,an ,elementary ,course in ,synthetic ,projective ,geometry by ,lehmer1 ,derrick ,norman ,edition #a1 7,november #d1 #bjje7 ,,preface ,the following course is intended to give1 in as simple a way as possible1 the essentials of synthetic projective geometry4 ,while1 in the main1 the theory is developed along the well-beaten track laid out by the great masters of the subject1 it is believed that there has been a slight smoothing of the road in some places4 ,especially will this be observed in the chapter on ,involution4 ,the author has never felt satisfied with the usual treatment of that subject by means of circles and anharmonic ratios4 ,a purely projective notion ought not to be based on metrical foundations4 ,metrical developments should be made there1 as elsewhere in the theory1 by the introduction of infinitely distant elements4 ,the author has departed from the century-old custom of writing in parallel columns each theorem and its dual4 ,he has not found that it conduces to sharpness of vision to try to focus his eyes on two things at once4 ,those who prefer the usual method of procedure can1 of course1 develop the two sets of theorems side by side2 the author has not found this the better plan in actual teaching4 ,as regards nomenclature1 the author has followed the lead of the earlier writers in ,english1 and has called the system of lines in a plane which all pass through a point a pencil of rays instead of a bundle of rays1 as later writers seem inclined to do4 ,for a point considered as made up of all the lines and planes through it he has ventured to use the term point system1 as being the natural dualization of the usual term plane system4 ,he has also rejected the term foci of an involution1 and has not used the customary terms for classifying involutions--hyperbolic involution1 elliptic involution and parabolic involution4 ,he has found that all these terms are very confusing to the student1 who inevitably tries to connect them in some way with the conic sections4 ,enough examples have been provided to give the student a clear grasp of the theory4 ,many are of sufficient generality to serve as a basis for individual investigation on the part of the student4 ,thus1 the third example at the end of the first chapter will be found to be very fruitful in interesting results4 ,a correspondence is there indicated between lines in space and circles through a fixed point in space4 ,if the student will trace a few of the consequences of that correspondence1 and determine what configurations of circles correspond to intersecting lines1 to lines in a plane1 to lines of a plane pencil1 to lines cutting three skew lines1 etc41 he will have acquired no little practice in picturing to himself figures in space4 ,the writer has not followed the usual practice of inserting historical notes at the foot of the page1 and has tried instead1 in the last chapter1 to give a consecutive account of the history of pure geometry1 or1 at least1 of as much of it as the student will be able to appreciate who has mastered the course as given in the preceding chapters4 ,one is not apt to get a very wide view of the history of a subject by reading a hundred biographical footnotes1 arranged in no sort of sequence4 ,the writer1 moreover1 feels that the proper time to learn the history of a subject is after the student has some general ideas of the subject itself4 ,the course is not intended to furnish an illustration of how a subject may be developed1 from the smallest possible number of fundamental assumptions4 ,the author is aware of the importance of work of this sort1 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 students4 ,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 triangles4 ,no solid geometry is needed beyond the little used in the proof of ,desargues' theorem 7#be71 and1 except in certain metrical developments of the general theory1 there will be no call for a knowledge of trigonometry or analytical geometry4 ,naturally the student who is equipped with these subjects as well as with the calculus will be a little more mature1 and may be expected to follow the course all the more easily4 ,the author has had no difficulty1 however1 in presenting it to students in the freshman class at the ,university of ,california4 ,the subject of synthetic projective geometry is1 in the opinion of the writer1 destined shortly to force its way down into the secondary schools2 and if this little book helps to accelerate the movement1 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 university4 ,the material for the course has been drawn from many sources4 ,the author is chiefly indebted to the classical works of ,reye1 ,cremona1 ,steiner1 ,poncelet1 and ,von ,staudt4 ,acknowledgments and thanks are also due to ,professor ,walter ,c4 ,eells1 of the ,u4,s4 ,naval ,academy at ,annapolis1 for his searching examination and keen criticism of the manuscript2 also to ,professor ,herbert ,ellsworth ,slaught1 of ,the ,university of ,chicago1 for his many valuable suggestions1 and to ,professor ,b4 ,m4 ,woods and ,dr4 ,h4 ,n4 ,wright1 of the ,university of ,california1 who have tried out the methods of presentation1 in their own classes4 ,d4 ,n4 ,,lehmer ,,berkeley1 ,,california ,,contents ,preface ,contents ,,chapter ,i - ,,one-,,to-,,one ,,correspondence #a4 ,definition of one-to-one correspondence #b4 ,consequences of one-to-one correspondence #c4 ,applications in mathematics #d4 ,one-to-one correspondence and enumeration #e4 ,correspondence between a part and the whole #f4 ,infinitely distant point #g4 ,axial pencil2 fundamental forms #h4 ,perspective position #i4 ,projective relation #aj4 ,infinity-to-one correspondence #aa4 ,infinitudes of different orders #ab4 ,points in a plane #ac4 ,lines through a point #ad4 ,planes through a point #ae4 ,lines in a plane #af4 ,plane system and point system #ag4 ,planes in space #ah4 ,points of space #ai4 ,space system #bj4 ,lines in space #ba4 ,correspondence between points and numbers #bb4 ,elements at infinity ,,problems ,,chapter ,,ii - ,,relations ,,between ,,fundamental ,,forms ,,in ,,one-,,to-,,one ,,correspondence ,,with ,,each ,,other #bc4 ,seven fundamental forms #bd4 ,projective properties #be4 ,desargues's theorem #bf4 ,fundamental theorem concerning two complete quadrangles #bg4 ,importance of the theorem #bh4 ,restatement of the theorem #bi4 ,four harmonic points #cj4 ,harmonic conjugates #ca4 ,importance of the notion of four harmonic points #cb4 ,projective invariance of four harmonic points #cc4 ,four harmonic lines #cd4 ,four harmonic planes #ce4 ,summary of results #cf4 ,definition of projectivity #cg4 ,correspondence between harmonic conjugates #ch4 ,separation of harmonic conjugates #ci4 ,harmonic conjugate of the point at infinity #dj4 ,projective theorems and metrical theorems4 ,linear construction #da4 ,parallels and mid-points #db4 ,division of segment into equal parts #dc4 ,numerical relations #dd4 ,algebraic formula connecting four harmonic points #de4 ,further formulae #df4 ,anharmonic ratio ,,problems ,,chapter ,,iii - ,,combination ,,of ,,two ,,projectively ,,related ,,fundamental ,,forms #dg4 ,superposed fundamental forms4 ,self-corresponding elements #dh4 ,special case #di4 ,fundamental theorem4 ,postulate of continuity #ej4 ,extension of theorem to pencils of rays and planes #ea4 ,projective point-rows having a self-corresponding point in common #eb4 ,point-rows in perspective position #ec4 ,pencils in perspective position #ed4 ,axial pencils in perspective position #ee4 ,point-row of the second order #ef4 ,degeneration of locus #eg4 ,pencils of rays of the second order #eh4 ,degenerate case #ei4 ,cone of the second order ,,problems ,,chapter ,,iv - ,,point-,,rows ,,of ,,the ,,second ,,order #fj4 ,point-row of the second order defined #fa4 ,tangent line #fb4 ,determination of the locus #fc4 ,restatement of the problem #fd4 ,solution of the fundamental problem #fe4 ,different constructions for the figure #ff4 ,lines joining four points of the locus to a fifth #fg4 ,restatement of the theorem #fh4 ,further important theorem #fi4 ,pascal's theorem #gj4 ,permutation of points in ,pascal's theorem #ga4 ,harmonic points on a point-row of the second order #gb4 ,determination of the locus #gc4 ,circles and conics as point-rows of the second order #gd4 ,conic through five points #ge4 ,tangent to a conic #gf4 ,inscribed quadrangle #gg4 ,inscribed triangle #gh4 ,degenerate conic ,,problems ,,chapter ,v - ,,pencils ,,of ,,rays ,,of ,,the ,,second ,,order #gi4 ,pencil of rays of the second order defined #hj4 ,tangents to a circle #ha4 ,tangents to a conic #hb4 ,generating point-rows lines of the system #hc4 ,determination of the pencil #hd4 ,brianchon's theorem #he4 ,permutations of lines in ,brianchon's theorem #hf4 ,construction of the penvil by ,brianchon's theorem #hg4 ,point of contact of a tangent to a conic #hh4 ,circumscribed quadrilateral #hi4 ,circumscribed triangle #ij4 ,use of ,brianchon's theorem #ia4 ,harmonic tangents #ib4 ,projectivity and perspectivity #ic4 ,degenerate case #id4 ,law of duality ,,problems ,,chapter ,,vi - ,,poles ,,and ,,polars #ie4 ,inscribed and circumscribed quadrilaterals #if4 ,definition of the polar line of a point #ig4 ,further defining properties #ih4 ,definition of the pole of a line #ii4 ,fundamental theorem of poles and polars #ajj4 ,conjugate points and lines #aja4 ,construction of the polar line of a given point #ajb4 ,self-polar triangle #ajc4 ,pole and polar projectively related #ajd4 ,duality #aje4 ,self-dual theorems #ajf4 ,other correspondences ,,problems ,,chapter ,,vii - ,,metrical ,,properties ,,of ,,the ,,conic ,,sections #ajg4 ,diameters4 ,center #ajh4 ,various theorems #aji4 ,conjugate diameters #aaj4 ,classification of conics #aaa4 ,asymptotes #aab4 ,various theorems #aac4 ,theorems concerning asymptotes #aad4 ,asymptotes and conjugate diameters #aae4 ,segments cut off on a chord by hyperbola and its asymptotes #aaf4 ,application of the theorem #aag4 ,triangle formed by the two asymptotes and a tangent #aah4 ,equation of hyperbola referred to the asymptotes #aai4 ,equation of parabola #abj4 ,equation of central conics referred to conjugate diameters ,,problems ,,chapter ,,viii - ,,involution #aba4 ,fundamental theorem #abb4 ,linear construction #abc4 ,definition of involution of points on a line #abd4 ,double-points in an involution #abe4 ,desargues's theorem concerning conics through four points #abf4 ,degenerate conics of the system #abg4 ,conics through four points touching a given line #abh4 ,double correspondence #abi4 ,steiner's construction #acj4 ,application of ,steiner's construction to double correspondence #aca4 ,involution of points on a point-row of the second order4 #acb4 ,involution of rays #acc4 ,double rays #acd4 ,conic through a fixed point touching four lines #ace4 ,double correspondence #acf4 ,pencils of rays of the second order in involution #acg4 ,theorem concerning pencils of the second order in involution #ach4 ,involution of rays determined by a conic #aci4 ,statement of theorem #adj4 ,dual of the theorem ,,problems ,,chapter ,,ix - ,,metrical ,,properties ,,of ,,involutions #ada4 ,introduction of infinite point2 center of involution #adb4 ,fundamental metrical theorem #adc4 ,existence of double points #add4 ,existence of double rays #ade4 ,construction of an involution by means of circles #adf4 ,circular points #adg4 ,pairs in an involution of rays which are at right angles4 ,circular involution #adh4 ,axes of conics #adi4 ,points at which the involution determined by a conic is circular #aej4 ,properties of such a point #aea4 ,position of such a point #aeb4 ,discovery of the foci of the conic #aec4 ,the circle and the parabola #aed4 ,focal properties of conics #aee4 ,case of the parabola #aef4 ,parabolic reflector #aeg4 ,directrix4 ,principal axis4 ,vertex #aeh4 ,another definition of a conic #aei4 ,eccentricity #afj4 ,sum or difference of focal distances ,,problems ,,chapter ,x - ,,on ,,the ,,history ,,of ,,synthetic ,,projective ,,geometry #afa4 ,ancient results #afb4 ,unifying principles #afc4 ,desargues #afd4 ,poles and polars #afe4 ,desargues's theorem concerning conics through four points #aff4 ,extension of the theory of poles and polars to space #afg4 ,desargues's method of describing a conic #afh4 ,reception of ,desargues's work #afi4 ,conservatism in ,desargues's time #agj4 ,desargues's style of writing #aga4 ,lack of appreciation of ,desargues #agb4 ,pascal and his theorem #agc4 ,pascal's essay #agd4 ,pascal's originality #age4 ,de la ,hire and his work #agf4 ,descartes and his influence #agg4 ,newton and ,maclaurin #agh4 ,maclaurin's construction #agi4 ,descriptive geometry and the second revival #ahj4 ,duality1 homology1 continuity1 contingent relations #aha4 ,poncelet and ,cauchy #ahb4 ,the work of ,poncelet #ahc4 ,the debt which analytic geometry owes to synthetic geometry #ahd4 ,steiner and his work #ahe4 ,von ,staudt and his work #ahf4 ,recent developments ,,index ,,chapter ,i - ,,one-,,to-,,one ,,correspondence 99#a4 ,definition of one-to-one correspondence499 ,given any two sets of individuals1 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 other1 then the two sets are said to be in one-to-one correspondence with each other4 ,this notion1 simple as it is1 is of fundamental importance in all branches of science4 ,the process of counting is nothing but a setting up of a one-to-one correspondence between the objects to be counted and certain words1 'one1' 'two1' 'three1' etc41 in the mind4 ,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 fingers4 ,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 serve1 not as a means of counting the objects1 but of listing them in a convenient way4 ,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 themselves4 ,indeed1 the whole notion of language springs from this idea of one-to-one correspondence4 99#b4 ,consequences of one-to-one correspondence499 ,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 it4 ,it is a favorite error to assume that whatever holds for one set must also hold for the other4 ,magicians are apt to assign magic properties to many of the words and symbols which they are in the habit of using1 and scientists are constantly confusing objective things with the subjective formulas for them4 ,after the physicist has set up correspondences between physical facts and mathematical formulas1 the interpretation of these formulas is his most important and difficult task4 99#c499 ,in mathematics1 effort is constantly being made to set up one-to-one correspondences between simple notions and more complicated ones1 or between the well-explored fields of research and fields less known4 ,thus1 by means of the mechanism employed in analytic geometry1 algebraic theorems are made to yield geometric ones1 and vice versa4 ,in geometry we get at the properties of the conic sections by means of the properties of the straight line1 and cubic surfaces are studied by means of the plane4 ,figure #a ,,fig4 #a ,figure #b ,,fig4 #b 99#d4 ,one-to-one correspondence and enumeration499 ,if a one-to-one correspondence has been set up between the objects of one set and the objects of another set1 then the inference may usually be drawn that they have the same number of elements4 ,if1 however1 there is an infinite number of individuals in each of the two sets1 the notion of counting is necessarily ruled out4 ,it may be possible1 nevertheless1 to set up a one-to-one correspondence between the elements of two sets even when the number is infinite4 ,thus1 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 long4 ,for let the lines 7,fig4 #a7 be ,,ab and ,a',b'4 ,join ,,aa' and ,,bb'1 and let these joining lines meet in ,s4 ,for every point ,c on ,,ab a point ,c' may be found on ,a',b' by joining ,c to ,s and noting the point ,c' where ,,cs meets ,a',b'4 ,similarly1 a point ,c may be found on ,,ab for any point ,c' on ,a',b'4 ,the correspondence is clearly one-to-one1 but it would be absurd to infer from this that there were just as many points on ,,ab as on ,a',b'4 ,in fact1 it would be just as reasonable to infer that there were twice as many points on ,a',b' as on ,,ab4 ,for if we bend ,a',b' into a circle with center at ,s 7,fig4 #b71 we see that for every point ,c on ,,ab there are two points on ,a',b'4 ,thus it is seen that the notion of one-to-one correspondence is more extensive than the notion of counting1 and includes the notion of counting only when applied to finite assemblages4 99#e4 ,correspondence between a part and the whole of an infinite assemblage499 ,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 elements4 ,a moment's reflection will convince one that this is never possible when there is a finite number of elements in the assemblage4--,indeed1 we may take this property as our definition of an infinite assemblage1 and say that an infinite assemblage is one that may be put into one-to-one correspondence with part of itself4 ,this has the advantage of being a positive definition1 as opposed to the usual negative definition of an infinite assemblage as one that cannot be counted4 99#f4 ,infinitely distant point499 ,we have illustrated above a simple method of setting the points of two lines into one-to-one correspondence4 ,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 point4 ,thus1 for any point ,c on the line ,,ab there is a line ,,sc through ,s4 ,we must assume the line ,,ab extended indefinitely in both directions1 however1 if we are to have a point on it for every line through ,s2 and even with this extension there is one line through ,s1 according to ,euclid's postulate1 which does not meet the line ,,ab and which therefore has no point on ,,ab to correspond to it4 ,in order to smooth out this discrepancy we are accustomed to assume the existence of an infinitely distant point on the line ,,ab and to assign this point as the corresponding point of the exceptional line of ,s4 ,with this understanding1 then1 we may say that we have set the lines through a point and the points on a line into one-to-one correspondence4 ,this correspondence is of such fundamental importance in the study of projective geometry that a special name is given to it4 ,calling the totality of points on a line a point-row1 and the totality of lines through a point a pencil of rays1 we say that the point-row and the pencil related as above are in perspective position1 or that they are perspectively related4 99#g4 ,axial pencil2 fundamental forms499 ,a similar correspondence may be set up between the points on a line and the planes through another line which does not meet the first4 ,such a system of planes is called an axial pencil1 and the three assemblages--the point-row1 the pencil of rays1 and the axial pencil--are called fundamental forms4 ,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 order4 ,it is usual also to speak of them as of the first order4 ,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 line1 and that they will very reasonably be said to be of a higher order4 99#h4 ,perspective position499 ,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 it4 ,two pencils of rays are also said to be in perspective position if corresponding rays meet on a straight line which is called the axis of perspectivity4 ,also1 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 perspectivity4 ,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 it1 and an axial pencil and a pencil of rays are in perspective position if each ray lies in the plane which corresponds to it2 and1 finally1 two axial pencils are perspectively related if corresponding planes meet in a plane4 99#i4 ,projective relation499 ,it is easy to imagine a more general correspondence between the points of two point-rows than the one just described4 ,if we take two perspective pencils1 ,a and ,s1 then a point-row a perspective to ,a will be in one-to-one correspondence with a point-row b perspective to ,b1 but corresponding points will not1 in general1 lie on lines which all pass through a point4 ,two such point-rows are said to be projectively related1 or simply projective to each other4 ,similarly1 two pencils of rays1 or of planes1 are projectively related to each other if they are perspective to two perspective point-rows4 ,this idea will be generalized later on4 ,it is important to note that between the elements of two projective fundamental forms there is a one-to-one correspondence1 and also that this correspondence is in general continuous2 that is1 by taking two elements of one form sufficiently close to each other1 the two corresponding elements in the other form may be made to approach each other arbitrarily close4 ,in the case of point-rows this continuity is subject to exception in the neighborhood of the point at infinity4 99#aj4 ,infinity-to-one correspondence499 ,it might be inferred that any infinite assemblage could be put into one-to-one correspondence with any other4 ,such is not the case1 however1 if the correspondence is to be continuous1 between the points on a line and the points on a plane4 ,consider two lines which lie in different planes1 and take m points on one and n points on the other4 ,the number of lines joining the m points of one to the n points jof the other is clearly mn4 ,if we symbolize the totality of points on a line by infinity1 then a reasonable symbol for the totality of lines drawn to cut two lines would be infinity#b4 ,clearly1 for every point on one line there are infinity lines cutting across the other1 so that the correspondence might be called infinity-to-one4 ,thus the assemblage of lines cutting across two lines is of higher order than the assemblage of points on a line2 and as we have called the point-row an assemblage of the first order1 the system of lines cutting across two lines ought to be called of the second order4 99#aa4 ,infinitudes of different orders499 ,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 planes4 ,in fact1 each line of the system of lines meets the plane in one point1 and each point in the plane determines one and only one line cutting across the two given lines--namely1 the line of intersection of the two planes determined by the given point with each of the given lines4 ,the assemblage of points in the plane is thus of the same order as that of the lines cutting across two lines which lie in different planes1 and ought therefore to be spoken of as of the second order4 ,we express all these results as follows3 99#ab499 ,if the infinitude of points on a line is taken as the infinitude of the first order1 then the infinitude of lines in a pencil of rays and the infinitude of planes in an axial pencil are also of the first order1 while the infinitude of lines cutting across two skew lines1 as well as the infinitude of points in a plane1 are of the second order4 99#ac499 ,if we join each of the points of a plane to a point not in that plane1 we set up a one-to-one correspondence between the points in a plane and the lines through a point in space4 ,thus the infinitude of lines through a point in space is of the second order4 99#ad499 ,if to each line through a point in space we make correspond that plane at right angles to it and passing through the same point1 we see that the infinitude of planes through a point in space is of the second order4 99#ae499 ,if to each plane through a point in space we make correspond the line in which it intersects a given plane1 we see that the infinitude of lines in a plane is of the second order4 ,this may also be seen by setting up a one-to-one correspondence between the points on a plane and the lines of that plane4 ,thus1 take a point ,s not in the plane4 ,join any point ,m of the plane to ,s4 ,through ,s draw a plane at right angles to ,,ms4 ,this meets the given plane in a line m which may be taken as corresponding to the point ,m4 ,another very important method of setting up a one-to-one correspondence between lines and points in a plane will be given later1 and many weighty consequences will be derived from it4 99#af4 ,plane system and point system499 ,the plane1 considered as made up of the points and lines in it1 is called a plane system and is a fundamental form of the second order4 ,the point1 considered as made up of all the lines and planes passing through it1 is called a point system and is also a fundamental form of the second order4 99#ag499 ,if now we take three lines in space all lying in different planes1 and select l points on the first1 m points on the second1 and n points on the third1 then the total number of planes passing through one of the selected points on each line will be lmn4 ,it is reasonable1 therefore1 to symbolize the totality of planes that are determined by the infinity points on each of the three lines by infinity#c1 and to call it an infinitude of the third order4 ,but it is easily seen that every plane in space is included in this totality1 so that the totality of planes in space is an infinitude of the third order4 99#ah499 ,consider now the planes perpendicular to these three lines4 ,every set of three planes so drawn will determine a point in space1 and1 conversely1 through every point in space may be drawn one and only one set of three planes at right angles to the three given lines4 ,it follows1 therefore1 that the totality of points in space is an infinitude of the third order4 99#ai4 ,space system499 ,space of three dimensions1 considered as made up of all its planes and points1 is then a fundamental form of the third order1 which we shall call a space system4 99#bj4 ,lines in space499 ,if we join the twofold infinity of points in one plane with the twofold infinity of points in another plane1 we get a totality of lines of space which is of the fourth order of infinity4 ,the totality of lines in space gives1 then1 a fundamental form of the fourth order4 99#ba4 ,correspondence between points and numbers499 ,in the theory of analytic geometry a one-to-one correspondence is assumed to exist between points on a line and numbers4 ,in order to justify this assumption a very extended definition of number must be made use of4 ,a one-to-one correspondence is then set up between points in the plane and pairs of numbers1 and also between points in space and sets of three numbers4 ,a single constant will serve to define the position of a point on a line2 two1 a point in the plane2 three1 a point in space2 etc4 ,in the same theory a one-to-one correspondence is set up between loci in the plane and equations in two variables2 between surfaces in space and equations in three variables2 etc4 ,the equation of a line in a plane involves two constants1 either of which may take an infinite number of values4 ,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 first4 ,in the same way a circle is determined by three conditions2 a sphere by four2 etc4 ,we might then expect to be able to set up a one-to-one correspondence between circles in a plane and points1 or planes in space1 or between spheres and lines in space4 ,such1 indeed1 is the case1 and it is often possible to infer theorems concerning spheres from theorems concerning lines1 and vice versa4 ,it is possibilities such as these that1 give to the theory of one-to-one correspondence its great importance for the mathematician4 ,it must not be forgotten1 however1 that we are considering only continuous correspondences4 ,it is perfectly possible to set1 up a one-to-one correspondence between the points of a line and the points of a plane1 or1 indeed1 between the points of a line and the points of a space of any finite number of dimensions1 if the correspondence is not restricted to be continuous4 99#bb4 ,elements at infinity499 ,a final word is necessary in order to explain a phrase which is in constant use in the study of projective geometry4 ,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 point4 ,we speak of it as a point and not as points1 because in the geometry studied by ,euclid we assume only one line through a point parallel to a given line4 ,in the same sense we speak of all the points at infinity in a plane as lying on a line1 the line at infinity1 because the straight line is the simplest locus we can imagine which has only one point in common with any line in the plane4 ,likewise we speak of the plane at infinity1 because that seems the most convenient way of imagining the points at infinity in space4 ,it must not be inferred that these conceptions have any essential connection with physical facts1 or that other means of picturing to ourselves the infinitely distant configurations are not possible4 ,in other branches of mathematics1 notably in the theory of functions of a complex variable1 quite different assumptions are made and quite different conceptions of the elements at infinity are used4 ,as we can know nothing experimentally about such things1 we are at liberty to make any assumptions we please1 so long as they are consistent and serve some useful purpose4 ,,problems #a4 ,since there is a threefold infinity of points in space1 there must be a sixfold infinity of pairs of points in space4 ,each pair of points determines a line4 ,why1 then1 is there not a sixfold infinity of lines in space8 #b4 ,if there is a fourfold infinity of lines in space1 why is it that there is not a fourfold infinity of planes through a point1 seeing that each line in space determines a plane through that point8 #c4 ,show that there is a fourfold infinity of circles in space that pass through a fixed point4 7,set up a one-to-one correspondence between the axes of the circles and lines in space47 #d4 ,find the order of infinity of all the lines of space that cut across a given line2 across two given lines2 across three given lines2 across four given lines4 #e4 ,find the order of infinity of all the spheres in space that pass through a given point2 through two given points2 through three given points2 through four given points4 #f4 ,find the order of infinity of all the circles on a sphere2 of all the circles on a sphere that pass through a fixed point2 through two fixed points2 through three fixed points2 of all the circles in space2 of all the circles that cut across a given line4 #g4 ,find the order of infinity of all lines tangent to a sphere2 of all planes tangent to a sphere2 of lines and planes tangent to a sphere and passing through a fixed point4 #h4 ,set up a one-to-one correspondence between the series of numbers #a1 #b1 #c1 #d1 444 and the series of even numbers #b1 #d1 #f1 #h 4444 ,are we justified in saying that there are just as many even numbers as there are numbers altogether8 #i4 ,is the axiom ,the whole is greater than one of its parts applicable to infinite assemblages8 #aj4 ,make out a classified list of all the infinitudes of the first1 second1 third1 and fourth orders mentioned in this chapter4 ,,chapter ,,ii - ,,relations ,,between ,,fundamental ,,forms ,,in ,,one-,,to-,,one ,,correspondence ,,with ,,each ,,other 99#bc4 ,seven fundamental forms499 ,in the preceding chapter we have called attention to seven fundamental forms3 the point-row1 the pencil of rays1 the axial pencil1 the plane system1 the point system1 the space system1 and the system of lines in space4 ,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 case4 ,we shall be concerned1 not with measurement of angles and areas or line segments as in the study of ,euclid1 but in combining and comparing these fundamental forms and in generating new forms by means of them4 ,in problems of construction we shall make no use of measurement1 either of angles or of segments1 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 points1 or to find the point of intersections of two lines1 or the line of intersection of two planes1 or1 in general1 the common elements of two fundamental forms4 99#bd4 ,projective properties499 ,our chief interest in this chapter will be the discovery of relations between the elements of one form which hold between the corresponding elements of any other form in one-to-one correspondence with it4 ,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 it4 ,this false assumption is the basis of the so-called proof by analogy so much in vogue among speculative theorists4 ,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 it1 we should view with suspicion any application of the proof by analogy in realms of thought where accurate judgments are not so easily made4 ,for example1 if in a given point-row u three points1 ,a1 ,b1 and ,c1 are taken such that ,b is the middle point of the segment ,,ac1 it does not follow that the three points ,a'1 ,b'1 ,c' in a point-row perspective to u will be so related4 ,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 projective relations4 ,relations involving measurement of lines or of angles are not projective4 99#be4 ,desargues's theorem499 ,we consider first the following beautiful theorem1 due to ,desargues and called by his name4 ,if two triangles1 ,a1 ,b1 ,c and ,a'1 ,b'1 ,c'1 are so situated that the lines ,,aa'1 ,,bb'1 and ,,cc' all meet in a point1 then the pairs of sides ,,ab and ,a',b'1 ,,bc and ,b',c'1 ,,ca and ,c',a' all meet on a straight line1 and conversely4 ,figure #c ,,fig4 #c ,let the lines ,,aa'1 ,,bb'1 and ,,cc' meet in the point ,m 7,fig4 #c74 ,conceive of the figure as in space1 so that ,m is the vertex of a trihedral angle of which the given triangles are plane sections4 ,the lines ,,ab and ,a',b' are in the same plane and must meet when produced1 their point of intersection being clearly a point in the plane of each triangle and therefore in the line of intersection of these two planes4 ,call this point ,p4 ,by similar reasoning the point ,q of intersection of the lines ,,bc and ,b',c' must lie on this same line as well as the point ,r of intersection of ,,ca and ,c',a'4 ,therefore the points ,p1 ,q1 and ,r all lie on the same line m4 ,if now we consider the figure a plane figure1 the points ,p1 ,q1 and ,r still all lie on a straight line1 which proves the theorem4 ,the converse is established in the same manner4 99#bf4 ,fundamental theorem concerning two complete quadrangles499 ,this theorem throws into our hands the following fundamental theorem concerning two complete quadrangles1 a complete quadrangle being defined as the figure obtained by joining any four given points by straight lines in the six possible ways4 ,given two complete quadrangles1 ,k1 ,l1 ,m1 ,n and ,k'1 ,l'1 ,m'1 ,n'1 so related that ,,kl1 ,k',l'1 ,,mn1 ,m',n' all meet in a point ,a2 ,,lm1 ,l',m'1 ,,nk1 ,n',k' all meet in a point ,q2 and ,,ln1 ,l',n' meet in a point ,b on the line ,,ac2 then the lines ,,km and ,k',m' also meet in a point ,d on the line ,,ac4 ,figure #d ,,fig4 #d ,for1 by the converse of the last theorem1 ,,kk'1 ,,ll'1 and ,,nn' all meet in a point ,s 7,fig4 #d74 ,also ,,ll'1 ,,mm'1 and ,,nn' meet in a point1 and therefore in the same point ,s4 ,thus ,,kk'1 ,,ll'1 and ,,mm' meet in a point1 and so1 by ,desargues's theorem itself1 ,a1 ,b1 and ,d are on a straight line4 99#bg4 ,importance of the theorem499 ,the importance of this theorem lies in the fact that1 ,a1 ,b1 and ,c being given1 an indefinite number of quadrangles ,k'1 ,l'1 ,m'1 ,n' my be found such that ,k',l' and ,m',n' meet in ,a1 ,k',n' and ,l',m' in ,c1 with ,l',n' passing through ,b4 ,indeed1 the lines ,,ak' and ,,am' may be drawn arbitrarily through ,a1 and any line through ,b may be used to determine ,l' and ,n'4 ,by joining these two points to ,c the points ,k' and ,m' are determined4 ,then the line joining ,k' and ,m'1 found in this way1 must pass through the point ,d already determined by the quadrangle ,k1 ,l1 ,m1 ,n4 ,the three points ,a1 ,b1 ,c1 given in order1 serve thus to determine a fourth point ,d4 99#bh499 ,in a complete quadrangle the line joining any two points is called the opposite side to the line joining the other two points4 ,the result of the preceding paragraph may then be stated as follows3 ,given three points1 ,a1 ,b1 ,c1 in a straight line1 if a pair of opposite sides of a complete quadrangle pass through ,a1 and another pair through ,c1 and one of the remaining two sides goes through ,b1 then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed4 99#bi4 ,four harmonic points499 ,four points1 ,a1 ,b1 ,c1 ,d1 related as in the preceding theorem are called four harmonic points4 ,the point ,d is called the fourth harmonic of ,b with respect to ,a and ,c4 ,since ,b and ,d play exactly the same rle in the above construction1 ,b is also the fourth harmonic of ,d with respect to ,a and ,c4 ,b and ,d are called harmonic conjugates with respect to ,a and ,c4 ,we proceed to show that ,a and ,c are also harmonic conjugates with respect to ,b and ,d--that is1 that it is possible to find a quadrangle of which two opposite sides shall pass through ,b1 two through ,d1 and of the remaining pair1 one through ,a and the other through ,c4 ,figure #e ,,fig4 #e ,let ,o be the intersection of ,,km and ,,ln 7,fig4 #e74 ,join ,o to ,a and ,c4 ,the joining lines cut out on the sides of the quadrangle four points1 ,p1 ,q1 ,r1 ,s4 ,consider the quadrangle ,p1 ,k1 ,q1 ,o4 ,one pair of opposite sides passes through ,a1 one through ,c1 and one remaining side through ,d2 therefore the other remaining side must pass through ,b4 ,similarly1 ,,rs passes through ,b and ,,ps and ,,qr pass through ,d4 ,the quadrangle ,p1 ,q1 ,r1 ,s therefore has two opposite sides through ,b1 two through ,d1 and the remaining pair through ,a and ,c4 ,a and ,c are thus harmonic conjugates with respect to ,b and ,d4 ,we may sum up the discussion1 therefore1 as follows3 99#cj499 ,if ,a and ,c are harmonic conjugates with respect to ,b and ,d1 then ,b and ,d are harmonic conjugates with respect to ,a and ,c4 99#ca4 ,importance of the notion499 ,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 u to the four points that correspond to them in any point-row u' perspective to u4 ,to prove this statement we construct a quadrangle ,k1 ,l1 ,m1 ,n such that ,,kl and ,,mn pass through ,a1 ,,kn and ,,lm through ,c1 ,,ln through ,b1 and ,,km through ,d4 ,take now any point ,s not in the plane of the quadrangle and construct the planes determined by ,s and all the seven lines of the figure4 ,cut across this set of planes by another plane not passing through ,s4 ,this plane cuts out on the set of seven planes another quadrangle which determines four new harmonic points1 ,a'1 ,b'1 ,c'1 ,d'1 on the lines joining ,s to ,a1 ,b1 ,c1 ,d4 ,but ,s may be taken as any point1 since the original quadrangle may be taken in any plane through ,a1 ,b1 ,c1 ,d2 and1 further1 the points ,a'1 ,b'1 ,c'1 ,d' are the intersection of ,,sa1 ,,sb1 ,,sc1 ,,sd by any line4 ,we have1 then1 the remarkable theorem3 99#cb499 ,if any point is joined to four harmonic points1 and the four lines thus obtained are cut by any fifth1 the four points of intersection are again harmonic4 99#cc4 ,four harmonic lines499 ,we are now able to extend the notion of harmonic elements to pencils of rays1 and indeed to axial pencils4 ,for if we define four harmonic rays as four rays which pass through a point and which pass one through each of four harmonic points1 we have the theorem ,four harmonic lines are cut by any transversal in four harmonic points4 99#cd4 ,four harmonic planes499 ,we also define four harmonic planes as four planes through a line which pass one through each of four harmonic points1 and we may show that ,four harmonic planes are cut by any plane not passing through their common line in four harmonic lines1 and also by any line in four harmonic points4 ,for let the planes 1 1 1 1 which all pass through the line g1 pass also through the four harmonic points ,a1 ,b1 ,c1 ,d1 so that passes through ,a1 etc4 ,then it is clear that any plane through ,a1 ,b1 ,c1 ,d will cut out four harmonic lines from the four planes1 for they are lines through the intersection ,p of g with the plane 1 and they pass through the given harmonic points ,a1 ,b1 ,c1 ,d4 ,any other plane cuts g in a point ,s and cuts 1 1 1 in four lines that meet in four points ,a'1 ,b'1 ,c'1 ,d' lying on ,,pa1 ,,pb1 ,,pc1 and ,,pd respectively1 and are thus four harmonic hues4 ,further1 any ray cuts 1 1 1 in four harmonic points1 since any plane through the ray gives four harmonic lines of intersection4 99#ce499 ,these results may be put together as follows3 ,given any two assemblages of points1 rays1 or planes1 perspectively related to each other1 four harmonic elements of one must correspond to four elements of the other which are likewise harmonic4 ,if1 now1 two forms are perspectively related to a third1 any four harmonic elements of one must correspond to four harmonic elements in the other4 ,we take this as our definition of projective correspondence1 and say3 99#cf4 ,definition of projectivity499 ,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 other4 ,figure #f ,,fig4 #f 99#cg4 ,correspondence between harmonic conjugates499 ,given four harmonic points1 ,a1 ,b1 ,c1 ,d2 if we fix ,a and ,c1 then ,b and ,d vary together in a way that should be thoroughly understood4 ,to get a clear conception of their relative motion we may fix the points ,l and ,m of the quadrangle ,k1 ,l1 ,m1 ,n 7,fig4 #f74 ,then1 as ,b describes the point-row ,,ac1 the point ,n describes the point-row ,,am perspective to it4 ,projecting ,n again from ,c1 we get a point-row ,k on ,,al perspective to the point-row ,n and thus projective to the point-row ,b4 ,project the point-row ,k from ,m and we get a point-row ,d on ,,ac again1 which is projective to the point-row ,b4 ,for every point ,b we have thus one and only one point ,d1 and conversely4 ,in other words1 we have set up a one-to-one correspondence between the points of a single point-row1 which is also a projective correspondence because four harmonic points ,b correspond to four harmonic points ,d4 ,we may note also that the correspondence is here characterized by a feature which does not always appear in projective correspondences3 namely1 the same process that carries one from ,b to ,d will carry one back from ,d to ,b again4 ,this special property will receive further study in the chapter on ,involution4 99#ch499 ,it is seen that as ,b approaches ,a1 ,d also approaches ,a4 ,as ,b moves from ,a toward ,c1 ,d moves from ,a in the opposite direction1 passing through the point at infinity on the line ,,ac1 and returns on the other side to meet ,b at ,c again4 ,in other words1 as ,b traverses ,,ac1 ,d traverses the rest of the line from ,a to ,c through infinity4 ,in all positions of ,b1 except at ,a or ,c1 ,b and ,d are separated from each other by ,a and ,c4 99#ci4 ,harmonic conjugate of the point at infinity499 ,it is natural to inquire what position of ,b corresponds to the infinitely distant position of ,d4 ,we have proved 7 #bg7 that the particular quadrangle ,k1 ,l1 ,m1 ,n employed is of no consequence4 ,we shall therefore avail ourselves of one that lends itself most readily to the solution of the problem4 ,we choose the point ,l so that the triangle ,,alc is isosceles 7,fig4 #g74 ,since ,d is supposed to be at infinity1 the line ,,km is parallel to ,,ac4 ,therefore the triangles ,,kac and ,,mac are equal1 and the triangle ,,anc is also isosceles4 ,the triangles ,,cnl and ,,anl are therefore equal1 and the line ,,lb bisects the angle ,,alc4 ,b is therefore the middle point of ,,ac1 and we have the theorem ,the harmonic conjugate of the middle point of ,,ac is at infinity4 ,figure #g ,,fig4 #g 99#dj4 ,projective theorems and metrical theorems4 ,linear construction499 ,this theorem is the connecting link between the general protective theorems which we have been considering so far and the metrical theorems of ordinary geometry4 ,up to this point we have said nothing about measurements1 either of line segments or of angles4 ,desargues's theorem and the theory of harmonic elements which depends on it have nothing to do with magnitudes at all4 ,not until the notion of an infinitely distant point is brought in is any mention made of distances or directions4 ,we have been able to make all of our constructions up to this point by means of the straightedge1 or ungraduated ruler4 ,a construction made with such an instrument we shall call a linear construction4 ,it requires merely that we be able to draw the line joining two points or find the point of intersection of two lines4 99#da4 ,parallels and mid-points499 ,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 points4 ,indeed1 it consists only in drawing a line through the given point and through the infinitely distant point on the given line4 ,it must be remembered1 however1 that the expression infinitely distant point must not be taken literally4 ,when we say that two parallel lines meet at infinity1 we really mean that they do not meet at all1 and the only reason for using the expression is to avoid tedious statement of exceptions and restrictions to our theorems4 ,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 points4 ,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 data4 ,for the construction is reversible1 and if we are given the middle point of a given segment1 we can construct linearly a line parallel to that segment4 ,thus1 given that ,b is the middle point of ,,ac1 we may draw any two lines through ,a1 and any line through ,b cutting them in points ,n and ,l4 ,join ,n and ,l to ,c and get the points ,k and ,m on the two lines through ,a4 ,then ,,km is parallel to ,,ac4 ,the bisection of a given segment and the drawing of a line parallel to the segment are equivalent data when linear construction is used4 99#db499 ,it is not difficult to give a linear construction for the problem to divide a given segment into n equal parts1 given only a parallel to the segment4 ,this is simple enough when n is a power of #b4 ,for any other number1 such as #bi1 divide any segment on the line parallel to ,,ac into #cb equal parts1 by a repetition of the process just described4 ,take #bi of these1 and join the first to ,a and the last to ,c4 ,let these joining lines meet in ,s4 ,join ,s to all the other points4 ,other problems1 of a similar sort1 are given at the end of the chapter4 99#dc4 ,numerical relations499 ,since three points1 given in order1 are sufficient to determine a fourth1 as explained above1 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 numbers2 a correspondence which1 to be sure1 we have not established here1 but which is discussed in any treatise on the theory of point sets4 ,we proceed to discover what relation between four numbers corresponds to the harmonic relation between four points4 ,figure #h ,,fig4 #h 99#dd499 ,let ,a1 ,b1 ,c1 ,d be four harmonic points 7,fig4 #h71 and let ,,sa1 ,,sb1 ,,sc1 ,,sd be four harmonic lines4 ,assume a line drawn through ,b parallel to ,,sd1 meeting ,,sa in ,a' and ,,sc in ,c'4 ,then ,a'1 ,b'1 ,c'1 and the infinitely distant point on ,a',c' are four harmonic points1 and therefore ,b is the middle point of the segment ,a',c'4 ,then1 since the triangle ,,das is similar to the triangle ,,baa'1 we may write the proportion ,,ab 3 ,,ad ,,ba' 3 ,,sd4 ,also1 from the similar triangles ,,dsc and ,,bcc'1 we have ,,cd 3 ,,cb ,,sd 3 ,b',c4 ,from these two proportions we have1 remembering that ,,ba' ,,bc'1 formula the minus sign being given to the ratio on account of the fact that ,a and ,c are always separated from ,b and ,d1 so that one or three of the segments ,,ab1 ,,cd1 ,,ad1 ,,cb must be negative4 99#de499 ,writing the last equation in the form ,,cb 3 ,,ab -,,cd 3 ,,ad1 and using the fundamental relation connecting three points on a line1 ,,pr ,,rq ,,pq1 which holds for all positions of the three points if account be taken of the sign of the segments1 the last proportion may be written 7,,cb - ,,ba7 3 ,,ab -7,,ca - ,,da7 3 ,,ad1 or 7,,ab - ,,ac7 3 ,,ab 7,,ac - ,,ad7 3 ,,ad2 so that ,,ab1 ,,ac1 and ,,ad are three quantities in hamonic progression1 since the difference between the first and second is to the first as the difference between the second and third is to the third4 ,also1 from this last proportion comes the familiar relation formula which is convenient for the computation of the distance ,,ad when ,,ab and ,,ac are given numerically4 99#df4 ,anharmonic ratio499 ,the corresponding relations between the trigonometric functions of the angles determined by four harmonic lines are not difficult to obtain1 but as we shall not need them in building up the theory of projective geometry1 we will not discuss them here4 ,students who have a slight acquaintance with trigonometry may read in a later chapter 7 #afa7 a development of the theory of a more general relation1 called the anharmonic ratio1 or cross ratio1 which connects any four points on a line4 ,,problems 99#a994 ,draw through a given point a line which shall pass through the inaccessible point of intersection of two given lines4 ,the following construction may be made to depend upon ,desargues's theorem3 ,through the given point ,p draw any two rays cutting the two lines in the points ,,ab' and ,a',b1 ,a1 ,b1 lying on one of the given lines and ,a'1 ,b'1 on the other4 ,join ,,aa' and ,,bb'1 and find their point of intersection ,s4 ,through ,s draw any other ray1 cutting the given lines in ,,cc'4 ,join ,,bc' and ,b',c1 and obtain their point of intersection ,q4 ,,pq is the desired line4 ,justify this construction4 99#b499 ,to draw through a given point ,p a line which shall meet two given lines in points ,a and ,b1 equally distant from ,p4 ,justify the following construction3 ,join ,p to the point ,s of intersection of the two given lines4 ,construct the fourth harmonic of ,,ps with respect to the two given lines4 ,draw through ,p a line parallel to this line4 ,this is the required line4 99#c499 ,given a parallelogram in the same plane with a given segment ,,ac1 to construct linearly the middle point of ,,ac4 99#d499 ,given four harmonic lines1 of which one pair are at right angles to each other1 show that the other pair make equal angles with them4 ,this is a theorem of which frequent use will be made4 99#e499 ,given the middle point of a line segment1 to draw a line parallel to the segment and passing through a given point4 99#f499 ,a line is drawn cutting the sides of a triangle ,,abc in the points ,a'1 ,b'1 ,c' the point ,a' lying on the side ,,bc1 etc4 ,the harmonic conjugate of ,a' with respect to ,b and ,c is then constructed and called ,a4 ,similarly1 ,b and ,c are constructed4 ,show that ,a,b,c lie on a straight line4 ,find other sets of three points on a line in the figure4 ,find also sets of three lines through a point4 ,,chapter ,,iii - ,,combination ,,of ,,two ,,projectively ,,related ,,fundamental ,,forms ,figure #i ,,fig4 #i 99#dg4 ,superposed fundamental forms4 ,self-corresponding elements499 ,we have seen 7 #cg7 that two projective point-rows may be superposed upon the same straight line4 ,this happens1 for example1 when two pencils which are projective to each other are cut across by a straight line4 ,it is also possible for two projective pencils to have the same center4 ,this happens1 for example1 when two projective point-rows are projected to the same point4 ,similarly1 two projective axial pencils may have the same axis4 ,we examine now the possibility of two forms related in this way1 having an element or elements that correspond to themselves4 ,we have seen1 indeed1 that if ,b and ,d are harmonic conjugates with respect to ,a and ,c1 then the point-row described by ,b is projective to the point-row described by ,d1 and that ,a and ,c are self-corresponding points4 ,consider more generally the case of two pencils perspective to each other with axis of perspectivity u' 7,fig4 #i74 ,cut across them by a line u4 ,we get thus two projective point-rows superposed on the same line u1 and a moment's reflection serves to show that the point ,n of intersection u and u' corresponds to itself in the two point-rows4 ,also1 the point ,m1 where u intersects the line joining the centers of the two pencils1 is seen to correspond to itself4 ,it is thus possible for two projective point-rows1 superposed upon the same line1 to have two self-corresponding points4 ,clearly ,m and ,n may fall together if the line joining the centers of the pencils happens to pass through the point of intersection of the lines u and u'4 ,figure #aj ,,fig4 #aj 99#dh499 ,we may also give an illustration of a case where two superposed projective point-rows have no self-corresponding points at all4 ,thus we may take two lines revolving about a fixed point ,s and always making the same angle a with each other 7,fig4 #aj74 ,they will cut out on any line u in the plane two point-rows which are easily seen to be projective4 ,for1 given any four rays ,,sp which are harmonic1 the four corresponding rays ,,sp' must also be harmonic1 since they make the same angles with each other4 ,four harmonic points ,p correspond1 therefore1 to four harmonic points ,p'4 ,it is clear1 however1 that no point ,p can coincide with its corresponding point ,p'1 for in that case the lines ,,ps and ,p',s would coincide1 which is impossible if the angle between them is to be constant4 99#di4 ,fundamental theorem4 ,postulate of continuity499 ,we have thus shown that two projective point-rows1 superposed one on the other1 may have two points1 one point1 or no point at all corresponding to themselves4 ,we proceed to show that ,if two projective point-rows1 superposed upon the same straight line1 have more than two self-corresponding points1 they must have an infinite number1 and every point corresponds to itself2 that is1 the two point-rows are not essentially distinct4 ,if three points1 ,a1 ,b1 and ,c1 are self-corresponding1 then the harmonic conjugate ,d of ,b with respect to ,a and ,c must also correspond to itself4 ,for four harmonic points must always correspond to four harmonic points4 ,in the same way the harmonic conjugate of ,d with respect to ,b and ,c must correspond to itself4 ,combining new points with old in this way1 we may obtain as many self-corresponding points as we wish4 ,we show further that every point on the line is the limiting point of a finite or infinite sequence of self-corresponding points4 ,thus1 let a point ,p lie between ,a and ,b4 ,construct now ,d1 the fourth harmonic of ,c with respect to ,a and ,b4 ,d may coincide with ,p1 in which case the sequence is closed2 otherwise ,p lies in the stretch ,,ad or in the stretch ,,db4 ,if it lies in the stretch ,,db1 construct the fourth harmonic of ,c with respect to ,d and ,b4 ,this point ,d' may coincide with ,p1 in which case1 as before1 the sequence is closed4 ,if ,p lies in the stretch ,,dd'1 we construct the fourth harmonic of ,c with respect to ,,dd'1 etc4 ,in each step the region in which ,p lies is diminished1 and the process may be continued until two self-corresponding points are obtained on either side of ,p1 and at distances from it arbitrarily small4 ,we now assume1 explicitly1 the fundamental postulate that the correspondence is continuous1 that is1 that 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 other4 ,suppose now that ,p is not a self-corresponding point1 but corresponds to a point ,p' at a fixed distance d from ,p4 ,as noted above1 we can find self-corresponding points arbitrarily close to ,p1 and it appears1 then1 that we can take a point ,d as close to ,p as we wish1 and yet the distance between the corresponding points ,d' and ,p' approaches d as a limit1 and not zero1 which contradicts the postulate of continuity4 99#ej499 ,it follows also that two projective pencils which have the same center may have no more than two self-corresponding rays1 unless the pencils are identical4 ,for if we cut across them by a line1 we obtain two projective point-rows superposed on the same straight line1 which may have no more than two self-corresponding points4 ,the same considerations apply to two projective axial pencils which have the same axis4 99#ea4 ,projective point-rows having a self-corresponding point in common499 ,consider now two projective point-rows lying on different lines in the same plane4 ,their common point may or may not be a self-corresponding point4 ,if the two point-rows are perspectively related1 then their common point is evidently a self-corresponding point4 ,the converse is also true1 and we have the very important theorem3 99#eb499 ,if in two protective point-rows1 the point of intersection corresponds to itself1 then the point-rows are in perspective position4 ,figure #aa ,,fig4 #aa ,let the two point-rows be u and u' 7,fig4 #aa74 ,let ,a and ,a'1 ,b and ,b'1 be corresponding points1 and let also the point ,m of intersection of u and u' correspond to itself4 ,let ,,aa' and ,,bb' meet in the point ,s4 ,take ,s as the center of two pencils1 one perspective to u and the other perspective to u'4 ,in these two pencils ,,sa coincides with its corresponding ray ,,sa'1 ,,sb with its corresponding ray ,,sb'1 and ,,sm with its corresponding ray ,,sm'4 ,the two pencils are thus identical1 by the preceding theorem1 and any ray ,,sd must coincide with its corresponding ray ,,sd'4 ,corresponding points of u and u'1 therefore1 all lie on lines through the point ,s4 99#ec499 ,an entirely similar discussion shows that ,if in two projective pencils the line joining their centers is a self-corresponding ray1 then the two pencils are perspectively related4 99#ed499 ,a similar theorem may be stated for two axial pencils of which the axes intersect4 ,very frequent use will be made of these fundamental theorems4 99#ee4 ,point-row of the second order499 ,the question naturally arises1 ,what is the locus of points of intersection of corresponding rays of two projective pencils which are not in perspective position8 ,this locus1 which will be discussed in detail in subsequent chapters1 is easily seen to have at most two points in common with any line in the plane1 and on account of this fundamental property will be called a point-row of the second order4 ,for any line u in the plane of the two pencils will be cut by them in two projective point-rows which have at most two self-corresponding points4 ,such a self-corresponding point is clearly a point of intersection of corresponding rays of the two pencils4 99#ef499 ,this locus degenerates in the case of two perspective pencils to a pair of straight lines1 one of which is the axis of perspectivity and the other the common ray1 any point of which may be considered as the point of intersection of corresponding rays of the two pencils4 99#eg4 ,pencils of rays of the second order499 ,similar investigations may be made concerning the system of lines joining corresponding points of two projective point-rows4 ,if we project the point-rows to any point in the plane1 we obtain two projective pencils having the same center4 ,at most two pairs of self-corresponding rays may present themselves4 ,such a ray is clearly a line joining two corresponding points in the two point-rows4 ,the result may be stated as follows3 ,the system of rays joining corresponding points in two protective point-rows has at most two rays in common with any pencil in the plane4 ,for that reason the system of rays is called a pencil of rays of the second order4 99#eh499 ,in the case of two perspective point-rows this system of rays degenerates into two pencils of rays of the first order1 one of which has its center at the center of perspectivity of the two point-rows1 and the other at the intersection of the two point-rows1 any ray through which may be considered as joining two corresponding points of the two point-rows4 99#ei4 ,cone of the second order499 ,the corresponding theorems in space may easily be obtained by joining the points and lines considered in the plane theorems to a point ,s in space4 ,two projective pencils give rise to two projective axial pencils with axes intersecting4 ,corresponding planes meet in lines which all pass through ,s and through the points on a point-row of the second order generated by the two pencils of rays4 ,they are thus generating lines of a cone of the second order1 or quadric cone1 so called because every plane in space not passing through ,s cuts it in a point-row of the second order1 and every line also cuts it in at most two points4 ,if1 again1 we project two point-rows to a point ,s in space1 we obtain two pencils of rays with a common center but lying in different planes4 ,corresponding lines of these pencils determine planes which are the projections to ,s of the lines which join the corresponding points of the two point-rows4 ,at most two such planes may pass through any ray through ,s4 ,it is called a pencil of planes of the second order4 ,,problems 99#a4 99 ,a man ,a moves along a straight road u1 and another man ,b moves along the same road and walks so as always to keep sight of ,a in a small mirror ,m at the side of the road4 ,how many times will they come together1 ,a moving always in the same direction along the road8 #b4 ,how many times would the two men in the first problem see each other in two mirrors ,m and ,n as they walk along the road as before8 7,the planes of the two mirrors are not necessarily parallel to u47 #c4 ,as ,a moves along u1 trace the path of ,b so that the two men may always see each other in the two mirrors4 #d4 ,two boys walk along two paths u and u' each holding a string which they keep stretched tightly between them4 ,they both move at constant but different rates of speed1 letting out the string or drawing it in as they walk4 ,how many times will the line of the string pass over any given point in the plane of the paths8 #e4 ,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 intersect4 #f4 ,a ship is sailing on a straight course and keeps a gun trained on a point on the shore4 ,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 all4 7,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 gun47 #g4 ,two lines u and u' revolve about two points ,u and ,u' respectively in the same plane4 ,they go in the same direction and at the same rate of speed1 but one has an angle a the start of the other4 ,show that they generate a point-row of the second order4 #h4 ,discuss the question given in the last problem when the two lines revolve in opposite directions4 ,can you recognize the locus8 ,,chapter ,,iv - ,,point-,,rows ,,of ,,the ,,second ,,order 99#fj4 ,point-row of the second order defined499 ,we have seen that two fundamental forms in one-to-one correspondence may sometimes generate a form of higher order4 ,thus1 two point-rows 7 #ee7 generate a system of rays of the second order1 and two pencils of rays 7 #eg71 a system of points of the second order4 ,as a system of points is more familiar to most students of geometry than a system of lines1 we study first the point-row of the second order4 99#fa4 ,tangent line499 ,we have shown in the last chapter 7 #ee7 that the locus of intersection of corresponding rays of two projective pencils is a point-row of the second order2 that is1 it has at most two points in common with any line in the plane4 ,it is clear1 first of all1 that the centers of the pencils are points of the locus2 for to the line ,,ss'1 considered as a ray of ,s1 must correspond some ray of ,s' which meets it in ,s'4 ,s'1 and by the same argument ,s1 is then a point where corresponding rays meet4 ,any ray through ,s will meet it in one point besides ,s1 namely1 the point ,p where it meets its corresponding ray4 ,now1 by choosing the ray through ,s sufficiently close to the ray ,,ss'1 the point ,p may be made to approach arbitrarily close to ,s'1 and the ray ,s',p may be made to differ in position from the tangent line at ,s' by as little as we please4 ,we have1 then1 the important theorem ,the ray at ,s' which corresponds to the common ray ,,ss' is tangent to the locus at ,s'4 ,in the same manner the tangent at ,s may be constructed4 99#fb4 ,determination of the locus499 ,we now show that it is possible to assign arbitrarily the position of three points1 ,a1 ,b1 and ,c1 on the locus 7besides the points ,s and ,s'72 but1 these three points being chosen1 the locus is completely determined4 99#fc499 ,this statement is equivalent to the following3 ,given three pairs of corresponding rays in two projective pencils1 it is possible to find a ray of one which corresponds to any ray of the other4 99#fd499 ,we proceed1 then1 to the solution of the fundamental ,,problem3 ,given three pairs of rays1 aa'1 bb'1 and cc'1 of two protective pencils1 ,s and ,s'1 to find the ray d' of ,s' which corresponds to any ray d of ,s4 ,figure #ab ,,fig4 #ab ,call ,a the intersection of aa'1 ,b the intersection of bb'1 and ,c the intersection of cc' 7,fig4 #ab74 ,join ,,ab by the line u1 and ,,ac by the line u'4 ,consider u as a point-row perspective to ,s1 and u' as a point-row perspective to ,s'4 u and u' are projectively related to each other1 since ,s and ,s' are1 by hypothesis1 so related4 ,but their point of intersection ,a is a self-corresponding point1 since a and a' were supposed to be corresponding rays4 ,it follows 7 #eb7 that u and u' are in perspective position1 and that lines through corresponding points all pass through a point ,m1 the center of perspectivity1 the position of which will be determined by any two such lines4 ,but the intersection of a with u and the intersection of c' with u' are corresponding points on u and u'1 and the line joining them is clearly c itself4 ,similarly1 b' joins two corresponding points on u and u'1 and so the center ,m of perspectivity of u and u' is the intersection of c and b'4 ,to find d' in ,s' corresponding to a given line d of ,s we note the point ,l where d meets u4 ,join ,l to ,m and get the point ,n where this line meets u'4 ,l and ,n are corresponding points on u and u'1 and d' must therefore pass through ,n4 ,the intersection ,p of d and d' is thus another point on the locus4 ,in the same manner any number of other points may be obtained4 99#fe499 ,the lines u and u' might have been drawn in any direction through ,a 7avoiding1 of course1 the line a for u and the line a' for u'71 and the center of perspectivity ,m would be easily obtainable2 but the above construction furnishes a simple and instructive figure4 ,an equally simple one is obtained by taking a' for u and a for u'4 99#ff4 ,lines joining four points of the locus to a fifth499 ,suppose that the points ,s1 ,s'1 ,b1 ,c1 and ,d are fixed1 and that four points1 ,a1 ,a#a1 ,a#b1 and ,a#c1 are taken on the locus at the intersection with it of any four harmonic rays through ,b4 ,these four harmonic rays give four harmonic points1 ,l1 ,l#a etc41 on the fixed ray ,,sd4 ,these1 in turn1 project through the fixed point ,m into four harmonic points1 ,n1 ,n#a etc41 on the fixed line ,,ds'4 ,these last four harmonic points give four harmonic rays ,,ca1 ,,ca#a1 ,,ca#b1 ,,ca#c4 ,therefore the four points ,a which project to ,b in four harmonic rays also project to ,c in four harmonic rays4 ,but ,c may be any point on the locus1 and so we have the very important theorem1 ,four points which are on the locus1 and which project to a fifth point of the locus in four harmonic rays1 project to any point of the locus in four harmonic rays4 99#fg499 ,the theorem may also be stated thus3 ,the locus of points from which1 four given points are seen along four harmonic rays is a point-row of the second order through them4 99#fh499 ,a further theorem of prime importance also follows3 ,any two points on the locus may be taken as the centers of two projective pencils which will generate the locus4 99#fi4 ,pascal's theorem499 ,the points ,a1 ,b1 ,c1 ,d1 ,s1 and ,s' may thus be considered as chosen arbitrarily on the locus1 and the following remarkable theorem follows at once4 ,given six points1 #a1 #b1 #c1 #d1 #e1 #f1 on the point-row of the second order1 if we call ,l the intersection of #ab with #de1 ,m the intersection of #bc with #ef1 ,n the intersection of #cd with #fa1 then ,l1 ,m1 and ,n are on a straight line4 ,figure #ac ,,fig4 #ac 99#gj499 ,to get the notation to correspond to the figure1 we may take 7,fig4 #ac7 ,a #a1 ,b #b1 ,s' #c1 ,d #d1 ,s #e1 and ,c #f4 ,if we make ,a #a1 ,c#b1 ,s#c1 ,d #d1 ,s'#e1 and4 ,b #f1 the points ,l and ,n are interchanged1 but the line is left unchanged4 ,it is clear that one point may be named arbitrarily and the other five named in #e6 #abj different ways1 but since1 as we have seen1 two different assignments of names give the same line1 it follows that there cannot be more than #fj different lines ,,lmn obtained in this way from a given set of six points4 ,as a matter of fact1 the number obtained in this way is in general #fj4 ,the above theorem1 which is of cardinal importance in the theory of the point-row of the second order1 is due to ,pascal and was discovered by him at the age of sixteen4 ,it is1 no doubt1 the most important contribution to the theory of these loci since the days of ,apollonius4 ,if the six points be called the vertices of a hexagon inscribed in the curve1 then the sides #ab and #de may be appropriately called a pair of opposite sides4 ,pascal's theorem1 then1 may be stated as follows3 ,the three pairs of opposite sides of a hexagon inscribed in a point-row of the second order meet in three points on a line4 99#ga4 ,harmonic points on a point-row of the second order499 ,before proceeding to develop the consequences of this theorem1 we note another result of the utmost importance for the higher developments of pure geometry1 which follows from the fact that if four points on the locus project to a fifth in four harmonic rays1 they will project to any point of the locus in four harmonic rays4 ,it is natural to speak of four such points as four harmonic points on the locus1 and to use this notion to define projective correspondence between point-rows of the second order1 or between a point-row of the second order and any fundamental form of the first order4 ,thus1 in particular1 the point-row of the second order1 1 is said to be perspectively related to the pencil ,s when every ray on ,s goes through the point on which corresponds to it4 99#gb4 ,determination of the locus499 ,it is now clear that five points1 arbitrarily chosen in the plane1 are sufficient to determine a point-row of the second order through them4 ,two of the points may be taken as centers of two projective pencils1 and the three others will determine three pairs of corresponding rays of the pencils1 and therefore all pairs4 ,if four points of the locus are given1 together with the tangent at one of them1 the locus is likewise completely determined4 ,for if the point at which the tangent is given be taken as the center ,s of one pencil1 and any other of the points for ,s'1 then1 besides the two pairs of corresponding rays determined by the remaining two points1 we have one more pair1 consisting of the tangent at ,s and the ray ,,ss'4 ,similarly1 the curve is determined by three points and the tangents at two of them4 99#gc4 ,circles and conics as point-rows of the second order499 ,it is not difficult to see that a circle is a point-row of the second order4 ,indeed1 take any point ,s on the circle and draw four harmonic rays through it4 ,they will cut the circle in four points1 which will project to any other point of the curve in four harmonic rays2 for1 by the theorem concerning the angles inscribed in a circle1 the angles involved in the second set of four lines are the same as those in the first set4 ,if1 moreover1 we project the figure to any point in space1 we shall get a cone1 standing on a circular base1 generated by two projective axial pencils which are the projections of the pencils at ,s and ,s'4 ,cut across1 now1 by any plane1 and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils4 ,it thus appears that a conic section is a point-row of the second order4 ,it will later appear that a point-row of the second order is a conic section4 ,in the future1 therefore1 we shall refer to a point-row of the second order as a conic4 ,figure #ad ,,fig4 #ad 99#gd4 ,conic through five points499 ,pascal's theorem furnishes an elegant solution of the problem of drawing a conic through five given points4 ,to construct a sixth point on the conic1 draw through the point numbered #a an arbitrary line 7,fig4 #ad71 and let the desired point #f be the second point of intersection of this line with the conic4 ,the point ,l #ab-de is obtainable at once2 also the point ,n #cd-fa4 ,but ,l and ,n determine ,pascal's line1 and the intersection of #bc with #ef must be on this line4 ,intersect1 then1 the line ,,ln with #bc and obtain the point ,m4 ,join ,m to #e and intersect with #fa for the desired point #f4 ,figure #ae ,,fig4 #ae 99#ge4 ,tangent to a conic499 ,if two points of ,pascal's hexagon approach coincidence1 then the line joining them approaches as a limiting position the tangent line at that point4 ,pascal's theorem thus affords a ready method of drawing the tangent line to a conic at a given point4 ,if the conic is determined by the points #a1 #b1 #c1 #d1 #e 7,fig4 #ae71 and it is desired to draw the tangent at the point #a1 we may call that point #a1 #f4 ,the points ,l and ,m are obtained as usual1 and the intersection of #cd with ,,lm gives ,n4 ,join ,n to the point #a for the desired tangent at that point4 99#gf4 ,inscribed quadrangle499 ,two pairs of vertices may coalesce1 giving an inscribed quadrangle4 ,pascal's theorem gives for this case the very important theorem ,two pairs of opposite sides of any quadrangle inscribed in a conic meet on a straight line1 upon which line also intersect the two pairs of tangents at the opposite vertices4 ,figure #af ,,fig4 #af ,figure #ag ,,fig4 #ag ,for let the vertices be ,a1 ,b1 ,c1 and ,d1 and call the vertex ,a the point #a1 #f2 ,b1 the point #b2 ,c1 the point #c1 #d2 and ,d1 the point #e 7,fig4 #af74 ,pascal's theorem then indicates that ,l ,,ab-,,cd1 ,m ,,ad-,,bc1 and ,n1 which is the intersection of the tangents at ,a and ,c1 are all on a straight line u4 ,but if we were to call ,a the point #b1 ,b the point #f1 #a1 ,c the point #e1 and ,d the point #d1 #c1 then the intersection ,p of the tangents at ,b and ,d are also on this same line u4 ,thus ,l1 ,m1 ,n1 and ,p are four points on a straight line4 ,the consequences of this theorem are so numerous and important that we shall devote a separate chapter to them4 99#gg4 ,inscribed triangle499 ,finally1 three of the vertices of the hexagon may coalesce1 giving a triangle inscribed in a conic4 ,pascal's theorem then reads as follows 7,fig4 #ag7 for this case3 ,the three tangents at the vertices of a triangle inscribed in a conic meet the opposite sides in three points on a straight line4 ,figure #ah ,,fig4 #ah 99#gh4 ,degenerate conic499 ,if we apply ,pascal's theorem to a degenerate conic made up of a pair of straight lines1 we get the following theorem 7,fig4 #ah73 ,if three points1 ,a1 ,b1 ,c1 are chosen on one line1 and three points1 ,a'1 ,b'1 ,c'1 are chosen on another1 then the three points ,l ,,ab'-,a',b1 ,m ,,bc'-,b',c1 ,n ,,ca'-,c',a are all on a straight line4 ,,problems #a4 ,in ,fig4 #ab1 select different lines u and trace the locus of the center of perspectivity ,m of the lines u and u'4 #b4 ,given four points1 ,a1 ,b1 ,c1 ,d1 in the plane1 construct a fifth point ,p such that the lines ,,pa1 ,,pb1 ,,pc1 ,,pd shall be four harmonic lines4 ,suggestion4 ,draw a line a through the point ,a such that the four lines a1 ,,ab1 ,,ac1 ,,ad are harmonic4 ,construct now a conic through ,a1 ,b1 ,c1 and ,d having a for a tangent at ,a4 #c4 ,where are all the points ,p1 as determined in the preceding question1 to be found8 #d4 ,select any five points in the plane and draw the tangent to the conic through them at each of the five points4 #e4 ,given four points on the conic1 and the tangent at one of them1 to construct the conic4 7,to construct the conic means here to construct as many other points as may be desired47 #f4 ,given three points on the conic1 and the tangent at two of them1 to construct the conic4 #g4 ,given five points1 two of which are at infinity in different directions1 to construct the conic4 7,in this1 and in the following examples1 the student is supposed to be able to draw a line parallel to a given line47 #h4 ,given four points on a conic 7two of which are at infinity and two in the finite part of the plane71 together with the tangent at one of the finite points1 to construct the conic4 #i4 ,the tangents to a curve at its infinitely distant points are called its asymptotes if they pass through a finite part of the plane4 ,given the asymptotes and a finite point of a conic1 to construct the conic4 #aj4 ,given an asymptote and three finite points on the conic1 to determine the conic4 #aa4 ,given four points1 one of which is at infinity1 and given also that the line at infinity is a tangent line1 to construct the conic4 ,,chapter ,v - ,,pencils ,,of ,,rays ,,of ,,the ,,second ,,order 99#gi4 ,pencil of rays of the second order defined499 ,if the corresponding points of two projective point-rows be joined by straight lines1 a system of lines is obtained which is called a pencil of rays of the second order4 ,this name arises from the fact1 easily shown 7 #eg71 that at most two lines of the system may pass through any arbitrary point in the plane4 ,for if through any point there should pass three lines of the system1 then this point might be taken as the center of two projective pencils1 one projecting one point-row and the other projecting the other4 ,since1 now1 these pencils have three rays of one coincident with the corresponding rays of the other1 the two are identical and the two point-rows are in perspective position1 which was not supposed4 ,figure #ai ,,fig4 #ai 99#hj4 ,tangents to a circle499 ,to get a clear notion of this system of lines1 we may first show that the tangents to a circle form a system of this kind4 ,for take any two tangents1 u and u'1 to a circle1 and let ,a and ,b be the points of contact 7,fig4 #ai74 ,let now t be any third tangent with point of contact at ,c and meeting u and u' in ,p and ,p' respectively4 ,join ,a1 ,b1 ,p1 ,p'1 and ,c to ,o1 the center of the circle4 ,tangents from any point to a circle are equal1 and therefore the triangles ,,poa and ,,poc are equal1 as also are the triangles ,p',,ob and ,p',,oc4 ,therefore the angle ,,pop' is constant1 being equal to half the constant angle ,,aoc ,,cob4 ,this being true1 if we take any four harmonic points1 ,p#a1 ,p#b1 ,p#c1 ,p#d1 on the line u1 they will project to ,o in four harmonic lines1 and the tangents to the circle from these four points will meet u' in four harmonic points1 ,p'#a1 ,p'#b1 ,p'#c1 ,p'#d1 because the lines from these points to ,o inclose the same angles as the lines from the points ,p#a1 ,p#b1 ,p#c1 ,p#d on u4 ,the point-row on u is therefore projective to the point-row on u'4 ,thus the tangents to a circle are seen to join corresponding points on two projective point-rows1 and so1 according to the definition1 form a pencil of rays of the second order4 99#ha4 ,tangents to a conic499 ,if now this figure be projected to a point outside the plane of the circle1 and any section of the resulting cone be made by a plane1 we can easily see that the system of rays tangent to any conic section is a pencil of rays of the second order4 ,the converse is also true1 as we shall see later1 and a pencil of rays of the second order is also a set of lines tangent to a conic section4 99#hb499 ,the point-rows u and u' are1 themselves1 lines of the system1 for to the common point of the two point-rows1 considered as a point of u1 must correspond some point of u'1 and the line joining these two corresponding points is clearly u' itself4 ,similarly for the line u4 99#hc4 ,determination of the pencil499 ,we now show that it is possible to assign arbitrarily three lines1 a1 b1 and c1 of the system 7besides the lines u and u'72 but if these three lines are chosen1 the system is completely determined4 ,this statement is equivalent to the following3 ,given three pairs of corresponding points in two projective point-rows1 it is possible to find a point in one which corresponds to any point of the other4 ,we proceed1 then1 to the solution of the fundamental ,,problem4 ,given three pairs of points1 ,,aa'1 ,,bb'1 and ,,cc'1 of two projective point-rows u and u'1 to find the point ,d' of u' which corresponds to any given point ,d of u4 ,figure #bj ,,fig4 #bj ,on the line a1 joining ,a and ,a'1 take two points1 ,s and ,s'1 as centers of pencils perspective to u and u' respectively 7,fig4 #bj74 ,the figure will be much simplified if we take ,s on ,,bb' and ,s' on ,,cc'4 ,,sa and ,s',a' are corresponding rays of ,s and ,s'1 and the two pencils are therefore in perspective position4 ,it is not difficult to see that the axis of perspectivity m is the line joining ,b' and ,c4 ,given any point ,d on u1 to find the corresponding point ,d' on u' we proceed as follows3 ,join ,d to ,s and note where the joining line meets m4 ,join this point to ,s'4 ,this last line meets u' in the desired point ,d'4 ,we have now in this figure six lines of the system1 a1 b1 c1 d1 u1 and u'4 ,fix now the position of u1 u'1 b1 c1 and d1 and take four lines of the system1 a#a1 a#b1 a#c1 a#d1 which meet b in four harmonic points4 ,these points project to ,d1 giving four harmonic points on m4 ,these again project to ,d'1 giving four harmonic points on c4 ,it is thus clear that the rays a#a1 a#b1 a#c1 a#d cut out two projective point-rows on any two lines of the system4 ,thus u and u' are not special rays1 and any two rays of the system will serve as the point-rows to generate the system of lines4 99#hd4 ,brianchon's theorem499 ,from the figure also appears a fundamental theorem due to ,brianchon3 ,if #a1 #b1 #c1 #d1 #e1 #f are any six rays of a pencil of the second order1 then the lines l 7#ab1 #de71 m 7#bc1 #ef71 n 7#cd1 #fa7 all pass through a point4 ,figure #ba ,,fig4 #ba 99#he499 ,to make the notation fit the figure 7,fig4 #ba71 make a#a1 b #b1 u' #c1 d #d1 u #e1 c #f2 or1 interchanging two of the lines1 a #a1 c #b1 u #c1 d #d1 u' #e1 b #f4 ,thus1 by different namings of the lines1 it appears that not more than #fj different ,brianchon points are possible4 ,if we call #ab and #de opposite vertices of a circumscribed hexagon1 then ,brianchon's theorem may be stated as follows3 ,the three lines joining the three pairs of opposite vertices of a hexagon circumscribed about a conic meet in a point4 99#hf4 ,construction of the pencil by ,brianchon's theorem499 ,brianchon's theorem furnishes a ready method of determining a sixth line of the pencil of rays of the second order when five are given4 ,thus1 select a point in line #a and suppose that line #f is to pass through it4 ,then l 7#ab1 #de71 n 7#cd1 #fa71 and the line m 7#bc1 #ef7 must pass through 7l1 n74 ,then 7#bc1 ln7 meets #e in a point of the required sixth line4 ,figure #bb ,,fig4 #bb 99#hg4 ,point of contact of a tangent to a conic499 ,if the line #b approach as a limiting position the line #a1 then the intersection 7#a1 #b7 approaches as a limiting position the point of contact of #a with the conic4 ,this suggests an easy way to construct the point of contact of any tangent with the conic4 ,thus 7,fig4 #bb71 given the lines #a1 #b1 #c1 #d1 #e to construct the point of contact of #af4 ,draw l 7#ab1de71 m 7#bc1ef72 then 7#cd1 lm7 meets #a in the required point of contact ,t4 ,figure #bc ,,fig4 #bc 99#hh4 ,circumscribed quadrilateral499 ,if two pairs of lines in ,brianchon's hexagon coalesce1 we have a theorem concerning a quadrilateral circumscribed about a conic4 ,it is easily found to be 7,fig4 #bc7 ,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 point4 ,the consequences of this theorem will be deduced later4 ,figure #bd ,,fig4 #bd 99#hi4 ,circumscribed triangle499 ,the hexagon may further degenerate into a triangle1 giving the theorem 7,fig4 #bd7 ,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 point4 99#ij499 ,brianchon's theorem may also be used to solve the following problems3 ,given four tangents and the point of contact on any one of them1 to construct other tangents to a conic4 ,given three tangents and the points of contact of any two of them1 to construct other tangents to a conic4 99#ia4 ,harmonic tangents499 ,we have seen that a variable tangent cuts out on any two fixed tangents projective point-rows4 ,it follows that if four tangents cut a fifth in four harmonic points1 they must cut every tangent in four harmonic points4 ,it is possible1 therefore1 to make the following definition3 ,four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points4 99#ib4 ,projectivity and perspectivity499 ,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 discussed4 ,in particular1 the points on a tangent are said to be perspectively related to the tangents of a conic when each point lies on the tangent which corresponds to it4 ,these notions are of importance in the higher developments of the subject4 ,figure #be ,,fig4 #be 99#ic499 ,brianchon's theorem may also be applied to a degenerate conic made up of two points and the lines through them4 ,thus7,fig4 #be71 ,if a1 b1 c are three lines through a point ,s1 and a'1 b'1 c' are three lines through another point ,s'1 then the lines l 7ab'1 a'b71 m 7bc'1 b'c71 and n 7ca'1 c'a7 all meet in a point4 99#id4 ,law of duality499 ,the observant student will not have failed to note the remarkable similarity between the theorems of this chapter and those of the preceding4 ,he will have noted that points have replaced lines and lines have replaced points2 that points on a curve have been replaced by tangents to a curve2 that pencils have been replaced by point-rows1 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 line4 ,the theory upon which this wonderful law of duality is based will be developed in the next chapter4 ,,problems #a4 ,given four lines in the plane1 to construct another which shall meet them in four harmonic points4 #b4 ,where are all such lines found8 #c4 ,given any five lines in the plane1 construct on each the point of contact with the conic tangent to them all4 #d4 ,given four lines and the point of contact on one1 to construct the conic4 7,to construct the conic means here to draw as many other tangents as may be desired47 #e4 ,given three lines and the point of contact on two of them1 to construct the conic4 #f4 ,given four lines and the line at infinity1 to construct the conic4 #g4 ,given three lines and the line at infinity1 together with the point of contact at infinity1 to construct the conic4 #h4 ,given three lines1 two of which are asymptotes1 to construct the conic4 #i4 ,given five tangents to a conic1 to draw a tangent which shall be parallel to any one of them4 #aj4 ,the lines a1 b1 c are drawn parallel to each other4 ,the lines a'1 b'1 c' are also drawn parallel to each other4 ,show why the lines 7ab'1 a'b71 7bc'1 b'c71 7ca'1 c'a7 meet in a point4 7,in problems #f to #aj inclusive1 parallel lines are to be drawn47 ,,chapter ,,vi - ,,poles ,,and ,,polars 99#ie4 ,inscribed and circumscribed quadrilaterals499 ,the following theorems have been noted as special cases of ,pascal's and ,brianchon's theorems3 ,if a quadrilateral be inscribed in a conic1 two pairs of opposite sides and the tangents at opposite vertices intersect in four points1 all of which lie on a straight line4 ,if a quadrilateral be circumscribed about a conic1 the lines joining two pairs of opposite vertices and the lines joining two opposite points of contact are four lines which meet in a point4 ,figure #bf ,,fig4 #bf 99#if4 ,definition of the polar line of a point499 ,consider the quadrilateral ,k1 ,l1 ,m1 ,n inscribed in the conic 7,fig4 #bf74 ,it determines the four harmonic points ,a1 ,b1 ,c1 ,d which project from ,n in to the four harmonic points ,m1 ,b1 ,k1 ,o4 ,now the tangents at ,k and ,m meet in ,p1 a point on the line ,,ab4 ,the line ,,ab is thus determined entirely by the point ,o4 ,for if we draw any line through it1 meeting the conic in ,k and ,m1 and construct the harmonic conjugate ,b of ,o with respect to ,k and ,m1 and also the two tangents at ,k and ,m which meet in the point ,p1 then ,,bp is the line in question4 ,it thus appears that the line ,,lon may be any line whatever through ,o2 and since ,d1 ,l1 ,o1 ,n are four harmonic points1 we may describe the line ,,ab as the locus of points which are harmonic conjugates of ,o with respect to the two points where any line through ,o meets the curve4 99#ig499 ,furthermore1 since the tangents at ,l and ,n meet on this same line1 it appears as the locus of intersections of pairs of tangents drawn at the extremities of chords through ,o4 99#ih499 ,this important line1 which is completely determined by the point ,o1 is called the polar of ,o with respect to the conic2 and the point ,o is called the pole of the line with respect to the conic4 99#ii499 ,if a point ,b is on the polar of ,o1 then it is harmonically conjugate to ,o with respect to the two intersections ,k and ,m of the line ,,bc with the conic4 ,but for the same reason ,o is on the polar of ,b4 ,we have1 then1 the fundamental theorem ,if one point lies on the polar of a second1 then the second lies on the polar of the first4 99#ajj4 ,conjugate points and lines499 ,such a pair of points are said to be conjugate with respect to the conic4 ,similarly1 lines are said to be conjugate to each other with respect to the conic if one1 and consequently each1 passes through the pole of the other4 ,figure #bg ,,fig4 #bg 99#aja4 ,construction of the polar line of a given point499 ,given a point ,p1 if it is within the conic 7that is1 if no tangents may be drawn from ,p to the conic71 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 extremities4 ,if the point ,p is outside the conic1 we may draw the two tangents and construct the chord of contact 7,fig4 #bg74 99#ajb4 ,self-polar triangle499 ,in ,fig4 #bf it is not difficult to see that ,,aoc is a self-polar triangle1 that is1 each vertex is the pole of the opposite side4 ,for ,b1 ,m1 ,o1 ,k are four harmonic points1 and they project to ,c in four harmonic rays4 ,the line ,,co1 therefore1 meets the line ,,amn in a point on the polar of ,a1 being separated from ,a harmonically by the points ,m and ,n4 ,similarly1 the line ,,co meets ,,kl in a point on the polar of ,a1 and therefore ,,co is the polar of ,a4 ,similarly1 ,,oa is the polar of ,c1 and therefore ,o is the pole of ,,ac4 99#ajc4 ,pole and polar projectively related499 ,another very important theorem comes directly from ,fig4 #bf4 ,as a point ,a 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 ,a4 ,for1 fix the points ,l and ,n and let the point ,a move along the line ,,aq2 then the point-row ,a is projective to the pencil ,,lk1 and since ,k moves along the conic1 the pencil ,,lk is projective to the pencil ,,nk1 which in turn is projective to the point-row ,c1 which1 finally1 is projective to the pencil ,,oc1 which is the polar of ,a4 99#ajd4 ,duality499 ,we have1 then1 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 projective1 because to four harmonic points or lines correspond always four harmonic lines or points4 ,to every figure made up of points and lines will correspond a figure made up of lines and points4 ,to a point-row of the second order1 which is a conic considered as a point-locus1 corresponds a pencil of rays of the second order1 which is a conic considered as a line-locus4 ,the name 'duality' is used to describe this sort of correspondence4 ,it is important to note that the dual relation is subject to the same exceptions as the one-to-one correspondence is1 and must not be appealed to in cases where the one-to-one correspondence breaks down4 ,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 point2 namely1 the line parallel to the line of the point-row4 ,any theorem1 therefore1 that involves explicitly the point at infinity is not to be translated into a theorem concerning lines4 ,further1 in the pencil the angle between two lines has nothing to correspond to it in a point-row perspective to the pencil4 ,any theorem1 therefore1 that mentions angles is not translatable into another theorem by means of the law of duality4 ,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 words1 of measuring4 ,if1 therefore1 we call any theorem that has to do with the line at infinity or with the measurement of angles a metrical theorem1 and any other kind a projective theorem1 we may put the case as follows3 ,any projective theorem involves another theorem1 dual to it1 obtainable by interchanging everywhere the words 'point' and 'line4' 99#aje4 ,self-dual theorems499 ,the theorems of this chapter will be found1 upon examination1 to be self-dual2 that is1 no new theorem results from applying the process indicated in the preceding paragraph4 ,it is therefore useless to look for new results from the theorem on the circumscribed quadrilateral derived from ,brianchon's1 which is itself clearly the dual of ,pascal's theorem1 and in fact was first discovered by dualization of ,pascal's4 99#ajf499 ,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 relations4 ,a very important one may be easily found that leaves angles unaltered4 ,the relation called similarity leaves ratios between corresponding segments unaltered4 ,the above statements apply only to the particular one-to-one correspondence considered4 ,,problems #a4 ,given a quadrilateral1 construct the quadrangle polar to it with respect to a given conic4 #b4 ,a point moves along a straight line4 ,show that its polar lines with respect to two given conics generate a point-row of the second order4 #c4 ,given five points1 draw the polar of a point with respect to the conic passing through them1 without drawing the conic itself4 #d4 ,given five lines1 draw the polar of a point with respect to the conic tangent to them1 without drawing the conic itself4 #e4 ,dualize problems #c and #d4 #f4 ,given four points on the conic1 and the tangent at one of them1 draw the polar of a given point without drawing the conic4 ,dualize4 #g4 ,a point moves on a conic4 ,show that its polar line with respect to another conic describes a pencil of rays of the second order4 ,suggestion4 ,replace the given conic by a pair of protective pencils4 #h4 ,show that the poles of the tangents of one conic with respect to another lie on a conic4 #i4 ,the polar of a point ,a with respect to one conic is a1 and the pole of a with respect to another conic is ,a'4 ,show that as ,a travels along a line1 ,a' also travels along another line4 ,in general1 if ,a describes a curve of degree n1 show that ,a' describes another curve of the same degree n4 7,the degree of a curve is the greatest number of points that it may have in common with any line in the plane47 ,,chapter ,,vii - ,,metrical ,,properties ,,of ,,the ,,conic ,,sections 99#ajg4 ,diameters4 ,center499 ,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 plane4 ,entering into the theory of poles and polars with these elements1 we have the following definitions3 ,the polar line of an infinitely distant point is called a diameter1 and the pole of the infinitely distant line is called the center1 of the conic4 99#ajh499 ,from the harmonic properties of poles and polars1 ,the center bisects all chords through it 7 #ci74 ,every diameter passes through the center4 ,all chords through the same point at infinity 7that is1 each of a set of parallel chords7 are bisected by the diameter which is the polar of that infinitely distant point4 99#aji4 ,conjugate diameters499 ,we have already defined conjugate lines as lines which pass each through the pole of the other 7 #ajj74 ,any diameter bisects all chords parallel to its conjugate4 ,the tangents at the extremities of any diameter are parallel1 and parallel to the conjugate diameter4 ,diameters parallel to the sides of a circumscribed parallelogram are conjugate4 ,all these theorems are easy exercises for the student4 99#aaj4 ,classification of conics499 ,conics are classified according to their relation to the infinitely distant line4 ,if a conic has two points in common with the line at infinity1 it is called a hyperbola2 if it has no point in common with the infinitely distant line1 it is called an ellipse2 if it is tangent to the line at infinity1 it is called a parabola4 99#aaa499 ,in a hyperbola the center is outside the curve 7 #aja71 since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center4 ,as previously noted1 these two tangents are called the asymptotes of the curve4 ,the ellipse and the parabola have no asymptotes4 99#aab499 ,the center of the parabola is at infinity1 and therefore all its diameters are parallel1 for the pole of a tangent line is the point of contact4 ,the locus of the middle points of a series of parallel chords in a parabola is a diameter1 and the direction of the line of centers is the same for all series of parallel chords4 ,the center of an ellipse is within the curve4 ,figure #bh ,,fig4 #bh 99#aac4 ,theorems concerning asymptotes499 ,we derived as a consequence of the theorem of ,brianchon 7 #hi7 the proposition that if a triangle be circumscribed about a conic1 the lines joining the vertices to the points of contact of the opposite sides all meet in a point4 ,take1 now1 for two of the tangents the asymptotes of a hyperbola1 and let any third tangent cut them in ,a and ,b 7,fig4 #bh74 ,if1 then1 ,o is the intersection of the asymptotes1--and therefore the center of the curve1-- then the triangle ,,oab is circumscribed about the curve4 ,by the theorem just quoted1 the line through ,a parallel to ,,ob1 the line through ,b parallel to ,,oa1 and the line ,,op through the point of contact of the tangent ,,ab all meet in a point ,c4 ,but ,,oacb is a parallelogram1 and ,,pa ,,pb4 ,therefore ,the asymptotes cut off on each tangent a segment which is bisected by the point of contact4 99#aad499 ,if we draw a line ,,oq parallel to ,,ab1 then ,,op and ,,oq are conjugate diameters1 since ,,oq is parallel to the tangent at the point where ,,op meets the curve4 ,then1 since ,a1 ,p1 ,b1 and the point at infinity on ,,ab are four harmonic points1 we have the theorem ,conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes4 99#aae499 ,the chord ,a,b1 parallel to the diameter ,,oq1 is bisected at ,p' by the conjugate diameter ,,op4 ,if the chord ,a,b meet the asymptotes in ,a'1 ,b'1 then ,a'1 ,p'1 ,b'1 and the point at infinity are four harmonic points1 and therefore ,p' is the middle point of ,a',b'4 ,therefore ,a',a ,b',b and we have the theorem ,the segments cut off on any chord between the hyperbola and its asymptotes are equal4 99#aaf499 ,this theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given4 ,figure #bi ,,fig4 #bi 99#aag499 ,for the circumscribed quadrilateral1 ,brianchon's theorem gave 7 #hh7 ,the lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point4 ,take now for two of the tangents the asymptotes1 and let ,,ab and ,,cd be any other two 7,fig4 #bi74 ,if ,b and ,d are opposite vertices1 and also ,a and ,c1 then ,,ac and ,,bd are parallel1 and parallel to ,,pq1 the line joining the points of contact of ,,ab and ,,cd1 for these are three of the four lines of the theorem just quoted4 ,the fourth is the line at infinity which joins the point of contact of the asymptotes4 ,it is thus seen that the triangles ,,abc and ,,adc are equivalent1 and therefore the triangles ,,aob and ,,cod are also4 ,the tangent ,,ab may be fixed1 and the tangent ,,cd chosen arbitrarily2 therefore ,the triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area4 99#aah4 ,equation of hyperbola referred to the asymptotes499 ,draw through the point of contact ,p of the tangent ,,ab two lines1 one parallel to one asymptote and the other parallel to the other4 ,one of these lines meets ,,ob at a distance y from ,o1 and the other meets ,,oa at a distance x from ,o4 ,then1 since ,p is the middle point of ,,ab1 x is one half of ,,oa and y is one half of ,,ob4 ,the area of the parallelogram whose adjacent sides are x and y is one half the area of the triangle ,,aob1 and therefore1 by the preceding paragraph1 is constant4 ,this area is equal to xy sin 1 where is the constant angle between the asymptotes4 ,it follows that the product xy is constant1 and since x and y are the oblique cordinates of the point ,p1 the asymptotes being the axes of reference1 we have ,the equation of the hyperbola1 referred to the asymptotes as axes1 is xy constant4 ,this identifies the curve with the hyperbola as defined and discussed in works on analytic geometry4 ,figure #cj ,,fig4 #cj 99#aai4 ,equation of parabola499 ,we have defined the parabola as a conic which is tangent to the line at infinity 7 #aaj74 ,draw now two tangents to the curve 7,fig4 #cj71 meeting in ,a1 the points of contact being ,b and ,c4 ,these two tangents1 together with the line at infinity1 form a triangle circumscribed about the conic4 ,draw through ,b a parallel to ,,ac1 and through ,c a parallel to ,,ab4 ,if these meet in ,d1 then ,,ad is a diameter4 ,let ,,ad meet the curve in ,p1 and the chord ,,bc in ,q4 ,p is then the middle point of ,,aq4 ,also1 ,q is the middle point of the chord ,,bc1 and therefore the diameter ,,ad bisects all chords parallel to ,,bc4 ,in particular1 ,,ad passes through ,p1 the point of contact of the tangent drawn parallel to ,,bc4 ,draw now another tangent1 meeting ,,ab in ,b' and ,,ac in ,c'4 ,then these three1 with the line at infinity1 make a circumscribed quadrilateral4 ,but1 by ,brianchon's theorem applied to a quadrilateral 7 #hh71 it appears that a parallel to ,,ac through ,b'1 a parallel to ,,ab through ,c'1 and the line ,,bc meet in a point ,d'4 ,also1 from the similar triangles ,,bb',d' and ,,bac we have1 for all positions of the tangent line ,b',c1 ,b',d' 3 ,,bb' ,,ac 3 ,,ab1 or1 since ,b',d' ,,ac'1 ,,ac'3 ,,bb' ,,ac3,,ab constant4 ,if another tangent meet ,,ab in ,b and ,,ac in ,c1 we have ,,ac' 3 ,,bb' ,,ac 3 ,,bb1 and by subtraction we get ,c',c 3 ,b',b constant2 whence ,the segments cut off on any two tangents to a parabola by a variable tangent are proportional4 ,if now we take the tangent ,b',c' as axis of ordinates1 and the diameter through the point of contact ,o as axis of abscissas1 calling the coordinates of ,b7x1 y7 and of ,c7x'1 y'71 then1 from the similar triangles ,,bmd' and we have y 3 y' ,,bd' 3 ,d',c ,,bb' 3 ,,ab'4 ,also y 3 y' ,b',d' 3 ,c',c ,,ac' 3 ,c',c4 ,if now a line is drawn through ,a parallel to a diameter1 meeting the axis of ordinates in ,k1 we have ,,ak 3 ,,oq' ,,ac' 3 ,,cc' y 3 y'1 and ,,om 3 ,,ak ,,bb' 3 ,,ab' y 3 y'1 and1 by multiplication1 ,,om 3 ,,oq' y#b 3 y'#b1 or x 3 x' y#b 3 y'#b2 whence ,the abscissas of two points on a parabola are to each other as the squares of the corresponding cordinates1 a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference4 ,the last equation may be written y#b #bpx1 where #bp stands for y'#b 3 x'4 ,the parabola is thus identified with the curve of the same name studied in treatises on analytic geometry4 99#abj4 ,equation of central conics referred to conjugate diameters499 ,consider now a central conic1 that is1 one which is not a parabola and the center of which is therefore at a finite distance4 ,draw any four tangents to it1 two of which are parallel 7,fig4 #ca74 ,let the parallel tangents meet one of the other tangents in ,a and ,b and the other in ,c and ,d1 and let ,p and ,q be the points of contact of the parallel tangents ,r and ,s of the others4 ,then ,,ac1 ,,bd1 ,,pq1 and ,,rs all meet in a point ,w 7 #hh74 ,from the figure1 ,,pw 3 ,,wq ,,ap 3 ,,qc ,,pd 3 ,,bq1 or ,,ap ,,bq ,,pd ,,qc4 ,if now ,,dc is a fixed tangent and ,,ab a variable one1 we have from this equation ,,ap ,,bq constant4 ,this constant will be positive or negative according as ,,pa and ,,bq are measured in the same or in opposite directions4 ,accordingly we write ,,ap ,,bq b#b4 ,figure #ca ,,fig4 #ca ,since ,,ad and ,,bc are parallel tangents1 ,,pq is a diameter and the conjugate diameter is parallel to ,,ad4 ,the middle point of ,,pq is the center of the conic4 ,we take now for the axis of abscissas the diameter ,,pq1 and the conjugate diameter for the axis of ordinates4 ,join ,a to ,q and ,b to ,p and draw a line through ,s parallel to the axis of ordinates4 ,these three lines all meet in a point ,n1 because ,,ap1 ,,bq1 and ,,ab form a triangle circumscribed to the conic4 ,let ,,ns meet ,,pq in ,m4 ,then1 from the properties of the circumscribed triangle 7 #hi71 ,m1 ,n1 ,s1 and the point at infinity on ,,ns are four harmonic points1 and therefore ,n is the middle point of ,,ms4 ,if the cordinates of ,s are 7x1 y71 so that ,,om is x and ,,ms is y1 then ,,mn y_/#b4 ,now from the similar triangles ,,pmn and ,,pqb we have ,,bq 3 ,,pq ,,nm 3 ,,pm1 and from the similar triangles ,,pqa and ,,mqn1 ,,ap 3 ,,pq ,,mn 3 ,,mq1 whence1 multiplying1 we have b#b_/d a#b y#b_/d 7a x77a - x71 where formula or1 simplifying1 formula which is the equation of an ellipse when b#b has a positive sign1 and of a hyperbola when b#b has a negative sign4 ,we have thus identified point-rows of the second order with the curves given by equations of the second degree4 ,,problems #a4 ,draw a chord of a given conic which shall be bisected by a given point ,p4 #b4 ,show that all chords of a given conic that are bisected by a given chord are tangent to a parabola4 #c4 ,construct a parabola1 given two tangents with their points of contact4 #d4 ,construct a parabola1 given three points and the direction of the diameters4 #e4 ,a line u' is drawn through the pole ,u of a line u and at right angles to u4 ,the line u revolves about a point ,p4 ,show that the line u' is tangent to a parabola4 7,the lines u and u' are called normal conjugates47 #f4 ,given a circle and its center ,o1 to draw a line through a given point ,p parallel to a given line q4 ,prove the following construction3 ,let p be the polar of ,p1 ,q the pole of q1 and ,a the intersection of p with ,,oq4 ,the polar of ,a is the desired line4 ,,chapter ,,viii - ,,involution ,figure #cb ,,fig4 #cb 99#aba4 ,fundamental theorem499 ,the important theorem concerning two complete quadrangles 7 #bf71 upon which the theory of four harmonic points was based1 can easily be extended to the case where the four lines ,,kl1 ,k',l'1 ,,mn1 ,m',n' do not all meet in the same point ,a1 and the more general theorem that results may also be made the basis of a theory no less important1 which has to do with six points on a line4 ,the theorem is as follows3 ,given two complete quadrangles1 ,k1 ,l1 ,m1 ,n and ,k'1 ,l'1 ,m'1 ,n'1 so related that ,,kl and ,k',l' meet in ,a1 ,,mn and ,m',n' in ,a'1 ,,kn and ,k',n' in ,b1 ,,lm and ,l',m' in ,b'1 ,,ln and ,l',n' in ,c1 and ,,km and ,k',m' in ,c'1 then1 if ,a1 ,a'1 ,b1 ,b'1 and ,c are in a straight line1 the point ,c' also lies on that straight line4 ,the theorem follows from ,desargues's theorem 7,fig4 #cb74 ,it is seen that ,,kk'1 ,,ll'1 ,,mm'1 ,,nn' all meet in a point1 and thus1 from the same theorem1 applied to the triangles ,,klm and ,k',l',m'1 the point ,c' is on the same line with ,a and ,b'4 ,as in the simpler case1 it is seen that there is an indefinite number of quadrangles which may be drawn1 two sides of which go through ,a and ,a'1 two through ,b and ,b'1 and one through ,c4 ,the sixth side must then go through ,c'4 ,therefore1 99#abb499 ,two pairs of points1 ,a1 ,a' and ,b1 ,b'1 being given1 then the point ,c' corresponding to any given point ,c is uniquely determined4 ,the construction of this sixth point is easily accomplished4 ,draw through ,a and ,a' any two lines1 and cut across them by any line through ,c in the points ,l and ,n4 ,join ,n to ,b and ,l to ,b'1 thus determining the points ,k and ,m on the two lines through ,a and ,a'1 ,the line ,,km determines the desired point ,c'4 ,manifestly1 starting from ,c'1 we come in this way always to the same point ,c4 ,the particular quadrangle employed is of no consequence4 ,moreover1 since one pair of opposite sides in a complete quadrangle is not distinguishable in any way from any other1 the same set of six points will be obtained by starting from the pairs ,,aa' and ,,cc'1 or from the pairs ,,bb' and ,,cc'4 99#abc4 ,definition of involution of points on a line499 ,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 quadrangle4 ,if two pairs are fixed and one of the third pair describes the line1 then the other also describes the line1 and the points of the line are said to be paired in the involution determined by the two fixed pairs4 ,figure #cc ,,fig4 #cc 99#abd4 ,double-points in an involution499 ,the points ,c and ,c' describe projective point-rows1 as may be seen by fixing the points ,l and ,m4 ,the self-corresponding points1 of which there are two or none1 are called the double-points in the involution4 ,it is not difficult to see that the double-points in the involution are harmonic conjugates with respect to corresponding points in the involution4 ,for1 fixing as before the points ,l and ,m1 let the intersection of the lines ,,cl and ,c',m be ,p 7,fig4 #cc74 ,the locus of ,p is a conic which goes through the double-points1 because the point-rows ,c and ,c' are projective1 and therefore so are the pencils ,,lc and ,,mc' which generate the locus of ,p4 ,also1 when ,c and ,c' fall together1 the point ,p coincides with them4 ,further1 the tangents at ,l and ,m to this conic described by ,p are the lines ,,lb and ,,mb4 ,for in the pencil at ,l the ray ,,lm common to the two pencils which generate the conic is the ray ,,lb' and corresponds to the ray ,,mb of ,m1 which is therefore the tangent line to the conic at ,m4 ,similarly for the tangent ,,lb at ,l4 ,,lm is therefore the polar of ,b with respect to this conic1 and ,b and ,b' are therefore harmonic conjugates with respect to the double-points4 ,the same discussion applies to any other pair of corresponding points in the involution4 ,figure #cd ,,fig4 #cd 99#abe4 ,desargues's theorem concerning conics through four points499 ,let ,,dd' be any pair of points in the involution determined as above1 and consider the conic passing through the five points ,k1 ,l1 ,m1 ,n1 ,d4 ,we shall use ,pascal's theorem to show that this conic also passes through ,d'4 ,the point ,d' is determined as follows3 ,fix ,l and ,m as before 7,fig4 #cd7 and join ,d to ,l1 giving on ,,mn the point ,n'4 ,join ,n' to ,b1 giving on ,,lk the point ,k'4 ,then ,,mk' determines the point ,d' on the line ,,aa'1 given by the complete quadrangle ,k'1 ,l1 ,m1 ,n'4 ,consider the following six points1 numbering them in order3 ,d #a1 ,d' #b1 ,m #c1 ,n #d1 ,k #e1 and ,l #f4 ,we have the following intersections3 ,b 7#ab-de71 ,k' 7#bc-ef71 ,n' 7#cd-fa72 and since by construction ,b1 ,n1 and ,k' are on a straight line1 it follows from the converse of ,pascal's theorem1 which is easily established1 that the six points are on a conic4 ,we have1 then1 the beautiful theorem due to ,desargues3 ,the system of conics through four points meets any line in the plane in pairs of points in involution4 99#abf499 ,it appears also that the six points in involution determined by the quadrangle through the four fixed points belong also to the same involution with the points cut out by the system of conics1 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 system4 99#abg4 ,conics through four points touching a given line499 ,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 line4 ,we may therefore infer the theorem ,through four fixed points in the plane two conics or none may be drawn tangent to any given line4 ,figure #ce ,,fig4 #ce 99#abh4 ,double correspondence499 ,we have seen that corresponding points in an involution form two projective point-rows superposed on the same straight line4 ,two projective point-rows superposed on the same straight line are1 however1 not necessarily in involution1 as a simple example will show4 ,take two lines1 a and a'1 which both revolve about a fixed point ,s and which always make the same angle with each other 7,fig4 #ce74 ,these lines cut out on any line in the plane which does not pass through ,s two projective point-rows1 which are not1 however1 in involution unless the angle between the lines is a right angles4 ,for a point ,p may correspond to a point ,p'1 which in turn will correspond to some other point than ,p4 ,the peculiarity of point-rows in involution is that any point will correspond to the same point1 in whichever point-row it is considered as belonging4 ,in this case1 if a point ,p corresponds to a point ,p'1 then the point ,p' corresponds back again to the point ,p4 ,the points ,p and ,p' are then said to correspond doubly4 ,this notion is worthy of further study4 ,figure #cf ,,fig4 #cf 99#abi4 ,steiner's construction499 ,it will be observed that the solution of the fundamental problem given in #hc1 ,given three pairs of points of two protective point-rows1 to construct other pairs1 cannot be carried out if the two point-rows lie on the same straight line4 ,of course the method may be easily altered to cover that case also1 but it is worth while to give another solution of the problem1 due to ,steiner1 which will also give further information regarding the theory of involution1 and which may1 indeed1 be used as a foundation for that theory4 ,let the two point-rows ,a1 ,b1 ,c1 ,d1 444 and ,a'1 ,b'1 ,c'1 ,d'1 444 be superposed on the line u4 ,project them both to a point ,s and pass any conic through ,s4 ,we thus obtain two projective pencils1 a1 b1 c1 d1 444 and a'1 b'1 c'1 d'1 444 at ,s1 which meet the conic in the points 1 1 1 1 444 and '1 '1 '1 '1 444 7,fig4 #cf74 ,take now as the center of a pencil projecting the points '1 '1 '1 4441 and take ' as the center of a pencil projecting the points 1 1 1 4444 ,these two pencils are projective to each other1 and since they have a self-correspondin ray in common1 they are in perspective position and corresponding rays meet on the line joining 7'1 '7 to 7'1 '74 ,the correspondence between points in the two point-rows on u is now easily traced4 99#acj4 ,application of ,steiner's construction to double correspondence499 ,steiner's construction throws into our hands an important theorem concerning double correspondence3 ,if two projective point-rows1 superposed on the same line1 have one pair of points which correspond to each other doubly1 then all pairs correspond to each other doubly1 and the line is paired in involution4 ,to make this appear1 let us call the point ,a on u by two names1 ,a and ,p'1 according as it is thought of as belonging to the one or to the other of the two point-rows4 ,if this point is one of a pair which correspond to each other doubly1 then the points ,a' and ,p must coincide 7,fig4 #cg74 ,take now any point ,c1 which we will also call ,r'4 ,we must show that the corresponding point ,c' must also coincide with the point ,b4 ,join all the points to ,s1 as before1 and it appears that the points and ' coincide1 as also do the points ' and '4 ,by the above construction the line ' must meet ' on the line joining 7'1 '7 with 7'1 '74 ,but these four points form a quadrangle inscribed in the conic1 and we know by #ie that the tangents at the opposite vertices and ' meet on the line v4 ,the line ' is thus a tangent to the conic1 and ,c' and ,r are the same point4 ,that two projective point-rows superposed on the same line are also in involution when one pair1 and therefore all pairs1 correspond doubly may be shown by taking ,s at one vertex of a complete quadrangle which has two pairs of opposite sides going through two pairs of points4 ,the details we leave to the student4 ,figure #cg ,,fig4 #cg ,figure #ch ,,fig4 #ch 99#aca4 ,involution of points on a point-row of the second order499 ,it is important to note also1 in ,steiner's construction1 that we have obtained two point-rows of the second order superposed on the same conic1 and have paired the points of one with the points of the other in such a way that the correspondence is double4 ,we may then extend the notion of involution to point-rows of the second order and say that the points of a conic are paired in involution when they are corresponding points of two projective point-rows superposed on the conic1 and when they correspond to each other doubly4 ,with this definition we may prove the theorem3 ,the lines joining corresponding points of a point-row of the second order in involution all pass through a fixed point ,u1 and the line joining any two points ,a1 ,b meets the line joining the two corresponding points ,a'1 ,b' in the points of a line u1 which is the polar of ,u with respect to the conic4 ,for take ,a and ,a' as the centers of two pencils1 the first perspective to the point-row ,a'1 ,b'1 ,c' and the second perspective to the point-row ,a1 ,b1 ,c4 ,then1 since the common ray of the two pencils corresponds to itself1 they are in perspective position1 and their axis of perspectivity u 7,fig4 #ch7 is the line which joins the point 7,,ab'1 ,a',b7 to the point 7,,ac'1 ,a',c74 ,it is then immediately clear1 from the theory of poles and polars1 that ,,bb' and ,,cc' pass through the pole ,u of the line u4 99#acb4 ,involution of rays499 ,the whole theory thus far developed may be dualized1 and a theory of lines in involution may be built up1 starting with the complete quadrilateral4 ,thus1 ,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 involution4 ,if the pairs aa' and bb' are fixed1 and the line c describes a pencil1 the corresponding line c' also describes a pencil1 and the rays of the pencil are said to be paired in the involution determined by aa' and bb'4 99#acc4 ,double rays499 ,the self-corresponding rays1 of which there are two or none1 are called double rays of the involution4 ,corresponding rays of the involution are harmonic conjugates with respect to the double rays4 ,to the theorem of ,desargues 7 #abe7 which has to do with the system of conics through four points we have the dual3 ,the tangents from a fixed point to a system of conics tangent to four fixed lines form a pencil of rays in involution4 99#acd499 ,if a conic of the system should go through the fixed point1 it is clear that the two tangents would coincide and indicate a double ray of the involution4 ,the theorem1 therefore1 follows3 ,two conics or none may be drawn through a fixed point to be tangent to four fixed lines4 99#ace4 ,double correspondence499 ,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 doubly4 ,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 involution4 ,while this would have been entirely in accord with the treatment which was given the corresponding problem in the theory of harmonic points and lines1 it is more satisfactory1 from an aesthetic point of view1 to build the theory of lines in involution on its own base4 ,the student can show1 by methods entirely analogous to those used in the second chapter1 that involution is a projective property2 that is1 six rays in involution are cut by any transversal in six points in involution4 99#acf4 ,pencils of rays of the second order in involution499 ,we may also extend the notion of involution to pencils of rays of the second order4 ,thus1 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 conic1 and when they correspond to each other doubly4 ,we have then the theorem3 99#acg499 ,the intersections of corresponding rays of a pencil of the second order in involution are all on a straight line u1 and the intersection of any two tangents ab1 when joined to the intersection of the corresponding tangents a'b'1 gives a line which passes through a fixed point ,u1 the pole of the line u with respect to the conic4 99#ach4 ,involution of rays determined by a conic499 ,we have seen in the theory of poles and polars 7 #ajc7 that if a point ,p moves along a line m1 then the polar of ,p revolves about a point4 ,this pencil cuts out on m another point-row ,p'1 projective also to ,p4 ,since the polar of ,p passes through ,p'1 the polar of ,p' also passes through ,p1 so that the correspondence between ,p and ,p' is double4 ,the two point-rows are therefore in involution1 and the double points1 if any exist1 are the points where the line m meets the conic4 ,a similar involution of rays may be found at any point in the plane1 corresponding rays passing each through the pole of the other4 ,we have called such points and rays conjugate with respect to the conic 7 #ajj74 ,we may then state the following important theorem3 99#aci499 ,a conic determines on every line in its plane an involution of points1 corresponding points in the involution being conjugate with respect to the conic4 ,the double points1 if any exist1 are the points where the line meets the conic4 99#adj499 ,the dual theorem reads3 ,a conic determines at every point in the plane an involution of rays1 corresponding rays being conjugate with respect to the conic4 ,the double rays1 if any exist1 are the tangents from the point to the conic4 ,,problems #a4 ,two lines are drawn through a point on a conic so as always to make right angles with each other4 ,show that the lines joining the points where they meet the conic again all pass through a fixed point4 #b4 ,two lines are drawn through a fixed point on a conic so as always to make equal angles with the tangent at that point4 ,show that the lines joining the two points where the lines meet the conic again all pass through a fixed point4 #c4 ,four lines divide the plane into a certain number of regions4 ,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 lines4 #d4 ,if a variable quadrangle move in such a way as always to remain inscribed in a fixed conic1 while three of its sides turn each around one of three fixed collinear points1 then the fourth will also turn around a fourth fixed point collinear with the other three4 #e4 ,state and prove the dual of problem #d4 #f4 ,extend problem #d as follows3 ,if a variable polygon of an even number of sides move in such a way as always to remain inscribed in a fixed conic1 while all its sides but one pass through as many fixed collinear points1 then the last side will also pass through a fixed point collinear with the others4 #g4 ,if a triangle ,,qrs be inscribed in a conic1 and if a transversal s meet two of its sides in ,a and ,a'1 the third side and the tangent at the opposite vertex in ,b and ,b'1 and the conic itself in ,c and ,c'1 then ,,aa'1 ,,bb'1 ,,cc' are three pairs of points in an involution4 #h4 ,use the last exercise to solve the problem3 ,given five points1 ,q1 ,r1 ,s1 ,c1 ,c'1 on a conic1 to draw the tangent at any one of them4 #i4 ,state and prove the dual of problem #g and use it to prove the dual of problem #h4 #aj4 ,if a transversal cut two tangents to a conic in ,b and ,b'1 their chord of contact in ,a1 and the conic itself in ,p and ,p'1 then the point ,a is a double point of the involution determined by ,,bb' and ,,pp'4 #aa4 ,state and prove the dual of problem #aj4 #ab4 ,if a variable conic pass through two given points1 ,p and ,p'1 and if it be tangent to two given lines1 the chord of contact of these two tangents will always pass through a fixed point on ,,pp'4 #ac4 ,use the last theorem to solve the problem3 ,given four points1 ,p1 ,p'1 ,q1 ,s1 on a conic1 and the tangent at one of them1 ,q1 to draw the tangent at any one of the other points1 ,s4 #ad4 ,apply the theorem of problem #i to the case of a hyperbola where the two tangents are the asymptotes4 ,show in this way that if a hyperbola and its asymptotes be cut by a transversal1 the segments intercepted by the curve and by the asymptotes respectively have the same middle point4 #ae4 ,in a triangle circumscribed about a conic1 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 sides4 ,,chapter ,,ix - ,,metrical ,,properties ,,of ,,involutions ,figure #ci ,,fig4 #ci 99#ada4 ,introduction of infinite point2 center of involution499 ,we connect the projective theory of involution with the metrical1 as usual1 by the introduction of the elements at infinity4 ,in an involution of points on a line the point which corresponds to the infinitely distant point is called the center of the involution4 ,since corresponding points in the involution have been shown to be harmonic conjugates with respect to the double points1 the center is midway between the double points when they exist4 ,to construct the center 7,fig4 #ci7 we draw as usual through ,a and ,a' any two rays and cut them by a line parallel to ,,aa' in the points ,k and ,m4 ,join these points to ,b and ,b'1 thus determining on ,,ak and ,,an the points ,l and ,n4 ,,ln meets ,,aa' in the center ,o of the involution4 99#adb4 ,fundamental metrical theorem499 ,from the figure we see that the triangles ,,olb' and ,,plm are similar1 ,p being the intersection of ,,km and ,,ln4 ,also the triangles ,,kpn and ,,bon are similar4 ,we thus have ,,ob 3 ,,pk ,,on 3 ,,pn and ,,ob' 3 ,,pm ,,ol 3 ,,pl2 whence ,,ob ,,ob' 3 ,,pk ,,pm ,,on ,,ol 3 ,,pn ,,pl4 ,in the same way1 from the similar triangles ,,oal and ,,pkl1 and also ,,oa',n and ,,pmn1 we obtain ,,oa ,,oa' 3 ,,pk ,,pm ,,on ,,ol 3 ,,pn ,,pl1 and this1 with the preceding1 gives at once the fundamental theorem1 which is sometimes taken also as the definition of involution3 ,,oa ,,oa' ,,ob ,,ob' constant1 or1 in words1 ,the product of the distances from the center to two corresponding points in an involution of points is constant4 99#adc4 ,existence of double points499 ,clearly1 according as the constant is positive or negative the involution will or will not have double points4 ,the constant is the square root of the distance from the center to the double points4 ,if ,a and ,a' lie both on the same side of the center1 the product ,,oa ,,oa' is positive2 and if they lie on opposite sides1 it is negative4 ,take the case where they both lie on the same side of the center1 and take also the pair of corresponding points ,,bb'4 ,then1 since ,,oa ,,oa' ,,ob ,,ob'1 it cannot happen that ,b and ,b' are separated from each other by ,a and ,a'4 ,this is evident enough if the points are on opposite sides of the center4 ,if the pairs are on the same side of the center1 and ,b lies between ,a and ,a'1 so that ,,ob is greater1 say1 than ,,oa1 but less than ,,oa'1 then1 by the equation ,,oa ,,oa' ,,ob ,,ob'1 we must have ,,ob' also less than ,,oa' and greater than ,,oa4 ,a similar discussion may be made for the case where ,a and ,a' lie on opposite sides of ,o4 ,the results may be stated as follows1 without any reference to the center3 ,given two pairs of points in an involution of points1 if the points of one pair are separated from each other by the points of the other pair1 then the involution has no double points4 ,if the points of one pair are not separated from each other by the points of the other pair1 then the involution has two double points4 99#add499 ,an entirely similar criterion decides whether an involution of rays has or has not double rays1 or whether an involution of planes has or has not double planes4 ,figure #dj ,,fig4 #dj 99#ade4 ,construction of an involution by means of circles499 ,the equation just derived1 ,,oa ,,oa' ,,ob ,,ob'1 indicates another simple way in which points of an involution of points may be constructed4 ,through ,a and ,a' draw any circle1 and draw also any circle through ,b and ,b' to cut the first in the two points ,g and ,g' 7,fig4 #dj74 ,then any circle through ,g and ,g' will meet the line in pairs of points in the involution determined by ,,aa' and ,,bb'4 ,for if such a circle meets the line in the points ,,cc'1 then1 by the theorem in the geometry of the circle which says that if any chord is drawn through a fixed point within a circle1 the product of its segments is constant in whatever direction the chord is drawn1 and if a secant line be drawn from a fixed point without a circle1 the product of the secant and its external segment is constant in whatever direction the secant line is drawn1 we have ,,oc ,,oc' ,,og ,,og' constant4 ,so that for all such points ,,oa ,,oa' ,,ob ,,ob' ,,oc ,,oc'4 ,further1 the line ,,gg' meets ,,aa' in the center of the involution4 ,to find the double points1 if they exist1 we draw a tangent from ,o to any of the circles through ,,gg'4 ,let ,t be the point of contact4 ,then lay off on the line ,,oa a line ,,of equal to ,,ot4 ,then1 since by the above theorem of elementary geometry ,,oa ,,oa' ,,ot#b ,,of#b1 we have one double point ,f4 ,the other is at an equal distance on the other side of ,o4 ,this simple and effective method of constructing an involution of points is often taken as the basis for the theory of involution4 ,in projective geometry1 however1 the circle1 which is not a figure that remains unaltered by projection1 and is essentially a metrical notion1 ought not to be used to build up the purely projective part of the theory4 99#adf499 ,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 infinity1 called the circular points and usually denoted by ,i and ,j4 ,the above method of obtaining a point-row in involution is1 then1 nothing but a special case of the general theorem of the last chapter 7 #abe71 which asserted that a system of conics through four points will cut any line in the plane in a point-row in involution4 ,figure #da ,,fig4 #da 99#adg4 ,pairs in an involution of rays which are at right angles4 ,circular involution499 ,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 line4 ,there is one pair of rays1 however1 which does differ from all the others in that for this particular pair the angle is a right angle4 ,this is most easily shown by using the construction that employs circles1 as indicated above4 ,the centers of all the circles through ,g and ,g' lie on the perpendicular bisector of the line ,,gg'4 ,let this line meet the line ,,aa' in the point ,c 7,fig4 #da71 and draw the circle with center ,c which goes through ,g and ,g'4 ,this circle cuts out two points ,m and ,m' in the involution4 ,the rays ,,gm and ,,gm' are clearly at right angles1 being inscribed in a semicircle4 ,if1 therefore1 the involution of points is projected to ,g1 we have found two corresponding rays which are at right angles to each other4 ,given now any involution of rays with center ,g1 we may cut across it by a straight line and proceed to find the two points ,m and ,m'4 ,clearly there will be only one such pair unless the perpendicular bisector of ,,gg' coincides with the line ,,aa'4 ,in this case every ray is at right angles to its corresponding ray1 and the involution is called circular4 99#adh4 ,axes of conics499 ,at the close of the last chapter 7 #adj7 we gave the theorem3 ,a conic determines at every point in its plane an involution of rays1 corresponding rays being conjugate with respect to the conic4 ,the double rays1 if any exist1 are the tangents from the point to the conic4 ,in particular1 taking the point as the center of the conic1 we find that conjugate diameters form a system of rays in involution1 of which the asymptotes1 if there are any1 are the double rays4 ,also1 conjugate diameters are harmonic conjugates with respect to the asymptotes4 ,by the theorem of the last paragraph1 there are two conjugate diameters which are at right angles to each other4 ,these are called axes4 ,in the case of the parabola1 where the center is at infinity1 and on the curve1 there are1 properly speaking1 no conjugate diameters4 ,while the line at infinity might be considered as conjugate to all the other diameters1 it is not possible to assign to it any particular direction1 and so it cannot be used for the purpose of defining an axis of a parabola4 ,there is one diameter1 however1 which is at right angles to its conjugate system of chords1 and this one is called the axis of the parabola4 ,the circle also furnishes an exception in that every diameter is an axis4 ,the involution in this case is circular1 every ray being at right angles to its conjugate ray at the center4 99#adi4 ,points at which the involution determined by a conic is circular499 ,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 circular4 ,we shall proceed to the curious problem of proving the existence of such points and of determining their number and situation4 ,we shall then develop the important properties of such points4 99#aej499 ,it is clear1 in the first place1 that such a point cannot be on the outside of the conic1 else the involution would have double rays and such rays would have to be at right angles to themselves4 ,in the second place1 if two such points exist1 the line joining them must be a diameter and1 indeed1 an axis4 ,for if ,f and ,f' were two such points1 then1 since the conjugate ray at ,f to the line ,,ff' must be at right angles to it1 and also since the conjugate ray at ,f' to the line ,,ff' must be at right angles to it1 the pole of ,,ff' must be at infinity in a direction at right angles to ,,ff'4 ,the line ,,ff' is then a diameter1 and since it is at right angles to its conjugate diameter1 it must be an axis4 ,from this it follows also that the points we are seeking must all lie on one of the two axes1 else we should have a diameter which does not go through the intersection of all axes--the center of the conic4 ,at least one axis1 therefore1 must be free from any such points4 ,figure #db ,,fig4 #db 99#aea499 ,let now ,p be a point on one of the axes 7,fig4 #db71 and draw any ray through it1 such as q4 ,as q revolves about ,p1 its pole ,q moves along a line at right angles to the axis on which ,p lies1 describing a point-row p projective to the pencil of rays q4 ,the point at infinity in a direction at right angles to q also describes a point-row projective to q4 ,the line joining corresponding points of these two point-rows is always a conjugate line to q and at right angles to q1 or1 as we may call it1 a conjugate normal to q4 ,these conjugate normals to q1 joining as they do corresponding points in two projective point-rows1 form a pencil of rays of the second order4 ,but since the point at infinity on the point-row ,q corresponds to the point at infinity in a direction at right angles to q1 these point-rows are in perspective position and the normal conjugates of all the lines through ,p meet in a point4 ,this point lies on the same axis with ,p1 as is seen by taking q at right angles to the axis on which ,p lies4 ,the center of this pencil may be called ,p'1 and thus we have paired the point ,p with the point ,p'4 ,by moving the point ,p along the axis1 and by keeping the ray q parallel to a fixed direction1 we may see that the point-row ,p and the point-row ,p' are projective4 ,also the correspondence is double1 and by starting from the point ,p' we arrive at the point ,p4 ,therefore the point-rows ,p and ,p' are in involution1 and if only the involution has double points1 we shall have found in them the points we are seeking4 ,for it is clear that the rays through ,p and the corresponding rays through ,p' are conjugate normals2 and if ,p and ,p' coincide1 we shall have a point where all rays are at right angles to their conjugates4 ,we shall now show that the involution thus obtained on one of the two axes must have double points4 ,figure #dc ,,fig4 #dc 99#aeb4 ,discovery of the foci of the conic499 ,we know that on one axis no such points as we are seeking can lie 7 #aej74 ,the involution of points ,,pp' on this axis can therefore have no double points4 ,nevertheless1 let ,,pp' and ,,rr' be two pairs of corresponding points on this axis 7,fig4 #dc74 ,then we know that ,p and ,p' are separated from each other by ,r and ,r' 7 #adc74 ,draw a circle on ,,pp' as a diameter1 and one on ,,rr' as a diameter4 ,these must intersect in two points1 ,f and ,f'1 and since the center of the conic is the center of the involution ,,pp'1 ,,rr'1 as is easily seen1 it follows that ,f and ,f' are on the other axis of the conic4 ,moreover1 ,,fr and ,,fr' are conjugate normal rays1 since ,,rfr' is inscribed in a semicircle1 and the two rays go one through ,r and the other through ,r'4 ,the involution of points ,,pp'1 ,,rr' therefore projects to the two points ,f and ,f' in two pencils of rays in involution which have for corresponding rays conjugate normals to the conic4 ,we may1 then1 say3 ,there are two and only two points of the plane where the involution determined by the conic is circular4 ,these two points lie on one of the axes1 at equal distances from the center1 on the inside of the conic4 ,these points are called the foci of the conic4 99#aec4 ,the circle and the parabola499 ,the above discussion applies only to the central conics1 apart from the circle4 ,in the circle the two foci fall together at the center4 ,in the case of the parabola1 that part of the investigation which proves the existence of two foci on one of the axes will not hold1 as we have but one axis4 ,it is seen1 however1 that as ,p moves to infinity1 carrying the line q with it1 q becomes the line at infinity1 which for the parabola is a tangent line4 ,its pole ,q is thus at infinity and also the point ,p'1 so that ,p and ,p' fall together at infinity1 and therefore one focus of the parabola is at infinity4 ,there must therefore be another1 so that ,a parabola has one and only one focus in the finite part of the plane4 ,figure #dd ,,fig4 #dd 99#aed4 ,focal properties of conics499 ,we proceed to develop some theorems which will exhibit the importance of these points in the theory of the conic section4 ,draw a tangent to the conic1 and also the normal at the point of contact ,p4 ,these two lines are clearly conjugate normals4 ,the two points ,t and ,n1 therefore1 where they meet the axis which contains the foci1 are corresponding points in the involution considered above1 and are therefore harmonic conjugates with respect to the foci 7,fig4 #dd72 and if we join them to the point ,p1 we shall obtain four harmonic lines4 ,but two of them are at right angles to each other1 and so the others make equal angles with them 7,problem #d1 ,chapter ,,ii74 ,therefore ,the lines joining a point on the conic to the foci make equal angles with the tangent4 ,it follows that rays from a source of light at one focus are reflected by an ellipse to the other4 99#aee499 ,in the case of the parabola1 where one of the foci must be considered to be at infinity in the direction of the diameter1 we have ,figure #de ,,fig4 #de ,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 7,fig4 #de74 99#aef499 ,this last theorem is the basis for the construction of the parabolic reflector4 ,a ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector4 99#aeg4 ,directrix4 ,principal axis4 ,vertex499 ,the polar of the focus with respect to the conic is called the directrix4 ,the axis which contains the foci is called the principal axis1 and the intersection of the axis with the curve is called the vertex of the curve4 ,the directrix is at right angles to the principal axis4 ,in a parabola the vertex is equally distant from the focus and the directrix1 these three points and the point at infinity on the axis being four harmonic points4 ,in the ellipse the vertex is nearer to the focus than it is to the directrix1 for the same reason1 and in the hyperbola it is farther from the focus than it is from the directrix4 ,figure #df ,,fig4 #df 99#aeh4 ,another definition of a conic499 ,let ,p be any point on the directrix through which a line is drawn meeting the conic in the points ,a and ,b 7,fig4 #df74 ,let the tangents at ,a and ,b meet in ,t1 and call the focus ,f4 ,then ,,tf and ,,pf are conjugate lines1 and as they pass through a focus they must be at right angles to each other4 ,let ,,tf meet ,,ab in ,c4 ,then ,p1 ,a1 ,c1 ,b are four harmonic points4 ,project these four points parallel to ,,tf upon the directrix1 and we then get the four harmonic points ,p1 ,m1 ,q1 ,n4 ,since1 now1 ,,tfp is a right angle1 the angles ,,mfq and ,,nfq are equal1 as well as the angles ,,afc and ,,bfc4 ,therefore the triangles ,,maf and ,,nfb are similar1 and ,,fa 3 ,,fm ,,fb 3 ,,bn4 ,dropping perpendiculars ,,aa and ,,bb' upon the directrix1 this becomes ,,fa 3 ,,aa' ,,fb 3 ,,bb'4 ,we have thus the property often taken as the definition of a conic3 ,the ratio of the distances from a point on the conic to the focus and the directrix is constant4 ,figure #dg ,,fig4 #dg 99#aei4 ,eccentricity499 ,by taking the point at the vertex of the conic1 we note that this ratio is less than unity for the ellipse1 greater than unity for the hyperbola1 and equal to unity for the parabola4 ,this ratio is called the eccentricity4 ,figure #dh ,,fig4 #dh 99#afj4 ,sum or difference of focal distances499 ,the ellipse and the hyperbola have two foci and two directrices4 ,the eccentricity1 of course1 is the same for one focus as for the other1 since the curve is symmetrical with respect to both4 ,if the distances from a point on a conic to the two foci are r and r'1 and the distances from the same point to the corresponding directrices are d and d' 7,fig4 #dg71 we have r 3 d r' 3 d'2 7r r'7 3 7d d'74 ,in the ellipse 7d d'7 is constant1 being the distance between the directrices4 ,in the hyperbola this distance is 7d - d'74 ,it follows 7,fig4 #dh7 that ,in the ellipse the sum of the focal distances of any point on the curve is constant1 and in the hyperbola the difference between the focal distances is constant4 ,,problems #a4 ,construct the axis of a parabola1 given four tangents4 #b4 ,given two conjugate lines at right angles to each other1 and let them meet the axis which has no foci on it in the points ,a and ,b4 ,the circle on ,,ab as diameter will pass through the foci of the conic4 #c4 ,given the axes of a conic in position1 and also a tangent with its point of contact1 to construct the foci and determine the length of the axes4 #d4 ,given the tangent at the vertex of a parabola1 and two other tangents1 to find the focus4 #e4 ,the locus of the center of a circle touching two given circles is a conic with the centers of the given circles for its foci4 #f4 ,given the axis of a parabola and a tangent1 with its point of contact1 to find the focus4 #g4 ,the locus of the center of a circle which touches a given line and a given circle consists of two parabolas4 #h4 ,let ,f and ,f' be the foci of an ellipse1 and ,p any point on it4 ,produce ,,pf to ,g1 making ,,pg equal to ,,pf'4 ,find the locus of ,g4 #i4 ,if the points ,g of a circle be folded over upon a point ,f1 the creases will all be tangent to a conic4 ,if ,f is within the circle1 the conic will be an ellipse2 if ,f is without the circle1 the conic will be a hyperbola4 #aj4 ,if the points ,g in the last example be taken on a straight line1 the locus is a parabola4 #aa4 ,find the foci and the length of the principal axis of the conics in problems #i and #aj4 #ab4 ,in problem #aj a correspondence is set up between straight lines and parabolas4 ,as there is a fourfold infinity of parabolas in the plane1 and only a twofold infinity of straight lines1 there must be some restriction on the parabolas obtained by this method4 ,find and explain this restriction4 #ac4 ,state and explain the similar problem for problem #i4 #ad4 ,the last four problems are a study of the consequences of the following transformation3 ,a point ,o is fixed in the plane4 ,then to any point ,p is made to correspond the line p at right angles to ,,op and bisecting it4 ,in this correspondence1 what happens to p when ,p moves along a straight line8 ,what corresponds to the theorem that two lines have only one point in common8 ,what to the theorem that the angle sum of a triangle is two right angles8 ,etc4 ,,chapter ,x - ,,on ,,the ,,history ,,of ,,synthetic ,,projective ,,geometry 99#afa4 ,ancient results499 ,the theory of synthetic projective geometry as we have built it up in this course is less than a century old4 ,this is not to say that many of the theorems and principles involved were not discovered much earlier1 but isolated theorems do not make a theory1 any more than a pile of bricks makes a building4 ,the materials for our building have been contributed by many different workmen from the days of ,euclid down to the present time4 ,thus1 the notion of four harmonic points was familiar to the ancients1 who considered it from the metrical point of view as the division of a line internally and externally in the same ratio7#a7 the involution of six points cut out by any transversal which intersects the sides of a complete quadrilateral as studied by ,pappus7#b72 but these notions were not made the foundation for any general theory4 ,taken by themselves1 they are of small consequence2 it is their relation to other theorems and sets of theorems that gives them their importance4 ,the ancients were doubtless familiar with the theorem1 ,two lines determine a point1 and two points determine a line1 but they had no glimpse of the wonderful law of duality1 of which this theorem is a simple example4 ,the principle of projection1 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 ,greeks4 ,the ellipse1 the hyperbola1 and the parabola were to them entirely different curves1 to be treated separately with methods appropriate to each4 ,thus the focus of the ellipse was discovered some five hundred years before the focus of the parabola6 ,it was not till #aebb that ,verner7#c7 of ,nrnberg undertook to demonstrate the properties of the conic sections by means of the circle4 99#afb4 ,unifying principles499 ,in the early years of the seventeenth century--that wonderful epoch in the history of the world which produced a ,galileo1 a ,kepler1 a ,tycho ,brahe1 a ,descartes1 a ,desargues1 a ,pascal1 a ,cavalieri1 a ,wallis1 a ,fermat1 a ,huygens1 a ,bacon1 a ,napier1 and a goodly array of lesser lights1 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 ancients4 ,thus1 in #afjd the great astronomer ,kepler7#d7 introduced the notion that parallel lines should be considered as meeting at an infinite distance1 and that a parabola is at once the limiting case of an ellipse and of a hyperbola4 ,he also attributes to the parabola a blind focus 7caecus focus7 at infinity on the axis4 99#afc4 ,desargues499 ,in #afci ,desargues17#e7 an architect of ,lyons1 published a little treatise on the conic sections1 in which appears the theorem upon which we have founded the theory of four harmonic points 7 #be74 ,desargues1 however1 does not make use of it for that purpose4 ,four harmonic points are for him a special case of six points in involution when two of the three pairs coincide giving double points4 ,his development of the theory of involution is also different from the purely geometric one which we have adopted1 and is based on the theorem 7 #adb7 that the product of the distances of two conjugate points from the center is constant4 ,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 involution1 then any transversal must meet them in six points which are also in involution4 99#afd4 ,poles and polars499 ,in this little treatise is also contained the theory of poles and polars4 ,the polar line is called a traversal47#f7 ,the harmonic properties of poles and polars are given1 but ,desargues seems not to have arrived at the metrical properties which result when the infinite elements of the plane are introduced4 ,thus he says1 ,when the traversal is at an infinite distance1 all is unimaginable4 99#afe4 ,desargues's theorem concerning conics through four points499 ,we find in this little book the beautiful theorem concerning a quadrilateral inscribed in a conic section1 which is given by his name in #ach4 ,the theorem is not given in terms of a system of conics through four points1 for ,desargues had no conception of any such system4 ,he states the theorem1 in effect1 as follows3 ,given a simple quadrilateral inscribed in a conic section1 every transversal meets the conic and the four sides of the quadrilateral in six points which are in involution4 99#aff4 ,extension of the theory of poles and polars to space499 ,as an illustration of his remarkable powers of generalization1 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 degree4 ,this is a matter which has not been touched on in this book1 but the notion is not difficult to grasp4 ,if we draw through any point ,p in space a line to cut a sphere in two points1 ,a and ,s1 and then construct the fourth harmonic of ,p with respect to ,a and ,b1 the locus of this fourth harmonic1 for various lines through ,p1 is a plane called the polar plane of ,p with respect to the sphere4 ,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 plane4 ,desargues closes his discussion of this matter with the remark1 ,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 circle4 ,it should not be inferred from this remark1 however1 that he was acquainted with all the different varieties of surfaces of the second order4 ,the ancients were well acquainted with the surfaces obtained by revolving an ellipse or a parabola about an axis4 ,even the hyperboloid of two sheets1 obtained by revolving the hyperbola about its major axis1 was known to them1 but probably not the hyperboloid of one sheet1 which results from revolving a hyperbola about the other axis4 ,all the other solids of the second degree were probably unknown until their discovery by ,euler47#g7 99#afg499 ,desargues had no conception of the conic section of the locus of intersection of corresponding rays of two projective pencils of rays4 ,he seems to have tried to describe the curve by means of a pair of compasses1 moving one leg back and forth along a straight line instead of holding it fixed as in drawing a circle4 ,he does not attempt to define the law of the movement necessary to obtain a conic by this means4 99#afh4 ,reception of ,desargues's work499 ,strange to say1 ,desargues's immortal work was heaped with the most violent abuse and held up to ridicule and scorn6 ,incredible errors6 ,enormous mistakes and falsities6 ,really it is impossible for anyone who is familiar with the science concerning which he wishes to retail his thoughts1 to keep from laughing6 ,such were the comments of reviewers and critics4 ,nor were his detractors altogether ignorant and uninstructed men4 ,in spite of the devotion of his pupils and in spite of the admiration and friendship of men like ,descartes1 ,fermat1 ,mersenne1 and ,roberval1 his book disappeared so completely that two centuries after the date of its publication1 when the ,french geometer ,chasles wrote his history of geometry1 there was no means of estimating the value of the work done by ,desargues4 ,six years later1 however1 in #ahde1 ,chasles found a manuscript copy of the ,bruillon-project1 made by ,desargues's pupil1 ,de la ,hire4 99#afi4 ,conservatism in ,desargues's time499 ,it is not necessary to suppose that this effacement of ,desargues's work for two centuries was due to the savage attacks of his critics4 ,all this was in accordance with the fashion of the time1 and no man escaped bitter denunciation who attempted to improve on the methods of the ancients4 ,those were days when men refused to believe that a heavy body falls at the same rate as a lighter one1 even when ,galileo made them see it with their own eyes at the foot of the tower of ,pisa4 ,could they not turn to the exact page and line of ,aristotle which declared that the heavier body must fall the faster6 ,i have read ,aristotle's writings from end to end1 many times1 wrote a ,jesuit provincial to the mathematician and astronomer1 ,christoph ,scheiner1 at ,ingolstadt1 whose telescope seemed to reveal certain mysterious spots on the sun1 and ,i can assure you ,i have nowhere found anything similar to what you describe4 ,go1 my son1 and tranquilize yourself2 be assured that what you take for spots on the sun are the faults of your glasses1 or of your eyes4 ,the dead hand of ,aristotle barred the advance in every department of research4 ,physicians would have nothing to do with ,harvey's discoveries about the circulation of the blood4 ,nature is accused of tolerating a vacuum6 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 atmosphere4 99#agj4 ,desargues's style of writing499 ,nevertheless1 authority counted for less at this time in ,paris than it did in ,italy1 and the tragedy enacted in ,rome when ,galileo was forced to deny his inmost convictions at the bidding of a brutal ,inquisition could not have been staged in ,france4 ,moreover1 in the little company of scientists of which ,desargues was a member the utmost liberty of thought and expression was maintained4 ,one very good reason for the disappearance of the work of ,desargues is to be found in his style of writing4 ,he failed to heed the very good advice given him in a letter from his warm admirer ,descartes47#h7 ,you may have two designs1 both very good and very laudable1 but which do not require the same method of procedure3 ,the one is to write for the learned1 and show them some new properties of the conic sections which they do not already know2 and the other is to write for the curious unlearned1 and to do it so that this matter which until now has been understood by only a very few1 and which is nevertheless very useful for perspective1 for painting1 architecture1 etc41 shall become common and easy to all who wish to study them in your book4 ,if you have the first idea1 then it seems to me that it is necessary to avoid using new terms2 for the learned are already accustomed to using those of ,apollonius1 and will not readily change them for others1 though better1 and thus yours will serve only to render your demonstrations more difficult1 and to turn away your readers from your book4 ,if you have the second plan in mind1 it is certain that your terms1 which are ,french1 and conceived with spirit and grace1 will be better received by persons not preoccupied with those of the ancients4444 ,but1 if you have that intention1 you should make of it a great volume2 explain it all so fully and so distinctly that those gentlemen who cannot study without yawning2 who cannot distress their imaginations enough to grasp a proposition in geometry1 nor turn the leaves of a book to look at the letters in a figure1 shall find nothing in your discourse more difficult to understand than the description of an enchanted palace in a fairy story4 ,the point of these remarks is apparent when we note that ,desargues introduced some seventy new terms in his little book1 of which only one1 involution1 has survived4 ,curiously enough1 this is the one term singled out for the sharpest criticism and ridicule by his reviewer1 ,de ,beaugrand47#i7 ,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 book4 99#aga4 ,lack of appreciation of ,desargues499 ,desargues's methods1 entirely different from the analytic methods just then being developed by ,descartes and ,fermat1 seem to have been little understood4 ,between you and me1 wrote ,descartes7#aj7 to ,mersenne1 ,i can hardly form an idea of what he may have written concerning conics4 ,desargues seems to have boasted that he owed nothing to any man1 and that all his results had come from his own mind4 ,his favorite pupil1 ,de la ,hire1 did not realize the extraordinary simplicity and generality of his work4 ,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 years6 99#agb4 ,pascal and his theorem499 ,one does not have to believe all the marvelous stories of ,pascal's admiring sisters to credit him with wonderful precocity4 ,we have the fact that in #afdj1 when he was sixteen years old1 he published a little placard1 or poster1 entitled ,essay pour les conique17#aa7 in which his great theorem appears for the first time4 ,his manner of putting it may be a little puzzling to one who has only seen it in the form given in this book1 and it may be worth while for the student to compare the two methods of stating it4 ,it is given as follows3 ,if in the plane of ,m1 ,s1 ,q we draw through ,m the two lines ,,mk and ,,mv1 and through the point ,s the two lines ,,sk and ,,sv1 and let ,k be the intersection of ,,mk and ,,sk2 ,v the intersection of ,,mv and ,,sv2 ,a the intersection of ,,ma and ,,sa 7,a is the intersection of ,,sv and ,,mk71 and the intersection of ,,mv and ,,sk2 and if through two of the four points ,a1 ,k1 1 ,v1 which are not in the same straight line with ,m and ,s1 such as ,k and ,v1 we pass the circumference of a circle cutting the lines ,,mv1 ,,mp1 ,,sv1 ,,sk in the points ,o1 ,p1 ,q1 ,n2 ,i say that the lines ,,ms1 ,,no1 ,,pq are of the same order4 7,by lines of the same order ,pascal means lines which meet in the same point or are parallel47 ,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 section4 99#agc499 ,it must be understood that the ,essay was only a rsum of a more extended treatise on conics which1 owing partly to ,pascal's extreme youth1 partly to the difficulty of publishing scientific works in those days1 and also to his later morbid interest in religious matters1 was never published4 ,leibniz7#ab7 examined a copy of the complete work1 and has reported that the great theorem on the mystic hexagram was made the basis of the whole theory1 and that ,pascal had deduced some four hundred corollaries from it4 ,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-day4 ,unfortunately for science1 ,pascal's early death prevented the further development of the subject at his hands4 99#agd499 ,in the ,essay ,pascal gives full credit to ,desargues1 saying of one of the other propositions1 ,we prove this property also1 the original discoverer of which is ,m4 ,desargues1 of ,lyons1 one of the greatest minds of this age 444 and ,i wish to acknowledge that ,i owe to him the little which ,i have discovered4 ,this acknowledgment led ,descartes to believe that ,pascal's theorem should also be credited to ,desargues4 ,but in the scientific club which the young ,pascal attended in company with his father1 who was also a scientist of some reputation1 the theorem went by the name of 'la ,pascalia1' and ,descartes's remarks do not seem to have been taken seriously1 which indeed is not to be wondered at1 seeing that he was in the habit of giving scant credit to the work of other scientific investigators than himself4 99#age4 ,de la ,hire and his work499 ,de la ,hire added little to the development of the subject1 but he did put into print much of what ,desargues had already worked out1 not fully realizing1 perhaps1 how much was his own and how much he owed to his teacher4 ,writing in #afgi1 he says17#ac7 ,i have just read for the first time ,m4 ,desargues's little treatise1 and have made a copy of it in order to have a more perfect knowledge of it4 ,it was this copy that saved the work of his master from oblivion4 ,de la ,hire should be credited1 among other things1 with the invention of a method by which figures in the plane may be transformed into others of the same order4 ,his method is extremely interesting1 and will serve as an exercise for the student in synthetic projective geometry4 ,it is as follows3 ,draw two parallel lines1 a and b1 and select a point ,p in their plane4 ,through any point ,m of the plane draw a line meeting a in ,a and b in ,b4 ,draw a line through ,b parallel to ,,ap1 and let it meet ,,mp in the point ,m'4 ,it may be shown that the point ,m' thus obtained does not depend at all on the particular ray ,,mab used in determining it1 so that we have set up a one-to-one correspondence between the points ,m and ,m' in the plane4 ,the student may show that as ,m describes a point-row1 ,m' describes a point-row projective to it4 ,as ,m describes a conic1 ,m' describes another conic4 ,this sort of correspondence is called a collineation4 ,it will be found that the points on the line b transform into themselves1 as does also the single point ,p4 ,points on the line a transform into points on the line at infinity4 ,the student should remove the metrical features of the construction and take1 instead of two parallel lines a and b1 any two lines which may meet in a finite part of the plane4 ,the collineation is a special one in that the general one has an invariant triangle instead of an invariant point and line4 99#agf4 ,descartes and his influence499 ,the history of synthetic projective geometry has little to do with the work of the great philosopher ,descartes1 except in an indirect way4 ,the method of algebraic analysis invented by him1 and the differential and integral calculus which developed from it1 attracted all the interest of the mathematical world for nearly two centuries after ,desargues1 and synthetic geometry received scant attention during the rest of the seventeenth century and for the greater part of the eighteenth century4 ,it is difficult for moderns to conceive of the richness and variety of the problems which confronted the first workers in the calculus4 ,to come into the possession of a method which would solve almost automatically problems which had baffled the keenest minds of antiquity2 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 experimentally2 such was the happy experience of mathematicians for a century and a half after ,descartes1 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 ancients1 so that in his preface to his ,mchanique ,analytique1 published in #aghh1 ,lagrange boasts1 ,one will find no figures in this work4 ,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 of investigation began to attract attention4 ,lagrange himself1 in his later years1 turned in weariness from analysis and mechanics1 and applied himself to chemistry1 physics1 and philosophical speculations4 ,this state of mind1 says ,darboux17#ad7 we find almost always at certain moments in the lives of the greatest scholars4 ,at any rate1 after lying fallow for almost two centuries1 the field of pure geometry was attacked with almost religious enthusiasm4 99#agg4 ,newton and ,maclaurin499 ,but in hastening on to the epoch of ,poncelet and ,steiner we should not omit to mention the work of ,newton and ,maclaurin4 ,although their results were obtained by analysis for the most part1 nevertheless they have given us theorems which fall naturally into the domain of synthetic projective geometry4 ,thus ,newton's organic method7#ae7 of generating conic sections is closely related to the method which we have made use of in ,chapter ,,iii4 ,it is as follows3 ,if two angles1 ,,aos and ,,ao',s1 of given magnitudes turn about their respective vertices1 ,o and ,o'1 in such a way that the point of intersection1 ,s1 of one pair of arms always lies on a straight line1 the point of intersection1 ,a1 of the other pair of arms will describe a conic4 ,the proof of this is left to the student4 99#agh499 ,another method of generating a conic is due to ,maclaurin47#af7 ,the construction1 which we also leave for the student to justify1 is as follows3 ,if a triangle ,c',,pq move in such a way that its sides1 ,,pq1 ,,qc'1 and ,c',p1 turn around three fixed points1 ,r1 ,a1 ,b1 respectively1 while two of its vertices1 ,p1 ,q1 slide along two fixed lines1 ,,cb' and ,,ca'1 respectively1 then the remaining vertex will describe a conic4 99#agi4 ,descriptive geometry and the second revival499 ,the second revival of pure geometry was again to take place at a time of great intellectual activity4 ,the period at the close of the eighteenth and the beginning of the nineteenth century is adorned with a glorious list of mighty names1 among which are ,gauss1 ,lagrange1 ,legendre1 ,laplace1 ,monge1 ,carnot1 ,poncelet1 ,cauchy1 ,fourier1 ,steiner1 ,von ,staudt1 ,mbius1 ,abel1 and many others4 ,the renaissance may be said to date from the invention by ,monge7#ag7 of the theory of descriptive geometry4 ,descriptive geometry is concerned with the representation of figures in space of three dimensions by means of space of two dimensions4 ,the method commonly used consists in projecting the space figure on two planes 7a vertical and a horizontal plane being most convenient71 the projections being made most simply for metrical purposes from infinity in directions perpendicular to the two planes of projection4 ,these two planes are then made to coincide by revolving the horizontal into the vertical about their common line4 ,such is the method of descriptive geometry which in the hands of ,monge acquired wonderful generality and elegance4 ,problems concerning fortifications were worked so quickly by this method that the commandant at the military school at ,mzires1 where ,monge was a draftsman and pupil1 viewed the results with distrust4 ,monge afterward became professor of mathematics at ,mzires and gathered around him a group of students destined to have a share in the advancement of pure geometry4 ,among these were ,hachette1 ,brianchon1 ,dupin1 ,chasles1 ,poncelet1 and many others4 99#ahj4 ,duality1 homology1 continuity1 contingent relations499 ,analytic geometry had left little to do in the way of discovery of new material1 and the mathematical world was ready for the construction of the edifice4 ,the activities of the group of men that followed ,monge were directed toward this end1 and we now begin to hear of the great unifying notions of duality1 homology1 continuity1 contingent relations1 and the like4 ,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 analysts4 ,their dream was the building up of a system of geometry which should be independent of analysis4 ,monge1 and after him ,poncelet1 spent much thought on the so-called principle of continuity1 afterwards discussed by ,chasles under the name of the principle of contingent relations4 ,to get a clear idea of this principle1 consider a theorem in geometry in the proof of which certain auxiliary elements are employed4 ,these elements do not appear in the statement of the theorem1 and the theorem might possibly be proved without them4 ,in drawing the figure for the proof of the theorem1 however1 some of these elements may not appear1 or1 as the analyst would say1 they become imaginary4 ,no matter1 says the principle of contingent relations1 the theorem is true1 and the proof is valid whether the elements used in the proof are real or imaginary4 99#aha4 ,poncelet and ,cauchy499 ,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 ,cauchy4 ,in his review of ,poncelet's great work on the projective properties of figures7#ah7 ,cauchy says1 ,in his preliminary discourse the author insists once more on the necessity of admitting into geometry what he calls the 'principle of continuity4' ,we have already discussed that principle 444 and we have found that that principle is1 properly speaking1 only a strong induction1 which cannot be indiscriminately applied to all sorts of questions in geometry1 nor even in analysis4 ,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 ,trait des ,proprits ,projectives des ,figures4 ,although this principle is constantly made use of at the present day in all sorts of investigations1 careful geometricians are in agreement with ,cauchy in this matter1 and use it only as a convenient working tool for purposes of exploration4 ,the one-to-one correspondence between geometric forms and algebraic analysis is subject to many and important exceptions4 ,the field of analysis is much more general than the field of geometry1 and while there may be a clear notion in analysis to1 correspond to every notion in geometry1 the opposite is not true4 ,thus1 in analysis we can deal with four cordinates as well as with three1 but the existence of a space of four dimensions to correspond to it does not therefore follow4 ,when the geometer speaks of the two real or imaginary intersections of a straight line with a conic1 he is really speaking the language of algebra4 ,apart from the algebra involved1 it is the height of absurdity to try to distinguish between the two points in which a line fails to meet a conic6 99#ahb4 ,the work of ,poncelet499 ,but ,poncelet's right to the title ,the ,father of ,modern ,geometry does not stand or fall with the principle of contingent relations4 ,in spite of the fact that he considered this principle the most important of all his discoveries1 his reputation rests on more solid foundations4 ,he was the first to study figures in homology1 which is1 in effect1 the collineation described in #age1 where corresponding points lie on straight lines through a fixed point4 ,he was the first to give1 by means of the theory of poles and polars1 a transformation by which an element is transformed into another of a different sort4 ,point-to-point transformations will sometimes generalize a theorem1 but the transformation discovered by ,poncelet may throw a theorem into one of an entirely different aspect4 ,the principle of duality1 first stated in definite form by ,gergonne17#ai7 the editor of the mathematical journal in which ,poncelet published his researches1 was based by ,poncelet on his theory of poles and polars4 ,he also put into definite form the notions of the infinitely distant elements in space as all lying on a plane at infinity4 99#ahc4 ,the debt which analytic geometry owes to synthetic geometry499 ,the reaction of pure geometry on analytic geometry is clearly seen in the development of the notion of the class of a curve1 which is the number of tangents that may be drawn from a point in a plane to a given curve lying in that plane4 ,if a point moves along a conic1 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 conic4 ,this may be expressed by the statement that the conic is of the second order and also of the second class4 ,it might be thought that if a point moved along a cubic curve1 its polar line with respect to a conic would remain tangent to another cubic curve4 ,this is not the case1 however1 and the investigations of ,poncelet and others to determine the class of a given curve were afterward completed by ,plcker4 ,the notion of geometrical transformation led also to the very important developments in the theory of invariants1 which1 geometrically1 are the elements and configurations which are not affected by the transformation4 ,the anharmonic ratio of four points is such an invariant1 since it remains unaltered under all projective transformations4 99#ahd4 ,steiner and his work499 ,in the work of ,poncelet and his contemporaries1 ,chasles1 ,brianchon1 ,hachette1 ,dupin1 ,gergonne1 and others1 the anharmonic ratio enjoyed a fundamental rle4 ,it is made also the basis of the great work of ,steiner17#bj7 who was the first to treat of the conic1 not as the projection of a circle1 but as the locus of intersection of corresponding rays of two projective pencils4 ,steiner not only related to each other1 in one-to-one correspondence1 point-rows and pencils and all the other fundamental forms1 but he set into correspondence even curves and surfaces of higher degrees4 ,this new and fertile conception gave him an easy and direct route into the most abstract and difficult regions of pure geometry4 ,much of his work was given without any indication of the methods by which he had arrived at it1 and many of his results have only recently been verified4 99#ahe4 ,von ,staudt and his work499 ,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 ratio4 ,this work was accomplished by ,von ,staudt17#ba7 who applied himself to the restatement of the theory of geometry in a form independent of analytic and metrical notions4 ,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 ,staudt4 ,his work is characterized by a most remarkable generality1 in that he is able to discuss real and imaginary forms with equal ease4 ,thus he assumes a one-to-one correspondence between the points and lines of a plane1 and defines a conic as the locus of points which lie on their corresponding lines1 and a pencil of rays of the second order as the system of lines which pass through their corresponding points4 ,the point-row and pencil of the second order may be real or imaginary1 but his theorems still apply4 ,an illustration of a correspondence of this sort1 where the conic is imaginary1 is given in #ae of the first chapter4 ,in defining conjugate imaginary points on a line1 ,von ,staudt made use of an involution of points having no double points4 ,his methods1 while elegant and powerful1 are hardly adapted to an elementary course1 but ,reye7#bb7 and others have done much toward simplifying his presentation4 99#ahf4 ,recent developments499 ,it would be only confusing to the student to attempt to trace here the later developments of the science of protective geometry4 ,it is concerned for the most part with curves and surfaces of a higher degree than the second4 ,purely synthetic methods have been used with marked success in the study of the straight line in space4 ,the struggle between analysis and pure geometry has long since come to an end4 ,each has its distinct advantages1 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 hand4 ,pure geometry has to its credit some of the finest discoveries in mathematics1 and need not apologize for having been born4 ,the day of its usefulness has not passed with the invention of abridged notation and of short methods in analysis4 ,while we may be certain that any geometrical problem may always be stated in analytic form1 it does not follow that that statement will be simple or easily interpreted4 ,for many mathematicians the geometric intuitions are weak1 and for such the method will have little attraction4 ,on the other hand1 there will always be those for whom the subject will have a peculiar glamor--who will follow with delight the curious and unexpected relations between the forms of space4 ,there is a corresponding pleasure1 doubtless1 for the analyst in tracing the marvelous connections between the various fields in which he wanders1 and it is as absurd to shut one's eyes to the beauties in one as it is to ignore those in the other4 ,let us cultivate geometry1 then1 says ,darboux17#bc7 without wishing in all points to equal it to its rival4 ,besides1 if we were tempted to neglect it1 it would not be long in finding in the applications of mathematics1 as once it has already done1 the means of renewing its life and of developing itself anew4 ,it is like the ,giant ,antaeus1 who renewed1 his strength by touching the earth4 ,,index 7,the numbers refer to the paragraphs7 ,abel 7#ahjb-ahbi71 #agi ,analogy1 #bd ,analytic geometry1 #ba1 #aah1 #aai1 #abj1 #adf1 #agf1 #ahj ,anharmonic ratio1 #df1 #afa1 #ahd1 #ahe ,apollonius 7second half of third century ,b4,c471 #gj ,archimedes 7#bhg-bab ,b4,c471 #agf ,aristotle 7#chd-cbb ,b4,c471 #afi ,asymptotes1 #aaa1 #aac1 #aad1 #aae1 #aaf1 #aag1 #aah1 #adh ,axes of a conic1 #adh ,axial pencil1 #g1 #h1 #bc1 #ej1 #ed ,axis of perspectivity1 #h1 #dg ,bacon 7#aefa-afbf71 #afb ,bisection1 #da1 #aji ,brianchon 7#aghe-ahfd71 #hd1 #he1 #hf1 #hh1 #hi1 #ij1 #ie1 #aje1 #aac1 #agd1 #ahd ,calculus1 #agf ,carnot 7#agif-ahcb71 #agi ,cauchy 7#aghi-aheg71 #agi1 #aha ,cavalieri 7#aeih-afdg71 #afb ,center of a conic1 #ajg1 #aab1 #adh ,center of involution1 #ada1 #adb ,center of perspectivity1 #h ,central conic1 #abj ,chasles 7#agic-ahhj71 #afh1 #agi1 #ahj1 #ahd ,circle1 #ba1 #gc1 #hj1 #ade1 #adf1 #adg ,circular involution1 #adg1 #adi1 #aej1 #aea ,circular points1 #adf ,class of a curve1 #ahc ,classification of conics1 #aaj ,collineation1 #age ,concentric pencils1 #ej ,cone of the second order1 #ei ,conic1 #gc1 #ha ,conjugate diameters1 #aad1 #adh ,conjugate normal1 #aea ,conjugate points and lines1 #ajj1 #aji1 #ach1 #aci1 #adj ,constants in an equation1 #ba ,contingent relations1 #ahj1 #aha ,continuity1 #ahj1 #aha ,continuous correspondence1 #i1 #aj1 #ba1 #di ,corresponding elements1 #fd ,counting1 #a1 #d ,cross ratio1 #df ,darboux1 #agf1 #ahf ,de ,beaugrand1 #agj ,degenerate pencil of rays of the second order1 #eh1 #ic ,degenerate point-row of the second order1 #ef1 #gh ,de la ,hire 7#afdj-agah71 #afh1 #aga1 #age ,desargues 7#aeic-affb71 #be1 #bf1 #dj1 #aba1 #abe1 #afb1 #afc1 #afd1 #afe1 #aff1 #afg1 #afh1 #afi1 #agj1 #aga1 #agd1 #age ,descartes 7#aeif-afej71 #afb1 #agj1 #aga1 #agd1 #agf ,descriptive geometry1 #agi ,diameter1 #ajg ,directrix1 #aeg1 #aeh1 #aei1 #afj ,double correspondence1 #abh1 #acj ,double points of an involution1 #abd ,double rays of an involution1 #acc1 #acd ,duality1 #id1 #ajd1 #afa1 #ahj1 #ahb ,dupin 7#aghd-ahgc71 #agd1 #ahd ,eccentricity of conic1 #aei ,ellipse1 #aaj1 #aaa1 #afb ,equation of conic1 #aah1 #aai1 #abj ,euclid 7ca4 #cjj ,b4,c471 #f1 #bb1 #ajd ,euler 7#agjg-aghc71 #aff ,fermat 7#afja-affe71 #afb1 #aga ,foci of a conic1 #aeb1 #aec1 #aed1 #aee1 #aef1 #aeg1 #aeh1 #aei1 #afj1 #afa1 #afb ,fourier 7#agfh-ahcj71 #agi ,fourth harmonic1 #bi ,fundamental form1 #g1 #af1 #bc1 #cf1 #dg1 #fj1 #ahd ,galileo 7#aefd-afdb71 #afb1 #afi1 #agj1 #agf ,gauss 7#aggg-ahee71 #agi ,gergonne 7#agga-ahei71 #ahb1 #ahd ,greek geometry1 #afa ,hachette 7#agfi-ahcd71 #agi1 #ahd ,harmonic conjugates1 #bi1 #cj1 #ci ,harmonic elements1 #hf1 #di1 #ia1 #afc1 #ahe ,harmonic lines1 #cc1 #cd1 #ce1 #ff1 #fg ,harmonic planes1 #cd1 #ce ,harmonic points1 #bi1 #ca1 #cb1 #cc1 #cd1 #ce1 #cf1 #dc1 #ga1 #afa ,harmonic tangents to a conic1 #ia1 #ib ,harvey 7#aegh-afeg71 #afi ,homology1 #ahj1 #ahb ,huygens 7#afbi-afie71 #afb ,hyperbola1 #aaj1 #aaa1 #aac1 #aad1 #aae1 #aaf1 #aag1 #aah1 #afb ,imaginary elements1 #adf1 #ahj1 #aha1 #ahb1 #ahe ,infinitely distant elements1 #f1 #i1 #bb1 #ci1 #dj1 #da1 #ajd1 #ajg1 #aaj ,infinity1 #d1 #e1 #aj1 #ab1 #ac1 #ad1 #ae1 #ag1 #ah1 #ai1 #bj1 #ba1 #bb1 #da ,involution1 #cg1 #abc1 #abd1 #abe1 #abf1 #abg1 #abh1 #abi1 #acj1 #aca1 #acb1 #acc1 #acd1 #ace1 #acf1 #acg1 #ach1 #aci1 #adj1 #afa1 #afc1 #agj ,kepler 7#aega-afcj71 #afb ,lagrange 7#agcf-ahac71 #agf1 #agi ,laplace 7#agdi-ahbg71 #agi ,legendre 7#ageb-ahcc71 #agi ,leibniz 7#afdf-agaf71 #agc ,linear construction1 #dj1 #da1 #db ,maclaurin 7#afih-agdf71 #agg1 #agh ,measurements1 #bc1 #dj1 #da1 #ajd ,mersenne 7#aehh-afdh71 #afh1 #aga ,metrical theorems1 #dj1 #ajd1 #ajf1 #ajg1 #ada ,middle point1 #ci1 #da ,mbius 7#agij-ahfh71 #agi ,monge 7#agdf-ahah71 #agi1 #ahj ,napier 7#aeej-afag71 #afb ,newton 7#afdb-agbg71 #agg ,numbers1 #d1 #ba1 #dc ,numerical computations1 #dc1 #dd1 #df ,one-to-one correspondence1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #i1 #aj1 #aa1 #bd1 #cf1 #hg1 #dc1 #fj1 #ajd1 #ajf1 #ahd ,opposite sides of a hexagon1 #gj ,opposite sides of a quadrilateral1 #bh1 #bi ,order of a form1 #g1 #aj1 #aa1 #ab1 #ac1 #ad1 #ae1 #af1 #ag1 #ah1 #ai1 #bj1 #ba ,pappus 7fourth century ,a4,d471 #afa ,parabola1 #aaj1 #aaa1 #aab1 #aai1 #afb ,parallel lines1 #ci1 #da1 #afb ,pascal 7#afbc-affb71 #fi1 #gj1 #gd1 #ge1 #gf1 #gg1 #gh1 #ie1 #aje1 #abe1 #afb1 #afi1 #aga1 #agb1 #agc ,pencil of planes of the second order1 #ei ,pencil of rays1 #f1 #g1 #h1 #bc2 of the second order1 #eg1 #fj1 #gi1 #ha ,perspective position1 #f1 #h1 #ce1 #cg1 #ea1 #ec1 #ga ,plane system1 #af1 #bc ,planes on space1 #ag ,point of contact1 #hg1 #hh1 #hi1 #ij ,point system1 #af1 #bc ,point-row1 #f1 #g1 #h1 #i1 #bc2 of the second order1 #ee1 #fj1 #fa1 #ff1 #fg1 #gb ,points in space1 #ah ,pole and polar1 #ih1 #ii1 #ajj1 #aja1 #ach1 #afd1 #aff ,poncelet 7#aghh-ahfg71 #agg1 #agi1 #ahj1 #aha1 #ahb1 #ahc1 #ahd ,principal axis of a conic1 #aeg ,projection1 #afa ,protective axial pencils1 #ei ,projective correspondence1 #i1 #ce1 #cf1 #cg1 #dg1 #ga1 #ib1 #ajd ,projective pencils1 #ec1 #fd1 #fh ,projective point-rows1 #ea1 #gi ,projective properties1 #bd ,projective theorems1 #dj1 #ajd ,quadrangle1 #bf1 #bg1 #bh1 #bi ,quadric cone1 #ei ,quadrilateral1 #hh1 #ie1 #if ,roberval 7#afjb-afge71 #afh ,ruler construction1 #dj ,scheiner1 #afi ,self-corresponding elements1 #dg1 #dh1 #di1 #ej1 #ea ,self-dual1 #aje ,self-polar triangle1 #ajb ,separation of elements in involution1 #adh ,separation of harmonic conjugates1 #ch ,sequence of points1 #di ,sign of segment1 #dd1 #de ,similarity1 #ajf ,skew lines1 #ab ,space system1 #ai1 #bc ,sphere1 #ba ,steiner 7#agif-ahfc71 #abi1 #acj1 #aca1 #agg1 #agi1 #ahd ,steiner's construction1 #abi1 #acj1 #aca ,superposed point-rows1 #dg1 #dh1 #di ,surfaces of the second degree1 #aff ,system of lines in space1 #bj1 #bc ,systems of conics1 #abe ,tangent line1 #fa1 #hj1 #ha1 #hg1 #hh1 #hi1 #ij1 #ia1 #ib ,tycho ,brahe 7#aedf-afja71 #afb ,verner1 #afa ,vertex of conic1 #aeg1 #aei ,von ,staudt 7#agih-ahfg71 #agi1 #ahe ,wallis 7#afaf-agjc71 #afb ,,footnotes #a ,the more general notion of anharmonic ratio1 which includes the harmonic ratio as a special case1 was also known to the ancients4 ,while we have not found it necessary to make use of the anharmonic ratio in building up our theory1 it is so frequently met with in treatises on geometry that some account of it should be given4 ,consider any four points1 ,a1 ,b1 ,c1 ,d1 on a line1 and join them to any point ,s not on that line4 ,then the triangles ,,asb1 ,,gsd1 ,,asd1 ,,csb1 having all the same altitude1 are to each other as their bases4 ,also1 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 them1 we have formula ,now the fraction on the right would be unchanged if instead of the points ,a1 ,b1 ,c1 ,d we should take any other four points ,a'1 ,b'1 ,c'1 ,d' lying on any other line cutting across ,,sa1 ,,sb1 ,,sc1 ,,sd4 ,in other words1 the fraction on the left is unaltered in value if the points ,a1 ,b1 ,c1 ,d are replaced by any other four points perspective to them4 ,again1 the fraction on the left is unchanged if some other point were taken instead of ,s4 ,in other words1 the fraction on the right is unaltered if we replace the four lines ,,sa1 ,,sb1 ,,sc1 ,,sd by any other four lines perspective to them4 ,the fraction on the left is called the anharmonic ratio of the four points ,a1 ,b1 ,c1 ,d2 the fraction on the right is called the anharmonic ratio of the four lines ,,sa1 ,,sb1 ,,sc1 ,,sd4 ,the anharmonic ratio of four points is sometimes written 7,,abcd71 so that formula ,if we take the points in different order1 the value of the anharmonic ratio will not necessarily remain the same4 ,the twenty-four different ways of writing them will1 however1 give not more than six different values for the anharmonic ratio1 for by writing out the fractions which define them we can find that 7,,abcd7 7,,badc7 7,,cdab7 7,,dcba74 ,if we write 7,,abcd7 a1 it is not difficult to show that the six values are formula ,the proof of this we leave to the student4 ,if ,a1 ,b1 ,c1 ,d are four harmonic points 7see ,fig4 #f1 p4 99#bb71 and a quadrilateral ,,klmn is constructed such that ,,kl and ,,mn pass through ,a1 ,,kn and ,,lm through ,c1 ,,ln through ,b1 and ,,km through ,d1 then1 projecting ,a1 ,b1 ,c1 ,d from ,l upon ,,km1 we have 7,,abcd7 7,,komd71 where ,o is the intersection of ,,km with ,,ln4 ,but1 projecting again the points ,k1 ,o1 ,m1 ,d from ,n back upon the line ,,ab1 we have 7,,komd7 7,,cbad74 ,from this we have 7,,abcd7 7,,cbad71 or formula whence a #j or a #b4 ,but it is easy to see that a #j implies that two of the four points coincide4 ,for four harmonic points1 therefore1 the six values of the anharmonic ratio reduce to three1 namely1 #b1 formula1 and -#a4 ,incidentally we see that if an interchange of any two points in an anharmonic ratio does not change its value1 then the four points are harmonic4 ,figure #di ,,fig4 #di ,many theorems of projective geometry are succinctly stated in terms of anharmonic ratios4 ,thus1 the 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 it4 ,the anharmonic ratio of the lines joining any four fixed points on a conic to a variable fifthpoint on the conic is constant4 ,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 points4 ,we leave these theorems for the student1 who may also justify the following solution of the problem3 ,given three points and a certain anharmonic ratio1 to find a fourth point which shall have with the given three the given anharmonic ratio4 ,let ,a1 ,b1 ,d be the three given points 7,fig4 #di74 ,on any convenient line through ,a take two points ,b' and ,d' such that ,,ab'_/,,ad' is equal to the given anharmonic ratio4 ,join ,,bb' and ,,dd' and let the two lines meet in ,s4 ,draw through ,s a parallel to ,,ab'4 ,this line will meet ,,ab in the required point ,c4 #b ,pappus1 ,mathematicae ,collectiones1 vii1 #abi4 #c ,j4 ,verneri1 ,libellus super vigintiduobus elementis conicis1 etc4 #aebb4 #d ,kepler1 ,ad ,vitellionem paralipomena quibus astronomiae pars optica traditur4 #afjd4 #e ,desargues1 ,bruillon-project d'une atteinte aux vnements des rencontres d'un cne avec un plan4 #afci4 ,edited and analyzed by ,poudra1 #ahfd4 #f ,the term 'pole' was first introduced1 in the sense in which we have used it1 in #ahaj1 by a ,french mathematician named ,servois 7,gergonne1 ,annales des ,mathmatiques1 ,i1 #ccg71 and the corresponding term 'polar' by the editor1 ,gergonne1 of this same journal three years later4 #g ,euler1 ,introductio in analysin infinitorum1 ,appendix1 cap4 ,v4 #agdh4 #h uvres de ,desargues1 t4 ,,ii1 #acb4 #i uvres de ,desargues1 t4 ,,ii1 #cgj4 #aj uvres de ,descartes1 t4 ,,ii1 #dii4 #aa uvres de ,pascal1 par ,brunsehvig et ,boutroux1 t4 ,i1 #beb4 #ab ,chasles1 ,histoire de la ,gomtrie1 #gj4 #ac uvres de ,desargues1 t4 ,i1 #bca4 #ad ,see ,ball1 ,history of ,mathematics1 ,french edition1 t4 ,,ii1 #bcc4 #ae ,newton1 ,principia1 lib4 i1 lemma ,,xxi4 #af ,maclaurin1 ,philosophical ,transactions of the ,royal ,society of ,london1 #agce4 #ag ,monge1 ,gomtrie ,descriptive4 #ahjj4 #ah ,poncelet1 ,trait des ,proprits ,projectives des ,figures4 #ahbb4 7,see p4 #ceg1 ,vol4 ,,ii1 of the edition of #ahff47 #ai ,gergonne1 ,annales de ,mathmatiques1 ,,xvi1 #bji4 #ahbf4 #bj ,steiner1 ,systematische ,ehtwickelung der ,abhngigkeit geometrischer ,gestalten von einander4 #ahcb4 #ba ,von ,staudt1 ,geometrie der ,lage4 #ahdg4 #bb ,reye1 ,geometrie der ,lage4 ,translated by ,holgate1 #ahig4 #bc ,ball1 loc4 cit4 p4 #bfa4