[Math] - The Axioms of Descriptive Geometry [Whitehead](1)

8/2/2019 [Math] - The Axioms of Descriptive Geometry [Whitehead](1)

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barnrmage I ncts in Mathematicsand Mathematical Physics

GENERAL DITORS9. G. LEATHEM, M.A .

E. %. WHITTAKER, M.A., F.R.S.

No. 5

T H E A X I O M S OFD E S C R I P T I V E GEOMETRY

byA. N . WHITEHEAD, Sc.D., F.R.S.

Cambridge University Press WarehouseC. F. CL AY,Manager

London : Fetter Lane, E.C.Glasgow : 50, 'Wrellington Street

= 907


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Cambridge Tracts in Mathematicsand Mathematical Physics

GENERALDITORSJ. G. LEATHEM , M.A.

E. T. T J TH I TTAKER , M.A., F.R.S.

No. 5T h e Axioms of Descriptive Geometry.


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CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,C. F. CLAY, MANAGER.

ZonBon: FETTER LANE, E.C.@Iaegoh : 60, WELLINGTON STREET.

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THE AXIOMS OFDESCRIPTIVE GEOMETRY

A. N. WHITEHEAD, Sc.D., F.R.S.Fellow of Trinity College.

CAMBRIDGE:at the University Press

I907


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Qtambribge:PRINTED BY JOHN CLAY, M.A.AT THE UNIVERSITY PRESS*


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PREFACE.

T IS tract is written in coililectioil with the previous tract, No. 4 oftthis series, on Projective Geometry, an d w ith th e same generalaims. In t h at tract, after th e statenieilt of th e axioms, th e ideasconsidered were those concerning harmonic ranges, projectivity, order,th e introduc tion of coordinates, an d cross-ratio. In the present tract,after the statement of the axioms, the ideas considered are thoseconcerning the association of Pr~ject~ivend Descriptive Geometry byriieans of ideal points, point to point correspondence, congruence,distance, and metrical geometry. It has been my object in bothtracts to extend th e investigations jus t far enough t o assure the readerthat the whole of Geometry is really secured by the axioms stat'ed.My hopes for a comparative freedom from typographical errors arebased upon my experience of the excellence of the University Press.


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CONTENTS.PAGE

Sect. 1.2.3.4.5.6.7.8.9.10.

The Introduction of Order . 1Undefined Ideas . 2Peano's Axioms . . 3Definition of a Plane . 4Axioms of a Plane . 4Axioms of Three Dimensioils . 5Various Definitions . 6VeblenJs Axioms . 7Dedekind's Axiom 9The Euclidean Axiom . . 10Sect. 11.

12.13.14.15.16.17.18.19.20.

Convex Regions . . 12Convex Regions as Descriptive Spaces . . 13Existence Theorems . . 14Further Existence Theorems . . 14Advantages of Ideal Points. . 15A Lemma . . 15Desargues' Perspective Theorerns . . 16Trihedrons . . 17Theorem on Sheaves of Planes . . 18Theorem on Coplanarity of Lines . . 19

Sect. 21. Definitions of Ideal Elements . . 2222. A Property of Ideal Points . . 2323. A Property of Ideal Lines . . 2324. Ideal Points and Lines . . 26


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CONTENTS v iiPAGE

Sect. 25. Axioms of Projective Geometry . . 2726. Theorems on Ideal Elements . . 2727. Axioms of the Projective Plane . . 2928. Harmonic Ranges . 3029. Order of Ideal Points . . 3130. The Dedekind Property . . 32

CHAPTER V. GENERAL HEORY F CORRESPONDENCE.Sect. 31.

32.33.34.35.36.37.38.39.40.

Definition of a Relation .The Transformation of Coordinates .One-one Point Correspondences .Finite Continuous Groups .Infinitesimal Transformations .The Second Fundamental Theorem .Latent Elements .The Correspondence of NeighbourlloodsThe General Projective Group .Infinitesimal Projective Transformations

Sect. 41. Congruence as a Fundamental Idea . . 4442. Lie's Procedure . . 4543. Lie's First Solution . . 4644. Lie's Second Solution . . 4745. A Third Definition of Congruence . . 47

CHAPTER I. INFINITESIMALOTATIONS.Sect. 46.

47.48.49.50.51.52.53.54.55.56.57.58.

The Uniqueness of Rotations .Latent Planes through. the Axis .Monodromy .The Latency of Points on Axis .Latent Planes .Latent Planes (continued) .The Axes of Rotations.The Axes of Rotations (conti?zued) .Rectangular AxesThe Axes of Rotations (co?ltinzced) .The Common Latent PlaneThe Axes of Rotations (continued) .General Form for Infinitesimal Rotations round Origin


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CONTENTSPAGE

CHAPTER II . THE ABSOLUTE.Sect. 59.

60.61.62.63.64.65.66.67.

Surfaces of Revolution . 61Spheres . 61Quadrics of Revolution . 62Families of Concentric Spl~eres . . 63The Absolute Sphere . . 64The Absolute . . 65The Absolute (continued) . . 65The Three Types of Congruence Groups . . 67Parabolic Groups . . 67

CHAPTER III. METRICALGEOMETRY.Sect. 68.69.

$0.Definition of Distance .Analytical Expression for Distance in Elliptic Groups .Analytical Expressio~i for Dist;~ilce in HyperbolicGroups .Analytical Expression for Distance in Pamholic GroupsAngles between Planes .Angles between Lines .Metrical Geoinetry and Congruence Groups .Special Metrical Axioms .


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CHAPTER I.F O R M U L A T I O N S O F THE AXIOMS.

1. THEgeneral considerations which m ust govern a m athematicalinvestigation on th e foundations of Geom etry have been explained inChapter I of the previous tr ac t of this series, on the Axioms ofProjective Geometry*. It is explained there that 'DescriptiveGeometry' is here used as a generic term for any Geometry in whichtwo straight lines in a plane do no t necessarily intersect. Also i t ispointed ou t tha t the purely classificatory portions of a D escriptiveGeometry are clumsy and uninteresting, and th a t accordingly the ideaof order is introduced from th e very beginning.There are three main ways by which th is in troduction of ordercan be conveniently managed. In one way, which is represented byYeano's axioms given below ($ 3-6), the class of points which lie

betwean any two points is taken as a fundamental idea. It is theneasy to define the class of points collinear with the two points andlying beyond one of them . Th us these three classes of points, namelythe two classes lying beyond the two points respectively and the classlying between the two points, together with the two points themselvesform the straight line defined by th e two points. Then a set of axiomsof th e straig ht line are required, concerned with t he idea of ' between,'and also axioms are required respecting coplanar lines.Another way, which was pointed out by Vailati t and Russell 1, isto conceive a straigh t line as essentially a serial relation involving twoterms. The whole field of such a relation, namely th e term s which arethus ranged in order by it, forms the class of points on th e straigh tline. Thu s the Geometry sta rts with the fundamental conception of a

* In the sequel this tract will be referred to as ' Proj. Geom.'1- Cf. Riv i s ta d i Matemat ica , vol. IV.f Cf . Principles of &1athematics, 5 376 ,


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2 THE INTRODUCTION OF ORDER [CH. Iclass of relations. The axioms of the straight line are the axiomswhich secure that ' each of these relations is a serial relation. Thepoints are th e entities occurring in th e fields of any of these relations.The axioms of th e plane are th e same as in the previous mode ofdevelopment.The third way, recently developed by Prof. 0. Veblen*, is toconsider th e science of Desc riptive Geom etry as the stu dy of theproperties of one single three-te rm ed relation of order. The entitiesforming th e field of th is relation are th e points. When this relationholds between three points A , B, and C, i t is said that ' the pointsA, B,and C, are in the linear order ABC.' This method of conceivingth e subject resu lts in a notable simplification, and combines advantagesfrom the two previous metho ds. Veblen's axioms will be sta ted in full(cf. S 8).

2. Th e enunciation of th e axioms of Descrip tive Geometry, whichis given in the sections (S 3-6) immediately following, is that due toPe an ot. H is formulation is based upon th at of Pasch r , to whom is dueth e first satisfactory systematic exposition of th e subject. The unde-fined fund am ental ideas are two i n number, namely th a t of a class ofentities called ' points,' and th at of the ' class of poin ts lying betweenany two given points.' It has already been explained that thisundeterm ined class of points is in fact any class of en tities with in ter -relations, such th a t the axioms are satisfied when considered as referringto them.The symbol A B will represent the class of points lying betweenthe points A and B. This class will be called th e segm ent AB .The first group of axioms, eleven in nu mber, secure th e ordinaryproperties of a straig ht line with respect to th e order of po ints on it,an d also with respect to th e division of a line ./] nto three parts by any

* Cf. A Sys tem of Axioms for Geom etry, Trans. of the Amer. Math. Soc., vol. v.,1904..t. Cf. I principii di Geontetria, Turin, 1889. These axioms are repeated byhim in an article, S ui fondam enti della Geometria, Rivista di Matematica, vol. IV .1894. In this latter article the minute mathematical deductions are omitted, andtheir place is taken by valuable observations on the main points to be considered.Also a treatment of congruence is given which does not appear in the earliertract. This article should be studied carefully by every student of the subject.

$ Cf. Vo;lesungen uber neuere Geometrie, Leipzig, 1882. This treatise is theclassic work on the subject.

5 Cf. Proj. Geom. 2.11 Note that 'line' will be habitually used for 'straight line.'


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two points on it, and into two parts by any single point on it. Th eDedekind property* is not secured by them, but, ' compactnesst issecured by axiom IV.3. Peano's axioms of the straight line are as follows :I. There is a t least one point.11. If A is any point, there is a point distinct from A .111. If A is a point, there is no point lying between A and A.It follows that the class dA possesses no members.IV. If A and B are distinct points, there is at least one pointlying between A and B.Thu s the class AB is not th e null class.V. If the point C lies between A and B , i t also lies betweenB and A.It easily follows th at the classes A B and BA are identical.VI . The point A does not lie between the points A and B.Thus the class, or segment, A B does not include its end-pointsA and B.Definition. If A and B are points, the symbol A'B represents th eclass of points, such as C, with the property that B lies betweenA and C. Thu s A'B is th e prolongation of the line beyond B, and

B'A is its prolongation beyond A .VII. If A and B are distinct points, there exists a t least onemember of A'B.VIII. If A and D are distinct points, and C i s a member of AD,and B of A C, then B is a member of AD.IX. If A and D are distinct points, and B and C a re members ofA D , then either B is a member of AC , or U is identical with C, or Bis a member of CD.X. If A and B are distinct points, and C nd D are members ofA'B, then either C is identical with D, or C is a member of BD, or Dis a member of BC.XI. 1f A, B, C, D are points, and B is a member of AC, and Cof BD, then C is a member of AD.Definition. The straight line possessing A and B, symbolized bys tr (A , R), is composed of th e three classes A'B, AB, B'A, togetherwith the points A and B themselves.Then by the aid of the previous axioms the usual theorems,

* Cf . Proj. Geom. 5 19, and 9 of the present tract.9 Cf. Proj. Geom. 9 16 ,

1-2


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4 DEFINITION OF A PLANE [CH. Iexcluding the Dedekiud property, respecting the order of points on aline can be proved. Also any two points are both contained by oneand only one line.

4. Peano uses the following useful notation which is an extensionof h is nota tion for segments an d prolongations. If A is a point andu is a class of points, then Au is th e class of points ly ing on th esegments between A and points of u, and A'u is th e class of poin ts onth e prolongations of these segm ents beyond th e points of u.Then in conformity with this notation the seven regions into whicha plane is divided by three lines are a s in t he figure.

The plane determined by the three non-collinear points A, B, C-written ple (A, B, C)-is defined to be th e class of poin ts consisting ofthe points on the three lines str (BC), str (CA), and s tr (A B ), andof th e points in t he seven regions A (BC), A' (BC), B (CA), C' (AB) ,A' (B'C), B' (CIA), C' (A'B).

5. Three axioms are required to establish the Geometry ofa plane.XII. If r is a straight line, there exists a point which does notlie on r .Note that it would be sufficient to enunciate this axiom for onestraight line.XIII. If A, B, 0 re three non-collinear points, and B lies onthe segment BC, and E on the segment AD, there exists a point Fon both the segment d C nd the prolongation B'E (cf. fig. i, p. 5).XIV. If A , B, C are three non-collinear points, ap d D lies 011 the


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3-81 AXIOMS OF A PLANE 5segment BC, and F on the segment AC, there exists a point E lyingon both th e segments AD and BF (cf. fig. ii).

Fig. i.W ith these axioms all the usua l properties of th e division of aplane by a line, and of th e inside and outside of a plane closed figure,

Fig. ii.can be proved. Thus if A B C form a triangle and a coplanar lineintersect the segment BC, i t must intersect one and only one of t hesegments CA and A B .Also any three non-collinear points lie in one and only one plane ;and the line determined by any two points lying in a plane lies entirelyin th a t plane. Bu t, as the case of Euclidean Geom etry shews, wecannot prove from these axioms th at any two lines in a , laneintersect.

6. For three-dimensional Geometry two other axioms are required.XV. A point can be found external to any plane. The enuncia-tion of this axiom can be restricted to a particular plane.


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6 AXIOMS OF THREE DIMENSIONS [CH. IXVI. Given any plane p, and any point A outside it, and anypoint Q on it, and any point B on the prolongation A'&, then , if X i sany other point, either X lies on the plane p , or A X ntersects the

plane p , or BX intersects th e plane p.The annexed figure illustrates the axiom, the points XI,X,, X3being positions of X which illustrate the three alternatives contem-plated in the axiom. Th us X, lies on the plane p ; X, lies on the

same side of p as B, so that dX,must cut p in some point L ;X3lies on the same side of p as A, so that BX3 must cut p in somepoint $1;Axiom XVI secures the limitation t o three dimensions, and thedivision of space by a plane. It can also be proved from the axiomsth at , if two planes intersect in a t least one point, they intersect ina straight line.7. A point will be said to divide a line into two half-rays whichemanate from it.A line will be said to divide a plane into two half-planes whichare bounded by it.A plane will be said to divide space into two half-spaces which arebounded by it.


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6-81 VARIOUS DEFINITIONS 7A sheaf of lines is a complete se t of cop lanar lines concurrentat one point (the vertex). A sheaf of half-rays is a complete set ofcoplanar half-rays emanating from one point (th e vertex).A bundle of lines is a complete se t of lines concurrent at one point(the vertex). A bundle of half-ray s is a complete set of half-ray semanating from one point (the vertex).If p, q, r are three half-rays belonging to a sheaf of half-rays, then

r is said to 'lie between' p and q, if points A and B can be found onp and q respectively, such th a t th e segment AB intersects r.It can be proved that if r lies between p and q, then p does notlie between r and q.The complete set of planes throug h a given line (th e axis) is calleda sheaf of planes. Th e axis divides each plane into two half-planes.These half-planes form a sheaf of half-planes.If p, q, r are three half-planes belonging t o a sheaf of half-planes,then I- is said to ' ie between 'p and p, if points A and B can be foundon p and q respectively, such tha t th e segm ent A B intersects 9-.It can be proved that if r lies between p and q, then p does not liebetween r an d p.The theorems indicated in this and in the preceding sections, andallied theorems, are not always very easy t o prove. But their proofsdepend so largely upon the particu lar mode of formulation of th eaxioms, th at i t would be ou tside the scope of this tra ct to enter intoa consideration of them. In the sequel we shall assume th at th e wholeclass of theorems of the types, which have been thus generally indi-cated, can be proved from the axioms stated.

8. Form ulations of th e axioms of Desc riptive Geom etry have alsobeen given by H ilber tm , and by E. H. Moorei, and by B. Russell:,and by 0. Veblens. Veblen's memoir represents th e final outcome ofthese successive labou rs, and his form ulation will be given now. Th eaxioms are stated in terms of 'po ints ' and of a relation among threepoints called ' order.' Po ints and order are no t defined.I. There exist a t least two distin ct points.

* Grundlagen der Geometrie, Leipzig, 1899, English Translation by E. 3.Townsend, Chicago, 1902.+ On the Projective Axioms of Geometry, Trans. of the Amer. Math. Soc.,vol. III., 1902.

," T he Principles of Mathemat ics , Cambridge, 1903, ch. XLVI.5 A System of Axioms for Geometry, Trans. of the Amer. Math. Soc., vol. v. ,1904.


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11. If the points A, B, C are in th e order A B C , they are in theorder CBA.111. If the points A, B, C are in the order AR C, they are not

in the order BC A.IV. If the points A , B, C are in the order ABC, then A isdistinct from C.V. If A and B are any two dis tinct points, there exists a point Csuch tha t A, B , C are in the order ABC .Definition . The line A B ( A$IB ) consists of A and B, and ofall points X in one of the possible orders A B X , A X B , XA B . Thepoints X in the order A X B constitute the ' segment' AB. A andB are the ' end-points ' of th e segment, bu t are not included in it.VI. If points C and D C =/= D) ie on the line AB, then A lies onthe line CD.VII. If there exist three distinct points, there exist three pointsA , B, C not in any of the orders d BC, BCA, or CAB.Dejn.ition. Three distinct points not lying on the same line arethe 'vertices ' of a ' riangle ' A B C , whose sides are the segments AB ,

BC, CA , and whose ' boundary ' consists of it s vertices and th e pointsof its sides.VIII. If three distinct points A , B , C do not lie 011 the same line,and 13 and E are two points in the orders BCD and CEA, then a

point P exists in the order AFB and such that D, E, F lie on thesame line.Definition . A point 0 is ' n th e interior o f ' a triangle, if it lieson a segm ent, the end-points of which are points of differen t sides of atriangle. The set of such points 0 s ' he interior ' of the triangle.Definition 6. If A , B, C form a triangle, the 'plane ' A B Cconsists of all points collinear with any tw o points of the sides of the

triangle.


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8, 91 DEDEKIND'S AXIOM 9IX. If there exist three points not lying in the same line, thereexists a p lane A B C such tha t there i s a point D not lying in theplane ABC.Definition 7. If A , B, C; an d D are four points n ot lying in th esame plane, they form a ' etrahedron ' ,4BCD, whose ' aces ' are theinteriors of th e triangles A B C, B C D , CD A , D A B , whose 'vertices 'are th e four points A , B, C, and D, and whose 'edges' are thesegments AB, BC, C D, D A , A C , BD. The points of faces, edges,

and vertices constitute the ' surface ' of the tetrahedron.Dejnit ion 8. If A, l?, C, D are the vertices of.a tetrahedron, thespace ABCD consists of all points collinear with any two points of the

faces of th e tetra hed ron .X. If there exist four points, neith er lying i n t he same line, norlying in the same plane, the re exists a space A B C D , such th at there isno point E n o t collinear with two points of the spaceABCD.Th e above axioms of Veblen ar e equivalent t o th e axioms of Pea.nowhich have been previously given. Both Peano and Veblen give anaxiom securilig th e Dedekind property (cf. $ 9). Also Veblen gives anaxiom securing th e ' Euclidean ' property (cf. 5 10).9. Dedekind's original formulation* of his famous property appliesdirectly to the case of a clescriptive line a nd is a s follows :"If all points of the straight line fall into two classes such thatevery point of the first class lies to the left of every point of the secondclass, then there exists one and only one poitrt which produces thisdivision of all points in to two classes, this severing of th e stra igh t lineinto two portions."It is of course to be understood that the dividing point itselfbelongs to one of the two classes.It follows immediately that the boundary of a triangle consists ofpoints in a compact closed order possessing the Dedekind property asalready formulated for closed seriesi .This definition may be repeated here to exhibit its essential in-dependence of the special definition of projective segments upon whichthe previous formnlation rests.L et A , I?, C be any three poi~ltsof a closed series. Then by* Cf. his Continuity and Irratio?zal Nqcnzbers, ch . III. ; the quotation here is

from Beman's translation, Chicago, 1901,.i. Cf. Proj. Geom. 19 (a).


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10 DEDEKIND'S AXIOM [CH. Ihypothesis the series is such that there are two ways round from Ato C, namely, one through R aud one not through B . Let segm (ARC)

denote the points, excluding A and C, which are traversed froinA to C through B, and let segm (ABC) denote the remainingpoin ts of th e series. Again le t a class u of the points of the seriesbe called a segment of the series, when (1) there is a point B of theseries which does not belong to u, nd (2) if P and Q be any twopoint3 of u hen segm ( P ~ Q ) elongs ent irely to u.Then the series possesses the Dedekind property if any segnientsuch as u (which excludes more than one point of the series) mustpossess two boundary points, that is to say, if there must exist pointsA and C such that segm AS^, with th e possible exception of eith eror both of A and C, is identical with u. Here--as above-B is apoint which does not belong to u.Hence a sheaf of half-rays can also be considered as a closedcompact series with the Dedekind property. This is made imm ediatelyevident by surrounding the vertex by a triangle in th e plane of tliesheaf. Then each half-ray of th e sheaf intersects the boundary of th etriangle in one and only one point. Also the order of the points onth e boundary is th e order of th e corresponding half-rays of the sheaf.Bu t the boundary of th e triangle is a closed series with the Dedekindproperty.

10. By t he aid of the Dedekind axiom and of the precedingaxioms, it can now be proved that, if I be any line and A be any point,not incident in I , then in the plane Al a t least one line can be drawnthrough A, which does not intersect I.


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9, 101 THE EUCLIDEAN AXIOM 11For take any point B on I, and l e t p and q be the two supplementaryhalf-rays of I which emanate from B. Consider the sheaf of half-rays,

vertex A , in the plane Al. Some of these half-rays intersect p andsome intersect p, and these classes are mutually exclusive.Also, from the Dedekind property, there exist two semi-rays whichare limits of t he semi-rays intersecting p. AB is one of the semi-rays,let r be th e other. Now th e semi-ray p has no end-point. Hence r isnot among the semi-rays intersecting p. Again by similar reasoningthere is a semi-ray s which is the limit of the semi-rays inter-secting p, and s does not intersect p.Now first let rr and s be not collineur, and let I.' and s' be thehalf-rays supplementary to r and s respectively. Consider the set(a, say) of lines through A with one set of their half-rays betweenr and st, and therefore with their supplementary half-rays betweenr' and s. There are an infinite num ber of such lines. Now all half-rays emanating from A and lying between the half-ray AB and I*intersect p, and no other half-rays from A intersect p. Siinilarly forth e half-rays AB and s and q. Also if st lie between the half-ray ABand r, then r' lies between the half-ray AB and s ; and in this caseevery line of the se t a intersects the line I twice, namely once for eachof it s pair of supp lementary half-rays emanating from A. But this isimpossible. Hence neither the supplement of r nor that of s canintersect I. Secondly if r and s are collinear, then the complete lineformed by r and s cannot intersect I. For neither r nor s intersects I.Thus takiilg ally point A and any line I, the sheaf of lines,vertex A and in the plane Al, falls into two parts, namely the lineswhich intersect I , called t he lines ' secant ' to I, and the lines which donot intersect I , called the lines 'non-secant ' to I. The non-secant linesof the sheaf may reduce to one line. The supposition th at this is thecase is th e ' Euclidean Axiom.'


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CHAPTER 11.THE ASSOCIATED PROJECTIVE SPACE.

11. WE have now to establish the relation of Descriptive Geometryto an associated Projective Geometry. In a Projective Space le t a' convex region ' be defined to be a region which (1) does not includethe whole of any line, and (2) includes the whole of one of the twosegments between any two points within it. It is easy to prove th atsuch regions exist. For remem bering* th a t we can employ th eordinary theory of homogeneous coordinates, the surfaceis well known to enclose such a region. L et a quadric enclosing aconvex region be called a ' convex quadric.' Again in two dim easio l~slet A , B, C be any three non-collinear points, and let P be any pointnot collinear with any two of them . Le t AP meet the line BC in L,BP meet CA in M, CP meet A B in N.

Define the triangular region (BBCIIP) to be th e se t of po intsformed by th e collection of segm ents such as segm ( AQR )., whereQ is any point on segm (BYM),nd R s the point where the line

* Cf. Proj. Geom. 37 and 42..t. Cf. Proj. Geom. 8 13.


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11, 121 CONVEX REGIONS 13AQ intersects BC. The points A, B, C can be interchanged in thisdefinition without altering the region obtained.Similarly in three dimensions, if A , B, C, D are the vertices of anon-degenerate tetrahedron, and P be any point not on any of theplanes, ABC, BCD, eto., the ' etrahedral region ' (AR CD IP) can besimilarly defined. From the ordinary theory of homogeneous coord inates,it is well known that a triangular region in two dimensions, and atetrahedral region in three dimensions, are both convex regions.Again the triangular region (A B C /P ) considered above has as its' boundary ' the segments (BLC ) , (CMA ), (A NB), together with thepoints A, B, C. Also considering the tetrahedral region (ARCDIP),l et A P in tersect BC D in L, BP intersect GDA in PF,C P intersectD A B in N, DP intersect A R C in 0. Then the 'boundary' of thetetrahedral region ( A R C D IP ) consists of t he triang ular regions(BCDIL), (CDAIM), (DABIN), (ABCIO), together with theboundaries of these triangular regions.

It is now a well-known re sult from t he use of coo rdinates th a t intwo dimensions any line through a point in a triangular region cutsthe boundary in two points only; and that in three dimensions anyline through a point in a tetrahedral region cu ts the boundary in twopoints only.12. Now consider a convex region, let it be either the regionwithin a convex quadric, or a tetrahedral region. Call the pointswithin it ' Descriptive points' ; and call the portions of lines within it' Descriptive lines.' Th e projective order of points on a line becomesan open order for Descriptive points on Descriptive lines. Then bythe use of coordinate Geometry it is easy to prove that all theDescriptive axioms of the present tract, either in Peano's form or in

Veblen's form, are satisfied, including th e Dedekind axiom , bu t exclud-ing the Euclidean axiom. Th us in the figure the lilies d B and CD


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14 EXISTENCE THEOREMS [CH. I1intersect a t a point H in Descriptive Space ; ut the lines A B and EFdo not intersect in Descriptive Space, since K lies outside it. Also itis evident tha t through any point Y n infinite number of lines can bedrawn, coplanar with AB , and not intersecting it in Descriptive Space.

13. The previous article ( 5 12) proves* the existence theorem forDescriptive Space with the negation of th e ~ o c l i d e a n xiom ; in otherwords, i t proves th e independence of th e Euclidean axiom.The existence theorem for Descriptive Space with the Euclideanaxiom is immediately proved by considering th e region of ProjectiveSpace found by excluding all th e points on one projective plane. Theregion is convex according to the above definition ; also all theDescriptive axioms, together with the Dedek ind axiom and t,heEuclideanaxiom, hold for it t .

14. T'he independence o f' th e Dedekind axiom of th e other axioms,combined with th e negation of the Euclidean axiom, is proved by con-sidering, as in $ 12, Descriptive Space t o be a tetrahedral region inProjective Space, but confining ourselves to the points whose co-ordinates are algebraic numbers:, as in th e corresponding proof forProjective Geometry.The independen ce of th e Dedekind axiom of th e o ther axioms,combined with t he Euclidean axiom , is similarly proved by consideringDescriptive Space to be the region in Projective Space found by* Cf. Proj. Geom. 43.f In the later Greek period, and during the seventeenth and eighteenth centuries,

the discussion of th e foundations of Geometry was almost entirely confined toattempts to prove the Euclidean axiom. Th e explicit recognitions of i ts inde-pendence by Lobatschefskij (1828), and by J. Bolgai (1832) laid t he foundation ofth e existing theories of non-Euclidean Geometry. For the literature of th e wholequestion cf. StLckel and Engel, Die T heor ie de r Paral l e l l in i en won Eu kl i d h i s a ,ufG aus s , Leipzig, 1895, and also their U r k t l nden zur Geschichte der Nichteukl id-i s chen Geometri e , I. Lobatsclzefski j , Leipzig, 1898.

$ Cf. Proj. Geom. $ 4 3 (a). An oversight i n t hi s proof ma y be here corrected.The proof, as printed, proceeds by considering only points with rat ional coordinates.But then a difficulty ar ises as to th e theory of segments given in Chapter IV . ofProj. Geom. Fo r it is necessary t ha t the real double points of a hyperbolicinvolution should belong to the points considered. Bu t these double points aregiven by a quadratic equation. Thu s algebraic numbers (i.e. numbers which canoccur as the roots of equations with integral coefficients) should be subst ituted forrational numbers. Th e proof proceeds without other alteration. I am indebted t oMr G. G. Berry of the Bodleian Library for this correction.


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12-1 61 ADVANTAGES OF IDEAL POINTS 15excluding a particular plane ; and further, as before, we confine ourconsideration to th e points whose coordinates are algebraic numbers.

15. It has been proved in 12 and 13 t lhata convex region of aProjective Space is a Descriptive Space. The converse problem hasnow to be considered in th is and in t he next chapter ; namely, givena Descriptive Space, to construct a Projective Space of which theDescriptive Space is part. This effects a very considerable siniplifica-tion in the investigation of the properties of Descriptive Space owingto the superior generality of the analogous properties of ProjectiveSpace. Thus a Projective Space affords a coinplete interpretation ofall the entities indicated in coordinate geometry. It is in order togain this simplification that the 'plane at infinity ' is introdnced intoordinary Euclidean Geometry. We have in effect to seek the logicaljustification for th is procedure by indicating the exact na ture of t heentities which are vaguely defined as t he 'points a t infinity ' ; and theprocedure is extended by shewing that it is not necessarily connectedwith th e assumption of th e Euclidean axiom. This investigation isthe Theory of Ideal Points*, or of the generation of 'Proper andImproper Projective Points ' in Descriptive Geometry. The Ellelideanaxiom will not be assumed except when it is explicitly introduced.The remainder of this chapter will be occupied with the generaltheorems which are required for the investigation.

16. If A be any point and I be any line not containing A , thenthe plane A l divides the bundle of half-rays emanating from A intothree sets, (I) the half-rays in the plane A l , (2) the half-rays on oneside of the plane, (3) the half-rays on the other side of the plane.The sets (2) and (3) are formed of ha lf-rays supplementary one tothe other.Lemma. With the above notation, it is possible to find a planethrough the line l and in tersecting an y finite num ber of the half-rayseither of set (2 ) or of set (3).For let a,, . .a , be n half-rays of one of th e two sets. Let B, beany point on a,, and B, be any point on q. Then either the plane* Originally suggested by Klei i~ extending an earlier s~iggestion f von Staudt ),

;?lath. Annnl. vols. IV . and VI., 1871 and 187'2 ; first worked out in detail by Pasch,loc. cit., 5s 6-9. In the text I have followed very closely a simplification of theargument given by R. Bonola, Sulltc Ilztroduzione degli Enti Impropri i i n GeometriuProjectiva, Giornale di Matematiche, vol. XXXVIII., 1900.


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B, I lies between the planes B,Z and AI, or the plane BB,Zies betweenthe planes B, and AZ, or the planes BIZ and B,Z are identical. B utin ei ther of t he first two cases the inte rmediate plane intersects bothsemi-rays a , atid a,. Hence a plane is found through I, intersectingboth a, and u,. Call it the plane Biz. Again take any point B, on a,;and the same argument shews that at least one of B,'I and B,I inter-sects a,, a,, and a,. Proceeding in thi s w ay, a plane is finally foundwhich iiitersects each of the 12 semi-rays.

17. Desargues' Perspective theorems* can be enunciated in thefollowing modified forms :(1) If two coplaliar triangles ABC and A'B'C' are such that th e

lines AA', BB', CC' are concurrent in a point 0, then the three inter-

sections of BC and B'C', CA and C'A',AB and A'B', if they exist, arecollinear. Cf. Proj. Geom. $ 7.


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16-1 81 TRIHEDRONS 17(2) If th e pairs of homologous sides of th e two coplanar trianglesA B C and A'B'C', namely, B C and B'C', CA and C'A', A B and A'B',intersect in three collinear points, then the lines AA' , BB', CC', if any

two intersect, are concurrent in the same point.Considering the first proposition let A B and A'B' intersect in L,B C and B 'C ' in &f, A and C'A' in N. Now it is not possible bothfor A' to lie on th e segm ent OA, and for A to lie on the segment OA'.Assume that 4.' does not lie on the segment OA. Le t R be any pointexternal to the given plane (a, say). Now by the lemma of 16, it ispossible to find a plane th roug h L&& lying between th e planes LMR andL M A ( i .e . the plane a), and intersecting the three lines RA , R B , RC ,say, in the points A", B", C" (in the figure C." is not shewn). Thenevidently A" m ust lie in the segment R A . Hence A'A", since A 'does not lie in the segment OA, must intersect OR in th e segment OR.Th us the intersection of the lines A'A " and OR is secured. Le t itbe the point 0'. Again the lines O'A' and O'B' are the projectionsfrom R on the plane A'O'B' of the lines 08' and OB'. Now A"B'passes through L. Hence B" lies on the plane A'AMB', i.e. on theplane OA'B'. Hence B" is on the projection of the line OB' on theplane O'A'B', i.e. B" lies on O'B'. Thus B'B" passes throu gh 0'.Reasoning in exactly th e same way for RC and B 'C', i t follows th a tC'C" passes through 0'. The same figure has now been constructed asin the proof of the corresponding theorem for Projective Geometry*.Accordingly the theorem follows by the same reasoning.In order to demonstrate the converse theorem, we proceed exactlyas above, except that, L, M , N are now assumed to be collinear, 0 sth e point of intersection of A A ' and BB '. Then th e same construc-tion is made as before, and it is successively proved by similar reasoningth a t every pair of the lines A'A", B'B", C'C" intersect. Bu t th e linesare not coplanar. Hence they intersect in the same point 0'. But 0'm ust lie on RO. Thus CC' passes through 0.

Cwollary. The enunc iation of t he first theorem can be modifiedby removing the assumption tha t A C and A'C' intersect, but by addingkhe assumption th at A C intersects LM.18. A trihedron is the figure formed by three lines concurrentin the same point, and not all coplanar. The three lines form theedges of the trihe dro n; the three planes containing th e lines, two by

* Cf . Proj. Geom. 7.W.


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18 SHEAVES OF PLANES [CH. I1two, form th e faces of th e tr ihedro n ; the point of concurrence of thethree edges is the vertex of the trihedron.

It follows (cf. 16) tha t, if two trihedrons h ave the same vertex, anytwo faces, one from each trihedron, must intersect in a line throughthe vertex ; also that any two planes each containing two edges, oneedge from each trihedron , m ust in tersect i11 a line throu gh the vertex .Desargues' theorems can be applied to two trihedrons with thesame vertex ; only in this case, as in Projective Geometry, there areno exceptional cases depend ing on non-intersection.The enunciations are as follows :

(1) If a, b, c and a', b', e' are the edges of two trihedrons with thesame vertex, sllch that the planes containing a and a', b an d b', c and c',are concurrent in a line s ( i . e . belong to th e same sheaf), then the threeintersections of the planes bc and b'c', ca and c'a', a b and arb' arecoplanar.

(2) If a , b, c and a', b', c' are the edges of two trihedrons with thesame vertex, such th a t the three intersections of th e planes bc andb'c', ca and c'a', a b and arb', are coplanar, then the three planescontaining a and a', b and b', c and c' belong to the same sheaf.These propositions imm ediately follow from the case of trianglesby noticing that, by the lemma of 5 16, th e six edges of th e trihedronscan be cut by a plane, not through th e vertex. Hence by th e previousremarks on trihedrons, Desarguesian triangles are obtained without theexceptions due to non-intersection.

19. The two theorems of th e present an d next articles are thecentral theorems of the whole theory of Ideal Points.

If the lines a, b, c are the intersections of thre e planes a,P, y of asheaf with a plane x , not belonging to the sheaf, and if 0 be anypoint not incident in T , then the three planes Oa, Ob, Oc belong toone sheaf.


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18-20] COPLANARITY OF LINES 19If the axis ( r )of the sheaf intersect the plane a n a point S, thenthe three lines a, b, c pass through S, and the line 05' is evidentlycontailled in Oa, Oh, Oc, and thus forms the axis of the new sheaf.Consider now the case when the axis ( r ) of the sheaf does notintersect T . Then a, b, c are not concurrent, and no two of themintersect. Hence one of th e thr ee (6, say) m ust lie between [ i . e . anysegment, joining a point on a and a point on c, intersects 61 the othertwo. Take an y two points L and N on a and c respectively, then the

segneilt LN intersects b in a point M. Take two other points P ndQ on LN so th at we have th e order P, , &I,N, Q. Take any pointD on c ; hen the segment PD must intersect a and b in two pointsA and B respectively; and the segment A & must intersect th esegment DN in a point C. Then the line B C must intersect thesegment ATQin a point R. Thus a triangle A B C has been formed,whose vertices lie on a, b, c, and whose sides A B , AC, B C passthrough Y, , R respectively.By taking another point D' n c, another triangle A'B'C' can besimilarly formed, whose vertices lie on a , 6, c, and whose sides A'B'and A'C' pa-ss thro ugh P and Q respectively.We have first to shew that B'C' passes through B. For takingany point T on the axis (r) of t h e sheaf, the l ines T A , T B , T C formthe edges of one trihedron, and the lines TA', TB', TC' form theedges of another trihedron with the same vertex.Also the planes TAA', TBB', TCC' belong to the same sheaf.Hence th e three intersections of th e pairs of planes TAB and TA'B',TA C and TA'C', TBC and THC' are coplanar ; hence they lie inthe plane TY&. Hence B'C' passes through R.NOWconsidering th e two trihedrons with edges O d , OB, OC, an dOA', 0% OC', th e intersections of the p airs of faces O A B and OA'B',OA C and OB'C', O B C and OB'C', are respectively O P , OQ, OR ;and these are coplanar. Hen ce by the converse part of Desargues'theorem for trihedrons, the planes OAA', OBB', OCC' belong to thesame sheaf. Hence Oa, Ob, Oc belong to th e same sheaf (;.a. havea common line of intersection).

20. If any two of th e lines a , 6, c ar e coplanar, b ut th e thr eelines are not coplanar, and similarly for the lines a, b, d, then c and dare coplanar.

If a and b intersect, the theorem is evident ; for n, 6, c are con-current, and a, h, d are concurrent, Hence c and d are concurrent.- 2


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20 COPLANARITP OF LINES PH.1Assume that a and b do not intersect. Then it is easy to provethat no two of the lines intersect. It follows tha t no one of th e lines

c, b, d can intersect any of the planes ab, ac, ad in which it doesnot lie.

Fig. 1.

Fig. 2.


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2ol COPLANARITY OF LINES 21Hence it follows that either e and d lie on opposite sides of theplane ah, or d and b on opposite sides of the plane ae, or b and c onopposite sides of the plane ad.First, let e and cd lie on opposite sides of th e plane mb (cf. fig. 1).Take any point C on e. Then the plane Cd must intersect the plane

a b in a line, d ' , say. Then the lines a, 6, d' are th e intersections ofthe three planes da, dh, dC with the plane ah ; and these three planesbelong to the same sheaf. Hence (cf. 119) the three planes throughthe lines a, b, d ' respectively and through any point not on ab belongto th e same sheaf. Bu t C is such a point. Hence the planes Ca,Cb, Cd' belong to the same sheaf. But c is the common line of Caand Cb. Hence Cd' contains the line c. Hence c and d are coplanar.Secondly, le t the plane ad lie between b and c. Then th e plane bcmust intersect the plane ad in some line, d' say. Thus th e three lines

h, d ' , c are the intersections of the three planes ab, ad, ac with theplane he. These three planes belong to the same sheaf. Hence(cf. f i 19), if D is any point on d, no t on be, the planes Db, Dc, Dd'belong to the same sheaf. But Db and Dd' intersect in the line d ;hence DG passes throug h th e line d. Thus c aild d are coplanar.Thirdly, let the plane ac lie between b and d. Then the proof isas i n the second case, interchanging e and d.


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CHAPTER 111,IDEAL POINTS.

21. Definition. An ' Associated* Projective Point,' or an ' dealPoint,' is the class of lines which is composed of two coplanar lines,a and b, say, and of the lines formed by the intersections of pairsof distinct planes through n. and b respectively, and of th e lines in th eplane ab which are coplanar with any of the lines of the projectivepoint not lying in the plane ab.

It follo~vs mm ediately from 20 th at th e lines forming a projectivepoint are two by two coplanar; and further that (with the notationof the definition) th e lines of th e projective point lying in the planeab are the lines in al, coplanar with any o?2e of t he lines of t heprojective point not lying in ab.

Dej~~itirm.A projective point is termed 'proper,' if the linescomposing it intersect. Their point of intersection will be called the' vertex ' of the point.Thus a proper projective point is simply a bundle of lines, andevery bundle is n proper projective point.

Definition. A projective point is termed ' mproper,' if the linescomposing it do not intersect.It is proved (cf. S 24-30) that Projective Geometry holds good ofprojective points as thus defined, when a fitting definition has beengiven of a ' projective line.'Definition. A projective point will be said to be ' coherent with '

a plane, if any of the lines composing it lie in the plane.De$nition. A 'projective line' is the class of those projectivepoints which are coherent with tw o given planes. If the planes

" The word 'Associated ' will usually be omitted.


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21-23] DEFINITIONS OF IDEAL ELEMENTS 23intersect, the projective line is called 'proper' ; and the line of inter-section is the 'axis.' If th e planes do not intersect, th e projectiveline is called ' mproper.'

Since Projective Geometry has been developed* from the twofundamental ideas of 'point ' and 'straight line, ' the other definitionsof projective elements must simply be those which have been givenin considering Projective Geometry. Th us t a projective plane is theclass of those projective (ideal) points, which lie on a ny projectiveline joining an y given projective point A to any projective point onan y given projective line no t possessing th e given projective point A .

Definition. If a projective plane possesses any proper projectivepoints, i t will be called a 'proper projective plane.' Otherwise it i s anI improper projective plane.'

The vertices of all the proper projective points on a proper pro-jective plane will be seen to form a plane (cf. $ 26 (a)).Definition.. A proper projective point a nd its vertex a re said to be'associated,' so likewise are a proper projective line and its axis, andalso a proper projective plane and the plane constituted by the vertices

of it s proper projective po ints.22. Since any two lines belonging to a projective point arecoplanar, it easily follows that any two lines of the projective pointcan be used in place of the two special lines (a and 6) used in thedefinition (cf. $ 21). Hence i t can easily be proved th a t a ny plane,containing one line of a projective point, contains an infinite numberof such lines. I n other words, if a projective point is coherent witha plane, an infinite number of the lines of the projective point liein the plane. In fact it follows that, through each point of the plane,one line passes which belongs to th e projective point.23. If three projective points are incident in th e same projectiveline, then with any plane, with which two of the projective pointscohere, the third projective point also coheres.First, if the three projective points are proper, the theorem isimmediately evident.Secondly, let two of the projective points, M and N, say, beproper, and let the third projective point, L, ay, be improper. L et

" Cf . Proj. Geom. I- f. Proj. Geom. $ 4.


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24 A PROPERTY OF IDEAL LINES [CH. TIIthe projective line possessing L, M, N be defined by the two planesa and at (cf. Definition of S 21). Then the three projective points coherewith n and n'. Let T" be any third plane with which two of the three,

Fig. 1.L, M, and N, cohere. Let Nl and Nl be the vertices of the properpoints, M and N. Then MINl is a line in the plane n (cf. fig.* I) ;also the line MINl belongs to L. Again (cf. 1 22) another line rexists in a belonging to L; nd l)fl and N1 must lie on the sameside of r. Let rt be any line in n belonging to L, nd on the oppositeside of the line r to MI and N, ; such a line exists (cf. 8 22). L et 0be any point of a on the side of r' remote from Ml and Nl. Thenthe segment OM; intersects r and r', in A and A', say; and the

* Note in drawing an illustrative figure, it is convenient to make the assump-tion of 3 12, and to consider Descriptive space as a convex region in a largerProjective Space. This region is marked off by an oval curve in the figure, and anideal point, such as L, is a point outside the oval. Note that the existence of L,as an analogous entity to Ml and N l , must not be assumed in the presentreasoning.


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23] A PROPERTY O F IDEAL LINES 25segment O N , intersects r and r', in B and B', say. The segmentsA N , and BMl intersect, in C, say ; the seginents B'M, and A tN 1intersect, in C', say. Now project frorn any point 0' in 4 , and twotrihedrons are formed, namely O'A, O'B, O'C, and O'A', O'B', O'C',with the same vertex 0'. Also the homologous faces intersect in thethree coplanar lines O'L, O'M,, O'N,. Hence the three planes O'A A',O'BB', O'CC' are concurrent in a line. Hence the plane O'CC' con-tains the line 0'0. Therefore CC' passes through 0. Again projectfrom any point 0 in i;: and consider the trihedrons Ot'A,O B , C;and O"Af,O"B', 0C'. Then the planes OVAA', OBB', O"CCf areconcurrent in the same line 0"O. Thus the three l ines O L , OM,,O N , are coplanar. Hence if two of them lie in T", the third must doso also. Hence if two of L, M, N cohere with d', he third alsodoes so.

Fig. 2.Thirdly, le t eithe r two or three of L, M, N be improper. Thuslet L and M be certainly improper (cf. figs. 2 and 3) . In the plane rr


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26 A PROPERTY O F IDEAL LINES [CH. 111form a triangle ABC, such that its sides A R , BC, CA belong toL, 3f, N respectively. Thus if N is a proper point, CA passesthrough N,, the vertex of N: also since the lines B C and A B do notintersect the line Xln/IL, the points A , B, C lie on the same sideof this line (cf. fig. 2) : also since the lines B C and B A do notintersect the line N,ML, either C lies on the same side of th e line A Bas N,, or A lies on the same side of the line B C as N,. Assume tha tC lies on the same side of A B its N , (cf. fig. 2). The rest of the prooffor figures 2 and 3 is now identical. In t4heplane T, le t r be any line

\

Fig. 3.belonging to L, on the side of A B remote from C. 111 the plane n,take any point 0 n the side of q* remote from C. Then th e segmentsOA and 0.B intersect r, say in A' an d B'. Also the line A ' Nintersects the segment 0 8 , and does not intersect the segment A C ;hence it must intersect the segment OC, say in C'. Then, by pro-jecting from any point Of in the plane ?; and by similar reasoningto that in the second case, it is proved that the line B'C' belongst o M. Then, as in the second case, by projecting from any point0" in T", it follows th a t, if an y two of th e p rojective points L, , Ncohere with x", so also does th e third .

24. (a) It follows from 5 23 that any two planes, with whichboth of two given projective points cohere, define the same projective


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23-26] AXIOMS OF PROJECTIVE GEOMETRY 27line as a ny other pair of such planes. Hence two projective pointsdetermine llot more than one projective line.(p ) Two projective points determ ine a t least one projective line.Far if the points are proper, this is immediately evident. But in anycase let the projective points be A and B, and let 0 be any point.There are at least two lines, a, m d a,, which are members of A andsuch that the plane a,u, does not contain 0. Then tlre planes Oa,and Oa, intersect in a line which passes through 0 a i d i s a member ofA . Hence thro ugh any p oint th ere passes a line wllich is a m ember ofa projective point. Hence through 0 there are lines belonging to Aand B respectively. B ut these lines determ ine a plane, with which Aand B both cohere. Similarly a second such plane can be determined.Hence there is s projective line possessing both A and B.

25. Th e Axioms of Projective Geom etry* can now be seen t o betru e for the ' Projective Elements' as thus defined. Th us we have thefollowing theorem s correspondiiig to th ose ax ioms of t h e previous trac t,of which the numbers are enumerated in the initial brackets.(I, 11 111. Th ere is a class of Projective Po ints, possessing a tleast two members.1 V, I 1 1 I I ) If A and B are Projective Points, thereis a definite projective line AB, which (1) is a class of projectivepoints, and (2) is the same a s th e projective lineB d , and (3) possessesA and B, and (4) possesses a t lea st one projective po int d istinct froinA and B.

Note that two improper projective points may possess no conlmonline.(IX and X.) If A and R are projective points, and C is a pro-jective point belonging to the projective line Al l , an d is not identicalwith A , then ( 1 ) R belongs to the projective line 80' and (2) theprojective line AC is contained in t he projective line AR.(XI.) If A and B are distinct projective points, there existsat least one projective point not belonging t o th e project'ive line A B .

26. Before considering the proof of the ' axioms of th e projectiveplane t, some furth er propositions are required.(a) Since a line exists through any given point and belongingto an y given projective point, i t easily follows th a t t he set of projective

" Cf. Psoj. Geom. 4, 7 , 8, 14..j. Cf. Proj. Geom., Axioms XII , XI I I , XIV.


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28 THEOREMS ON IDEAL ELEMENTS [CH. I11points cohering with a p lane form a proper projective plane ; and thatconversely, an y proper projective plane is the set of projective pointscohering with some plane.

(p ) Any projective line intersects any given proper projectiveplane. For through th e vertex of any proper projective point on theprojective plane, a plane passes w ith which every point of theprojective line coheres (cf. 23). This plane intersects the planeassociated with the projective plane in a line. Two such planes canbe found . The two lines in the plane associated with th e projectiveplane define a projective point which lies both in the projective lineand the projective plane.

( y ) Two projective lines in a proper projective plane necessarilyintersect.For let m and n be the projective lines and a be the properprojective plane, and a, its associated plane. Take any point 0outside a,. Then two planes Om and On exist, with which re-spectively all projective points of m and n cohere. These planesintersect in a line through 0, I, say. Let A be any point in a,.The plane Al intersects a, in a line, 1', say. The two lines 1 and 1'define a projective point which lies in both the projective lines mand 12.

(6) Desargues' Theorem holds for triangles formed by projectivelines and projective points in a proper projective plane.By ( y ) imm ediately above, no exception arises from non-intersec-tion. Then by taking a point external to the associated plane, twotrihedrons can be formed for which th e theorem holds. Hence th etheorem holds for the proper projective plane.(E) The projections upon a proper projective plane of threeprojective points belonging to th e same projective line also belong t oa projective line.The theorem is immediately evident, if th e centre of projection,or if an y one of th e three projective points, is proper. Assume th a tall the projective points are improper. L et L, M, N be the three pro-jective points, and S the projective point which is the centre ofprojection. Let rr be th e proper projective plane on to which L,M, Nare to be projected. Let a be any plane with which 4 M, allcohere. On a construct figure 3 of 5 23. Project (remembering ( P )above) th e whole figure of associated projec tive points from S on t o theplane T. Then by th e first case of th e presen t theorem, all collineargroups of projective po ints which possess a proper projective point are


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26, 271 AXIOMS OF THE PROJECTIVE PLANE 29projected into collinear groups. L e t A , B , . M, N be projectedillto A,, B,, . Ml,Nl.Thus, in the plane rr, two homological triangles A IBICl andAllB,'Cl' ar e obtain ed, A1A l1, BIB,', CIC;' being con curren t in 0, ;also B IA 1 an d BltA,', B,Cl an d B,'C,', AICl and Al'Cl' are concurrentrespectively in &, Ml, N,. Hence, by (6) above, Ll, M I, Nl belongto th e same projective line.

27. Th e ne xt group of propositions correspoud t o the th ree axiomsconceriiing th e projective plane.(XII.) If A, B, C are three projective points, which do notbelong to the same projective line, and A.' belongs to the projectiveline RC, an d B' to the projective line CA , then tlie projective linesA A ' and BB' possess a projective p oint i n common.If th e projective plane A B C is proper, t he theorem follows fromS 26 ( y ) . If t he projective plane A B C is improper, consider any planewith which all the projective points of the projective line BB' cohere.

Such planes exist. Thus the associated projective plane of such aplane is a proper projective plane containing the line BB'. But by5 26 ( P ) the projective line AA' intersects this proper projective plane,in the projective point D, say. Also by 26 (c) the projections ofB, A', C from A on to this proper projective plane belong to thesame projective line. Hence D belongs to BB'. T hus A d ' a nd BB'intersect.

(XIII.) If A, B, C are three projective points, not belonging tothe same projective line, then there exists a projective point notbelonging to the projective plane ABC.This follows immediately from Peano's Axiom XV given in 5 6above.


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30 H A R M O N I C R A N G E S [CH, 11128. The theory of Harmonic Ranges must now be considered.Let A, B be any two points, C a point in the segment AB . Take

P any point outside th e line A B , and H any point on th e segnientFC , and let E G be as in figure I. Then the point D, if it exist, is

Fig. 1.th e harnionic conjug ate of C with respect to A and B. By consideringthe associated projective points and the associated projective lines, therequisite harmonic coiljugate (as a projective point) always exists.Th us, on the basis of the axioms of Projective G eometry alreadyproved, the proof for the uniqueness of the harmonic conjugate in theassociated projective geometry holds good*. Th us in t he originalDescriptive Geometry, the harmonic conjugate, if i t exist, is unique.FurtI.lermore, since H is on the segment F c E' and G are re-spectively on the segments AB 'and F B . Hence D, if i t exist, cannotlie 011 the segment A B . Conversely, if D is any point on the l ine A B ,say on the side of H emote from A , take E any point on the segmentA$', then Dl$' must intersect the segment BE H en ce A G a n d BEmust intersect in H, on the segments A G and BE. Therefore FHintersects the segment A B . Thu s t h e harmonic conjugate with respectto A and B of any point on the line AB, not on the segment AB,must exist, and lies on the segment AB.Furthermore, if D lies on the side of H emote from A, and C'lies on the segment BC, let F C ' and EB intersect in H ' ; and A H 'intersect the segment FB in G'. Then since C' lies in th e segmentBC, II' lies in the segment BH, and G' lies in the segment BG.Hence D' exists and lies in the segment BD. Thus as C movestowards B, D also moves in t h e opp osite direction towards Bt.

* Cf . Proj. Geom. $9 6 and 7 . Jy Cf. Proj. Geom. $ 17 (8) .


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28, 291 ORDER OF IDEAL POINTS 31Hence it is easily seen that the segment A B is divided into threepar ts by reference to th e harm onic conjugates of p oints in it withrespect to A and B. The p art formed by th e segment A K , (cf. fig. 2),

Fig. 2.exclusive of A and K , , coiitains th e p oints whose harmonic conjugateslie on the side of A remote from B ; the segment BK2, excl~lsiveofB and K,, contains the points whose harmonic conjugates lie on theside of B remote from A ; he segment K,K2,nclusive of K, and K,,contains the points for which the harmonic conjugates do not exist.It is not necessary that the points K, and K, be distinc t. If the ycoincide, the segment K1K2,nclusive of El and K;,, shrinks into asingle point K. Thus in Euclidean Geometry the middle point of anysegment AB is this degenerate portion of th e segment.It immediately follows that Faao's axiom* is satisfied for properprojective lines. Hence, remembering th at th e harmonic relation isprojective t , we have :(XIV.) If A and B are distinct projective points, aiitl C is aprojective point of the projective line AB, distinct froill A and B,then the harnlonic coiljugate of (7, with respect to d and B, isdistinct from C.Also tlie restrictioii to th ree dimeiisio~ is follows at, onc,e fromPeano's Axiom XVI of 6, giving the same restriction for DescriptiveGeom etry. Hence we find :

(XV.) If a be an y projective plane, an d A be an y projective pointnot lying in a,any projective point Y ies on some line joining A tosome projective point on a.29. The order of the projective points on a projective line mustnow be considered.If the projective line is proper, the order of the proper projectivepoints on it will be defined to correspond to the order of the associatedpoints. Thus (cf. fig. 2 of S 28) if th e points marked in th e figure a reprojective points, as C moves from A to K,, excluding K,, the projec-tive point D, which is t he harmonic conjugate to C with respect to dand B, moves from C through all the proper projective points on the" Cf. Proj. Geom. 9 8. f Cf. Proj. Georn. $ 9 (6).


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32 THE DEDEKIND PROPERTY [CH. IIIside of A remote from B ; aud as C moves from K,, excluding K,,to B, D moves towards B through all the proper projective pointson th e side of B remote from A .

Now let t he order of the improper projective po ints be defined soas to make the above theorem hold generally : thu s a s C moves fromK, to K,, including Kl and K,, let the order of the improperprojective points through which D moves be such that D passescontinually in the same direction round the line from the properprojective points on the side of A remote from B to the properprojective points on the side of B remote from A.Then by theorem (a) of 11 7 of th e T rac t on Projective Geometry,the order as thus defined agrees with the order as defined in 14and 15 of th a t Tract. Also th e order on the improper projectivelines is obtained from the order on the proper projective lines byprojection. Since th e harmonic property is projective, the ordersobtained thu s by different projections mus t agree. Also from Peano'saxioms of the segments of the Descriptive line given in 9 3 above,i t follows tha t th e Pro jective axioms of order* ar e satisfied, namely :

(XVI.) If A , B, C are distinct projective points on the sameAprojective line, and D is a projective point on segm (ABC)t,notidentical either with A or C, then D belongs t o segm (B CA ).

(XVII.) If A, B , C are distinct projective points on the sameprojective line, and D is a projective point belonging to bothsegm (BC A ) and segm (CA B), then D cannot belong to segm (ABC).(XVIII .) I f A , B, C are distinct projective points on the sameprojective line, and D is a projective point, d istinct from B , an dbelonging t o segm (A B C ) [which excludes A an d C'l, and E belongs

to segm (AD C), then E belongs to segm (A BC ).30. Finally, the Dedekind property 1 for the projective line

follows immediately from its assumption for Descriptive Geometry(cf. $ 9 above).Th us all the axioms for Projective Geometry, including th e axiomsof order and the Dedekind property, are satisfied by the ProjectivePoin ts and the Projective Lines. Furth erm ore th e proper projectivepoints ev idently form a convex region in th e projective space formedby th e projective points. Also th e geometry of th is convex region of

* Cf. Proj. Geom. 3 14../. i .e . on the segment between A and C not possessing B, cf. Proj. Geom. 3 13.$ Cf. Proj. Geom. 3 19 (a).


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29, 301 THE DEDEKIND PROPERTY 33proper projective points corresponds step by step with the geometry ofthe original descriptive space. Thus the geometry of descriptive spacecan always be investigated by considering it as a convex region in aprojective space. This simply amounts to considering the associatedproper projective points and adding thereto the improper projectivepoints. A particular case arises when the Euclidean axiom (cf. S 10,above) is assumed. The improper projective points then lie on a singleimproper projective plane. Thus in Euclidean Geometry when the'plane at infinity ' is considered, the associated projective geometry hasbeen introduced, and this plane is the single improper projective plane.


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CHAPTER IV.GENERAL THEORY OF CORRESPONDENCE.

31. IN th is chapter the general ideas of Correspondences, orTransformations, and of groups of transformations are explained, an dthus the idea of continuous motion is led up to.Consider any proposition respecting two entities p and q ; let it bedenoted by 4 (p, q). The proposition may be varied by replacingp and q by two other entities, say u and it, so that the new propositionis (zc, v). Thus we arrive at the notion of a constant form commonto all the propositions of the type #J (x, y), where x and y are any twoentities such th a t a significant proposition results when x an d y replacep and q in $ ( p , 9). Note that a false proposition is significant ; aninsignificant proposition is not in truth a proposition at all, it is asequence of ideas lacking the ty pe of un ity proper to a proposition.The cons tant form of th e proposition 4 (x, y), as x and y vary, maybe said to constitute a relation between x and y, in those special casesfor which + (x, y) is a tru e proposition. The order of x and y inrespect to this relation represents the special roles of x and yrespectively in the proposition 9 (x , y). Th us if thi s relation is calledR , 'x as the relation 22 to y,' or more briefly xBy, is equivalent tothe proposition 4 (x,y), however x .and y be varied. It is evident thatwe might have considered the relation indicated by the proposition insuch form th at , if it be denoted by R', yR'x represents 4 (x,y). ThenR and R' are called mutually converse relations. It is evident thateach relation has one and only one corresponding converse relation.When xRy holds, x is called the referent and y t.he relatum. Prelation is said to be a one-one relation when to each referent there ionly one relatum, and to each relatum there is only one referent. Foexample, if aRb and aRc both hold, where b and c are distinct entities,then th e relation R is not one-one.


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31, 32) DEFINITION OF A RELATION 35The class of all the referents in respect to a relation is called thedomatn of th e relation, and th e class of all relata is the conversedomain. . n mathem atics a one-one relation is often spoken of a s a trans-

form ation (or correspondence) of the members of i ts domain into (orwith) th e corresponding members of the converse domain. Th e corre-spondence is definite and reversible, and constitutes a rule by whichwe can pass from an y member of one class to a corresponding definitemember of the other class.For example, the equation,

constitutes a one-one relation of all real numbers, positive or negative,to th e same class of real numbers. This brings ou t th e fact th a t thedomain and the converse domain can be identical.Again, a projective relation between all the points on one line ofprojective space and all the points on another (or the same) lineconst itutes a one-one relation, or transformation, or correspondence,between th e points of the two lines. Any one-one relation of whichboth the domain and the converse domain are each of them all thepoints of a projective space will be called a one-one point corre-spondence.

32. By reasoning* based upon the axioms of Projective Geometry,without reference to any idea of distan ce or of congruence, coordina tescan be introduced, so that the ratios of four coordinates characterizeeach point, aud th e equation of a plane is a homogeneous equation ofthe first degree. Let X, Y , 2, U be the four coordinates of anypoint ; then it will be more convenient for us to work with non-homogeneous coordinates found by putting a for X/U , y for Y /U,a for WU. Accordingly th e actu al values of x, y, z are, as usual, thecoordinates characterizing a point. All points can th us be representedby finite values of x, y, z, except points on th e plane, U=0. For thesepoints some or all of x, y, and a are infinite. In order to deal withthis plane either recourse must be had to the original homogeneouscoordinates, or the limiting values of x to y to G must be considered asthey become infinite.The plane, x = 0, is called the p plane, the line, g = 0, z = 0, iscalled the axis of x, and the plane, U= 0, is called the infiniteplane.

* Cf. Proj. Geom. chs. VI, and VII .


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36 ONE-ONE POIN T CORRESPONDEN CES [CH. IVWhen the fundamental tetrahedron is changed, th e coordinates arechanged according to t he formula

X 1 = u { a , , X +a,, Y + a 1 3Z + a , , U ) ,Y ' = u { a , , X + a, Y +a, , Z + a,, U),Z ' = u { a 3 1 X + a 3 2 + a 33Z + a3,771,Uf=a{a1 + u , Y + a 3 Z + a 4 U).

Hence th e non-homogeneou s coordinates a re transformed by th eformulaX' = (allx + q , y + alga+ a14)/(ulx+ a2y+ a3x+ a,),y' = (a2,$ + aB y+ aa3z+ aJ/(a,x + a 2 y+ a3 z+ rc,),z '= (a3,$+ a,y + a,z +a,)/(a,x + a2 y+ a3Z+a4) .But if the infinite plane is th e same in both cases, th e formula fortransformation becomes

X' = a l lx + a,,y + al3z+ a,,,with two similar equations.

33. A one-one point correspondence can be characterized byformula? giving t he coordinates of any relatum in terms of those ofthe corresponding referent. It must be .remembered tha t every pointis both a relatum and a referent. Let the correspondence underconsideration be called T, then th e coordinates of the relatum of an ypoint x, y, s will be written Tx, Ty, TZ. Thus we have,3 = 1 ($ 9 y 1 , TY= 4 2 (2, , 4 , T z = $3 (x, y , 4 ,

where the functions +,, +3 ar e defined for every poin t of space andare single-valued. Furtherm ore, since th e correspondence, being one-one, is reversible, it must be possible to solve these equations for x, y,and z, obtainingx = $1 (Tx , Ty, Tz), y = $2 (Tx,y, Tz), z = $3 (T x, Ty, Tz).Let this converse relation be written T,, and let th e coordinates ofth e relatum of any point x, y, z be written T lx , T,y, Tlz. Then

x = TlTx = TTlx, y = TITy = TTly, z = TT,Tx= TTlz.Then, remembering that by properly choosing x, y, z we can takeTx, Ty, Tz to be any point of space, we find

T lx =$ l( x, y, 4 , T1?/=$2(x, y, z), T1z=$3(x, y, s),where $,, 9,, q3 are defined for every point of space and are single-valued.


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32-35] FINITE CONTINUOUS GROUPS 3'134. Consider* the one-one point correspondence ( T ) defined by

T x = +1(x, y, z, a ,, a, ....a,), 7'y = 4, ( ~ , y ,, a,, a2,...a1.),Tz = +3 (z, , 5, a1, a,, ..a,),..here a,, a,, .a, are r parameters. Let the parameters be assumed tobe effective, so that two different choices of special values for themnecessarily produce different correspondences. Then by varying th eparam eters an assemblage of correspondences is p roduced, each corre-spondence being defined by a p articu lar choice of th e parameters

Now let 8 and T be an y two members of thi s assemblage. ThenXTx, i .e . S ( T x) , STY,and S T z obviously are the coordinates of a pointwhich is related to th e point (x, y, z) by a one-one point correspondence.This correspondence is denoted by ST. Now, if XT is necessarily amember of the assemblage whenever S and T are both members of it,the assemblage is called a group. When each correspondence of th egroup is defined by r effective parameters, where r is a finite number,the group is called finite and r-limbed. The group is said to becontinuous, if, A!?and T being an y two different transformations of th egroup, whenever the parameters of 8 vary continuously and ultimatelyapproach those of T as their limits, then, for every value of x, y, z,also Sx , Sy, Sz vary continuously and approach T x, Ty, Tz as theirlimits.The assumption that +,, +,, 9, are analytical functions of theirarguments, x, y, z, a ,... a,, secures th a t the group is continuous.

35. The identical one-one point correspondence, n say, is suchthat, for every value of x, y, z,........................x = x , spy, ~ Z = Z (I).Finite Continuous Transformation G roups exist which do no tcontain the identical transformation. But the chief interest of thesubject is concerned with those which do contain it. Let alO, a,.0...be the value-system of th e parameters for which t,he correspondingtransformation of the group is th e identical transformation a , so th a t

with two similar equations." Cf. Vorlesungen uber Continuierliche Gruppen, by Lie, ch . VI. $ 2.


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I I I , I I ,3 & &


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35-37] LATENT ELEMENTS 39x,, yo, z0 be th e values of x, y, z when t = 0. Then from the aboveequatioiis it can be proved to be possible, owing to the properties ofthe con tinuous group of one-one transformations, to solve for el,C2,4in term s of xo , yo, x,. Thus we obtain again equations (4) of f i 35,namely

x = yo,zO,alo+ elt, ...at!+ e,t),with two similar equations, where xo , yo , z0 now correspond to x, y, zin those equations, and x, y, z to Tx, Ty, x.

B ut it is not true that, if any equations of the same form asequations (6) of 35 be written down, where tl, , t are anyarbitrarily chosen functions of x, y, 2, h e integ ral forms give the finitetransformations of a n r-limbed finite continuo us group. For inequations (6 ) of S 35, tl, &. are derived from equations (3) of 35,th a t is to say, they are partia l differential coefficients of fuilctions withspecial properties. The ennnc iation of th e conditions to be satisfiedby fl, . ., 5,) so that a fillite conti?~uous roup of transformations mayresu lt from the integration of th e corresponding equations, is called th eSecond Funda men tal Theorem of the subject. It is not required here.

Also if q, e, , . . ,. are kept unchanged, then the assemblage oftransformations found by the variation of t in equations (4) of 5 35form a one-limbed continuous group, which is defined by the singleinfinitesimal transformation which it contains, namely that one corre-sponding to th e given value-system of el, s, ...er. Also every finitetransformation is a member of the one-limbed group produced by theindefinite repetition of some infinitesimal transformation.

37. A latent point of a transformation is a point which is trans-formed into itself. A late nt curve or surface is such th at a ny point ofit is transformed into some point on the same curve or surface.It is evident that the latent points, lines, and surfaces of anyinfinitesimal transformation are also latent for every finite transfor-mation belonging to the one-limbed group defined by it. They arecalled the lat en t points, lines, and surfaces of th e group.

The transformations which leave a given surface late nt must forma group, for the successive application of two such transformations stillleaves th e surface late nt. Also th e assemblage of th e infinitesimaltransformations which leave a surface latent must be the infinitesimaltransform ations of a continuous transformation group.


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40 THE CORRESPONDENCE OF NEIGHBOURHOODS [CH. IV38. If a reentrant single-branched curve a (which may be a straig htline) is transformed by an infinitesimal transformation of a continuousgroup into a cnrve /I,hen th e senses* round, or directions round, the

curves correspond in a perfectly definite manner, the same for all suchinfinitesimal transformations.In order to make clear th e correspondence of directions round anytwo reentrant single-branched curves a and P , let OPIL and OP2Ldefine two complem entary segments round a, and let O'QIMand O'Q,Mdefine two complementary segments round /?. Now consider anyone-one point transformation which (I) transforms a into P, (2) trans-forms segments of a into segments of /?, 3) transforms 0 into 0'.Then one of the tw o following mu tually exclusive cases m ust hold, e i t h e pone of the two, O'QIM nd the relatum of OPIL,contains th e other,one of the two, OfQ,Mand the relatum of OPIL contains th e other.If one of the two, OfQIMnd the relatum of OPJ, contains the other,then the segments OPILand O'&,Mwill be said to correspond in sensewhere 0 nd 0' are corresponding origins. Also we shall consider anarbitrary small portion of a containing 0 as the neighbourhood of 0thus 0 divides its neighbourhood into two parts, one lying in thesegment OPIL, and the other in the segment OP2L. Similarly 0'divides its neighbourhood on /? into two parts. Then the case con-templated above, when the segments OPIL and O'QIM correspond insense with 0 and 0' as corresponding origins, will also be expressed bysaying that 0 corresponds to 0' and the neighbourhood of 0 in thesegment OPILcorresponds to the neighbourhood of Of in the segmentO'Ql2k.fNow considering th e case of an infinitesimal transformation, thecurve p must lie infinitesimally near to the curve a, so tha t the pointQl may be assumed to be a point infinitesimally near to the point P,and the point Q2 to be a point infinitesimally near t o th e point P2.Then no point of the segment OP,L which is infinitesim ally near to Plis infinitesimally near to any point on the segment 01Q2M. Hence thesegments OPILand OrQ,Mmust correspond in sense with 0 nd 0' ascorresponding origins. Th us only one of th e two cases of corre-spondence in sense is now possible.Notice th a t for this theorem the curves a and p need no t be distinct,nor need the points 0 and 0'.If a straight line I is l at en t for a transformation, and 0 is a latentpoint on it, and segments with origin 0 correspond in sense with

* Cf. Proj. Geom. 15, extended to any reentrant lines .


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381 THE CORRESPONDENCE OF NEIGHBOURHOODS 41themselves, then the line is said to be transformed directly in theneighbourhood of 0 , in the o ther case i t is said to be transformedinversely in the neighbourhood of 0.

Thus it follows as a corollary from the above proposition that aninfinitesimal transformation, which leaves lat,ent a line and also apoint 0 on it, transforms the line directly in the neighbonrhood of 0.Hence also it follows that any finite transformation of the one-limbedgroup defined by the infinitesimal transformation, transforms the linedirectly in the neighbourhood of 0.Similar theorems hold with respect to surfaces. It is sufficient forus to consider a transformation for which (1) a given straigh t line I islatent a nd also a point O on it, and (2) the relata of planes through Iare planes and th e relata of straigh t lines through 0 are stra igh t lines.The general extension is obvious.The portion of a plane through 0, hich lies within an arbitrarilysmall convex surface (cf. 5 11) which contains 0 within it, will be

called the neighbourhood of 0. The axis I divides into two parts th eneighbonrhood of 0 on a plane p through I ; call them p, and p,. L etthe plane q be the relatum of p with respect to th e transformation, andlet the two parts of its neighbourhood, as divided by I , be q, and q,.Let a line through 0 in p cut the convex surface in P,,P, and letthe relatum of the line in q cut the surface in Q,, Q, ; also let OP,stand for th e segment of th e line in the neighbourhood p ,, and so on.Then (assuming that continuous lines are transformed into con-tinuous lines) if O P, an d 0&,correspond in sense, th e same must holdfor all similar parts of corresponding lines through 0 n the neighbour-


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42 THE GENERAL PROJECTIVE GROUP [CH. IVhoods p1 and q l . The neighbourhoods p1 and q , will then be said tocorrespond in sense. Also if p is latent, it will be said to be trans-formed directly in the neighbourhood of 0 with I as axis, if theneighbourhood p, corresponds to itself in sense.Now, if the transformation is infinitesimal, it follows at once froillthe case of curves, th a t a definite one of th e two neighbourhoodsq, and q, niust correspond in sense with p l , and that, if the plane p islaten t, i t must be transformed directly in the neighbourhood of 0 withI as axis.

39. The general projective group of one-one point correspondencesis the group of those transform ations which transform planes into planes.Such transformations must therefore transform straight lines intostraigh t lines, and must leave unaltered all projective relations betweensets of points on lines.Now, if in such a transformation three points A, B, C on a line Iare knowli to be transformed into A', B', C' on a line Z', the relatrimon 1 of every point on I is determined. For, by the Fundam entalTheorem* one and only one projective relation can be establishedbetween the points on I and those on Z such th at A corresponds t o A',B to B', and C to C'. Thus the given transformation must transformI into Z' according to this relation.Hence it follows th a t if four points, A , B, C, D on a plane p, nothree of which are collinear, are known to be transformed into A', H,C',B' on a plane p', the relatum on p' of every point o n p is determined.For let A D meet BC in & and A'D' meet B'C' in E". Then E'corresponds to E. Hence A, B, E o n AB correspond to A', B', E' onA'R'. Hence the relatum on A'B' of every point on A B s deterinined,and similarly for B C and B'C', and for CA and FA'. But through anypoint P on p a line I can be drawn cutting B c CA, A U in L, , N.Thus the relata on p' of L, M, N, namely L',M, N' on E', are deter-mined. Thus the relatum of every point on I is determined. Henceth e relatum of P s determined.Similarly, if A , B, C, D, E are five points, no four of which arecoplanar, and if for any projective transformation their relata aredetermined, then th e relatum of every point is determined. Accordinglya p rojective transformation is com pletely determined when th e relataof five points, 110 four of which are coplanar, are de termined .

" Cf.Proj. Geom. 0 9 (y).


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38-40] INFINITESIMAL PROJECTIVE TRANSFORMATIONS 4340. Now consider transformations of th e typ e

Th ey obviously belong to th e general projective g roup as definedabove. Also the re ar e fifteen effective param eters. Bu t if wesubstitute for x, y, z the coordinates of a given point A , and forTx, Ty, Tz the coordinates of a given point A', three equations areobtained between th e parameters. L et the same be done forB and B',C and C', D and D', E and E'. Then in all fifteen equations arefound. Also if no four of A, B, 0, , E are coplanar, and no four ofA', B , a', Df,B are coplanar, these equations are consistent, anddefinitely determine the transformation T. Hence (cf. 139) th eequations (1) can, by a proper choice of parameters, be made torepresent an y assigned transformation of the general projective group.Hence the transformations represented by them are those of the wholegeneral projective group.

It is obvious froin the form of these equations that the group isa fifteen-limbed continuo us transformation-group. To find its infini-tesimal transformatioi~s,p u tall= 1 +a l l t , a12=a12 t , 13=a13 t, 14=a14 t , ,=a , t, az = az t ,a3=a3t, etc.

Then we find th a t th e analogues of equation s (6 ) of 5 35 are

These equations give the general form of an infinitesimal trans-formation of the general projective group.


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CHAPTER V.AXIOMS O F CONGRUENCE.

41. THE ogical analysis of the method of superposition as appliedto geometrical proofs is now to be undertaken. In this method afigure is said to move unchanged till it arrives at coincidence withsome other figure. But what moves? Certainly not th e points of th espace. For they remain where they are. If it is some physical bodyoccupying space which moves, then the assumption, that the bodyremains unchagzged in its motion, involves the very comparisonbetween the assemblage of points occupied in one position with thatoccupied in another position, which the supposition was designed toexplain. Accordingly we find th a t Pasch* in effect tre ats 'congruence'as a fundamental idea not definable in terms of th e geometrical conceptswhich we have already acquired. H e states ten axioms of congruencein a form applicable to Descriptive Geometry. They are as follows,where the single capital letters represent points, and the figures areth e ordered assemblages of th e points mentioned, ordered in th e orderof mention.

I. The figures A B and B A are congruent.11. To the figure ABC, one and only one point B' can be added,

so tha t A B and AB' are congruent figures and 8 ies in the segmentA C or C in the segment AB'.111. If the point C lies in the segment A B and the figuresA B C a nd A'B'C' are congruent, then the point C' lies in the segment

A'B'.IV. If the point Cl lies in the segment A B , and th e segment AC,is lengthened by the segment C1C2which is congruent to it, and A 4is lengthened by the segment C2C3,congruent to AC ,, an d so on, thenfinally a segment CnCn+l s arrived a t which contains th e point B.* loc. cit . $ 13.


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41, 421 CONGRUENCE A S A F U N D A M E N T A L I D E A 45V. If in th e figure A.BC the segments A C and BC are congruent,then the figures ABC and B A C are congruent.VI. If two figures are congruent, so also are their homologous

parts congruent.VII. If two figures are each con

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[Math] - The Axioms of Descriptive Geometry [Whitehead](1)