +points will meet <hi rend="font-style: italic">u'</hi> in four harmonic +points, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">4</hi></hi>, because +the lines from these points +to <hi rend="font-style: italic">O</hi> inclose the same angles as +the lines from the points <hi rend="font-style: italic">P<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">2</hi></hi>, +<hi rend="font-style: italic">P<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">4</hi></hi> on <hi rend="font-style: italic">u</hi>. The point-row on <hi rend="font-style: italic">u</hi> is therefore projective +to the point-row on <hi rend="font-style: italic">u'</hi>. 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> +<index index="toc" level1="81. Tangents to a conic" /><index index="pdf" /> +<head></head><p><anchor id="p81" /><hi rend="font-weight: bold">81. Tangents to a conic.</hi> 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> +<index index="toc" level1="82. Generating point-rows lines of the system" /><index index="pdf" /> +<head></head><p><anchor id="p82" /><hi rend="font-weight: bold">82.</hi> The point-rows <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are, themselves, lines of +the system, for to the common point of the two point-rows, +considered as a point of <hi rend="font-style: italic">u</hi>, must correspond some +point of <hi rend="font-style: italic">u'</hi>, and the line joining these two corresponding +points is clearly <hi rend="font-style: italic">u'</hi> itself. Similarly for the line <hi rend="font-style: italic">u</hi>.</p></div> + +<div> +<index index="toc" level1="83. Determination of the pencil" /><index index="pdf" /> +<head></head><p><anchor id="p83" /><hi rend="font-weight: bold">83. Determination of the pencil.</hi> We now show that +<hi rend="font-style: italic">it is possible to assign arbitrarily three lines, <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, and <hi rend="font-style: italic">c</hi>, of +<pb n="50" /><anchor id="Pg50" /> +the system (besides the lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>); but if these three +lines are chosen, the system is completely determined.</hi></p> + +<p>This statement is equivalent to the following:</p> + +<p><hi rend="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.</hi></p> + +<p>We proceed, then, to the solution of the fundamental</p> + +<p><hi rend="font-variant: small-caps">Problem.</hi> <hi rend="font-style: italic">Given three pairs of points, <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, and +<hi rend="font-style: italic">CC'</hi>, of two projective point-rows <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, to find the point +<hi rend="font-style: italic">D'</hi> of <hi rend="font-style: italic">u'</hi> which corresponds to any given point <hi rend="font-style: italic">D</hi> of <hi rend="font-style: italic">u</hi>.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image20.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 20</head> +<figDesc>Figure 20</figDesc> +</figure></p> + +<p>On the line <hi rend="font-style: italic">a</hi>, joining <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, take two points, <hi rend="font-style: italic">S</hi> +and <hi rend="font-style: italic">S'</hi>, as centers of pencils perspective to <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> +respectively (Fig. 20). The figure +will be much simplified if we take +<hi rend="font-style: italic">S</hi> on <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">S'</hi> on <hi rend="font-style: italic">CC'</hi>. <hi rend="font-style: italic">SA</hi> and +<hi rend="font-style: italic">S'A'</hi> are corresponding rays of <hi rend="font-style: italic">S</hi> +and <hi rend="font-style: italic">S'</hi>, and the two pencils are +therefore in perspective position. +It is not difficult to see that the +axis of perspectivity <hi rend="font-style: italic">m</hi> is the line +joining <hi rend="font-style: italic">B'</hi> and <hi rend="font-style: italic">C</hi>. Given any point +<hi rend="font-style: italic">D</hi> on <hi rend="font-style: italic">u</hi>, to find the corresponding +point <hi rend="font-style: italic">D'</hi> on <hi rend="font-style: italic">u'</hi> we proceed as +follows: Join <hi rend="font-style: italic">D</hi> to <hi rend="font-style: italic">S</hi> and note +where the joining line meets <hi rend="font-style: italic">m</hi>. Join this point to <hi rend="font-style: italic">S'</hi>. +This last line meets <hi rend="font-style: italic">u'</hi> in the desired point <hi rend="font-style: italic">D'</hi>.</p> + +<p>We have now in this figure six lines of the system, +<hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi>, <hi rend="font-style: italic">d</hi>, <hi rend="font-style: italic">u</hi>, and <hi rend="font-style: italic">u'</hi>. Fix now the position of <hi rend="font-style: italic">u</hi>, <hi rend="font-style: italic">u'</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi>, and +<hi rend="font-style: italic">d</hi>, and take four lines of the system, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">4</hi></hi>, which +meet <hi rend="font-style: italic">b</hi> in four harmonic points. These points project to +<pb n="51" /><anchor id="Pg51" /> +<hi rend="font-style: italic">D</hi>, giving four harmonic points on <hi rend="font-style: italic">m</hi>. These again project +to <hi rend="font-style: italic">D'</hi>, giving four harmonic points on <hi rend="font-style: italic">c</hi>. It is thus clear +that the rays <hi rend="font-style: italic">a<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">4</hi></hi> cut out two projective point-rows +on any two lines of the system. Thus <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> 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> +<index index="toc" level1="84. Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p84" /><hi rend="font-weight: bold">84. Brianchon's theorem.</hi> From the figure also appears +a fundamental theorem due to Brianchon:</p> + +<p><hi rend="font-style: italic">If <hi rend="font-style: italic">1</hi>, <hi rend="font-style: italic">2</hi>, <hi rend="font-style: italic">3</hi>, <hi rend="font-style: italic">4</hi>, <hi rend="font-style: italic">5</hi>, <hi rend="font-style: italic">6</hi> are any six rays of a pencil of the +second order, then the lines <hi rend="font-style: italic">l = (12, 45)</hi>, <hi rend="font-style: italic">m = (23, 56)</hi>, +<hi rend="font-style: italic">n = (34, 61)</hi> all pass through a point.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image21.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 21</head> +<figDesc>Figure 21</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="85. Permutations of lines in Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p85" /><hi rend="font-weight: bold">85.</hi> To make the notation fit the figure (Fig. 21), make +<hi rend="font-style: italic">a=1</hi>, <hi rend="font-style: italic">b = 2</hi>, <hi rend="font-style: italic">u' = 3</hi>, <hi rend="font-style: italic">d = 4</hi>, <hi rend="font-style: italic">u = 5</hi>, <hi rend="font-style: italic">c = 6</hi>; or, interchanging +two of the lines, <hi rend="font-style: italic">a = 1</hi>, +<hi rend="font-style: italic">c = 2</hi>, <hi rend="font-style: italic">u = 3</hi>, <hi rend="font-style: italic">d = 4</hi>, <hi rend="font-style: italic">u' = 5</hi>, +<hi rend="font-style: italic">b = 6</hi>. Thus, by different +namings of the +lines, it appears that +not more than 60 different +<hi rend="font-style: italic">Brianchon points</hi> +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><hi rend="font-style: italic">The three lines joining the three pairs of opposite vertices +of a hexagon circumscribed about a conic meet in a point.</hi></p></div> + +<div> +<index index="toc" level1="86. Construction of the penvil by Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p86" /><hi rend="font-weight: bold">86. Construction of the pencil by Brianchon's theorem.</hi> +Brianchon's theorem furnishes a ready method of determining +a sixth line of the pencil of rays of the second +<pb n="52" /><anchor id="Pg52" /> +order when five are given. Thus, select a point in line +1 and suppose that line 6 is to pass through it. Then +<hi rend="font-style: italic">l = (12, 45)</hi>, <hi rend="font-style: italic">n = (34, 61)</hi>, and the line <hi rend="font-style: italic">m = (23, 56)</hi> must +pass through <hi rend="font-style: italic">(l, n)</hi>. Then <hi rend="font-style: italic">(23, ln)</hi> meets 5 in a point of +the required sixth line.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image22.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 22</head> +<figDesc>Figure 22</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="87. Point of contact +of a tangent to a conic" /><index index="pdf" /> +<head></head><p><anchor id="p87" /><hi rend="font-weight: bold">87. Point of contact +of a tangent to a conic.</hi> +If the line 2 approach +as a limiting position the +line 1, then the intersection +<hi rend="font-style: italic">(1, 2)</hi> 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 <hi rend="font-style: italic">1=6</hi>. +Draw <hi rend="font-style: italic">l = (12,45)</hi>, <hi rend="font-style: italic">m =(23,56)</hi>; +then <hi rend="font-style: italic">(34, lm)</hi> meets 1 in the +required point of contact <hi rend="font-style: italic">T</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image23.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 23</head> +<figDesc>Figure 23</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="88. Circumscribed quadrilateral" /><index index="pdf" /> +<head></head><p><anchor id="p88" /><hi rend="font-weight: bold">88. Circumscribed quadrilateral.</hi> +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><hi rend="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.</hi> The +consequences of this theorem will be deduced later.</p> + +<pb n="53" /><anchor id="Pg53" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image24.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 24</head> +<figDesc>Figure 24</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="89. Circumscribed triangle" /><index index="pdf" /> +<head></head><p><anchor id="p89" /><hi rend="font-weight: bold">89. Circumscribed triangle.</hi> The hexagon may further +degenerate into a triangle, giving the theorem (Fig. 24) +<hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="90. Use of Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p90" /><hi rend="font-weight: bold">90.</hi> Brianchon's theorem may +also be used to solve the following +problems:</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="91. Harmonic tangents" /><index index="pdf" /> +<head></head><p><anchor id="p91" /><hi rend="font-weight: bold">91. Harmonic tangents.</hi> 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><hi rend="font-style: italic">Four tangents to a conic are said to be harmonic when +they meet every other tangent in four harmonic points.</hi></p></div> + +<div> +<index index="toc" level1="92. Projectivity and perspectivity" /><index index="pdf" /> +<head></head><p><anchor id="p92" /><hi rend="font-weight: bold">92. Projectivity and perspectivity.</hi> 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 <hi rend="font-style: italic">perspectively related</hi> 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> + +<pb n="54" /><anchor id="Pg54" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image25.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 25</head> +<figDesc>Figure 25</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="93. Degenerate case" /><index index="pdf" /> +<head></head><p><anchor id="p93" /><hi rend="font-weight: bold">93.</hi> 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><hi rend="font-style: italic">If <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi> are three lines +through a point <hi rend="font-style: italic">S</hi>, and <hi rend="font-style: italic">a'</hi>, <hi rend="font-style: italic">b'</hi>, +<hi rend="font-style: italic">c'</hi> are three lines through another +point <hi rend="font-style: italic">S'</hi>, then the lines +<hi rend="font-style: italic">l = (ab', a'b)</hi>, <hi rend="font-style: italic">m = (bc', b'c)</hi>, +and <hi rend="font-style: italic">n = (ca', c'a)</hi> all meet in +a point.</hi></p></div> + +<div> +<index index="toc" level1="94. Law of duality" /><index index="pdf" /> +<head></head><p><anchor id="p94" /><hi rend="font-weight: bold">94. Law of duality.</hi> 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 <hi rend="font-style: italic">law of duality</hi> is based +will be developed in the next chapter.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Given four lines in the plane, to construct another +which shall meet them in four harmonic points.</p> + +<p>2. Where are all such lines found?</p> + +<p>3. Given any five lines in the plane, construct on each +the point of contact with the conic tangent to them all.</p> + +<pb n="55" /><anchor id="Pg55" /> + +<p>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>5. Given three lines and the point of contact on two of +them, to construct the conic.</p> + +<p>6. Given four lines and the line at infinity, to construct +the conic.</p> + +<p>7. Given three lines and the line at infinity, together +with the point of contact at infinity, to construct the conic.</p> + +<p>8. Given three lines, two of which are asymptotes, to +construct the conic.</p> + +<p>9. Given five tangents to a conic, to draw a tangent +which shall be parallel to any one of them.</p> + +<p>10. The lines <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi> are drawn parallel to each other. +The lines <hi rend="font-style: italic">a'</hi>, <hi rend="font-style: italic">b'</hi>, <hi rend="font-style: italic">c'</hi> are also drawn parallel to each other. +Show why the lines (<hi rend="font-style: italic">ab'</hi>, <hi rend="font-style: italic">a'b</hi>), (<hi rend="font-style: italic">bc'</hi>, <hi rend="font-style: italic">b'c</hi>), (<hi rend="font-style: italic">ca'</hi>, <hi rend="font-style: italic">c'a</hi>) meet in a +point. (In problems 6 to 10 inclusive, parallel lines are to +be drawn.)</p> +</div> +</div> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="56" /><anchor id="Pg56" /> +<head>CHAPTER VI - POLES AND POLARS</head> + +<div> +<index index="toc" level1="95. Inscribed and circumscribed quadrilaterals" /><index index="pdf" /> +<head></head><p><anchor id="p95" /><hi rend="font-weight: bold">95. Inscribed and circumscribed quadrilaterals.</hi> The +following theorems have been noted as special cases of +Pascal's and Brianchon's theorems:</p> + +<p><hi rend="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.</hi></p> + +<p><hi rend="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.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image26.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 26</head> +<figDesc>Figure 26</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="96. Definition of the polar line of a point" /><index index="pdf" /> +<head></head><p><anchor id="p96" /><hi rend="font-weight: bold">96. Definition of the polar line of a point.</hi> Consider +the quadrilateral <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> inscribed in the conic +(Fig. 26). It +determines the +four harmonic +points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, +<hi rend="font-style: italic">D</hi> which project +from <hi rend="font-style: italic">N</hi> in to +the four harmonic +points <hi rend="font-style: italic">M</hi>, +<hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">O</hi>. Now +the tangents at <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> meet in <hi rend="font-style: italic">P</hi>, a point on the +line <hi rend="font-style: italic">AB</hi>. The line <hi rend="font-style: italic">AB</hi> is thus determined entirely by +<pb n="57" /><anchor id="Pg57" /> +the point <hi rend="font-style: italic">O</hi>. For if we draw any line through it, meeting +the conic in <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi>, and construct the harmonic +conjugate <hi rend="font-style: italic">B</hi> of <hi rend="font-style: italic">O</hi> with respect to <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi>, and also +the two tangents at <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> which meet in the point +<hi rend="font-style: italic">P</hi>, then <hi rend="font-style: italic">BP</hi> is the line in question. It thus appears +that the line <hi rend="font-style: italic">LON</hi> may be any line whatever through <hi rend="font-style: italic">O</hi>; +and since <hi rend="font-style: italic">D</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">N</hi> are four harmonic points, we +may describe the line <hi rend="font-style: italic">AB</hi> as the locus of points which +are harmonic conjugates of <hi rend="font-style: italic">O</hi> with respect to the two +points where any line through <hi rend="font-style: italic">O</hi> meets the curve.</p></div> + +<div> +<index index="toc" level1="97. Further defining properties" /><index index="pdf" /> +<head></head><p><anchor id="p97" /><hi rend="font-weight: bold">97.</hi> Furthermore, since the tangents at <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> meet +on this same line, it appears as the locus of intersections +of pairs of tangents drawn at the extremities of chords +through <hi rend="font-style: italic">O</hi>.</p></div> + +<div> +<index index="toc" level1="98. Definition of the pole of a line" /><index index="pdf" /> +<head></head><p><anchor id="p98" /><hi rend="font-weight: bold">98.</hi> This important line, which is completely determined +by the point <hi rend="font-style: italic">O</hi>, is called the <hi rend="font-style: italic">polar</hi> of <hi rend="font-style: italic">O</hi> with +respect to the conic; and the point <hi rend="font-style: italic">O</hi> is called the <hi rend="font-style: italic">pole</hi> +of the line with respect to the conic.</p></div> + +<div> +<index index="toc" level1="99. Fundamental theorem of poles and polars" /><index index="pdf" /> +<head></head><p><anchor id="p99" /><hi rend="font-weight: bold">99.</hi> If a point <hi rend="font-style: italic">B</hi> is on the polar of <hi rend="font-style: italic">O</hi>, then it is harmonically +conjugate to <hi rend="font-style: italic">O</hi> with respect to the two intersections +<hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> of the line <hi rend="font-style: italic">BC</hi> with the conic. But +for the same reason <hi rend="font-style: italic">O</hi> is on the polar of <hi rend="font-style: italic">B</hi>. We have, +then, the fundamental theorem</p> + +<p><hi rend="font-style: italic">If one point lies on the polar of a second, then the +second lies on the polar of the first.</hi></p></div> + +<div> +<index index="toc" level1="100. Conjugate points and lines" /><index index="pdf" /> +<head></head><p><anchor id="p100" /><hi rend="font-weight: bold">100. Conjugate points and lines.</hi> Such a pair of points +are said to be <hi rend="font-style: italic">conjugate</hi> with respect to the conic. Similarly, +lines are said to be <hi rend="font-style: italic">conjugate</hi> to each other with +respect to the conic if one, and consequently each, +passes through the pole of the other.</p> + +<pb n="58" /><anchor id="Pg58" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image27.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 27</head> +<figDesc>Figure 27</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="101. Construction of the polar line of a given point" /><index index="pdf" /> +<head></head><p><anchor id="p101" /><hi rend="font-weight: bold">101. Construction of the polar line of a given point.</hi> +Given a point <hi rend="font-style: italic">P</hi>, if it is within the conic (that is, if no +tangents may be drawn from <hi rend="font-style: italic">P</hi> 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 <hi rend="font-style: italic">P</hi> is +outside the conic, we may draw the two tangents and +construct the chord of contact (Fig. 27).</p></div> + +<div> +<index index="toc" level1="102. Self-polar triangle" /><index index="pdf" /> +<head></head><p><anchor id="p102" /><hi rend="font-weight: bold">102. Self-polar triangle.</hi> In Fig. 26 it is not difficult +to see that <hi rend="font-style: italic">AOC</hi> is a <hi rend="font-style: italic">self-polar</hi> triangle, that is, each +vertex is the pole of the opposite side. For <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">K</hi> +are four harmonic points, and they project to <hi rend="font-style: italic">C</hi> in four +harmonic rays. The line <hi rend="font-style: italic">CO</hi>, therefore, meets the line +<hi rend="font-style: italic">AMN</hi> in a point on the polar of <hi rend="font-style: italic">A</hi>, being separated from +<hi rend="font-style: italic">A</hi> harmonically by the points <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">N</hi>. Similarly, the +line <hi rend="font-style: italic">CO</hi> meets <hi rend="font-style: italic">KL</hi> in a point on the polar of <hi rend="font-style: italic">A</hi>, and +therefore <hi rend="font-style: italic">CO</hi> is the polar of <hi rend="font-style: italic">A</hi>. Similarly, <hi rend="font-style: italic">OA</hi> is the +polar of <hi rend="font-style: italic">C</hi>, and therefore <hi rend="font-style: italic">O</hi> is the pole of <hi rend="font-style: italic">AC</hi>.</p></div> + +<div> +<index index="toc" level1="103. Pole and polar projectively related" /><index index="pdf" /> +<head></head><p><anchor id="p103" /><hi rend="font-weight: bold">103. Pole and polar projectively related.</hi> Another very +important theorem comes directly from Fig. 26.</p> + +<p><hi rend="font-style: italic">As a point <hi rend="font-style: italic">A</hi> 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 <hi rend="font-style: italic">A</hi>.</hi></p> + +<p>For, fix the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> and let the point <hi rend="font-style: italic">A</hi> move +along the line <hi rend="font-style: italic">AQ</hi>; then the point-row <hi rend="font-style: italic">A</hi> is projective +to the pencil <hi rend="font-style: italic">LK</hi>, and since <hi rend="font-style: italic">K</hi> moves along the conic, +the pencil <hi rend="font-style: italic">LK</hi> is projective to the pencil <hi rend="font-style: italic">NK</hi>, which in +turn is projective to the point-row <hi rend="font-style: italic">C</hi>, which, finally, is +projective to the pencil <hi rend="font-style: italic">OC</hi>, which is the polar of <hi rend="font-style: italic">A</hi>.</p> + +<pb n="59" /><anchor id="Pg59" /></div> + +<div> +<index index="toc" level1="104. Duality" /><index index="pdf" /> +<head></head><p><anchor id="p104" /><hi rend="font-weight: bold">104. Duality.</hi> 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 <hi rend="font-style: italic">measuring</hi>. If, therefore, we call any +theorem that has to do with the line at infinity or with +<pb n="60" /><anchor id="Pg60" /> +the measurement of angles a <hi rend="font-style: italic">metrical</hi> theorem, and any +other kind a <hi rend="font-style: italic">projective</hi> theorem, we may put the case +as follows:</p> + +<p><hi rend="font-style: italic">Any projective theorem involves another theorem, dual to +it, obtainable by interchanging everywhere the words 'point' +and 'line.'</hi></p></div> + +<div> +<index index="toc" level1="105. Self-dual theorems" /><index index="pdf" /> +<head></head><p><anchor id="p105" /><hi rend="font-weight: bold">105. Self-dual theorems.</hi> The theorems of this chapter +will be found, upon examination, to be <hi rend="font-style: italic">self-dual</hi>; +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> +<index index="toc" level1="106. Other correspondences" /><index index="pdf" /> +<head></head><p><anchor id="p106" /><hi rend="font-weight: bold">106.</hi> 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 <hi rend="font-style: italic">similarity</hi> +leaves ratios between corresponding segments unaltered. +The above statements apply only to the particular one-to-one +correspondence considered.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Given a quadrilateral, construct the quadrangle polar +to it with respect to a given conic.</p> + +<p>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> + +<pb n="61" /><anchor id="Pg61" /> + +<p>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>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>5. Dualize problems 3 and 4.</p> + +<p>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>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><hi rend="font-style: italic">Suggestion.</hi> Replace the given conic by a pair of protective pencils.</p> + +<p>8. Show that the poles of the tangents of one conic with +respect to another lie on a conic.</p> + +<p>9. The polar of a point <hi rend="font-style: italic">A</hi> with respect to one conic is <hi rend="font-style: italic">a</hi>, +and the pole of <hi rend="font-style: italic">a</hi> with respect to another conic is <hi rend="font-style: italic">A'</hi>. Show +that as <hi rend="font-style: italic">A</hi> travels along a line, <hi rend="font-style: italic">A'</hi> also travels along another +line. In general, if <hi rend="font-style: italic">A</hi> describes a curve of degree <hi rend="font-style: italic">n</hi>, show +that <hi rend="font-style: italic">A'</hi> describes another curve of the same degree <hi rend="font-style: italic">n</hi>. (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> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="62" /><anchor id="Pg62" /> + +<head>CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS</head> + +<div> +<index index="toc" level1="107. Diameters. Center" /><index index="pdf" /> +<head></head><p><anchor id="p107" /><hi rend="font-weight: bold">107. Diameters. Center.</hi> 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>The polar line of an infinitely distant point is called +a <hi rend="font-style: italic">diameter</hi>, and the pole of the infinitely distant line is +called the <hi rend="font-style: italic">center</hi>, of the conic.</p></div> + +<div> +<index index="toc" level1="108. Various theorems" /><index index="pdf" /> +<head></head><p><anchor id="p108" /><hi rend="font-weight: bold">108.</hi> From the harmonic properties of poles and polars,</p> + +<p><hi rend="font-style: italic">The center bisects all chords through it (§ 39).</hi></p> + +<p><hi rend="font-style: italic">Every diameter passes through the center.</hi></p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="109. Conjugate diameters" /><index index="pdf" /> +<head></head><p><anchor id="p109" /><hi rend="font-weight: bold">109. Conjugate diameters.</hi> We have already defined +conjugate lines as lines which pass each through the +pole of the other (§ 100).</p> + +<p><hi rend="font-style: italic">Any diameter bisects all chords parallel to its conjugate.</hi></p> + +<p><hi rend="font-style: italic">The tangents at the extremities of any diameter are +parallel, and parallel to the conjugate diameter.</hi></p> + +<p><hi rend="font-style: italic">Diameters parallel to the sides of a circumscribed parallelogram +are conjugate.</hi></p> + +<p>All these theorems are easy exercises for the student.</p> + +<pb n="63" /><anchor id="Pg63" /></div> + +<div> +<index index="toc" level1="110. Classification of conics" /><index index="pdf" /> +<head></head><p><anchor id="p110" /><hi rend="font-weight: bold">110. Classification of conics.</hi> 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 <hi rend="font-style: italic">hyperbola</hi>; if it has no point in +common with the infinitely distant line, it is called an +<hi rend="font-style: italic">ellipse</hi>; if it is tangent to the line at infinity, it is called +a <hi rend="font-style: italic">parabola</hi>.</p></div> + +<div> +<index index="toc" level1="111. Asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p111" /><hi rend="font-weight: bold">111.</hi> <hi rend="font-style: italic">In a hyperbola the center is outside the curve</hi> +(§ 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 <hi rend="font-style: italic">asymptotes</hi> of the curve. The +ellipse and the parabola have no asymptotes.</p></div> + +<div> +<index index="toc" level1="112. Various theorems" /><index index="pdf" /> +<head></head><p><anchor id="p112" /><hi rend="font-weight: bold">112.</hi> <hi rend="font-style: italic">The center of the parabola is at infinity, and therefore +all its diameters are parallel,</hi> for the pole of a tangent +line is the point of contact.</p> + +<p><hi rend="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.</hi></p> + +<p><hi rend="font-style: italic">The center of an ellipse is within the curve.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image28.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 28</head> +<figDesc>Figure 28</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="113. Theorems concerning asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p113" /><hi rend="font-weight: bold">113. Theorems concerning asymptotes.</hi> 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 <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">B</hi> (Fig. 28). If, then, <hi rend="font-style: italic">O</hi> is the intersection of the +asymptotes,—and therefore the center of the curve,— +<pb n="64" /><anchor id="Pg64" /> +then the triangle <hi rend="font-style: italic">OAB</hi> is circumscribed about the curve. +By the theorem just quoted, the line through <hi rend="font-style: italic">A</hi> parallel +to <hi rend="font-style: italic">OB</hi>, the line through <hi rend="font-style: italic">B</hi> parallel to <hi rend="font-style: italic">OA</hi>, and the +line <hi rend="font-style: italic">OP</hi> through the point of +contact of the tangent <hi rend="font-style: italic">AB</hi> +all meet in a point <hi rend="font-style: italic">C</hi>. But +<hi rend="font-style: italic">OACB</hi> is a parallelogram, and +<hi rend="font-style: italic">PA = PB</hi>. Therefore</p> + +<p><hi rend="font-style: italic">The asymptotes cut off on +each tangent a segment which is +bisected by the point of contact.</hi></p></div> + +<div> +<index index="toc" level1="114. Asymptotes and conjugate diameters" /><index index="pdf" /> +<head></head><p><anchor id="p114" /><hi rend="font-weight: bold">114.</hi> If we draw a line <hi rend="font-style: italic">OQ</hi> +parallel to <hi rend="font-style: italic">AB</hi>, then <hi rend="font-style: italic">OP</hi> and <hi rend="font-style: italic">OQ</hi> are conjugate diameters, +since <hi rend="font-style: italic">OQ</hi> is parallel to the tangent at the point +where <hi rend="font-style: italic">OP</hi> meets the curve. Then, since <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">B</hi>, and +the point at infinity on <hi rend="font-style: italic">AB</hi> are four harmonic points, +we have the theorem</p> + +<p><hi rend="font-style: italic">Conjugate diameters of the hyperbola are harmonic +conjugates with respect to the asymptotes.</hi></p></div> + +<div> +<index index="toc" level1="115. Segments cut off on a chord by hyperbola and its asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p115" /><hi rend="font-weight: bold">115.</hi> The chord <hi rend="font-style: italic">A"B"</hi>, parallel to the diameter <hi rend="font-style: italic">OQ</hi>, is +bisected at <hi rend="font-style: italic">P'</hi> by the conjugate diameter <hi rend="font-style: italic">OP</hi>. If the +chord <hi rend="font-style: italic">A"B"</hi> meet the asymptotes in <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, then <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">P'</hi>, <hi rend="font-style: italic">B'</hi>, +and the point at infinity are four harmonic points, and +therefore <hi rend="font-style: italic">P'</hi> is the middle point of <hi rend="font-style: italic">A'B'</hi>. Therefore +<hi rend="font-style: italic">A'A" = B'B"</hi> and we have the theorem</p> + +<p><hi rend="font-style: italic">The segments cut off on any chord between the hyperbola +and its asymptotes are equal.</hi></p></div> + +<div> +<index index="toc" level1="116. Application of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p116" /><hi rend="font-weight: bold">116.</hi> 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> + +<pb n="65" /><anchor id="Pg65" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image29.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 29</head> +<figDesc>Figure 29</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="117. Triangle formed by the two asymptotes and a tangent" /><index index="pdf" /> +<head></head><p><anchor id="p117" /><hi rend="font-weight: bold">117.</hi> For the circumscribed quadrilateral, Brianchon's +theorem gave (§ 88) <hi rend="font-style: italic">The lines joining opposite vertices +and the lines joining opposite points of contact are four +lines meeting in a point.</hi> Take now for two of the +tangents the asymptotes, and let <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">CD</hi> be any +other two (Fig. 29). +If <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are opposite +vertices, and +also <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>, then +<hi rend="font-style: italic">AC</hi> and <hi rend="font-style: italic">BD</hi> are parallel, +and parallel to +<hi rend="font-style: italic">PQ</hi>, the line joining +the points of contact +of <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">CD</hi>, +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 <hi rend="font-style: italic">ABC</hi> and <hi rend="font-style: italic">ADC</hi> are +equivalent, and therefore the triangles <hi rend="font-style: italic">AOB</hi> and <hi rend="font-style: italic">COD</hi> +are also. The tangent AB may be fixed, and the tangent +<hi rend="font-style: italic">CD</hi> chosen arbitrarily; therefore</p> + +<p><hi rend="font-style: italic">The triangle formed by any tangent to the hyperbola +and the two asymptotes is of constant area.</hi></p></div> + +<div> +<index index="toc" level1="118. Equation of hyperbola referred to the asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p118" /><hi rend="font-weight: bold">118. Equation of hyperbola referred to the asymptotes.</hi> +Draw through the point of contact <hi rend="font-style: italic">P</hi> of the tangent +<hi rend="font-style: italic">AB</hi> two lines, one parallel to one asymptote and the +other parallel to the other. One of these lines meets +<hi rend="font-style: italic">OB</hi> at a distance <hi rend="font-style: italic">y</hi> from <hi rend="font-style: italic">O</hi>, and the other meets <hi rend="font-style: italic">OA</hi> at +a distance <hi rend="font-style: italic">x</hi> from <hi rend="font-style: italic">O</hi>. Then, since <hi rend="font-style: italic">P</hi> is the middle point +<pb n="66" /><anchor id="Pg66" /> +of <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">x</hi> is one half of <hi rend="font-style: italic">OA</hi> and <hi rend="font-style: italic">y</hi> is one half of <hi rend="font-style: italic">OB</hi>. +The area of the parallelogram whose adjacent sides are +<hi rend="font-style: italic">x</hi> and <hi rend="font-style: italic">y</hi> is one half the area of the triangle <hi rend="font-style: italic">AOB</hi>, and +therefore, by the preceding paragraph, is constant. This +area is equal to <hi rend="font-style: italic">xy · <hi rend="font-style: normal">sin</hi> α</hi>, where α is the constant angle +between the asymptotes. It follows that the product <hi rend="font-style: italic">xy</hi> +is constant, and since <hi rend="font-style: italic">x</hi> and <hi rend="font-style: italic">y</hi> are the oblique coördinates +of the point <hi rend="font-style: italic">P</hi>, the asymptotes being the axes +of reference, we have</p> + +<p><hi rend="font-style: italic">The equation of the hyperbola, referred to the asymptotes +as axes, is <hi rend="font-style: italic">xy =</hi> constant.</hi></p> + +<p>This identifies the curve with the hyperbola as defined +and discussed in works on analytic geometry.</p></div> + +<div> +<index index="toc" level1="119. Equation of parabola" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image30.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 30</head> +<figDesc>Figure 30</figDesc> +</figure></p> + +<p><anchor id="p119" /><hi rend="font-weight: bold">119. Equation of +parabola.</hi> 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 +<hi rend="font-style: italic">A</hi>, the points of contact +being <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">C</hi>. +These two tangents, +together with the +line at infinity, form +a triangle circumscribed +about the +conic. Draw through <hi rend="font-style: italic">B</hi> a parallel to <hi rend="font-style: italic">AC</hi>, and through +<hi rend="font-style: italic">C</hi> a parallel to <hi rend="font-style: italic">AB</hi>. If these meet in <hi rend="font-style: italic">D</hi>, then <hi rend="font-style: italic">AD</hi> is a +<pb n="67" /><anchor id="Pg67" /> +diameter. Let <hi rend="font-style: italic">AD</hi> meet the curve in <hi rend="font-style: italic">P</hi>, and the chord +<hi rend="font-style: italic">BC</hi> in <hi rend="font-style: italic">Q</hi>. <hi rend="font-style: italic">P</hi> is then the middle point of <hi rend="font-style: italic">AQ</hi>. Also, <hi rend="font-style: italic">Q</hi> +is the middle point of the chord <hi rend="font-style: italic">BC</hi>, and therefore the +diameter <hi rend="font-style: italic">AD</hi> bisects all chords parallel to <hi rend="font-style: italic">BC</hi>. In particular, +<hi rend="font-style: italic">AD</hi> passes through <hi rend="font-style: italic">P</hi>, the point of contact of +the tangent drawn parallel to <hi rend="font-style: italic">BC</hi>.</p> + +<p>Draw now another tangent, meeting <hi rend="font-style: italic">AB</hi> in <hi rend="font-style: italic">B'</hi> and <hi rend="font-style: italic">AC</hi> +in <hi rend="font-style: italic">C'</hi>. 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 <hi rend="font-style: italic">AC</hi> through <hi rend="font-style: italic">B'</hi>, a parallel to <hi rend="font-style: italic">AB</hi> through <hi rend="font-style: italic">C'</hi>, +and the line <hi rend="font-style: italic">BC</hi> meet in a point <hi rend="font-style: italic">D'</hi>. Also, from the similar +triangles <hi rend="font-style: italic">BB'D'</hi> and <hi rend="font-style: italic">BAC</hi> we have, for all positions of the +tangent line <hi rend="font-style: italic">B'C</hi>,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">B'D' : BB' = AC : AB,</hi> +</p> + +<p>or, since <hi rend="font-style: italic">B'D' = AC'</hi>,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AC': BB' = AC:AB =</hi> constant. +</p> + +<p>If another tangent meet <hi rend="font-style: italic">AB</hi> in <hi rend="font-style: italic">B"</hi> and <hi rend="font-style: italic">AC</hi> in <hi rend="font-style: italic">C"</hi>, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic"> +AC' : BB' = AC" : BB", +</hi></p> + +<p>and by subtraction we get</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">C'C" : B'B" =</hi> constant; +</p> + +<p>whence</p> + +<p><hi rend="font-style: italic">The segments cut off on any two tangents to a parabola +by a variable tangent are proportional.</hi></p> + +<p>If now we take the tangent <hi rend="font-style: italic">B'C'</hi> as axis of ordinates, +and the diameter through the point of contact <hi rend="font-style: italic">O</hi> as axis +of abscissas, calling the coordinates of <hi rend="font-style: italic">B(x, y)</hi> and of +<hi rend="font-style: italic">C(x', y')</hi>, then, from the similar triangles <hi rend="font-style: italic">BMD'</hi> and +we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">y : y' = BD' : D'C = BB' : AB'.</hi> +</p> + +<p>Also</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">y : y' = B'D' : C'C = AC' : C'C.</hi> +</p> + +<pb n="68" /><anchor id="Pg68" /> + +<p>If now a line is drawn through <hi rend="font-style: italic">A</hi> parallel to a diameter, +meeting the axis of ordinates in <hi rend="font-style: italic">K</hi>, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AK : OQ' = AC' : CC' = y : y',</hi> +</p> + +<p>and</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OM : AK = BB' : AB' = y : y',</hi> +</p> + +<p>and, by multiplication,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OM : OQ' = y<hi rend="vertical-align: super">2</hi> : y'<hi rend="vertical-align: super">2</hi>,</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">x : x' = y<hi rend="vertical-align: super">2</hi> : y'<hi rend="vertical-align: super">2</hi>;</hi> +</p> + +<p>whence</p> + +<p><hi rend="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.</hi></p> + +<p>The last equation may be written</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">y<hi rend="vertical-align: super">2</hi> = 2px,</hi> +</p> + +<p>where <hi rend="font-style: italic">2p</hi> stands for <hi rend="font-style: italic">y'<hi rend="vertical-align: super">2</hi> : x'</hi>.</p> + +<p>The parabola is thus identified with the curve of the +same name studied in treatises on analytic geometry.</p></div> + +<div> +<index index="toc" level1="120. Equation of central conics referred to conjugate +diameters" /><index index="pdf" /> +<head></head><p><anchor id="p120" /><hi rend="font-weight: bold">120. Equation of central conics referred to conjugate +diameters.</hi> Consider now a <hi rend="font-style: italic">central conic</hi>, 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 <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">B</hi> and the other in <hi rend="font-style: italic">C</hi> and <hi rend="font-style: italic">D</hi>, and let <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">Q</hi> be +the points of contact of the parallel tangents <hi rend="font-style: italic">R</hi> and <hi rend="font-style: italic">S</hi> +of the others. Then <hi rend="font-style: italic">AC</hi>, <hi rend="font-style: italic">BD</hi>, <hi rend="font-style: italic">PQ</hi>, and <hi rend="font-style: italic">RS</hi> all meet in +a point <hi rend="font-style: italic">W</hi> (§ 88). From the figure,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">PW : WQ = AP : QC = PD : BQ,</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP · BQ = PD · QC.</hi> +</p> + +<pb n="69" /><anchor id="Pg69" /> + +<p>If now <hi rend="font-style: italic">DC</hi> is a fixed tangent and <hi rend="font-style: italic">AB</hi> a variable one, +we have from this equation</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP · BQ = <hi rend="font-style: normal">constant.</hi></hi> +</p> + +<p>This constant will be positive or negative according as +<hi rend="font-style: italic">PA</hi> and <hi rend="font-style: italic">BQ</hi> are measured in the same or in opposite +directions. Accordingly we write</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP · BQ = ± b<hi rend="vertical-align: super">2</hi>.</hi> +</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image31.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 31</head> +<figDesc>Figure 31</figDesc> +</figure></p> + +<p>Since <hi rend="font-style: italic">AD</hi> and <hi rend="font-style: italic">BC</hi> are parallel tangents, <hi rend="font-style: italic">PQ</hi> is a diameter +and the conjugate diameter is parallel to <hi rend="font-style: italic">AD</hi>. The +middle point of <hi rend="font-style: italic">PQ</hi> is the +center of the conic. We take +now for the axis of abscissas +the diameter <hi rend="font-style: italic">PQ</hi>, and the +conjugate diameter for the +axis of ordinates. Join <hi rend="font-style: italic">A</hi> to +<hi rend="font-style: italic">Q</hi> and <hi rend="font-style: italic">B</hi> to <hi rend="font-style: italic">P</hi> and draw a +line through <hi rend="font-style: italic">S</hi> parallel to +the axis of ordinates. These +three lines all meet in a point +<hi rend="font-style: italic">N</hi>, because <hi rend="font-style: italic">AP</hi>, <hi rend="font-style: italic">BQ</hi>, and <hi rend="font-style: italic">AB</hi> +form a triangle circumscribed +to the conic. Let <hi rend="font-style: italic">NS</hi> meet +<hi rend="font-style: italic">PQ</hi> in <hi rend="font-style: italic">M</hi>. Then, from the properties of the circumscribed +triangle (§ 89), <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>, <hi rend="font-style: italic">S</hi>, and the point at infinity +on <hi rend="font-style: italic">NS</hi> are four harmonic points, and therefore <hi rend="font-style: italic">N</hi> is the +middle point of <hi rend="font-style: italic">MS</hi>. If the coördinates of <hi rend="font-style: italic">S</hi> are <hi rend="font-style: italic">(x, y)</hi>, +so that <hi rend="font-style: italic">OM</hi> is <hi rend="font-style: italic">x</hi> and <hi rend="font-style: italic">MS</hi> is <hi rend="font-style: italic">y</hi>, then <hi rend="font-style: italic">MN = y/2</hi>. Now +from the similar triangles <hi rend="font-style: italic">PMN</hi> and <hi rend="font-style: italic">PQB</hi> we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">BQ : PQ = NM : PM,</hi> +</p> + +<pb n="70" /><anchor id="Pg70" /> + +<p>and from the similar triangles <hi rend="font-style: italic">PQA</hi> and <hi rend="font-style: italic">MQN</hi>,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP : PQ = MN : MQ,</hi> +</p> + +<p>whence, multiplying, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">±b<hi rend="vertical-align: super">2</hi>/4 a<hi rend="vertical-align: super">2</hi> = y<hi rend="vertical-align: super">2</hi>/4 (a + x)(a - x),</hi> +</p> + +<p>where</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +a=\frac{PQ}{2}, +\] +</formula> +</p> + +<p>or, simplifying,</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +x^2/a^2 + y^2/\pm b^2 = 1, +\] +</formula> +</p> + +<p>which is the equation of an ellipse when <hi rend="font-style: italic">b<hi rend="vertical-align: super">2</hi></hi> has a positive +sign, and of a hyperbola when <hi rend="font-style: italic">b<hi rend="vertical-align: super">2</hi></hi> 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> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Draw a chord of a given conic which shall be bisected +by a given point <hi rend="font-style: italic">P</hi>.</p> + +<p>2. Show that all chords of a given conic that are bisected +by a given chord are tangent to a parabola.</p> + +<p>3. Construct a parabola, given two tangents with their +points of contact.</p> + +<p>4. Construct a parabola, given three points and the direction +of the diameters.</p> + +<p>5. A line <hi rend="font-style: italic">u'</hi> is drawn through the pole <hi rend="font-style: italic">U</hi> of a line <hi rend="font-style: italic">u</hi> and +at right angles to <hi rend="font-style: italic">u</hi>. The line <hi rend="font-style: italic">u</hi> revolves about a point <hi rend="font-style: italic">P</hi>. +Show that the line <hi rend="font-style: italic">u'</hi> is tangent to a parabola. (The lines <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">u'</hi> are called normal conjugates.)</p> + +<p>6. Given a circle and its center <hi rend="font-style: italic">O</hi>, to draw a line through +a given point <hi rend="font-style: italic">P</hi> parallel to a given line <hi rend="font-style: italic">q</hi>. Prove the following +construction: Let <hi rend="font-style: italic">p</hi> be the polar of <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi> the pole of +<hi rend="font-style: italic">q</hi>, and <hi rend="font-style: italic">A</hi> the intersection of <hi rend="font-style: italic">p</hi> with <hi rend="font-style: italic">OQ</hi>. The polar of <hi rend="font-style: italic">A</hi> is +the desired line.</p> +</div> +</div> + +<div rend="page-break-before: always"> +<pb n="71" /><anchor id="Pg71" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER VIII - INVOLUTION</head> + +<div> +<index index="toc" level1="121. Fundamental theorem" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image32.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 32</head> +<figDesc>Figure 32</figDesc> +</figure></p> + +<p><anchor id="p121" /><hi rend="font-weight: bold">121. Fundamental theorem.</hi> 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 +<hi rend="font-style: italic">KL</hi>, <hi rend="font-style: italic">K'L'</hi>, <hi rend="font-style: italic">MN</hi>, +<hi rend="font-style: italic">M'N'</hi> do not +all meet in the +same point <hi rend="font-style: italic">A</hi>, +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><hi rend="font-style: italic">Given two complete quadrangles, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> and +<hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L'</hi>, <hi rend="font-style: italic">M'</hi>, <hi rend="font-style: italic">N'</hi>, so related that <hi rend="font-style: italic">KL</hi> and <hi rend="font-style: italic">K'L'</hi> meet in <hi rend="font-style: italic">A</hi>, +<hi rend="font-style: italic">MN</hi> and <hi rend="font-style: italic">M'N'</hi> in <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">KN</hi> and <hi rend="font-style: italic">K'N'</hi> in <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">LM</hi> and <hi rend="font-style: italic">L'M'</hi> +in <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">LN</hi> and <hi rend="font-style: italic">L'N'</hi> in <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">KM</hi> and <hi rend="font-style: italic">K'M'</hi> in <hi rend="font-style: italic">C'</hi>, then, +if <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">B'</hi>, and <hi rend="font-style: italic">C</hi> are in a straight line, the point <hi rend="font-style: italic">C'</hi> +also lies on that straight line.</hi></p> + +<p>The theorem follows from Desargues's theorem +(Fig. 32). It is seen that <hi rend="font-style: italic">KK'</hi>, <hi rend="font-style: italic">LL'</hi>, <hi rend="font-style: italic">MM'</hi>, <hi rend="font-style: italic">NN'</hi> all +<pb n="72" /><anchor id="Pg72" /> +meet in a point, and thus, from the same theorem, applied +to the triangles <hi rend="font-style: italic">KLM</hi> and <hi rend="font-style: italic">K'L'M'</hi>, the point <hi rend="font-style: italic">C'</hi> is on +the same line with <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B'</hi>. 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 +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, two through <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, and one through <hi rend="font-style: italic">C</hi>. +The sixth side must then go through <hi rend="font-style: italic">C'</hi>. Therefore,</p></div> + +<div> +<index index="toc" level1="122. Linear construction" /><index index="pdf" /> +<head></head><p><anchor id="p122" /><hi rend="font-weight: bold">122.</hi> <hi rend="font-style: italic">Two pairs of points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">A'</hi> and <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">B'</hi>, being given, +then the point <hi rend="font-style: italic">C'</hi> corresponding to any given point <hi rend="font-style: italic">C</hi> is +uniquely determined.</hi></p> + +<p>The construction of this sixth point is easily accomplished. +Draw through <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> any two lines, and +cut across them by any line through <hi rend="font-style: italic">C</hi> in the points +<hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi>. Join <hi rend="font-style: italic">N</hi> to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">L</hi> to <hi rend="font-style: italic">B'</hi>, thus determining +the points <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> on the two lines through <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, +The line <hi rend="font-style: italic">KM</hi> determines the desired point <hi rend="font-style: italic">C'</hi>. Manifestly, +starting from <hi rend="font-style: italic">C'</hi>, we come in this way always to the +same point <hi rend="font-style: italic">C</hi>. 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 <hi rend="font-style: italic">AA'</hi> and +<hi rend="font-style: italic">CC'</hi>, or from the pairs <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">CC'</hi>.</p></div> + +<div> +<index index="toc" level1="123. Definition of involution of points on a line" /><index index="pdf" /> +<head></head><p><anchor id="p123" /><hi rend="font-weight: bold">123. Definition of involution of points on a line.</hi></p> + +<p><hi rend="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.</hi></p> + +<pb n="73" /><anchor id="Pg73" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image33.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 33</head> +<figDesc>Figure 33</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="124. Double-points in an involution" /><index index="pdf" /> +<head></head><p><anchor id="p124" /><hi rend="font-weight: bold">124. Double-points in an involution.</hi> The points <hi rend="font-style: italic">C</hi> and +<hi rend="font-style: italic">C'</hi> describe projective point-rows, as may be seen by fixing +the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi>. The self-corresponding points, of +which there are two or none, are called the <hi rend="font-style: italic">double-points</hi> 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 <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi>, let the intersection +of the lines <hi rend="font-style: italic">CL</hi> and <hi rend="font-style: italic">C'M</hi> be <hi rend="font-style: italic">P</hi> (Fig. 33). The locus of <hi rend="font-style: italic">P</hi> is +a conic which goes through the double-points, because the +point-rows <hi rend="font-style: italic">C</hi> and +<hi rend="font-style: italic">C'</hi> are projective, +and therefore so +are the pencils +<hi rend="font-style: italic">LC</hi> and <hi rend="font-style: italic">MC'</hi> +which generate +the locus of <hi rend="font-style: italic">P</hi>. +Also, when <hi rend="font-style: italic">C</hi> +and <hi rend="font-style: italic">C'</hi> fall together, +the point +<hi rend="font-style: italic">P</hi> coincides with +them. Further, the tangents at <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> to this conic +described by <hi rend="font-style: italic">P</hi> are the lines <hi rend="font-style: italic">LB</hi> and <hi rend="font-style: italic">MB</hi>. For in the +pencil at <hi rend="font-style: italic">L</hi> the ray <hi rend="font-style: italic">LM</hi> common to the two pencils which +generate the conic is the ray <hi rend="font-style: italic">LB'</hi> and corresponds to the +ray <hi rend="font-style: italic">MB</hi> of <hi rend="font-style: italic">M</hi>, which is therefore the tangent line to the +conic at <hi rend="font-style: italic">M</hi>. Similarly for the tangent <hi rend="font-style: italic">LB</hi> at <hi rend="font-style: italic">L</hi>. <hi rend="font-style: italic">LM</hi> is +therefore the polar of <hi rend="font-style: italic">B</hi> with respect to this conic, and +<hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi> 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> + +<pb n="74" /><anchor id="Pg74" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image34.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 34</head> +<figDesc>Figure 34</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="125. Desargues's theorem concerning conics through +four points" /><index index="pdf" /> +<head></head><p><anchor id="p125" /><hi rend="font-weight: bold">125. Desargues's theorem concerning conics through +four points.</hi> Let <hi rend="font-style: italic">DD'</hi> be any pair of points in the involution +determined as above, and consider the conic +passing through the five points <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>, <hi rend="font-style: italic">D</hi>. We +shall use Pascal's theorem to show that this conic also +passes through <hi rend="font-style: italic">D'</hi>. The point <hi rend="font-style: italic">D'</hi> is determined as follows: +Fix <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> as before (Fig. 34) and join <hi rend="font-style: italic">D</hi> to <hi rend="font-style: italic">L</hi>, +giving on <hi rend="font-style: italic">MN</hi> +the point <hi rend="font-style: italic">N'</hi>. +Join <hi rend="font-style: italic">N'</hi> to <hi rend="font-style: italic">B</hi>, +giving on <hi rend="font-style: italic">LK</hi> +the point <hi rend="font-style: italic">K'</hi>. +Then <hi rend="font-style: italic">MK'</hi> determines +the +point <hi rend="font-style: italic">D'</hi> on +the line <hi rend="font-style: italic">AA'</hi>, +given by the +complete quadrangle +<hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N'</hi>. Consider the following six points, +numbering them in order: <hi rend="font-style: italic">D = 1</hi>, <hi rend="font-style: italic">D' = 2</hi>, <hi rend="font-style: italic">M = 3</hi>, <hi rend="font-style: italic">N = 4</hi>, +<hi rend="font-style: italic">K = 5</hi>, and <hi rend="font-style: italic">L = 6</hi>. We have the following intersections: +<hi rend="font-style: italic">B = (12-45)</hi>, <hi rend="font-style: italic">K' = (23-56)</hi>, <hi rend="font-style: italic">N' = (34-61)</hi>; and since by +construction <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">N</hi>, and <hi rend="font-style: italic">K'</hi> 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><hi rend="font-style: italic">The system of conics through four points meets any line +in the plane in pairs of points in involution.</hi></p></div> + +<div> +<index index="toc" level1="126. Degenerate conics of the system" /><index index="pdf" /> +<head></head><p><anchor id="p126" /><hi rend="font-weight: bold">126.</hi> It appears also that the six points in involution +determined by the quadrangle through the four fixed +<pb n="75" /><anchor id="Pg75" /> +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> +<index index="toc" level1="127. Conics through four points touching a given line" /><index index="pdf" /> +<head></head><p><anchor id="p127" /><hi rend="font-weight: bold">127. Conics through four points touching a given line.</hi> +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><hi rend="font-style: italic">Through four fixed points in the plane two conics or +none may be drawn tangent to any given line.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image35.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 35</head> +<figDesc>Figure 35</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="128. Double correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p128" /><hi rend="font-weight: bold">128. Double correspondence.</hi> 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, <hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">a'</hi>, which +both revolve about a fixed point <hi rend="font-style: italic">S</hi> +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 <hi rend="font-style: italic">S</hi> two projective point-rows, +which are not, however, in +involution unless the angle between the lines is a right +angles. For a point <hi rend="font-style: italic">P</hi> may correspond to a point <hi rend="font-style: italic">P'</hi>, +which in turn will correspond to some other point +<pb n="76" /><anchor id="Pg76" /> +than <hi rend="font-style: italic">P</hi>. 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 <hi rend="font-style: italic">P</hi> corresponds to a point <hi rend="font-style: italic">P'</hi>, then +the point <hi rend="font-style: italic">P'</hi> corresponds back again to the point <hi rend="font-style: italic">P</hi>. +The points <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> are then said to <hi rend="font-style: italic">correspond doubly</hi>. +This notion is worthy of further study.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image36.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 36</head> +<figDesc>Figure 36</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="129. Steiner's construction" /><index index="pdf" /> +<head></head><p><anchor id="p129" /><hi rend="font-weight: bold">129. Steiner's construction.</hi> It will be observed that +the solution of the fundamental problem given in § 83, +<hi rend="font-style: italic">Given three pairs of points of two protective point-rows, to +construct other pairs</hi>, 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 <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, ... and <hi rend="font-style: italic">A'</hi>, +<hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi>, ... be superposed on the line <hi rend="font-style: italic">u</hi>. Project +them both to a point <hi rend="font-style: italic">S</hi> and pass any conic <hi rend="font-style: italic">κ</hi> through <hi rend="font-style: italic">S</hi>. +We thus obtain two projective pencils, <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi>, <hi rend="font-style: italic">d</hi>, ... and +<pb n="77" /><anchor id="Pg77" /> +<hi rend="font-style: italic">a'</hi>, <hi rend="font-style: italic">b'</hi>, <hi rend="font-style: italic">c'</hi>, <hi rend="font-style: italic">d'</hi>, ... at <hi rend="font-style: italic">S</hi>, which meet the conic in the points +<hi rend="font-style: italic">α</hi>, <hi rend="font-style: italic">β</hi>, <hi rend="font-style: italic">γ</hi>, <hi rend="font-style: italic">δ</hi>, ... and +<hi rend="font-style: italic">α'</hi>, <hi rend="font-style: italic">β'</hi>, <hi rend="font-style: italic">γ'</hi>, <hi rend="font-style: italic">δ'</hi>, ... (Fig. 36). Take now +<hi rend="font-style: italic">γ</hi> as the center of a pencil projecting the points <hi rend="font-style: italic">α'</hi>, <hi rend="font-style: italic">β'</hi>, +<hi rend="font-style: italic">δ'</hi>, ..., and take <hi rend="font-style: italic">γ'</hi> as the center of a pencil projecting +the points <hi rend="font-style: italic">α</hi>, <hi rend="font-style: italic">β</hi>, <hi rend="font-style: italic">δ</hi>, .... 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 <hi rend="font-style: italic">(γα', γ'α)</hi> +to <hi rend="font-style: italic">(γβ', γ'β)</hi>. The correspondence between points in +the two point-rows on <hi rend="font-style: italic">u</hi> is now easily traced.</p></div> + +<div> +<index index="toc" level1="130. Application of Steiner's construction to double +correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p130" /><hi rend="font-weight: bold">130. Application of Steiner's construction to double +correspondence.</hi> Steiner's construction throws into our +hands an important theorem concerning double correspondence: +<hi rend="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.</hi> To +make this appear, let us call the point <hi rend="font-style: italic">A</hi> on <hi rend="font-style: italic">u</hi> by two +names, <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">P'</hi>, 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 <hi rend="font-style: italic">A'</hi> and <hi rend="font-style: italic">P</hi> must coincide +(Fig. 37). Take now any point <hi rend="font-style: italic">C</hi>, which we will +also call <hi rend="font-style: italic">R'</hi>. We must show that the corresponding +point <hi rend="font-style: italic">C'</hi> must also coincide with the point <hi rend="font-style: italic">B</hi>. Join all +the points to <hi rend="font-style: italic">S</hi>, as before, and it appears that the points +α and <hi rend="font-style: italic">π'</hi> coincide, as also do the points <hi rend="font-style: italic">α'π</hi> and <hi rend="font-style: italic">γρ'</hi>. +By the above construction the line <hi rend="font-style: italic">γ'ρ</hi> must meet <hi rend="font-style: italic">γρ'</hi> +on the line joining <hi rend="font-style: italic">(γα', γ'α)</hi> with +<hi rend="font-style: italic">(γπ', γ'π)</hi>. But these +four points form a quadrangle inscribed in the conic, +and we know by § 95 that the tangents at the opposite +<pb n="78" /><anchor id="Pg78" /> +vertices <hi rend="font-style: italic">γ</hi> and <hi rend="font-style: italic">γ'</hi> meet on the line <hi rend="font-style: italic">v</hi>. The line <hi rend="font-style: italic">γ'ρ</hi> +is thus a tangent to the conic, and <hi rend="font-style: italic">C'</hi> and <hi rend="font-style: italic">R</hi> 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 <hi rend="font-style: italic">S</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image37.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 37</head> +<figDesc>Figure 37</figDesc> +</figure></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image38.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 38</head> +<figDesc>Figure 38</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="131. Involution of points on a point-row of the second order." /><index index="pdf" /> +<head></head><p><anchor id="p131" /><hi rend="font-weight: bold">131. Involution of points on a point-row of the second +order.</hi> 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 <hi rend="font-style: italic">the points of a conic +are paired in involution when they are corresponding +<pb n="79" /><anchor id="Pg79" /> +points of two projective point-rows superposed on the conic, +and when they correspond to each other doubly.</hi> With this +definition we may prove the theorem: <hi rend="font-style: italic">The lines joining +corresponding points of a point-row of the second order in +involution all pass through a fixed point <hi rend="font-style: italic">U</hi>, and the line +joining any two points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi> meets the line joining the +two corresponding points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi> in the +points of a line <hi rend="font-style: italic">u</hi>, which is the polar +of <hi rend="font-style: italic">U</hi> with respect to the conic.</hi> For +take <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> as the centers of two +pencils, the first perspective to the +point-row <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi> and the second +perspective to the point-row <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>. +Then, since the common ray of the +two pencils corresponds to itself, they are in perspective +position, and their axis of perspectivity <hi rend="font-style: italic">u</hi> (Fig. 38) +is the line which joins the point <hi rend="font-style: italic">(AB', A'B)</hi> to the +point <hi rend="font-style: italic">(AC', A'C)</hi>. It is then immediately clear, from +the theory of poles and polars, that <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">CC'</hi> pass +through the pole <hi rend="font-style: italic">U</hi> of the line <hi rend="font-style: italic">u</hi>.</p></div> + +<div> +<index index="toc" level1="132. Involution of rays" /><index index="pdf" /> +<head></head><p><anchor id="p132" /><hi rend="font-weight: bold">132. Involution of rays.</hi> 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><hi rend="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 <hi rend="font-style: italic">aa'</hi> and <hi rend="font-style: italic">bb'</hi> are fixed, and the line <hi rend="font-style: italic">c</hi> describes a pencil, +the corresponding line <hi rend="font-style: italic">c'</hi> also describes a pencil, and the +rays of the pencil are said to be paired in the involution +determined by <hi rend="font-style: italic">aa'</hi> and <hi rend="font-style: italic">bb'</hi>.</hi></p> + +<pb n="80" /><anchor id="Pg80" /></div> + +<div> +<index index="toc" level1="133. Double rays" /><index index="pdf" /> +<head></head><p><anchor id="p133" /><hi rend="font-weight: bold">133. Double rays.</hi> The self-corresponding rays, of +which there are two or none, are called <hi rend="font-style: italic">double rays</hi> 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><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="134. Conic through a fixed point touching four lines" /><index index="pdf" /> +<head></head><p><anchor id="p134" /><hi rend="font-weight: bold">134.</hi> 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><hi rend="font-style: italic">Two conics or none may be drawn through a fixed point +to be tangent to four fixed lines.</hi></p></div> + +<div> +<index index="toc" level1="135. Double correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p135" /><hi rend="font-weight: bold">135. Double correspondence.</hi> 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> + +<pb n="81" /><anchor id="Pg81" /></div> + +<div> +<index index="toc" level1="136. Pencils of rays of the second order in involution" /><index index="pdf" /> +<head></head><p><anchor id="p136" /><hi rend="font-weight: bold">136. Pencils of rays of the second order in involution.</hi> +We may also extend the notion of involution to pencils +of rays of the second order. Thus, <hi rend="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.</hi> We have then the theorem:</p></div> + +<div> +<index index="toc" level1="137. Theorem concerning pencils of the second +order in involution" /><index index="pdf" /> +<head></head><p><anchor id="p137" /><hi rend="font-weight: bold">137.</hi> <hi rend="font-style: italic">The intersections of corresponding rays of a pencil +of the second order in involution are all on a straight +line <hi rend="font-style: italic">u</hi>, and the intersection of any two tangents <hi rend="font-style: italic">ab</hi>, when +joined to the intersection of the corresponding tangents <hi rend="font-style: italic">a'b'</hi>, +gives a line which passes through a fixed point <hi rend="font-style: italic">U</hi>, the pole +of the line <hi rend="font-style: italic">u</hi> with respect to the conic.</hi></p></div> + +<div> +<index index="toc" level1="138. Involution of rays determined by a conic" /><index index="pdf" /> +<head></head><p><anchor id="p138" /><hi rend="font-weight: bold">138. Involution of rays determined by a conic.</hi> We +have seen in the theory of poles and polars (§ 103) +that if a point <hi rend="font-style: italic">P</hi> moves along a line <hi rend="font-style: italic">m</hi>, then the polar +of <hi rend="font-style: italic">P</hi> revolves about a point. This pencil cuts out on <hi rend="font-style: italic">m</hi> +another point-row <hi rend="font-style: italic">P'</hi>, projective also to <hi rend="font-style: italic">P</hi>. Since the +polar of <hi rend="font-style: italic">P</hi> passes through <hi rend="font-style: italic">P'</hi>, the polar of <hi rend="font-style: italic">P'</hi> also passes +through <hi rend="font-style: italic">P</hi>, so that the correspondence between <hi rend="font-style: italic">P</hi> and +<hi rend="font-style: italic">P'</hi> is double. The two point-rows are therefore in involution, +and the double points, if any exist, are the points +where the line <hi rend="font-style: italic">m</hi> 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 <hi rend="font-style: italic">conjugate</hi> with +respect to the conic (§ 100). We may then state the +following important theorem:</p></div> + +<div> +<index index="toc" level1="139. Statement of theorem" /><index index="pdf" /> +<head></head><p><anchor id="p139" /><hi rend="font-weight: bold">139.</hi> <hi rend="font-style: italic">A conic determines on every line in its plane an +involution of points, corresponding points in the involution +<pb n="82" /><anchor id="Pg82" /> +being conjugate with respect to the conic. The double points, +if any exist, are the points where the line meets the conic.</hi></p></div> + +<div> +<index index="toc" level1="140. Dual of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p140" /><hi rend="font-weight: bold">140.</hi> The dual theorem reads: <hi rend="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.</hi></p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>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>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>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>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>5. State and prove the dual of problem 4.</p> + +<p>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> + +<pb n="83" /><anchor id="Pg83" /> + +<p>7. If a triangle <hi rend="font-style: italic">QRS</hi> be inscribed in a conic, and if a +transversal <hi rend="font-style: italic">s</hi> meet two of its sides in <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, the third +side and the tangent at the opposite vertex in <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, and +the conic itself in <hi rend="font-style: italic">C</hi> and <hi rend="font-style: italic">C'</hi>, then <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, <hi rend="font-style: italic">CC'</hi> are three +pairs of points in an involution.</p> + +<p>8. Use the last exercise to solve the problem: Given five +points, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">S</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">C'</hi>, on a conic, to draw the tangent at any +one of them.</p> + +<p>9. State and prove the dual of problem 7 and use it to +prove the dual of problem 8.</p> + +<p>10. If a transversal cut two tangents to a conic in <hi rend="font-style: italic">B</hi> and +<hi rend="font-style: italic">B'</hi>, their chord of contact in <hi rend="font-style: italic">A</hi>, and the conic itself in <hi rend="font-style: italic">P</hi> +and <hi rend="font-style: italic">P'</hi>, then the point <hi rend="font-style: italic">A</hi> is a double point of the involution +determined by <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">PP'</hi>.</p> + +<p>11. State and prove the dual of problem 10.</p> + +<p>12. If a variable conic pass through two given points, +<hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi>, 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 <hi rend="font-style: italic">PP'</hi>.</p> + +<p>13. Use the last theorem to solve the problem: Given +four points, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">P'</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">S</hi>, on a conic, and the tangent at one of +them, <hi rend="font-style: italic">Q</hi>, to draw the tangent at any one of the other points, <hi rend="font-style: italic">S</hi>.</p> + +<p>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>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> + + +<div rend="page-break-before: always"> +<pb n="84" /><anchor id="Pg84" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS</head> + +<div> +<index index="toc" level1="141. Introduction of infinite point; center of involution" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image39.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 39</head> +<figDesc>Figure 39</figDesc> +</figure></p> + +<p><anchor id="p141" /><hi rend="font-weight: bold">141. Introduction of infinite point; center of involution.</hi> +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 <hi rend="font-style: italic">center</hi> 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 +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> any two rays and cut them by a line parallel +to <hi rend="font-style: italic">AA'</hi> in the points <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi>. Join these points to +<hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, thus determining on <hi rend="font-style: italic">AK</hi> and <hi rend="font-style: italic">AN</hi> the points <hi rend="font-style: italic">L</hi> +and <hi rend="font-style: italic">N</hi>. <hi rend="font-style: italic">LN</hi> meets <hi rend="font-style: italic">AA'</hi> in the center <hi rend="font-style: italic">O</hi> of the involution.</p> + +<pb n="85" /><anchor id="Pg85" /></div> + +<div> +<index index="toc" level1="142. Fundamental metrical theorem" /><index index="pdf" /> +<head></head><p><anchor id="p142" /><hi rend="font-weight: bold">142. Fundamental metrical theorem.</hi> From the figure +we see that the triangles <hi rend="font-style: italic">OLB'</hi> and <hi rend="font-style: italic">PLM</hi> are similar, <hi rend="font-style: italic">P</hi> +being the intersection of KM and LN. Also the triangles +<hi rend="font-style: italic">KPN</hi> and <hi rend="font-style: italic">BON</hi> are similar. We thus have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OB : PK = ON : PN</hi> +</p> + +<p>and</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OB' : PM = OL : PL;</hi> +</p> + +<p>whence</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OB · OB' : PK · PM = ON · OL : PN · PL.</hi> +</p> + +<p>In the same way, from the similar triangles <hi rend="font-style: italic">OAL</hi> and +<hi rend="font-style: italic">PKL</hi>, and also <hi rend="font-style: italic">OA'N</hi> and <hi rend="font-style: italic">PMN</hi>, we obtain</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OA · OA' : PK · PM = ON · OL : PN · PL,</hi> +</p> + +<p>and this, with the preceding, gives at once the fundamental +theorem, which is sometimes taken also as the +definition of involution:</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OA · OA' = OB · OB' = <hi rend="font-style: normal">constant</hi>,</hi> +</p> + +<p>or, in words,</p> + +<p><hi rend="font-style: italic">The product of the distances from the center to two corresponding +points in an involution of points is constant.</hi></p></div> + +<div> +<index index="toc" level1="143. Existence of double points" /><index index="pdf" /> +<head></head><p><anchor id="p143" /><hi rend="font-weight: bold">143. Existence of double points.</hi> 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 <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> lie both on the same side +of the center, the product <hi rend="font-style: italic">OA · OA'</hi> 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 <hi rend="font-style: italic">BB'</hi>. Then, +since <hi rend="font-style: italic">OA · OA' = OB · OB'</hi>, it cannot happen that <hi rend="font-style: italic">B</hi> and +<hi rend="font-style: italic">B'</hi> are separated from each other by <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>. This is +evident enough if the points are on opposite sides of +the center. If the pairs are on the same side of the +<pb n="86" /><anchor id="Pg86" /> +center, and <hi rend="font-style: italic">B</hi> lies between <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, so that <hi rend="font-style: italic">OB</hi> is +greater, say, than <hi rend="font-style: italic">OA</hi>, but less than <hi rend="font-style: italic">OA'</hi>, then, by the +equation <hi rend="font-style: italic">OA · OA' = OB · OB'</hi>, we must have <hi rend="font-style: italic">OB'</hi> also +less than <hi rend="font-style: italic">OA'</hi> and greater than <hi rend="font-style: italic">OA</hi>. A similar discussion +may be made for the case where <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> lie on +opposite sides of <hi rend="font-style: italic">O</hi>. The results may be stated as +follows, without any reference to the center:</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="144. Existence of double rays" /><index index="pdf" /> +<head></head><p><anchor id="p144" /><hi rend="font-weight: bold">144.</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image40.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 40</head> +<figDesc>Figure 40</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="145. Construction of an involution by means of circles" /><index index="pdf" /> +<head></head><p><anchor id="p145" /><hi rend="font-weight: bold">145. Construction of +an involution by means of circles.</hi> The equation just derived, <hi +rend="font-style: italic">OA · OA' = OB · OB'</hi>, indicates another +simple way in which points of an involution of points may be +constructed. Through <hi rend="font-style: italic">A</hi> and <hi +rend="font-style: italic">A'</hi> draw any circle, and draw also any +circle through <hi rend="font-style: italic">B</hi> and <hi +rend="font-style: italic">B'</hi> to cut the first in the two points <hi +rend="font-style: italic">G</hi> and <hi rend="font-style: +italic">G'</hi> (Fig. 40). Then any circle through <hi rend="font-style: +italic">G</hi> and <hi rend="font-style: italic">G'</hi> will meet the +line in pairs of points in the involution determined by <hi +rend="font-style: italic">AA'</hi> and <hi rend="font-style: +italic">BB'</hi>. For if such a circle meets the line in the points <hi +rend="font-style: italic">CC'</hi>, then, by the theorem in the geometry +of the circle which says that <hi rend="font-style: italic">if any chord +is +<pb n="87" /><anchor id="Pg87" /> +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</hi>, we have <hi rend="font-style: italic">OC · +OC' = OG · OG' =</hi> constant. So that for all such points +<hi rend="font-style: italic">OA · OA' = OB · OB' = OC · +OC'</hi>. Further, the line <hi rend="font-style: italic">GG'</hi> +meets <hi rend="font-style: italic">AA'</hi> in the center of the +involution. To find the double points, if they exist, we draw a tangent +from <hi rend="font-style: italic">O</hi> to any of the circles through +<hi rend="font-style: italic">GG'</hi>. Let <hi rend="font-style: +italic">T</hi> be the point of contact. Then lay off on the line <hi +rend="font-style: italic">OA</hi> a line <hi rend="font-style: +italic">OF</hi> equal to <hi rend="font-style: italic">OT</hi>. Then, +since by the above theorem of elementary geometry +<hi rend="font-style: italic">OA · OA' = OT<hi rend="vertical-align: super">2</hi> = OF<hi rend="vertical-align: super">2</hi></hi>, we have one double +point <hi rend="font-style: italic">F</hi>. The other is at an equal +distance on the other side of <hi rend="font-style: italic">O</hi>. 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> +<index index="toc" level1="146. Circular points" /><index index="pdf" /> +<head></head><p><anchor id="p146" /><hi rend="font-weight: bold">146.</hi> 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 +<hi rend="font-style: italic">circular points</hi> and usually denoted by <hi rend="font-style: italic">I</hi> and <hi rend="font-style: italic">J</hi>. 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> + +<pb n="88" /><anchor id="Pg88" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image41.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 41</head> +<figDesc>Figure 41</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="147. Pairs in an involution of rays which are at right +angles. Circular involution" /><index index="pdf" /> +<head></head><p><anchor id="p147" /><hi rend="font-weight: bold">147. Pairs in an involution of rays which are at right +angles. Circular involution.</hi> 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 <hi rend="font-style: italic">G</hi> and <hi rend="font-style: italic">G'</hi> lie on the perpendicular +bisector of the line <hi rend="font-style: italic">GG'</hi>. Let +this line meet the line <hi rend="font-style: italic">AA'</hi> +in the point <hi rend="font-style: italic">C</hi> (Fig. 41), and +draw the circle with center <hi rend="font-style: italic">C</hi> +which goes through <hi rend="font-style: italic">G</hi> and <hi rend="font-style: italic">G'</hi>. +This circle cuts out two points +<hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">M'</hi> in the involution. The rays <hi rend="font-style: italic">GM</hi> and <hi rend="font-style: italic">GM'</hi> are +clearly at right angles, being inscribed in a semicircle. +If, therefore, the involution of points is projected to +<hi rend="font-style: italic">G</hi>, we have found two corresponding rays which are +at right angles to each other. Given now any involution +of rays with center <hi rend="font-style: italic">G</hi>, we may cut across it +by a straight line and proceed to find the two points +<hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">M'</hi>. Clearly there will be only one such pair +unless the perpendicular bisector of <hi rend="font-style: italic">GG'</hi> coincides with +the line <hi rend="font-style: italic">AA'</hi>. In this case every ray is at right angles +to its corresponding ray, and the involution is called +<hi rend="font-style: italic">circular</hi>.</p></div> + +<div> +<index index="toc" level1="148. Axes of conics" /><index index="pdf" /> +<head></head><p><anchor id="p148" /><hi rend="font-weight: bold">148. Axes of conics.</hi> At the close of the last chapter +(§ 140) we gave the theorem: <hi rend="font-style: italic">A conic determines at every +point in its plane an involution of rays, corresponding rays +<pb n="89" /><anchor id="Pg89" /> +being conjugate with respect to the conic. The double rays, +if any exist, are the tangents from the point to the conic.</hi> +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 <hi rend="font-style: italic">axis</hi> 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> +<index index="toc" level1="149. Points at which the involution determined by +a conic is circular" /><index index="pdf" /> +<head></head><p><anchor id="p149" /><hi rend="font-weight: bold">149. Points at which the involution determined by +a conic is circular.</hi> 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> + +<pb n="90" /><anchor id="Pg90" /></div> + +<div> +<index index="toc" level1="150. Properties of such a point" /><index index="pdf" /> +<head></head><p><anchor id="p150" /><hi rend="font-weight: bold">150.</hi> 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 <hi rend="font-style: italic">F</hi> +and <hi rend="font-style: italic">F'</hi> were two such points, then, since the conjugate +ray at <hi rend="font-style: italic">F</hi> to the line <hi rend="font-style: italic">FF'</hi> must be at right angles to it, +and also since the conjugate ray at <hi rend="font-style: italic">F'</hi> to the line <hi rend="font-style: italic">FF'</hi> +must be at right angles to it, the pole of <hi rend="font-style: italic">FF'</hi> must +be at infinity in a direction at right angles to <hi rend="font-style: italic">FF'</hi>. +The line <hi rend="font-style: italic">FF'</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image42.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 42</head> +<figDesc>Figure 42</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="151. Position of such a point" /><index index="pdf" /> +<head></head><p><anchor id="p151" /><hi rend="font-weight: bold">151.</hi> Let now <hi rend="font-style: italic">P</hi> be a point on one of the axes (Fig. 42), +and draw any ray through it, such as <hi rend="font-style: italic">q</hi>. As <hi rend="font-style: italic">q</hi> revolves +about <hi rend="font-style: italic">P</hi>, its pole <hi rend="font-style: italic">Q</hi> moves along a line at right angles +to the axis on which <hi rend="font-style: italic">P</hi> lies, describing a point-row <hi rend="font-style: italic">p</hi> +projective to the pencil of rays <hi rend="font-style: italic">q</hi>. The point at infinity +in a direction at right angles to <hi rend="font-style: italic">q</hi> also describes a point-row +projective to <hi rend="font-style: italic">q</hi>. The line joining corresponding +points of these two point-rows is always a conjugate +line to <hi rend="font-style: italic">q</hi> and at right angles to <hi rend="font-style: italic">q</hi>, or, as we may call it, +a <hi rend="font-style: italic">conjugate normal</hi> to <hi rend="font-style: italic">q</hi>. These conjugate normals to <hi rend="font-style: italic">q</hi>, +joining as they do corresponding points in two projective +point-rows, form a pencil of rays of the second +<pb n="91" /><anchor id="Pg91" /> +order. But since the point at infinity on the point-row +<hi rend="font-style: italic">Q</hi> corresponds to the point at infinity in a direction +at right angles to <hi rend="font-style: italic">q</hi>, these point-rows are in perspective +position and the normal conjugates of all the lines +through <hi rend="font-style: italic">P</hi> meet in a point. This point lies on the +same axis with <hi rend="font-style: italic">P</hi>, as is seen by taking <hi rend="font-style: italic">q</hi> at right angles +to the axis on which <hi rend="font-style: italic">P</hi> lies. The center of this pencil +may be called <hi rend="font-style: italic">P'</hi>, and thus we have paired the point <hi rend="font-style: italic">P</hi> +with the point <hi rend="font-style: italic">P'</hi>. By moving the point <hi rend="font-style: italic">P</hi> along the +axis, and by keeping the +ray <hi rend="font-style: italic">q</hi> parallel to a fixed +direction, we may see that +the point-row <hi rend="font-style: italic">P</hi> and the +point-row <hi rend="font-style: italic">P'</hi> are projective. +Also the correspondence is +double, and by starting +from the point <hi rend="font-style: italic">P'</hi> we arrive +at the point <hi rend="font-style: italic">P</hi>. Therefore +the point-rows <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> 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 <hi rend="font-style: italic">P</hi> and the corresponding rays through +<hi rend="font-style: italic">P'</hi> are conjugate normals; and if <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image43.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 43</head> +<figDesc>Figure 43</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="152. Discovery of the foci of the conic" /><index index="pdf" /> +<head></head><p><anchor id="p152" /><hi rend="font-weight: bold">152. Discovery of the foci of the conic.</hi> We know +that on one axis no such points as we are seeking can +lie (§ 150). The involution of points <hi rend="font-style: italic">PP'</hi> on this axis +<pb n="92" /><anchor id="Pg92" /> +can therefore have no double points. Nevertheless, let +<hi rend="font-style: italic">PP'</hi> and <hi rend="font-style: italic">RR'</hi> be two pairs of corresponding points on +this axis (Fig. 43). Then we know that <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> are +separated from each other by <hi rend="font-style: italic">R</hi> and <hi rend="font-style: italic">R'</hi> (§ 143). Draw +a circle on <hi rend="font-style: italic">PP'</hi> as a diameter, and one on <hi rend="font-style: italic">RR'</hi> as a +diameter. These must intersect in +two points, <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi>, and since the +center of the conic is the center +of the involution <hi rend="font-style: italic">PP'</hi>, <hi rend="font-style: italic">RR'</hi>, as is +easily seen, it follows that <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi> +are on the other axis of the conic. +Moreover, <hi rend="font-style: italic">FR</hi> and <hi rend="font-style: italic">FR'</hi> are conjugate +normal rays, since <hi rend="font-style: italic">RFR'</hi> is +inscribed in a semicircle, and the +two rays go one through <hi rend="font-style: italic">R</hi> and the other through <hi rend="font-style: italic">R'</hi>. +The involution of points <hi rend="font-style: italic">PP'</hi>, <hi rend="font-style: italic">RR'</hi> therefore projects +to the two points <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi> in two pencils of rays in +involution which have for corresponding rays conjugate +normals to the conic. We may, then, say:</p> + +<p><hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="153. The circle and the parabola" /><index index="pdf" /> +<head></head><p><anchor id="p153" /><hi rend="font-weight: bold">153. The circle and the parabola.</hi> 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 +<pb n="93" /><anchor id="Pg93" /> +axis. It is seen, however, that as <hi rend="font-style: italic">P</hi> moves to infinity, +carrying the line <hi rend="font-style: italic">q</hi> with it, <hi rend="font-style: italic">q</hi> becomes the line at infinity, +which for the parabola is a tangent line. Its pole +<hi rend="font-style: italic">Q</hi> is thus at infinity and also the point <hi rend="font-style: italic">P'</hi>, so that <hi rend="font-style: italic">P</hi> +and <hi rend="font-style: italic">P'</hi> fall together at infinity, and therefore one focus +of the parabola is at infinity. There must therefore be +another, so that</p> + +<p><hi rend="font-style: italic">A parabola has one and only one focus in the finite +part of the plane.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image44.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 44</head> +<figDesc>Figure 44</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="154. Focal properties of conics" /><index index="pdf" /> +<head></head><p><anchor id="p154" /><hi rend="font-weight: bold">154. Focal properties of conics.</hi> 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 <hi rend="font-style: italic">P</hi>. These +two lines are clearly conjugate +normals. The two points <hi rend="font-style: italic">T</hi> and +<hi rend="font-style: italic">N</hi>, 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 <hi rend="font-style: italic">P</hi>, 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><hi rend="font-style: italic">The lines joining a point on the conic to the foci make +equal angles with the tangent.</hi></p> + +<p>It follows that rays from a source of light at one +focus are reflected by an ellipse to the other.</p> + +<pb n="94" /><anchor id="Pg94" /></div> + +<div> +<index index="toc" level1="155. Case of the parabola" /><index index="pdf" /> +<head></head><p><anchor id="p155" /><hi rend="font-weight: bold">155.</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image45.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 45</head> +<figDesc>Figure 45</figDesc> +</figure></p> + +<p><hi rend="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).</hi></p></div> + +<div> +<index index="toc" level1="156. Parabolic reflector" /><index index="pdf" /> +<head></head><p><anchor id="p156" /><hi rend="font-weight: bold">156.</hi> 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> +<index index="toc" level1="157. Directrix. Principal axis. Vertex" /><index index="pdf" /> +<head></head><p><anchor id="p157" /><hi rend="font-weight: bold">157. Directrix. Principal axis. Vertex.</hi> The polar of +the focus with respect to the conic is called the <hi rend="font-style: italic">directrix</hi>. +The axis which contains the foci is called the <hi rend="font-style: italic">principal +axis</hi>, and the intersection of the axis with the curve is +called the <hi rend="font-style: italic">vertex</hi> 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 rend="text-align: center"> +<figure rend="w95" url="images/image46.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 46</head> +<figDesc>Figure 46</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="158. Another definition of a conic" /><index index="pdf" /> +<head></head><p><anchor id="p158" /><hi rend="font-weight: bold">158. Another definition of a conic.</hi> Let <hi rend="font-style: italic">P</hi> be any point +on the directrix through which a line is drawn meeting +the conic in the points <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi> (Fig. 46). Let the tangents +at <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi> meet in <hi rend="font-style: italic">T</hi>, and call the focus <hi rend="font-style: italic">F</hi>. Then +<hi rend="font-style: italic">TF</hi> and <hi rend="font-style: italic">PF</hi> are conjugate lines, and as they pass through +a focus they must be at right angles to each other. Let +<pb n="95" /><anchor id="Pg95" /> +<hi rend="font-style: italic">TF</hi> meet <hi rend="font-style: italic">AB</hi> in <hi rend="font-style: italic">C</hi>. Then <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">B</hi> are four harmonic +points. Project these four points parallel to <hi rend="font-style: italic">TF</hi> upon +the directrix, and we then get +the four harmonic points <hi rend="font-style: italic">P</hi>, +<hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">N</hi>. Since, now, <hi rend="font-style: italic">TFP</hi> is +a right angle, the angles <hi rend="font-style: italic">MFQ</hi> +and <hi rend="font-style: italic">NFQ</hi> are equal, as well +as the angles <hi rend="font-style: italic">AFC</hi> and <hi rend="font-style: italic">BFC</hi>. +Therefore the triangles <hi rend="font-style: italic">MAF</hi> +and <hi rend="font-style: italic">NFB</hi> are similar, and +<hi rend="font-style: italic">FA : FM = FB : BN</hi>. Dropping +perpendiculars <hi rend="font-style: italic">AA</hi> and <hi rend="font-style: italic">BB'</hi> +upon the directrix, this becomes +<hi rend="font-style: italic">FA : AA' = FB : BB'</hi>. We +have thus the property often taken as the definition +of a conic:</p> + +<p><hi rend="font-style: italic">The ratio of the distances from a point on the conic to +the focus and the directrix is constant.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image47.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 47</head> +<figDesc>Figure 47</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="159. Eccentricity" /><index index="pdf" /> +<head></head><p><anchor id="p159" /><hi rend="font-weight: bold">159. Eccentricity.</hi> 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 +<hi rend="font-style: italic">eccentricity</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image48.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 48</head> +<figDesc>Figure 48</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="160. Sum or difference of focal +distances" /><index index="pdf" /> +<head></head><p><anchor id="p160" /><hi rend="font-weight: bold">160. Sum or difference of focal +distances.</hi> 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 +<pb n="96" /><anchor id="Pg96" /> +a point on a conic to the two foci are <hi rend="font-style: italic">r</hi> and <hi rend="font-style: italic">r'</hi>, and +the distances from the same point to the corresponding +directrices are <hi rend="font-style: italic">d</hi> and <hi rend="font-style: italic">d'</hi> +(Fig. 47), we have <hi rend="font-style: italic">r : d = r' : d'</hi>; +<hi rend="font-style: italic">(r ± r') : (d ± d')</hi>. In the +ellipse <hi rend="font-style: italic">(d + d')</hi> is constant, +being the distance between +the directrices. In the hyperbola +this distance is <hi rend="font-style: italic">(d - d')</hi>. +It follows (Fig. 48) that</p> + +<p><hi rend="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.</hi></p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Construct the axis of a parabola, given four tangents.</p> + +<p>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 <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>. The circle on <hi rend="font-style: italic">AB</hi> as diameter will +pass through the foci of the conic.</p> + +<p>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>4. Given the tangent at the vertex of a parabola, and +two other tangents, to find the focus.</p> + +<p>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>6. Given the axis of a parabola and a tangent, with its +point of contact, to find the focus.</p> + +<pb n="97" /><anchor id="Pg97" /> + +<p>7. The locus of the center of a circle which touches a +given line and a given circle consists of two parabolas.</p> + +<p>8. Let <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi> be the foci of an ellipse, and <hi rend="font-style: italic">P</hi> any +point on it. Produce <hi rend="font-style: italic">PF</hi> to <hi rend="font-style: italic">G</hi>, making <hi rend="font-style: italic">PG</hi> equal to <hi rend="font-style: italic">PF'</hi>. +Find the locus of <hi rend="font-style: italic">G</hi>.</p> + +<p>9. If the points <hi rend="font-style: italic">G</hi> of a circle be folded over upon a +point <hi rend="font-style: italic">F</hi>, the creases will all be tangent to a conic. If <hi rend="font-style: italic">F</hi> is +within the circle, the conic will be an ellipse; if <hi rend="font-style: italic">F</hi> is without +the circle, the conic will be a hyperbola.</p> + +<p>10. If the points <hi rend="font-style: italic">G</hi> in the last example be taken on a +straight line, the locus is a parabola.</p> + +<p>11. Find the foci and the length of the principal axis of +the conics in problems 9 and 10.</p> + +<p>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>13. State and explain the similar problem for problem 9.</p> + +<p>14. The last four problems are a study of the consequences +of the following transformation: A point <hi rend="font-style: italic">O</hi> is fixed +in the plane. Then to any point <hi rend="font-style: italic">P</hi> is made to correspond +the line <hi rend="font-style: italic">p</hi> at right angles to <hi rend="font-style: italic">OP</hi> and bisecting it. In this +correspondence, what happens to <hi rend="font-style: italic">p</hi> when <hi rend="font-style: italic">P</hi> 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> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="98" /><anchor id="Pg98" /> +<head>CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY</head> + +<div> +<index index="toc" level1="161. Ancient results" /><index index="pdf" /> +<head></head><p><anchor id="p161" /><hi rend="font-weight: bold">161. Ancient results.</hi> 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<note place="foot"><p>The +more general notion of <hi rend="font-style: italic">anharmonic ratio</hi>, 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>Consider any four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, on a line, and join them to +any point <hi rend="font-style: italic">S</hi> not on that line. Then the triangles <hi rend="font-style: italic">ASB</hi>, <hi rend="font-style: italic">GSD</hi>, <hi rend="font-style: italic">ASD</hi>, +<hi rend="font-style: italic">CSB</hi>, 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 rend="text-align: center"><formula notation="tex">\[ +\frac{AB \times CD}{AD \times CB} = \frac{AS \times BS \sin ASB \times CS +\times DS \sin CSD}{AS \times DS \sin ASD \times CS \times BS \sin CSB} = +\frac{\sin ASB \times \sin CSD}{\sin ASD \times \sin CSB} +\]</formula></p> + +<p>Now the fraction on the right would be unchanged if instead of the +points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> we should take any other four points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi> +lying on any other line cutting across <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi>. In other +words, <hi rend="font-style: italic">the fraction on the left is unaltered in value if the points +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> are replaced by any other four points perspective to them.</hi> +Again, the fraction on the left is unchanged if some other point were +taken instead of <hi rend="font-style: italic">S</hi>. In other words, <hi rend="font-style: italic">the fraction on the right is +unaltered if we replace the four lines <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi> by any other four +lines perspective to them.</hi> The fraction on the left is called the <hi rend="font-style: italic">anharmonic +ratio</hi> of the four points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>; the fraction on the right +is called the <hi rend="font-style: italic">anharmonic ratio</hi> of the four lines <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi>. The +anharmonic ratio of four points is sometimes written (<hi rend="font-style: italic">ABCD</hi>), so that</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +\frac{AB \times CD}{AD \times CB} = (ABCD). +\] +</formula> +</p> + +<p>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 <hi rend="font-style: italic">(ABCD) = (BADC) = (CDAB) = (DCBA)</hi>. +If we write <hi rend="font-style: italic">(ABCD) = a</hi>, it is not difficult to show that +the six values are</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +a; 1/a; 1-a; 1/(1-a); (a-1)/a; a/(a-1). +\] +</formula> +</p> + +<p>The proof of this we leave to the student.</p> + +<p>If <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> are four harmonic points (see Fig. 6, p. *22), and a quadrilateral +<hi rend="font-style: italic">KLMN</hi> is constructed such that <hi rend="font-style: italic">KL</hi> and <hi rend="font-style: italic">MN</hi> pass through +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">KN</hi> and <hi rend="font-style: italic">LM</hi> through <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">LN</hi> through <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">KM</hi> through <hi rend="font-style: italic">D</hi>, then, +projecting <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> from <hi rend="font-style: italic">L</hi> upon <hi rend="font-style: italic">KM</hi>, we have <hi rend="font-style: italic">(ABCD) = (KOMD)</hi>, +where <hi rend="font-style: italic">O</hi> is the intersection of <hi rend="font-style: italic">KM</hi> with <hi rend="font-style: italic">LN</hi>. But, projecting again +the points <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">D</hi> from <hi rend="font-style: italic">N</hi> back upon the line <hi rend="font-style: italic">AB</hi>, we have +<hi rend="font-style: italic">(KOMD) = (CBAD)</hi>. From this we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">(ABCD) = (CBAD),</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +a=a/(a-1); +\] +</formula> +</p> + +<p>whence <hi rend="font-style: italic">a = 0</hi> or <hi +rend="font-style: italic">a = 2</hi>. But it is easy to see that <hi +rend="font-style: italic">a = 0</hi> 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, <formula +notation="tex">$\frac{1}{2}$</formula>, 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 rend="text-align: center"> +<figure rend="w95" url="images/image49.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 49</head> +<figDesc>Figure 49</figDesc> +</figure></p> + +<p>Many theorems of projective geometry are succinctly stated in +terms of anharmonic ratios. Thus, the <hi rend="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.</hi> We leave these +theorems for the student, who may +also justify the following solution of +the problem: <hi rend="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.</hi> +Let <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">D</hi> be the three given points +(Fig. 49). On any convenient line +through <hi rend="font-style: italic">A</hi> take two points <hi rend="font-style: italic">B'</hi> and <hi rend="font-style: italic">D'</hi> +such that <hi rend="font-style: italic">AB'/AD'</hi> is equal to the +given anharmonic ratio. Join <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">DD'</hi> and let the two lines +meet in <hi rend="font-style: italic">S</hi>. Draw through <hi rend="font-style: italic">S</hi> a parallel to <hi rend="font-style: italic">AB'</hi>. This line will meet +<hi rend="font-style: italic">AB</hi> in the required point <hi rend="font-style: italic">C</hi>.</p></note> +the involution of six points cut out by any transversal +which intersects the sides of a complete quadrilateral +<pb n="100" /><anchor id="Pg100" /> +as studied by Pappus<note place="foot"><p> +Pappus, Mathematicae Collectiones, vii, 129.</p></note>; +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, <hi rend="font-style: italic">Two lines determine a +point, and two points determine a line</hi>, 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 +<pb n="101" /><anchor id="Pg101" /> +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<note place="foot"><p>J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522.</p></note> +of Nürnberg undertook to demonstrate +the properties of the conic sections by means of +the circle.</p></div> + +<div> +<index index="toc" level1="162. Unifying principles" /><index index="pdf" /> +<head></head><p><anchor id="p162" /><hi rend="font-weight: bold">162. Unifying principles.</hi> 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<note place="foot"><p>Kepler, Ad Vitellionem paralipomena quibus astronomiae pars +optica traditur. 1604.</p></note> 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" +(<hi rend="font-style: italic">caecus focus</hi>) at infinity on the axis.</p></div> + +<div> +<index index="toc" level1="163. Desargues" /><index index="pdf" /> +<head></head><p><anchor id="p163" /><hi rend="font-weight: bold">163. Desargues.</hi> In 1639 Desargues,<note place="foot"><p>Desargues, Bruillon-project d'une atteinte aux événements des +rencontres d'un cône avec un plan. 1639. Edited and analyzed by +Poudra, 1864.</p></note> 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). +<pb n="102" /><anchor id="Pg102" /> +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> +<index index="toc" level1="164. Poles and polars" /><index index="pdf" /> +<head></head><p><anchor id="p164" /><hi rend="font-weight: bold">164. Poles and polars.</hi> In this little treatise is also +contained the theory of poles and polars. The polar +line is called a <hi rend="font-style: italic">traversal</hi>.<note place="foot"><p>The term 'pole' was first introduced, in the sense in which we +have used it, in 1810, by a French mathematician named Servois +(Gergonne, <hi rend="font-style: italic">Annales des Mathéématiques</hi>, I, 337), and the corresponding +term 'polar' by the editor, Gergonne, of this same journal three years +later.</p></note> 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 <hi rend="font-style: italic">traversal</hi> is at an infinite +distance, all is unimaginable."</p></div> + +<div> +<index index="toc" level1="165. Desargues's theorem concerning conics through +four points" /><index index="pdf" /> +<head></head><p><anchor id="p165" /><hi rend="font-weight: bold">165. Desargues's theorem concerning conics through +four points.</hi> 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 +<pb n="103" /><anchor id="Pg103" /> +any such system. He states the theorem, in effect, as +follows: <hi rend="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.</hi></p></div> + +<div> +<index index="toc" level1="166. Extension of the theory of poles and polars to +space" /><index index="pdf" /> +<head></head><p><anchor id="p166" /><hi rend="font-weight: bold">166. Extension of the theory of poles and polars to +space.</hi> 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 <hi rend="font-style: italic">P</hi> in +space a line to cut a sphere in two points, <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">S</hi>, and +then construct the fourth harmonic of <hi rend="font-style: italic">P</hi> with respect to +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>, the locus of this fourth harmonic, for various +lines through <hi rend="font-style: italic">P</hi>, is a plane called the <hi rend="font-style: italic">polar plane</hi> of <hi rend="font-style: italic">P</hi> +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 +<pb n="104" /><anchor id="Pg104" /> +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.<note place="foot"><p> +Euler, Introductio in analysin infinitorum, Appendix, cap. V. +1748.</p></note></p></div> + +<div> +<index index="toc" level1="167. Desargues's method of describing a conic" /><index index="pdf" /> +<head></head><p><anchor id="p167" /><hi rend="font-weight: bold">167.</hi> 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> +<index index="toc" level1="168. Reception of Desargues's work" /><index index="pdf" /> +<head></head><p><anchor id="p168" /><hi rend="font-weight: bold">168. Reception of Desargues's work.</hi> 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> + +<pb n="105" /><anchor id="Pg105" /></div> + +<div> +<index index="toc" level1="169. Conservatism in Desargues's time" /><index index="pdf" /> +<head></head><p><anchor id="p169" /><hi rend="font-weight: bold">169. Conservatism in Desargues's time.</hi> 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> +<index index="toc" level1="170. Desargues's style of writing" /><index index="pdf" /> +<head></head><p><anchor id="p170" /><hi rend="font-weight: bold">170. Desargues's style of writing.</hi> Nevertheless, authority +counted for less at this time in Paris than it did in +Italy, and the tragedy enacted in Rome when Galileo +<pb n="106" /><anchor id="Pg106" /> +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.<note place="foot"><p> +Œuvres de Desargues, t. II, 132.</p></note> "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 +<pb n="107" /><anchor id="Pg107" /> +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, <hi rend="font-style: italic">involution</hi>, has survived. +Curiously enough, this is the one term singled +out for the sharpest criticism and ridicule by his reviewer, +De Beaugrand.<note place="foot"><p> +Œuvres de Desargues, t. II, 370.</p></note> 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> +<index index="toc" level1="171. Lack of appreciation of Desargues" /><index index="pdf" /> +<head></head><p><anchor id="p171" /><hi rend="font-weight: bold">171. Lack of appreciation of Desargues.</hi> 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<note place="foot"><p> +Œuvres de Descartes, t. II, 499.</p></note> 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> + +<pb n="108" /><anchor id="Pg108" /></div> + +<div> +<index index="toc" level1="172. Pascal and his theorem" /><index index="pdf" /> +<head></head><p><anchor id="p172" /><hi rend="font-weight: bold">172. Pascal and his theorem.</hi> 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,"<note place="foot"><p> +Œuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252. +</p></note> 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: <hi rend="font-style: italic">"If in the plane of <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">S</hi>, <hi rend="font-style: italic">Q</hi> we draw +through <hi rend="font-style: italic">M</hi> the two lines <hi rend="font-style: italic">MK</hi> and <hi rend="font-style: italic">MV</hi>, and through the +point <hi rend="font-style: italic">S</hi> the two lines <hi rend="font-style: italic">SK</hi> and <hi rend="font-style: italic">SV</hi>, and let <hi rend="font-style: italic">K</hi> be the intersection +of <hi rend="font-style: italic">MK</hi> and <hi rend="font-style: italic">SK</hi>; <hi rend="font-style: italic">V</hi> the intersection of <hi rend="font-style: italic">MV</hi> and +<hi rend="font-style: italic">SV</hi>; <hi rend="font-style: italic">A</hi> the intersection of <hi rend="font-style: italic">MA</hi> and <hi rend="font-style: italic">SA</hi> (<hi rend="font-style: italic">A</hi> is the intersection +of <hi rend="font-style: italic">SV</hi> and <hi rend="font-style: italic">MK</hi>), and <hi rend="font-style: italic">μ</hi> the intersection of <hi rend="font-style: italic">MV</hi> +and <hi rend="font-style: italic">SK</hi>; and if through two of the four points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">K</hi>, +<hi rend="font-style: italic">μ</hi>, <hi rend="font-style: italic">V</hi>, which are not in the same straight line with <hi rend="font-style: italic">M</hi> and +<hi rend="font-style: italic">S</hi>, such as <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">V</hi>, we pass the circumference of a circle +cutting the lines <hi rend="font-style: italic">MV</hi>, <hi rend="font-style: italic">MP</hi>, <hi rend="font-style: italic">SV</hi>, <hi rend="font-style: italic">SK</hi> in the points <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">P</hi>, +<hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">N</hi>; I say that the lines <hi rend="font-style: italic">MS</hi>, <hi rend="font-style: italic">NO</hi>, <hi rend="font-style: italic">PQ</hi> are of the same +order."</hi> (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> +<index index="toc" level1="173. Pascal's essay" /><index index="pdf" /> +<head></head><p><anchor id="p173" /><hi rend="font-weight: bold">173.</hi> 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 +<pb n="109" /><anchor id="Pg109" /> +days, and also to his later morbid interest in religious +matters, was never published. Leibniz<note place="foot"><p> +Chasles, Histoire de la Géométrie, 70.</p></note> 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> +<index index="toc" level1="174. Pascal's originality" /><index index="pdf" /> +<head></head><p><anchor id="p174" /><hi rend="font-weight: bold">174.</hi> 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> +<index index="toc" level1="175. De la Hire and his work" /><index index="pdf" /> +<head></head><p><anchor id="p175" /><hi rend="font-weight: bold">175. De la Hire and his work.</hi> De la Hire added +little to the development of the subject, but he did put +into print much of what Desargues had already worked +<pb n="110" /><anchor id="Pg110" /> +out, not fully realizing, perhaps, how much was his +own and how much he owed to his teacher. Writing in +1679, he says,<note place="foot"><p> +Œuvres de Desargues, t. I, 231.</p></note> "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: <hi rend="font-style: italic">Draw +two parallel lines, <hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">b</hi>, and select a point <hi rend="font-style: italic">P</hi> in their +plane. Through any point <hi rend="font-style: italic">M</hi> of the plane draw a line +meeting <hi rend="font-style: italic">a</hi> in <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">b</hi> in <hi rend="font-style: italic">B</hi>. Draw a line through <hi rend="font-style: italic">B</hi> +parallel to <hi rend="font-style: italic">AP</hi>, and let it meet <hi rend="font-style: italic">MP</hi> in the point <hi rend="font-style: italic">M'</hi>. It +may be shown that the point <hi rend="font-style: italic">M'</hi> thus obtained does not +depend at all on the particular ray <hi rend="font-style: italic">MAB</hi> used in determining +it, so that we have set up a one-to-one correspondence +between the points <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">M'</hi> in the plane.</hi> The student +may show that as <hi rend="font-style: italic">M</hi> describes a point-row, <hi rend="font-style: italic">M'</hi> describes +a point-row projective to it. As <hi rend="font-style: italic">M</hi> describes a conic, +<hi rend="font-style: italic">M'</hi> describes another conic. This sort of correspondence +is called a <hi rend="font-style: italic">collineation</hi>. It will be found that the +points on the line <hi rend="font-style: italic">b</hi> transform into themselves, as does +also the single point <hi rend="font-style: italic">P</hi>. Points on the line <hi rend="font-style: italic">a</hi> 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 <hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">b</hi>, any +two lines which may meet in a finite part of the plane. +<pb n="111" /><anchor id="Pg111" /> +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> +<index index="toc" level1="176. Descartes and his influence" /><index index="pdf" /> +<head></head><p><anchor id="p176" /><hi rend="font-weight: bold">176. Descartes and his influence.</hi> 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 +<pb n="112" /><anchor id="Pg112" /> +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,<note place="foot"><p> +See Ball, History of Mathematics, French edition, t. II, 233. +</p></note> "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> +<index index="toc" level1="177. Newton and Maclaurin" /><index index="pdf" /> +<head></head><p><anchor id="p177" /><hi rend="font-weight: bold">177. Newton and Maclaurin.</hi> 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"<note place="foot"><p> +Newton, Principia, lib. i, lemma XXI.</p></note> +of generating conic sections is closely related to the +method which we have made use of in Chapter III. +It is as follows: <hi rend="font-style: italic">If two angles, <hi rend="font-style: italic">AOS</hi> and <hi rend="font-style: italic">AO'S</hi>, of given +magnitudes turn about their respective vertices, <hi rend="font-style: italic">O</hi> and <hi rend="font-style: italic">O'</hi>, +in such a way that the point of intersection, <hi rend="font-style: italic">S</hi>, of one pair +of arms always lies on a straight line, the point of intersection, +<hi rend="font-style: italic">A</hi>, of the other pair of arms will describe a conic.</hi> +The proof of this is left to the student.</p></div> + +<div> +<index index="toc" level1="178. Maclaurin's construction" /><index index="pdf" /> +<head></head><p><anchor id="p178" /><hi rend="font-weight: bold">178.</hi> Another method of generating a conic is due to +Maclaurin.<note place="foot"><p> +Maclaurin, Philosophical Transactions of the Royal Society of +London, 1735.</p></note> The construction, which we also leave for +the student to justify, is as follows: <hi rend="font-style: italic">If a triangle <hi rend="font-style: italic">C'PQ</hi> +move in such a way that its sides, <hi rend="font-style: italic">PQ</hi>, <hi rend="font-style: italic">QC'</hi>, and <hi rend="font-style: italic">C'P</hi>, turn +<pb n="113" /><anchor id="Pg113" /> +around three fixed points, <hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, respectively, while two of +its vertices, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, slide along two fixed lines, <hi rend="font-style: italic">CB'</hi> and <hi rend="font-style: italic">CA'</hi>, +respectively, then the remaining vertex will describe a conic.</hi></p></div> + +<div> +<index index="toc" level1="179. Descriptive geometry and the second revival" /><index index="pdf" /> +<head></head><p><anchor id="p179" /><hi rend="font-weight: bold">179. Descriptive geometry and the second revival.</hi> +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<note place="foot"><p> +Monge, Géométrie Descriptive. 1800.</p></note> of the theory of <hi rend="font-style: italic">descriptive geometry</hi>. +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 +<pb n="114" /><anchor id="Pg114" /> +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> +<index index="toc" level1="180. Duality, homology, continuity, contingent relations" /><index index="pdf" /> +<head></head><p><anchor id="p180" /><hi rend="font-weight: bold">180. Duality, homology, continuity, contingent relations.</hi> +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> + +<pb n="115" /><anchor id="Pg115" /></div> + +<div> +<index index="toc" level1="181. Poncelet and Cauchy" /><index index="pdf" /> +<head></head><p><anchor id="p181" /><hi rend="font-weight: bold">181. Poncelet and Cauchy.</hi> 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<note place="foot"><p> +Poncelet, Traité des Propriétés Projectives des Figures. 1822. +(See p. 357, Vol. II, of the edition of 1866.)</p></note> +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 +<pb n="116" /><anchor id="Pg116" /> +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. <hi rend="font-style: italic">Apart from the +algebra involved</hi>, it is the height of absurdity to try to +distinguish between the two points in which a line +<hi rend="font-style: italic">fails to meet a conic!</hi></p></div> + +<div> +<index index="toc" level1="182. The work of Poncelet" /><index index="pdf" /> +<head></head><p><anchor id="p182" /><hi rend="font-weight: bold">182. The work of Poncelet.</hi> 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 <hi rend="font-style: italic">in homology</hi>, 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,<note place="foot"><p> +Gergonne, <hi rend="font-style: italic">Annales de Mathématiques, XVI, 209. 1826.</hi></p></note> +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> +<index index="toc" level1="183. The debt which analytic geometry owes to synthetic +geometry" /><index index="pdf" /> +<head></head><p><anchor id="p183" /><hi rend="font-weight: bold">183. The debt which analytic geometry owes to synthetic +geometry.</hi> The reaction of pure geometry on +<pb n="117" /><anchor id="Pg117" /> +analytic geometry is clearly seen in the development of +the notion of the <hi rend="font-style: italic">class</hi> 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> +<index index="toc" level1="184. Steiner and his work" /><index index="pdf" /> +<head></head><p><anchor id="p184" /><hi rend="font-weight: bold">184. Steiner and his work.</hi> 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,<note place="foot"><p> +Steiner, Systematische Ehtwickelung der Abhängigkeit geometrischer +Gestalten von einander. 1832.</p></note> +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, +<pb n="118" /><anchor id="Pg118" /> +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> +<index index="toc" level1="185. Von Staudt and his work" /><index index="pdf" /> +<head></head><p><anchor id="p185" /><hi rend="font-weight: bold">185. Von Staudt and his work.</hi> 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,<note place="foot"><p> +Von Staudt, Geometrie der Lage. 1847.</p></note> +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 +<pb n="119" /><anchor id="Pg119" /> +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<note place="foot"><p> +Reye, Geometrie der Lage. Translated by Holgate, 1897.</p></note> +and others have done much toward simplifying +his presentation.</p></div> + +<div> +<index index="toc" level1="186. Recent developments" /><index index="pdf" /> +<head></head><p><anchor id="p186" /><hi rend="font-weight: bold">186. Recent developments.</hi> 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 +<pb n="120" /><anchor id="Pg120" /> +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,<note place="foot"><p> +Ball, loc. cit. p. 261.</p></note> +"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> + + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<head>INDEX</head> + +<p rend="text-align: center">(The numbers refer to the paragraphs)</p> + +<p rend="text-indent: 0">Abel (1802-1829), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Analogy, <ref target="p24">24</ref><lb /></p> + +<p rend="text-indent: 0">Analytic geometry, <ref target="p21">21</ref>, <ref target="p118">118</ref>, <ref target="p119">119</ref>, +120, <ref target="p146">146</ref>, <ref target="p176">176</ref>, <ref target="p180">180</ref><lb /></p> + +<p rend="text-indent: 0">Anharmonic ratio, <ref target="p46">46</ref>, <ref target="p161">161</ref>, <ref target="p184">184</ref>, <ref target="p185">185</ref><lb /></p> + +<p rend="text-indent: 0">Apollonius (second half of third +century B.C.), <ref target="p70">70</ref><lb /></p> + +<p rend="text-indent: 0">Archimedes (287-212 B.C.), <ref target="p176">176</ref><lb /></p> + +<p rend="text-indent: 0">Aristotle (384-322 B.C.), <ref target="p169">169</ref><lb /></p> + +<p rend="text-indent: 0">Asymptotes, <ref target="p111">111</ref>, <ref target="p113">113</ref>, <ref target="p114">114</ref>, <ref target="p115">115</ref>, +116, <ref target="p117">117</ref>, <ref target="p118">118</ref>, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Axes of a conic, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Axial pencil, <ref target="p7">7</ref>, <ref target="p8">8</ref>, <ref target="p23">23</ref>, <ref target="p50">50</ref>, <ref target="p54">54</ref><lb /></p> + +<p rend="text-indent: 0">Axis of perspectivity, <ref target="p8">8</ref>, <ref target="p47">47</ref><lb /></p> + +<p rend="text-indent: 0">Bacon (1561-1626), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Bisection, <ref target="p41">41</ref>, <ref target="p109">109</ref><lb /></p> + +<p rend="text-indent: 0">Brianchon (1785-1864), <ref target="p84">84</ref>, <ref target="p85">85</ref>, <ref target="p86">86</ref>, +88, <ref target="p89">89</ref>, <ref target="p90">90</ref>, <ref target="p95">95</ref>, <ref target="p105">105</ref>, <ref target="p113">113</ref>, <ref target="p174">174</ref>, <ref target="p184">184</ref> <lb /></p> + +<p rend="text-indent: 0">Calculus, <ref target="p176">176</ref><lb /></p> + +<p rend="text-indent: 0">Carnot (1796-1832), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Cauchy (1789-1857), <ref target="p179">179</ref>, <ref target="p181">181</ref><lb /></p> + +<p rend="text-indent: 0">Cavalieri (1598-1647), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Center of a conic, <ref target="p107">107</ref>, <ref target="p112">112</ref>, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Center of involution, <ref target="p141">141</ref>, <ref target="p142">142</ref><lb /></p> + +<p rend="text-indent: 0">Center of perspectivity, <ref target="p8">8</ref><lb /></p> + +<p rend="text-indent: 0">Central conic, <ref target="p120">120</ref><lb /></p> + +<p rend="text-indent: 0">Chasles (1793-1880), <ref target="p168">168</ref>, <ref target="p179">179</ref>, <ref target="p180">180</ref>, +184<lb /></p> + +<p rend="text-indent: 0">Circle, <ref target="p21">21</ref>, <ref target="p73">73</ref>, <ref target="p80">80</ref>, <ref target="p145">145</ref>, <ref target="p146">146</ref>, <ref target="p147">147</ref><lb /></p> + +<p rend="text-indent: 0">Circular involution, <ref target="p147">147</ref>, <ref target="p149">149</ref>, <ref target="p150">150</ref>, +151<lb /></p> + +<p rend="text-indent: 0">Circular points, <ref target="p146">146</ref><lb /></p> + +<p rend="text-indent: 0">Class of a curve, <ref target="p183">183</ref><lb /></p> + +<p rend="text-indent: 0">Classification of conics, <ref target="p110">110</ref><lb /></p> + +<p rend="text-indent: 0">Collineation, <ref target="p175">175</ref><lb /></p> + +<p rend="text-indent: 0">Concentric pencils, <ref target="p50">50</ref><lb /></p> + +<p rend="text-indent: 0">Cone of the second order, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Conic, <ref target="p73">73</ref>, <ref target="p81">81</ref><lb /></p> + +<p rend="text-indent: 0">Conjugate diameters, <ref target="p114">114</ref>, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Conjugate normal, <ref target="p151">151</ref><lb /></p> + +<p rend="text-indent: 0">Conjugate points and lines, <ref target="p100">100</ref>, +109, <ref target="p138">138</ref>, <ref target="p139">139</ref>, <ref target="p140">140</ref><lb /></p> + +<p rend="text-indent: 0">Constants in an equation, <ref target="p21">21</ref><lb /></p> + +<p rend="text-indent: 0">Contingent relations, <ref target="p180">180</ref>, <ref target="p181">181</ref><lb /></p> + +<p rend="text-indent: 0">Continuity, <ref target="p180">180</ref>, <ref target="p181">181</ref><lb /></p> + +<p rend="text-indent: 0">Continuous correspondence, <ref target="p9">9</ref>, <ref target="p10">10</ref>, +21, <ref target="p49">49</ref><lb /></p> + +<p rend="text-indent: 0">Corresponding elements, <ref target="p64">64</ref><lb /></p> + +<p rend="text-indent: 0">Counting, <ref target="p1">1</ref>, <ref target="p4">4</ref><lb /></p> + +<p rend="text-indent: 0">Cross ratio, <ref target="p46">46</ref><lb /></p> + +<p rend="text-indent: 0">Darboux, <ref target="p176">176</ref>, <ref target="p186">186</ref><lb /></p> + +<p rend="text-indent: 0">De Beaugrand, <ref target="p170">170</ref><lb /></p> + +<p rend="text-indent: 0">Degenerate pencil of rays of the +second order, <ref target="p58">58</ref>, <ref target="p93">93</ref><lb /></p> + +<p rend="text-indent: 0">Degenerate point-row of the +second order, <ref target="p56">56</ref>, <ref target="p78">78</ref><lb /></p> + +<p rend="text-indent: 0">De la Hire (1640-1718), <ref target="p168">168</ref>, <ref target="p171">171</ref>, +175<lb /></p> + +<p rend="text-indent: 0">Desargues (1593-1662), <ref target="p25">25</ref>, <ref target="p26">26</ref>, <ref target="p40">40</ref>, +121, <ref target="p125">125</ref>, <ref target="p162">162</ref>, <ref target="p163">163</ref>, <ref target="p164">164</ref>, <ref target="p165">165</ref>, <ref target="p166">166</ref>, +167, <ref target="p168">168</ref>, <ref target="p169">169</ref>, <ref target="p170">170</ref>, <ref target="p171">171</ref>, <ref target="p174">174</ref>, <ref target="p175">175</ref><lb /></p> + +<p rend="text-indent: 0">Descartes (1596-1650), <ref target="p162">162</ref>, <ref target="p170">170</ref>, +171, <ref target="p174">174</ref>, <ref target="p176">176</ref><lb /></p> + +<p rend="text-indent: 0">Descriptive geometry, <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Diameter, <ref target="p107">107</ref><lb /></p> + +<p rend="text-indent: 0">Directrix, <ref target="p157">157</ref>, <ref target="p158">158</ref>, <ref target="p159">159</ref>, <ref target="p160">160</ref><lb /></p> + +<p rend="text-indent: 0">Double correspondence, <ref target="p128">128</ref>, <ref target="p130">130</ref><lb /></p> + +<p rend="text-indent: 0">Double points of an involution, <ref target="p124">124</ref><lb /></p> + +<p rend="text-indent: 0">Double rays of an involution, <ref target="p133">133</ref>, +134<lb /></p> + +<p rend="text-indent: 0">Duality, <ref target="p94">94</ref>, <ref target="p104">104</ref>, <ref target="p161">161</ref>, <ref target="p180">180</ref>, <ref target="p182">182</ref><lb /></p> + +<p rend="text-indent: 0">Dupin (1784-1873), <ref target="p174">174</ref>, <ref target="p184">184</ref> <lb /></p> + +<p rend="text-indent: 0">Eccentricity of conic, <ref target="p159">159</ref><lb /></p> + +<p rend="text-indent: 0">Ellipse, <ref target="p110">110</ref>, <ref target="p111">111</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Equation of conic, <ref target="p118">118</ref>, <ref target="p119">119</ref>, <ref target="p120">120</ref><lb /></p> + +<p rend="text-indent: 0">Euclid (ca. 300 B.C.), <ref target="p6">6</ref>, <ref target="p22">22</ref>, <ref target="p104">104</ref><lb /></p> + +<p rend="text-indent: 0">Euler (1707-1783), <ref target="p166">166</ref> <lb /></p> + +<p rend="text-indent: 0">Fermat (1601-1665), <ref target="p162">162</ref>, <ref target="p171">171</ref><lb /></p> + +<p rend="text-indent: 0">Foci of a conic, <ref target="p152">152</ref>, <ref target="p153">153</ref>, <ref target="p154">154</ref>, <ref target="p155">155</ref>, +156, <ref target="p157">157</ref>, <ref target="p158">158</ref>, <ref target="p159">159</ref>, <ref target="p160">160</ref>, <ref target="p161">161</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Fourier (1768-1830), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Fourth harmonic, <ref target="p29">29</ref><lb /></p> + +<p rend="text-indent: 0">Fundamental form, <ref target="p7">7</ref>, <ref target="p16">16</ref>, <ref target="p23">23</ref>, <ref target="p36">36</ref>, +47, <ref target="p60">60</ref>, <ref target="p184">184</ref> <lb /></p> + +<p rend="text-indent: 0">Galileo (1564-1642), <ref target="p162">162</ref>, <ref target="p169">169</ref>, <ref target="p170">170</ref>, +176<lb /></p> + +<p rend="text-indent: 0">Gauss (1777-1855), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Gergonne (1771-1859), <ref target="p182">182</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Greek geometry, <ref target="p161">161</ref> <lb /></p> + +<p rend="text-indent: 0">Hachette (1769-1834), <ref target="p179">179</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic conjugates, <ref target="p29">29</ref>, <ref target="p30">30</ref>, <ref target="p39">39</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic elements, <ref target="p86">86</ref>, <ref target="p49">49</ref>, <ref target="p91">91</ref>, +163, <ref target="p185">185</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic lines, <ref target="p33">33</ref>, <ref target="p34">34</ref>, <ref target="p35">35</ref>, <ref target="p66">66</ref>, <ref target="p67">67</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic planes, <ref target="p34">34</ref>, <ref target="p35">35</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic points, <ref target="p29">29</ref>, <ref target="p31">31</ref>, <ref target="p32">32</ref>, <ref target="p33">33</ref>, +34, <ref target="p35">35</ref>, <ref target="p36">36</ref>, <ref target="p43">43</ref>, <ref target="p71">71</ref>, <ref target="p161">161</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic tangents to a conic, +91, <ref target="p92">92</ref><lb /></p> + +<p rend="text-indent: 0">Harvey (1578-1657), <ref target="p169">169</ref><lb /></p> + +<p rend="text-indent: 0">Homology, <ref target="p180">180</ref>, <ref target="p182">182</ref><lb /></p> + +<p rend="text-indent: 0">Huygens (1629-1695), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Hyperbola, <ref target="p110">110</ref>, <ref target="p111">111</ref>, <ref target="p113">113</ref>, <ref target="p114">114</ref>, <ref target="p115">115</ref>, +116, <ref target="p117">117</ref>, <ref target="p118">118</ref>, <ref target="p162">162</ref> <lb /></p> + +<p rend="text-indent: 0">Imaginary elements, <ref target="p146">146</ref>, <ref target="p180">180</ref>, <ref target="p181">181</ref>, +182, <ref target="p185">185</ref><lb /></p> + +<p rend="text-indent: 0">Infinitely distant elements, <ref target="p6">6</ref>, <ref target="p9">9</ref>, +22, <ref target="p39">39</ref>, <ref target="p40">40</ref>, <ref target="p41">41</ref>, <ref target="p104">104</ref>, <ref target="p107">107</ref>, <ref target="p110">110</ref><lb /></p> + +<p rend="text-indent: 0">Infinity, <ref target="p4">4</ref>, <ref target="p5">5</ref>, <ref target="p10">10</ref>, <ref target="p12">12</ref>, <ref target="p13">13</ref>, <ref target="p14">14</ref>, <ref target="p15">15</ref>, +17, <ref target="p18">18</ref>, <ref target="p19">19</ref>, <ref target="p20">20</ref>, <ref target="p21">21</ref>, <ref target="p22">22</ref>, <ref target="p41">41</ref><lb /></p> + +<p rend="text-indent: 0">Involution, <ref target="p37">37</ref>, <ref target="p123">123</ref>, <ref target="p124">124</ref>, <ref target="p125">125</ref>, <ref target="p126">126</ref>, +127, <ref target="p128">128</ref>, <ref target="p129">129</ref>, <ref target="p130">130</ref>, <ref target="p131">131</ref>, <ref target="p132">132</ref>, <ref target="p133">133</ref>, +134, <ref target="p135">135</ref>, <ref target="p136">136</ref>, <ref target="p137">137</ref>, <ref target="p138">138</ref>, <ref target="p139">139</ref>, <ref target="p140">140</ref>, +161, <ref target="p163">163</ref>, <ref target="p170">170</ref> <lb /></p> + +<p rend="text-indent: 0">Kepler (1571-1630), <ref target="p162">162</ref> <lb /></p> + +<p rend="text-indent: 0">Lagrange (1736-1813), <ref target="p176">176</ref>, <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Laplace (1749-1827), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Legendre (1752-1833), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Leibniz (1646-1716), <ref target="p173">173</ref><lb /></p> + +<p rend="text-indent: 0">Linear construction, <ref target="p40">40</ref>, <ref target="p41">41</ref>, <ref target="p42">42</ref> <lb /></p> + +<p rend="text-indent: 0">Maclaurin (1698-1746), <ref target="p177">177</ref>, <ref target="p178">178</ref><lb /></p> + +<p rend="text-indent: 0">Measurements, <ref target="p23">23</ref>, <ref target="p40">40</ref>, <ref target="p41">41</ref>, <ref target="p104">104</ref><lb /></p> + +<p rend="text-indent: 0">Mersenne (1588-1648), <ref target="p168">168</ref>, <ref target="p171">171</ref><lb /></p> + +<p rend="text-indent: 0">Metrical theorems, <ref target="p40">40</ref>, <ref target="p104">104</ref>, <ref target="p106">106</ref>, +107, <ref target="p141">141</ref><lb /></p> + +<p rend="text-indent: 0">Middle point, <ref target="p39">39</ref>, <ref target="p41">41</ref><lb /></p> + +<p rend="text-indent: 0">Möbius (1790-1868), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Monge (1746-1818), <ref target="p179">179</ref>, <ref target="p180">180</ref> <lb /></p> + +<p rend="text-indent: 0">Napier (1550-1617), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Newton (1642-1727), <ref target="p177">177</ref><lb /></p> + +<p rend="text-indent: 0">Numbers, <ref target="p4">4</ref>, <ref target="p21">21</ref>, <ref target="p43">43</ref><lb /></p> + +<p rend="text-indent: 0">Numerical computations, <ref target="p43">43</ref>, <ref target="p44">44</ref>, +46 <lb /></p> + +<p rend="text-indent: 0">One-to-one correspondence, <ref target="p1">1</ref>, <ref target="p2">2</ref>, +3, <ref target="p4">4</ref>, <ref target="p5">5</ref>, <ref target="p6">6</ref>, <ref target="p7">7</ref>, <ref target="p9">9</ref>, <ref target="p10">10</ref>, <ref target="p11">11</ref>, <ref target="p24">24</ref>, <ref target="p36">36</ref>, +87, <ref target="p43">43</ref>, <ref target="p60">60</ref>, <ref target="p104">104</ref>, <ref target="p106">106</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Opposite sides of a hexagon, <ref target="p70">70</ref><lb /></p> + +<p rend="text-indent: 0">Opposite sides of a quadrilateral, +28, <ref target="p29">29</ref><lb /></p> + +<p rend="text-indent: 0">Order of a form, <ref target="p7">7</ref>, <ref target="p10">10</ref>, <ref target="p11">11</ref>, <ref target="p12">12</ref>, <ref target="p13">13</ref>, +14, <ref target="p15">15</ref>, <ref target="p16">16</ref>, <ref target="p17">17</ref>, <ref target="p18">18</ref>, <ref target="p19">19</ref>, <ref target="p20">20</ref>, <ref target="p21">21</ref> <lb /></p> + +<p rend="text-indent: 0">Pappus (fourth century A.D.), +161<lb /></p> + +<p rend="text-indent: 0">Parabola, <ref target="p110">110</ref>, <ref target="p111">111</ref>, <ref target="p112">112</ref>, <ref target="p119">119</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Parallel lines, <ref target="p39">39</ref>, <ref target="p41">41</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Pascal (1623-1662), <ref target="p69">69</ref>, <ref target="p70">70</ref>, <ref target="p74">74</ref>, <ref target="p75">75</ref>, +76, <ref target="p77">77</ref>, <ref target="p78">78</ref>, <ref target="p95">95</ref>, <ref target="p105">105</ref>, <ref target="p125">125</ref>, <ref target="p162">162</ref>, +169, <ref target="p171">171</ref>, <ref target="p172">172</ref>, <ref target="p173">173</ref><lb /></p> + +<p rend="text-indent: 0">Pencil of planes of the second +order, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Pencil of rays, <ref target="p6">6</ref>, <ref target="p7">7</ref>, <ref target="p8">8</ref>, <ref target="p23">23</ref>; + of the second order, <ref target="p57">57</ref>, <ref target="p60">60</ref>, <ref target="p79">79</ref>, <ref target="p81">81</ref><lb /></p> + +<p rend="text-indent: 0">Perspective position, <ref target="p6">6</ref>, <ref target="p8">8</ref>, <ref target="p35">35</ref>, <ref target="p37">37</ref>, +51, <ref target="p53">53</ref>, <ref target="p71">71</ref><lb /></p> + +<p rend="text-indent: 0">Plane system, <ref target="p16">16</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Planes on space, <ref target="p17">17</ref><lb /></p> + +<p rend="text-indent: 0">Point of contact, <ref target="p87">87</ref>, <ref target="p88">88</ref>, <ref target="p89">89</ref>, <ref target="p90">90</ref><lb /></p> + +<p rend="text-indent: 0">Point system, <ref target="p16">16</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Point-row, <ref target="p6">6</ref>, <ref target="p7">7</ref>, <ref target="p8">8</ref>, <ref target="p9">9</ref>, <ref target="p23">23</ref>; + of the second order, <ref target="p55">55</ref>, <ref target="p60">60</ref>, <ref target="p61">61</ref>, <ref target="p66">66</ref>, + <ref target="p67">67</ref>, <ref target="p72">72</ref><lb /></p> + +<p rend="text-indent: 0">Points in space, <ref target="p18">18</ref><lb /></p> + +<p rend="text-indent: 0">Pole and polar, <ref target="p98">98</ref>, <ref target="p99">99</ref>, <ref target="p100">100</ref>, <ref target="p101">101</ref>, +138, <ref target="p164">164</ref>, <ref target="p166">166</ref><lb /></p> + +<p rend="text-indent: 0">Poncelet (1788-1867), <ref target="p177">177</ref>, <ref target="p179">179</ref>, +180, <ref target="p181">181</ref>, <ref target="p182">182</ref>, <ref target="p183">183</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Principal axis of a conic, <ref target="p157">157</ref><lb /></p> + +<p rend="text-indent: 0">Projection, <ref target="p161">161</ref><lb /></p> + +<p rend="text-indent: 0">Protective axial pencils, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Projective correspondence, <ref target="p9">9</ref>, <ref target="p35">35</ref>, +36, <ref target="p37">37</ref>, <ref target="p47">47</ref>, <ref target="p71">71</ref>, <ref target="p92">92</ref>, <ref target="p104">104</ref><lb /></p> + +<p rend="text-indent: 0">Projective pencils, <ref target="p53">53</ref>, <ref target="p64">64</ref>, <ref target="p68">68</ref><lb /></p> + +<p rend="text-indent: 0">Projective point-rows, <ref target="p51">51</ref>, <ref target="p79">79</ref><lb /></p> + +<p rend="text-indent: 0">Projective properties, <ref target="p24">24</ref><lb /></p> + +<p rend="text-indent: 0">Projective theorems, <ref target="p40">40</ref>, <ref target="p104">104</ref> <lb /></p> + +<p rend="text-indent: 0">Quadrangle, <ref target="p26">26</ref>, <ref target="p27">27</ref>, <ref target="p28">28</ref>, <ref target="p29">29</ref><lb /></p> + +<p rend="text-indent: 0">Quadric cone, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Quadrilateral, <ref target="p88">88</ref>, <ref target="p95">95</ref>, <ref target="p96">96</ref> <lb /></p> + +<p rend="text-indent: 0">Roberval (1602-1675), <ref target="p168">168</ref><lb /></p> + +<p rend="text-indent: 0">Ruler construction, <ref target="p40">40</ref> <lb /></p> + +<p rend="text-indent: 0">Scheiner, <ref target="p169">169</ref><lb /></p> + +<p rend="text-indent: 0">Self-corresponding elements, <ref target="p47">47</ref>, +48, <ref target="p49">49</ref>, <ref target="p50">50</ref>, <ref target="p51">51</ref><lb /></p> + +<p rend="text-indent: 0">Self-dual, <ref target="p105">105</ref><lb /></p> + +<p rend="text-indent: 0">Self-polar triangle, <ref target="p102">102</ref><lb /></p> + +<p rend="text-indent: 0">Separation of elements in involution, +148<lb /></p> + +<p rend="text-indent: 0">Separation of harmonic conjugates, +38<lb /></p> + +<p rend="text-indent: 0">Sequence of points, <ref target="p49">49</ref><lb /></p> + +<p rend="text-indent: 0">Sign of segment, <ref target="p44">44</ref>, <ref target="p45">45</ref><lb /></p> + +<p rend="text-indent: 0">Similarity, <ref target="p106">106</ref><lb /></p> + +<p rend="text-indent: 0">Skew lines, <ref target="p12">12</ref><lb /></p> + +<p rend="text-indent: 0">Space system, <ref target="p19">19</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Sphere, <ref target="p21">21</ref><lb /></p> + +<p rend="text-indent: 0">Steiner (1796-1863), <ref target="p129">129</ref>, <ref target="p130">130</ref>, <ref target="p131">131</ref>, +177, <ref target="p179">179</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Steiner's construction, <ref target="p129">129</ref>, <ref target="p130">130</ref>, +131<lb /></p> + +<p rend="text-indent: 0">Superposed point-rows, <ref target="p47">47</ref>, <ref target="p48">48</ref>, <ref target="p49">49</ref><lb /></p> + +<p rend="text-indent: 0">Surfaces of the second degree, <ref target="p166">166</ref><lb /></p> + +<p rend="text-indent: 0">System of lines in space, <ref target="p20">20</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Systems of conics, <ref target="p125">125</ref> <lb /></p> + +<p rend="text-indent: 0">Tangent line, <ref target="p61">61</ref>, <ref target="p80">80</ref>, <ref target="p81">81</ref>, <ref target="p87">87</ref>, <ref target="p88">88</ref>, +89, <ref target="p90">90</ref>, <ref target="p91">91</ref>, <ref target="p92">92</ref><lb /></p> + +<p rend="text-indent: 0">Tycho Brahe (1546-1601), <ref target="p162">162</ref> <lb /></p> + +<p rend="text-indent: 0">Verner, <ref target="p161">161</ref><lb /></p> + +<p rend="text-indent: 0">Vertex of conic, <ref target="p157">157</ref>, <ref target="p159">159</ref><lb /></p> + +<p rend="text-indent: 0">Von Staudt (1798-1867), <ref target="p179">179</ref>, <ref target="p185">185</ref> <lb /></p> + +<p rend="text-indent: 0">Wallis (1616-1703), <ref target="p162">162</ref><lb /></p> + +</div> + +</body> + +<back rend="page-break-before: right"> +<div> +<pgIf output="pdf"> + <then> + <div> + <divGen type="pgfooter" rend="page-break-before: right" /> + </div> + </then> + <else> + <div> + <head>Footnotes</head> + <divGen type="footnotes" /> + </div> + + <div> + <divGen type="pgfooter" rend="page-break-before: right" /> + </div> + </else> +</pgIf> +</div> +</back> + + </text> +</TEI.2> + +<!-- +A WORD FROM PROJECT GUTENBERG + + +This file should be named 17001-0.txt or 17001-0.zip. + +This and all associated files of various formats will be found in: + + + http://www.gutenberg.org/dirs/1/7/0/0/17001/ + + +Updated editions will replace the previous one — the old editions will be +renamed. + +Creating the works from public domain print editions means that no one +owns a United States copyright in these works, so the 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