Table Of Content

THE ROLE OF THE IMAGINARY IN PROJECTIVE GEOMETRY by CHARLES LEMUEL CARROLL JR A thesis submitted to the Faculty of the University of North Carolina in partial fulfillment of the requirements for the degree of Master of Arts in the Department of Mathematics. Chapel Hill 193? J Approved byj Table of Contents page Chapter I. A Brief Historical Summary of The Imaginary In Geometry 1 Wallle 1 Kuhm 2 Wessel 3 Argand 3 Buee, Mourey, and '.Tar ran 3 Cause 4 Von Staudt 4 Uonge, Poncelet, and Chaeles 7 Reye 8 Marie 8 W4*rs trass, Van Uven, and Terry 9 Study 9 Mouchot and Davis 10 Duport 10 Xlf^n 10 Lie 10 Segre 11 Veblen and Toung 11 Chapter II. The Imaginary In Projective Geometry 15 Solution of An Algebraic Equation 16 Measurement of Distance In The Imaginary Domain Measurement of Angle In The Imaginary Domain 32 Chapter III. Some Properties of Imaginary Elements 26 List of Properties 36 Construction Problems 35 Bibliography 40 Chapter I A Brief Historical Summary of the Imaginary la Geometry The first person who gives evidenoe of having a geo metric intrepretation of the complex roots of a real quad ratic equation was John Wallis. He approached the problem in his book Algebra. Oxford, 1685 in somewhat this manner. An indicated square root of a negative number is, on its face value, an absurdity, since the square of any number postive or negative is itself positive. But this absurdity is similar to the more familiar one where we speak of negative numbers; for what can be more absurd than to speak of a number that is less than nothing? How it is well known that this latter contradiction disappears entirely when we represent our positive and negative numbers by points on two opposite scales. Hence a proper study of the geometry of the plane should solve our diff icAlty. for instances Suppose that in one place 30 square miles have been reclaim ed from the sea and that in another plaoe 14 square miles have been taken from us by the sea. What has been our gain? Xvidently 16 square miles, or a square which is four miles to a side. However if in another plaoe the sea has taken 20 square miles, what is our gain? We know this to be -4 square miles. We could not properly say that we had gained a square -2 miles on a side nor 2 miles on a side, but one whose side was |p4 - 2^PT. The question that Wallis concerned himself with was 3 He uses a vector whose initial point ie the origin, instead of using its terminal point only, Argand. It was perhaps fortunate for the progress of mathe matics if not for the fame of Vessel that during the hundred years that his hook went unnoticed, other writers independent of him attacked the problem and achieved the same results. The first of these was Jean Robert Argand, a humble book keeper in Paris, In 1806 he published a short memoir Entitled "Essai sur une manlere de reprlsenter lee quantities imaginaires dans les constructions geometrique". Argand begins by saying that negative numbers are related to positive ones, not onl$? through numerical ratio, but also through a reversal of direction. With this viewpoint the problem of finding a mean between two Quantities with opposite signs requires us to find the square root of the product of their numerical values and a direction which is the mean between their two directions (i.e. which is perpendioular to them). Buee, Mourey, and Warren. The year 1806 in which Argand*s first memoir appeared also gave birth to another longer and more ambitious. It was Abbe Buee’s "Memoirs 8ur les quanti ties imaginaires*1. It was presented to the Royal Society by William Morgan. The year 1828 was also blessed with two publications on the geometric representation of the complex numbers. First was 0. V. Mourey*s "La vraie thisorie des quantities negatives et des quantities pretendus imaginaires**, and then Rev. John Warren's "A Treatise on the Geometric Representatio of Square Roots of Negative Quantities**. 4 Gauss. On April 15, 1851 Gauss presented to the Royal Society of Gottingen a short essay entitled wTheoria residurum biemadvaticorura comraent&tis secunda**. Gauss said let a. set of objects A, B, C et cetera be arranged in such a scale that we can say the relation or transfer from A to B is the same as that from B to C or from C to 0 et cetera. Each of these relations may be expressed by the symbol-t 1, and if the inverse relations be considered as that from B to A, we represent it by the sym bol -1. If our system of objects extend indefinitely in either direction, then any one of our integers positive or negative will express the relation of any one of our objects ohosen as the first to some one other object of the series. Suppose next, that instead of having a single series of objects we have a series of series, and the relation of any object in one series to the corresponding object in the one or other of the adjacent series be expressed by the symbol i or —i. The four fundamental relations are (1, i, -1,-i ). We then look upon our system of objects as arranged like the points of s. plane lattice. The system will be carried into itself by a rotation through 90 about any one of its points. Gauss in contradistinction to Wessel, Buee, Mourey and Warren thinksof the point ( ) not of the vector from the orgin to that point as representing x-fyi. Yon Btaudt. Usually our modern treatment of complex numbers has a two-fold aspect (i.e. geometrical representation and arithmetical theory of operations. But it should be under stood that up uptil this time in the development of the com- 5 plex in geometry, the workers had been seeking a geometrical representation for complex numbers. Whereas we now are approaching the problem from a different viewpoint. We have the imaginary elements and are seeking an explanation or rationalization in the real geometry. The first person to have this viewpoint was Von Staudt. In 1856 Beltrage zur Geometric der Lage.Nuvr%berg was published by Von Staudt. His purpose was to define elements 1 which arose analytically through a. synthetic method and then to develop the whole theory behind these elements synthetically. Von Staudt starts off with the following definition; An ellip tic Involution on a line together with a sense of description for that line shall be called an Imaginary point. The same involution coupled with the opposite sense shall be called the conjugate imaginary point. That is the elliptic involution: i ^ L £ F £ ' ---- F 'e .-----^ defines on the line «*• two conjugate imaginary points and X - . E F b f ' An imaginary line of the first kind is defined by an elliptio involution of coplamaif lines on a point. That is, on a flat pencil O / a ' ^ the elliptio involution: L/ (* f *' — } A ^ U ' f - * : ----^ defines on the real point KJ and on the real plane of the I Von Staudt*s work is the classical work of the Synthetic Projective Geometry approach. His method is used almost invariable by the present day synthetic geometers as the representation of imaginary elements in geometry. 0 flat pencil two conjugate imaginery lines £ t x f ana 5t - i f *' /. We observe that X and £ are distinct points and that: l-.fFf'F'- re'F'E = E’F'EF - F'EFM- * - FE'f E = EFtr F' = Ft- FU ' Von Staudt said that «n imaginary point is incident with one and only one real line, the support of the invo- lutory point now defining the imaginary point, and with all the real planes on this line. An imaginary line of the first kind is incident with one and only one real point and one and & Leli the real line which connects two conjugate imaginery points be taken as the %- axis in a real plane through it. A real involution thereon will be given by the real bili>vea- ecu8tlonj A % x * + -+ <t - o when v ^ A c this involution is elliptic and the search for the imagina ry double points leads to the cued ratio epuation: A * 1"-* + C - o whose roots are conjugate imaginary points: 5 L -T > ~ < $ J L They are the double points of the elliptic involution: • U 1 - ?(.X-v s ’)-*-?* + Q.*- - O Therefore there is a perfect one to one correspondence between the elliptic involutions of oollinear points in a real plane and the pairs of conjugate imaginary points of the olane. The importance of this correspondence can not be over stated, For instance in several representations the con jugate Imaginary points (Pt-AQ ,0), (P-$1,0) are represented by the real points (P,Q) and (P,-Q). Let us pass a circle through these real points. The point (P,0) has the power with regard to this circle. If this circle the axisiw**^*' *. ov - o XX '-PC*'**')-' pV+ Q x — r— ^ which is the elliptic involution which has p 4-Qa and P-Qa as double points. Coolidge, Geometry of The Complex Domain, page 75. 7 only one real plane, (l.e. the center and the plane of the lnvolutory flat pencil defining the imaginary line. An imaginary point X — £ f £ 'F defined by: u. (£F/T' — " - U ' f ' f and an imaginary line S. a a ‘ ^ ' defined by: A - A i U ' 5 ^ y - 4 / pi are incident when the lnvolutory point Vow ia. If * 'J is perspective to the lnvolutory flat pencil ^ ~~ An Imaginary line of the second kind is defined by an elliptic involution of lines on a regulus with a sense of description. If Qt' and b, b* are two pairs of lines of a regulus, the elliptic involution: A. defines two conjugate imaginary lines of the second kind: ^ - Q. \o qJ )o and x - «• b ©J b An imaginary line of the second kind is Incident with no real point and no real plane. With these definitions Von Staudts developed the whole theory of projective geometry of imaginary elements. He proved the fundamental theorem of projective geometry, the theorems of concurrence, colllnearity, and coplanarity. Then he proves Pascals theorem, and the Polar theory follows He completely develops the theory with the above given defi nitions and the assumptions and postulates of real pro jective geometry. Monge, Poncelet and Ohaslee. About this same time Mottge and after him Poncelet spent much time on the «o called 8 “principle of continuity", as a means of development of the theorem of the imaginary in geometry. This same principle was afterwards discussed by Ohasles under the name of the “principle of contingent relations'’. “To get an idea of this principle consider a theorem in geometry in the proof of which certain auxiliary elements are employed. These elements do not appear in the statement of the theorem and the theorem might possibly be proved without them. In draw ing the figure for the proof of the theorem, however, some of these elements may not appear or as the analyst would say they become imaginary: No matter, says the principle of contingent relatione, the theorem Is true and the oroof is valid whether ‘ 3 the elements used are real or imaginary", Reye. Another interesting work on the representation of the imaginary is that of Reye»s Geometry der Lage. It had its principle virtue in that it is one of the simplest geometrical rationalizations of imaginary. His approach was purely a pure geometry approach. The distinction between real and imaginary elements arise when we attempt to picture elements. Elements 4 are divided into two classes, pietur&lle and non-picturable. Marie. In hie Theorle dee fonctions de variables imagine!res. Paris, 1874-6 and his Realisation et usage des formes im^-inaires en geoastrle. Paris, 1891 P. Maximilier Jiarie set fourth the method of representing the complex point V - ^ x i-' 'Clj 'y-CoX X * , V ” !. iehmer, Synthetic Projective Geometry. 4. See: C. A. Scott,.*rThe Siatus 'o f "tmaginary In Pure Geometry*. Bullltln of the Anerioan Mathematical Society Vol. ^ 1899.1, p. 163-1687