PROJECTIVE VECTOR ALGEBRA 
 
PROJECTIVE 
 VECTOR ALGEBRA 
 
 AN ALGEBRA OF VECTORS INDEPENDENT 
 
 OF THE AXIOMS OF CONGRUENCE 
 
 AND OF PARALLELS 
 
 L. SILBERSTEIN, PH.D. 
 
 LECTURER IN MATHEMATICAL PHYSICS AT THE UNIVERSITY OF ROME 
 
 LONDON 
 
 G. BELL AND SONS, LTD. 
 1919 
 
PSL 
 6)4 
 
 Si 
 
 COPYRIGHT 
 
TO 
 
 GEORGE PAUL 
 
CONTENTS 
 
 SECTION PAGE 
 
 1-3. INTRODUCTORY REMARKS i 
 
 4. ADDITION OF TWO COINITIAL VECTORS DEFINED - - 4 
 
 5. THE ASSOCIATIVE LAW 6 
 
 6. SUM OF COLLINEAR VECTORS. MULTIPLE OF A VECTOR 9 
 
 7. SUBMULTIPLE OF A VECTOR 1 8 
 
 8. THE CONCEPT OF EQUALITY OF NON-COINITIAL VECTORS - - 21 
 
 9. DIFFERENCE OF VECTORS. NEGATIVE SCALAR FACTORS - 27 
 
 10. ADDITION OF VECTORS WITH ' IDEAL ' TERMINI - 34 
 
 11. ON THE USE OF THE ABOVE VECTOR ALGEBRA 37 
 
 12. VECTOR EQUATIONS OF STRAIGHTS AND OF PLANES- - - 40 
 
 13. MISCELLANEOUS EXAMPLES 41 
 
 14. PASCAL'S THEOREM. 49 
 
 15. VECTOR SUM FOR VARIABLE *-LiNE. CONIC RANGES AND 
 
 PENCILS 52 
 
 APPENDIX 
 
 A. ON THE NON-METRICAL SUM OF ANGLES OF A TRIANGLE 67 
 
 B. LINEAR VECTOR OPERATORS. PROJECTIVE COLLINEATIONS 71 
 INDEX - - 77 
 
PROJECTIVE VECTOR ALGEBRA 
 
 AN ALGEBRA OF VECTORS INDEPENDENT OF THE AXIOMS OF CONGRUENCE 
 AND OF PARALLELS 
 
 1. The reader will find in the widely known memoir of Hilbert 
 on the Foundations of Geometry * various ' algebras of segments,' 
 independent of one or of another group of axioms, the purpose 
 of these algebras being, in Hilbert's case, to show the mutual 
 independence of his set of axioms. More recently, in an excellent 
 book, Schur f has taken up von Staudt's calculus of projective 
 segments (Wurfrechnung) in order to develop it analytically and 
 to build upon it a complete system of metrical, euclidean and 
 non-euclidean, geometry. This is admirably done in 4 and 5 
 of his work. Schur bases his definitions of equality, of addition 
 and multiplication of projective segments, upon the correspondence 
 known as ' prospectivity,' and, at first, avails himself only of 
 the axioms of connection and of order [Schur's postulates I. to 8.] ; 
 for the further development of the subject, however, he has 
 recourse [ 5] to the axioms of congruence or of ' motion,' postulates 
 9. to 13., and completes his investigation by adding an independent, 
 1 4th postulate concerning the use of compasses. The result 
 is a most charming and lucid structure of the complete system of 
 non-euclidean geometry (of an isotropic three-dimensional space 
 of any constant curvature), the last touch to this true masterpiece 
 being given in Schur's closing section by adding the archimedean 
 postulate. 
 
 2. The purpose of the present investigation (intended originally 
 as a paper, but ultimately shaped into the form of a little book) 
 is a more modest and much more restricted one, viz. to construct, 
 
 * D. Hilbert, Grundlagen der Geometric, Gottingen, 1899 ; translated 
 by E. T. Townsend, Chicago, 1910. 
 
 f F. Schur, Grundlagen der Geometric, Teubner, 1909. 
 P.V.A. A 
 
2 PROJECTIVE VECTOR ALGEBRA 
 
 by means of the axioms of connection and of order only, a very 
 simple algebra of vectors, directed, i.e. arrowed straight segments, 
 an algebra, that is, embracing only the equality, the addition, and 
 hence also the subtraction, of vectors. There is no essential diffi- 
 culty in introducing also an appropriate vector multiplication of 
 two vectors ; this, however, besides being superfluous, does not 
 share the remarkable simplicity which will be seen to belong to the 
 proposed vector addition, and did not, therefore, seem sufficiently 
 interesting to be given in this little book. The absence of ' multi- 
 plication ' will not be much felt, in view of the aim of this book.* 
 
 3. It will not be necessary to subdivide the contents of the 
 book into a formally logical array of numbered definitions, axioms, 
 lemmas, theorems and corollaries. 
 
 The reader will keep in mind that we are basing ourselves only 
 upon the axioms of order and of connection, let us say, Schur's 
 postulates I. to 6. (loc. cit.},\ with postulate 7., ensuring three- 
 dimensionality, and 8., excluding further dimensions. 
 
 These suffice for the full validity of Desargues* theorem, a theorem 
 whose aid will be most essential. (Readers who are unfamiliar 
 with this fundamental theorem can look up its proof in Prof. 
 
 * Note added January 3, 1919 (in reading the final proof sheets). A 
 comparatively simple non-metrical generalization of the multiplication of 
 vectors into vectors which I chanced to build up only in the last few days 
 could not, for technical reasons, be included in this volume. I hope to 
 be able to publish it as a separate paper, uniform with the present text, 
 in one of the next issues of the Phil. Mag. The reader will then have an 
 easy opportunity of supplementing the contents of some portions of this 
 book, notably of its last section, 15. 
 
 f These axioms are, in English version of Schur's wording : 
 
 1. There is an unlimited multitude of elements : points. 
 
 2. Any two distinct points determine uniquely an unlimited [infinite] 
 multitude of points, to which they themselves belong, and which is called 
 a segment (Strecke). Any two points of a segment determine another 
 segment whose points belong to the former. 
 
 3. If C be a point of the segment AB, then a fourth point D of this 
 segment belongs either to A C or to CB, but never to both at the same time. 
 
 4. If C be a point of AB, distinct from B, and B a point of CD, then C t 
 and therefore also B, belong to the segment AD. 
 
 5. Outside every straight there are still points. 
 
 6. It A, B, C be any three non-collinear points, D a point of BC, and 
 E a point of AD, then there is always a point F belonging to the segment 
 AB, such that E lies on CF. 
 
INTRODUCTORY REMARKS 3 
 
 Mathews's Projective Geometry, which may be warmly recom- 
 mended ; Longmans, 1914, Chap. V.) 
 
 Equivalently, we may say that the only constructions admissible 
 for us will be those requiring only a straight-edge, and possibly 
 a theoretical plane-layer [to draw a plane through 3 given points 
 or through a point and a straight], but neither a ' transferor, ' 
 of course, nor compasses. These latter remarks would after 
 what has been said certainly be superfluous for geometers 
 by profession ; but the book is chiefly intended for beginners in 
 geometry and for physicists, more especially those who employ in 
 their investigations the usual metric vector algebra, and who may 
 like to know how far one can go without having recourse to ' rigid 
 bodies ' (transferers) whose use involves, in the best of cases, a 
 host of physical conditions. Again, some conclusions easily deriv- 
 able from our subject may be helpful to readers interested in the 
 degree of soundness of the foundations of the modern theory of 
 relativity. 
 
 With regard to the said eight postulates only one more remark. 
 Postulate 2. (as quoted above), claiming uniqueness of a stretch 
 or vector, determined by two distinct points, will be understood 
 to apply to a restricted region of space, we will say, the region 
 under consideration. This will leave the possibility of our space 
 as a whole being closed, elliptic (polar or antipodal), as well as 
 hyperbolic or homaloidal (euclidean). Or else we may exclude 
 the antipodal form of elliptic geometry, leaving only the polar ; 
 then the said restriction will become superfluous. For in the 
 polar form of elliptic space (often called simply the elliptic space, 
 as distinguished from the antipodal or ' spherical ' kind) any two 
 points determine one and only one straight line. 
 
 Lastly, should, for the sake of some special corollaries or illustra- 
 tions, the archimedean postulate be required, its assumption 
 will be expressly stated. Otherwise the results will be independent 
 of that postulate.* 
 
 * We have here in mind not the proper archimedean postulate, but that 
 generalized by Schur (loc. cit. Postulat 15) : 
 
 If E, P be any two points of OU, and we construct successively the 
 fourth harmonic points : E. 2 to O with respect to E and U, then E a to E 
 with respect to 2 and U, E t to .E a with respect to 3 and U, and so on, 
 E n to n _ 2 with respect to E n-l and U, then this series contains always 
 a point E n such that P lies on OE n . 
 
4 PROJECTIVE VECTOR ALGEBRA 
 
 In order to follow freely the investigation, the reader will require 
 hardly anything more than the knowledge of Desargues' theorem 
 as a consequence of the said axioms of order and of connection. 
 
 4. Addition of two coinitial vectors defined. The reason, 
 why ' addition ' is taken up even before the concept of ' equality ' 
 of vectors, will become manifest a little later on. 
 
 Let us begin with the treatment of vectors having all a common 
 origin 0, such as OX and OY (Fig. i), which we will henceforth 
 denote by the Clarendons X, Y, etc. All vectors under con- 
 sideration being, for the present, coinitial, the idea of two of them 
 
 FIG. i. 
 
 being ' equal ' does not enter at all. The origin of all our vectors 
 being fixed, if their end-points, X, Y, etc., are distinct points of 
 space, the vectors X, Y, etc., are all distinct or different from 
 one another. They differ, to use popular language, in ' direction,' 
 as a matter of course, and they have also no other thing in common, 
 since ' size ' or ' length,' as any other metrical property is out of 
 question. And to insist, with the logicians, upon the equality 
 X = X is needless, and inane. In short, we shall not need the 
 concept of the equality of two vectors until the restriction to 
 coinitiality is given up. 
 
 Now, X = OX and Y = 0F being two vectors, as in Fig. I, let 
 us assume on the line of each of them some fixed points T x and T r 
 respectively. The two terminal points T, so to call them, being 
 thus conventionally fixed, the TMine or, as we will say, the terminal 
 straight in the plane OXY, is therewith fixed. 
 
 Draw the straights XT r and YT X , and let the point A be their 
 cross. Let us base our first definition upon this simple construction 
 
ADDITION OF TWO COINITIAL VECTORS 5 
 
 (familiar in connection with the usual definition of ' coordinates ' 
 in projective geometry). The origin of all our vectors being 
 already fixed, let us call the new vector A = OA the sum of the 
 vectors X and Y, writing 
 
 A = X + Y, 
 
 the specification : ' with respect to T x , T T as termini ' being 
 understood. It is needless to insist that, the said axioms being 
 valid at least for the triangle OT X T Y as the 'restricted' region, 
 A is a unique point (cross) and, therefore, A a unique vector. 
 Moreover, the above operation or construction being perfectly 
 symmetrical with respect to the two addends, we can equally well 
 write A = Y + X. Thus, 
 
 X + Y = Y + X, (i) 
 
 the commutative law of addition. 
 
 It will readily be seen that the usual, euclidean vector sum, obtain- 
 able by drawing through X, Y parallels to OT r , OT X , is but a special 
 case of the above sum. That special case follows from the present 
 more general definition of vector sum if for T If T T are taken ' points 
 at infinity,' and if, of course, the existence of unique parallels, i.e. 
 the ordinary parallel axiom is assumed. To put it only in its practical 
 aspect, draw our present Fig. i upon a large sheet of paper, and 
 take Tjr farther and farther away, i.e. without metrical concepts, 
 take TV beyond TV, TV beyond TV, and so on ; and similarly for T rt 
 each on its line. Then, even without the definite concept of 
 ' parallels,' the farther the process is pushed, the more will the 
 above approach the euclidean (parallelogram) construction of the 
 sum, the ' equality of the sides ' OX, YA and XA, OY being tested 
 either by a ' legitimate ' pair of compasses or by the eye. Needless to 
 say that no such considerations are of any avail in our present circle 
 of ideas. The above definition of vector sum is manifestly a generali- 
 zation of the euclidean one, and is valid also for such spaces in 
 which there are no euclidean parallels nor even lobatschevskyan 
 parallels or asymptotic lines. In short, the space may be elliptic 
 in which all coplanar straights intersect one another. 
 
 If, as in Fig. I, X, Y are on the segments OT f) OT r or the 
 ' sides ' of the triangle OT f T r , the resultant point A is within it. 
 The same construction, however, will be used if X or Y, or both, 
 are on the prolongation of the straight lines beyond the origin, the 
 7"-line being always the same. If only the whole figure continues 
 to belong to the said restricted region, our previous reasoning will 
 
6 PROJECTIVE VECTOR ALGEBRA 
 
 continue to be applicable. Similarly if X, Y are 'beyond' the 
 TMine. Meanwhile, to fix the ideas, let us take the case represented 
 in Fig. i. 
 
 Thus, whatever the addends, X and Y, their sum can always 
 be found by a perfectly definite straight-edge operation. With 
 one exception, however, viz. when X and Y are collinear vectors. 
 For if such be the case, the point A, the cross of the auxiliary 
 straights, is indeterminate. Thus our sum definition does not, 
 at this stage, enable us to construct the sum of any two collinear 
 vectors, and in particular the sum X + X or 2X, if X be given. 
 But we shall see how to do it after having dealt with sums of three 
 or more vectors. 
 
 The reader may feel puzzled to see the ' simplest ' addition X+X 
 thus made the more difficult one. Let him imagine, however, that 
 in dealing with his ordinary vector algebra, he is only able to draw 
 parallels but is deprived of compasses and of any transferer. Then 
 he will easily convince himself that the end-point of the required 
 vector 2X (generally, of the sum of collinear vectors) is exactly 
 as undetermined as in our case. For the parallelogram collapses, 
 for every pair of collinear vectors. 
 
 5. The Associative Law. Once more return to Fig. I. Having 
 found A, the sum of X and Y, we can, by the same prescription, 
 add A to another vector Z, always with as origin. To be as 
 general as is possible, let Z = OZ be not coplanar with X, Y, and 
 let it have T z for its terminus, so that T x , T r , T z define a fixed 
 plane, the T-plane, as we will say. In order to apply again the 
 former sum definition, we have only to prescribe but one more 
 thing, viz. the terminus of the line OA. As such we shall take 
 the cross T A of OA prolonged with the previous TMine. In short, 
 the latter line will be the locus of termini of all straight lines in 
 the plane XOY* and Z being not coplanar with X, Y, the plane 
 through T It T Y , T z will henceforth be our (conventionally) fixed 
 T-plane, enabling us to add to A = X+Y any third vector, to the 
 sum thus obtained any fourth vector, and so on, all coinitial. 
 
 Let B = OB be the sum of the two vectors A and Z, i.e. 
 
 the brackets, at present indispensable, being used as separators. 
 
 * It is not difficult to see that this prescription is being naturally suggested 
 by the previous ones. 
 
THE ASSOCIATIVE LAW 7 
 
 In order to find this new vector, we have, by the fundamental 
 definition, to join A, T z and Z, T A (Fig. 2) ; their cross being B 
 we shall have E = OB. But we might have added together first 
 Y and Z, obtaining the vector A' = Y + Z, and then X. Let the 
 result of this entirely different operation be B', i.e. let 
 
 The question arises : Is B' the same vector as B ? In other words : 
 Do the points B' and B coincide ? The answer turns out to be 
 in the affirmative. 
 
 Draw the straights ZT T and YT Zj crossing in A' ; draw OA' 
 and prolong it, obtaining its terminus T A , ; join T A > with X, 
 
 FIG. a. 
 
 and T x with A' ; then the latter two straights will be found each 
 to pass through the previous B, so that B' is B. In fact, to prove 
 that A'T X passes through B notice that ZA'B and OVA is a pair 
 of triangles in perspective (the joins of corresponding vertices 
 passing through T g ) t and that T x , T At T r are collinear, by con- 
 struction. Then, appealing to Desargues' theorem, B will readily 
 be seen to be on A'T X . Similarly for XT A ,, noticing that the 
 triangles XAB and OVA' are in perspective (T x ), and that T r , 
 T A , y T z are collinear. Thus the coincidence of B and B' is proved, 
 being a consequence of the theorem of Desargues, and of the 
 explicit prescription that all termini should be taken on the same 
 T-plane. Similarly for B* the end-point of B*=*(Z+X)+Y, the 
 corresponding system of lines being omitted from Fig. 2 in order 
 not to make it too complicated. The rdles of X, Y, Z and of their 
 corresponding auxiliaries are manifestly interchangeable. The 
 
8 PROJECTIVE VECTOR ALGEBRA 
 
 above proof remains valid also in the case of three coplanar vector- 
 addends, the only (entirely irrelevant) difference being that the five 
 points TX, T A , T r , T A ,, T f are then all collinear. 
 
 We thus have the remarkable result that, for any three distinct 
 vectors X, Y, Z, 
 
 (X+Y)+Z = (Y+Z)+X = (Z + X)+Y, (2) 
 
 or that the vector sum, as defined above, obeys the associative law. 
 Thus the brackets become superfluous, and since the commuta- 
 tive law (i) holds as well, the above sum can simply be written 
 
 X + Y + Z or X + Z + Y, etc. 
 
 The same is manifestly valid for four or more addends. 
 
 It will be kept in mind that the validity of the associative 
 law is most immediately based upon the theorem of Desargues. 
 
 We might have started from this law, (x+Y)+Z=X+(Y+z), 
 as a postulate, and from our first sum definition. Then, if, for 
 any reason, we did not know the theorem of Desargues, we could at 
 once deduce it from the associative postulate in conjunction with 
 the law of summation. We might imagine, for instance, that we 
 were bidimensional beings confined to a plane and did not know 
 anything of points outside it ; then, assuming only the first six 
 postulates of Schur * and (instead of the yth, ' outside every plane 
 there are still points ') the associative law of vector addition, we 
 could deduce Desargues' theorem. But the order of ideas adopted 
 above has seemed the more natural one For, apart from didactic 
 reasons, what would there be to suggest to us the associativity for 
 such a comparatively complicated operation ? 
 
 It is chiefly, or almost uniquely, due to the validity of the 
 associative law that the above definition of vector sum has seemed 
 to be of genuine interest. (For a non-associative operation would 
 not carry us very far.) As in ordinary arithmetic or algebra, this 
 law, together with the commutative property, makes the addition 
 of vectors a powerful operation. It is interesting to see the 
 associative law hold for such a sum of vectors (kept far from 
 any idea of measurement), exactly as if they were numbers 
 of pebbles or of oranges to be counted together in various 
 groupings. 
 
 The associative theorem enables us, among other things, to deal 
 
 * Which are not sufficient for Desargues' theorem. 
 
SUM OF COLLINEAR VECTORS 9 
 
 easily with the addition of collinear vectors, hitherto postponed 
 by necessity. 
 
 6. Sum of collinear vectors. Multiple of a vector. Two 
 collinear vectors X and X' (Fig. 3) being given, with origin and 
 common terminus T x (upon our T-plane), we require the vector 
 S = X + X' to be constructed as a certain vector having again for 
 its origin and collinear with these two addends. Let us imagine 
 for the moment we know what S or X + X' means ; that is to say, 
 let OS in Fig. 3 be this vector. If so, then we can add to it another 
 
 vector not collinear with it, say Y = OY. As before, draw ST r 
 and YT X ; let these cross in A. Then, writing OA=A., we have 
 
 Now, apart from its origin and terminus T x , the symbol X + X' 
 stands hitherto for an undetermined vector. It is, therefore, in 
 our power to give it some reasonable meaning. The only thing 
 that might embarrass us is the freedom of the choice. Here, how- 
 ever, the idea readily suggests itself to base the choice upon the 
 following consideration. 
 We do not know what A means. But we do know the meaning of 
 
 for the two addends X+Y and X' are definite vectors and not 
 collinear. Moreover, we know that the associative law holds when 
 no two of the three addends are collinear. 
 
10 
 
 PROJECTIVE VECTOR ALGEBRA 
 
 Let us, therefore, agree to fix the meaning of X+X' so that 
 the associative property may hold also when two of the addends are 
 
 FIG. 4. 
 
 collinear* In short, let us require A to be the same vector as A', i.e. 
 
 Here, A. = OA (Fig. 3) is a perfectly definite vector. Thus the 
 problem of finding the meaning of X+X' = S is reduced to finding 
 
 FIG. 4<z. 
 
 a vector S (independent of Y) which, when added to Y, gives A, a 
 requirement which we will write shortly 
 
 S = A-Y, i.e. X + X' = (X + Y) + X'-Y, (4) 
 
 calling it the difference of A and Y. Now, a glance at Fig. 4 will 
 
 * This is merely a fresh, and still permissible, act of convention. 
 
SUM OF COLLINEAR VECTORS 11 
 
 suffice to see that this ' reverse ' problem is solved as follows. 
 Draw YA obtaining T s on the given line of termini T T T A pro- 
 longed ; then the cross of T a O with T r A will be S, and S = OS will 
 be the required difference of the two given vectors A and Y. Notice 
 that 5 may lie either as in Fig. 4 or ' beyond ' the T-line, as in 
 Fig. 40, according to the nature of the two data. (These possi- 
 bilities will be dealt with in detail at a later opportunity.) 
 
 This leads at once to the required construction of the sum of 
 two collinear vectors X, X', i.e. to the solution of the problem 
 represented by (4) : 
 
 Draw Q = (X + Y) + X', where T is any vector not collinear with 
 the given X, X'. Then the join of T T and Q will cut OT X in a 
 point S (Fig. 5), and OS = S will be the required vector X+X'. 
 
 The proof that S thus obtained is independent of the choice 
 of the auxiliary Y may be left to the reader. It is not difficult 
 to see that if X and X', the end-points of both the vector 
 addends, fall within OT X , so also will S, the end-point of their 
 vector sum. 
 
 As a particularly interesting sub-case consider the doubling of 
 a given vector X, i.e. the construction of 
 
 This is obtained from the above by making X' to coincide with X. 
 Thus, draw the auxiliary Y with its T r (Fig. 6) ; join Y, T x and 
 X, T r crossing in P ; draw TpX crossing YT X in P 2 . Then the 
 join of T r with P 2 will cut OT X in a point X 2 , and OX 2 will be the 
 required vector 2X. 
 
 But this is exactly equivalent to the well-known construction 
 
 * By = will always be meant ' the synonym of,' as aqua = water, X = OX, 
 and so on. 
 
12 
 
 PROJECTIVE VECTOR ALGEBRA 
 
 of the fourth harmonic to 0, X, T It with X, T s and 0, X 2 as 
 conjugate pairs.* 
 
 Thus, as a consequence of the definition of vector sum and 
 of its determination for collinear addends (through the extension 
 of the associative law to collinear vectors) we have the elegant 
 result that the doubling of a vector X, i.e. the construction of 2X, is 
 equivalent to finding the fourth harmonic to 0, X, T x , with X, T r 
 as conjugates. 
 
 FIG. 6. 
 
 In order to construct 3X=2X+X treat 2X as X' of Fig. 5. 
 Thus, using again Fig. 6, draw OP 2 up to its terminus T z , and 
 join T^X crossing YT r in P 3 . Then T r P 3 will cut OT X in X s , the 
 required end-point of the vector 3X. Similarly for 4X, 5X, and so 
 on, generally for any positive multiple riX of the given vector X. 
 
 It can easily be shown that P 3 lies on TpX z , so that the above 
 construction of 3X is equivalent to the familiar construction f 
 of the fourth harmonic X 3 to X, X 2 , T x (with X, T x as conjugates). 
 Similarly for X, X$, and so forth. 
 
 If we start with any vector X = OX, such that X is on the seg- 
 ment OT X and distinct from T x , then, no matter how great the 
 integer n, the end-point X n of the vector wX = (w-i)X + X will 
 
 * We are thus free from the need of proving formally the independence 
 of .X" 2 (the end -point of 2X) of the choice of the auxiliary Y. For the 
 fourth harmonic is known to be wholly independent of the auxiliary 
 elements. This, by the way, is again an immediate consequence of 
 Desargues' theorem. 
 
 f Cf., for instance, J. L. Coolidge, Elements of Non-Euclidean Geometry, 
 Oxford, 1909, pp. 255-6. 
 
MULTIPLE OF A VECTOR 13 
 
 not fall beyond T Xl nor reach it. This circumstance may be 
 expressed shortly by writing J' x =X to or T x = J^ = oo X, or 
 symbolically X : T x = o. The reader will, however, attach no 
 mystical meaning to these ' infinities ' ; T x is for us as good a point 
 as any other ; it was raised to its position of dignity by our own 
 free act of convention. It is only not accessible by any finite 
 number (n) of the described operations or ' repetitions ' of X, i.e. 
 of adding X to itself, X to 2X, X to 3X, and so forth. 
 
 The end-points of the vectors X, 2X, etc., are all distinct points 
 and follow upon one another, each X n being beyond X n _ l . If to the 
 axioms of order and connection (assumed throughout) we add also 
 Schur's generalization of the archimedean postulate (cf. p. 3, 
 footnote), then every point of the segment X T x can be exceeded by 
 the end-point X n of the vector nX, where n is an appropriate 
 positive integer. 
 
 If we so desire, we may call OX, XX 2 , X^X Z , etc., ' equal steps,' 
 or equal collinear vectors in chain-arrangement, meant as another 
 expression of the identities 
 
 X=( 2 X)-X=( 3 X)-( 2 X),etc., 
 
 and these are but identities, for we have declared S=A-Y to be 
 a synonym of A = S + Y. But it is better not to anticipate. It will 
 be remembered that hitherto our reasoning is confined to coinitial 
 vectors. Chains of vectors will be dealt with later on. 
 
 There is not the slightest objection to compare these ' equal 
 steps ' with the usual, ' rigid ' equal steps. The reader may adjust 
 a pair of legitimate compasses to the points O, X of Fig. 6, and 
 fixing one leg of the compasses in X turn round the other ; then 
 he will be sure to hit over the mark X 2 , and the more so in the case 
 of X 3 , etc., if he continues his kinematical experiment. If, however, 
 having transferred the pair O, X upon a large sheet of paper, the 
 reader will take T x farther away, i.e. so as to require a very great 
 number of his equal steps OX in order to reach or to exceed the 
 new T It then he will find that the discrepancies do not become 
 sensible until the number of compass-steps grows very large. Or, 
 equivalently, if he continues the drawing of Fig. 6 as it is, but adjusts 
 his compasses to the pair X u , X lt , then, turning round the com- 
 passes as before, he will not exceed sensibly the point X a , and 
 so on. 
 
 At first sight one may be inclined to conclude that, after all, the 
 equal steps as marked by a ' legitimate ' or ' rigid ' transferor are 
 
14 PROJECTIVE VECTOR ALGEBRA 
 
 preferable to these projective steps. [And so, in fact, they are in 
 their proper domain of physical application. Thus, for instance, in 
 engineering the rigid equal steps and a geometry based upon them 
 are the very thing that is wanted, simply because one is much 
 interested to know how far the end-point of, say, an iron bar will 
 just reach when turned round, and so on. But in the pure study of 
 space properties we are not interested in such mechanical prevision.] 
 Yet, on considering the complicated nature of the testing of the 
 ' legitimacy ' of transferers, involving not only mechanics, but such 
 factors as temperature, etc., the reader will, theoretically at least, 
 not condemn the more general series of ' equal steps.' Their field 
 of application is just as broad as that of projective geometry. The 
 purpose of the proposed Vector Algebra is precisely to facilitate 
 the treatment of problems of projective geometry. 
 
 To close this small-type digression but one more remark. The 
 so-called cross-ratio, treated metrically, 
 
 has, for a harmonic range, the value -i. This property may be 
 put in an interesting form in connection with the above non-metrical 
 result that the end-point of the vector 2X is the fourth harmonic 
 to 0, X, T x . In fact, if the values of n belonging to X, 2X, T* are 
 denoted by n lt n t , n T , we may define the cross-ratio by 
 
 whatever the four collinear points. But in our case 
 
 Wj = i, 2 = 2 > 
 Thus, 
 
 appears as a most immediate property of a harmonic range. The 
 tacit attribution of n = o to the nil vector = OO, or simply to 
 the origin, may easily be justified. 
 
 It will be noted that if one of the two addends of a vector sum 
 has its end-point on the T-plane, the sum is equal to that vector 
 alone. In symbols, explained by Fig. 7, 
 
 Y+T^ = T Z . (5) 
 
 This is, in the case of Y, not collinear with T x , an obvious conse- 
 quence of the original definition of sum, the join XT r being now 
 replaced by T x T r itself. The proof that (5) holds also when Y is 
 collinear with T x may be left to the reader. We have seen before 
 
SUM OF COLLINEAR VECTORS 15 
 
 in what sense T x plays the part of an ' infinite ' vector. The 
 property (5) is of the same nature. It resembles the algebraic 
 or arithmetical truth 3 + 00 =00. 
 
 The relation (5) continues to hold, of course, if Y is beyond 
 T r . Notice in passing that if X or Y or both are beyond the 
 T-plane, so is also the end-point of the vector X+Y. (If the 
 straight lines are closed, as in elliptic geometry, this concept of 
 1 beyond ' requires manifestly some further qualifications, which 
 may occupy us later on. Meanwhile we can imagine that the 
 contemplated region of space is a restricted one.) Finally, if both 
 
 FIG. 7. 
 
 addends are T-vectors, T x and T r , say, then their sum T x +T r is 
 some T-vector, since its end-point is by the sum definition, some- 
 where on the line T x T r . But its position on that straight line 
 is, thus far, undetermined. And in the present case an appeal 
 to the associative law and to a third auxiliary vector would not 
 help us much to make it determinate. In quite the same way, 
 Tf t T Tt T z being any, generally not collinear, points of the T-plane, 
 the sum T x + T r + T, is hitherto an undetermined vector. 
 
 We might localize its end-point on the T-plane by some supple- 
 mentary convention as, for instance, by choosing on OT It OT Tt 
 OT Z some ' standard ' vectors x, y, z and declaring that 
 
 for n-*oo. Then the T-point in question would be the terminus 
 of the vector x + y + z. But the choice of these ' standards ' being 
 perfectly arbitrary, we prefer to say simply that the position of the 
 end-point in question on the T-plane is undetermined, and to 
 leave it so, until we find some reason for fixing it. 
 
16 PROJECTIVE VECTOR ALGEBRA 
 
 To close this section, let us note that, by the associative law, 
 
 2 X 
 that is to say, 
 
 and similarly, for any number of vector addends, 
 
 2(X + Y + Z+...) = 2X+2Y+2Z+... . 
 
 In words, the doubling of a vector sum is distributive. And so also 
 will be its trebling, and so on, generally, 
 
 (X + Y + Z+...) = X + Y + Z+... , (6) 
 
 where, thus far, n is any positive integer. 
 
 At a first sight the truth expressed by such identities as 
 2(X+Y)=2X + 2Y may seem very simple or even trivial, because 
 one involuntarily associates it with the doubling, say, of the sum 
 of 5 apples and 3 apples, and asserting that these are as many as 
 twice 5 and twice 3 apples. Yet, in our case, this formula expresses 
 a property of a very complicated maze of straights and of their 
 crosses. To see this, it is enough to translate this vector formula 
 into geometric language, remembering the adopted meaning of 
 the innocent-looking signs + (or ' and ') and 2 (or ' twice '). The 
 origin and the TMine being fixed, as always, the right-hand 
 member represents a certain point, say R, and the left-hand 
 member a point L (the end-points of the corresponding vectors 
 B and L), and the formula asserts the identity or the coincidence of 
 R and L* Now, this is, from the geometrical point of view, 
 certainly a rather complicated truth. 
 
 In fact, to obtain L, we have to construct A =X +Y (Fig. 8) and 
 to prolong OA up to T A . Next, we have to construct L =2 A, i.e. to 
 draw YT A , crossing AT r 'va. B and giving T s , and to draw AT 8 
 crossing YT A in C ; finally the join TjC will give us on OT A the 
 required point L, the end-point of the left-hand member. To obtain 
 the right-hand member, R, draw (with dotted lines, for distinction) 
 ax and 2Y by the previous prescription, thus getting the points 
 
 * Notice that as long as we are confined to coinitial vectors, any of our 
 ' equations,' or, better, identities, expresses only the coincidence of two points, 
 L and R, which are arrived at by some two different sets of operations or con- 
 structions. Equations in the proper sense of the word will make their 
 appearance after the concept of equality of non-coinitial vectors has been 
 introduced. 
 
DISTRIBUTIVITY 17 
 
 X 2 , Y 2 . The pouits A, T A being already there, draw T A X, obtaining 
 D ; then T r D will give X z . Similarly, T A Y gives the previous B, 
 
 and T X B (dotted) finds Y 2 . Now, R is, by definition, the cross of 
 XT r and Y 2 T X or T^B. Thus, what is asserted by the formula 
 
 2(X + Y) = 2X+2Y 
 
 is : the straight line T X B will pass through L ; and so it does. 
 
 FIG. Sa. 
 
 Now, a skilled projectionist will be able to prove the coincidence 
 of L and R on the figure. The reader may try it as an exercise.* 
 
 * As a hint : harmonic ranges remain so on being projected, etc., etc. 
 P.V.A. B 
 
18 PROJECTIVE VECTOR ALGEBRA 
 
 Even if he is successful, he will still agree that the constructional 
 proof is much more complicated than the above simple deduction 
 from the associative law. 
 
 Of course, the case illustrated by Fig. 8 is of no particular interest 
 for the geometer. But later on we shall have the opportunity 
 to show several examples of algebraic formulae, almost as simple 
 as 2(X + Y)=2X + 2Y, and yet expressing very interesting geome- 
 trical properties. If the numerous auxiliary lines are omitted, the 
 geometrical meaning of 
 
 is expressed by Fig. 8a, which hardly calls for lengthy explanations. 
 The equation asserts that the points , i, f, etc., marked in the 
 interior of the figure are all collinear, that these points are projected 
 from Tj- upon OT r as the end-points of the vectors , I, fY, and 
 so on. 
 
 7. Submultiple of a vector. Let m be a positive integer, 
 and X a given vector with as origin. Then we shall call the mth 
 
 part of X, and denote by X, a vector X' coinitial with X and such 
 
 m 
 that 
 
 mX'=X. 
 
 This definition of the submultiple X may be restated by writing 
 
 It is scarcely necessary to say that X will be collinear with X. 
 
 If m = 2, 3, etc., we shall speak, without fear of a misunder- 
 standing, of one half, one third of X, etc. a fixed terminus T x being 
 always understood. 
 
 The geometrical problem is to construct any of these sub- 
 multiples of a given vector by means of a straight-edge. 
 
 Before treating it, however, let us note the immediate conse- 
 quences of the above definition together with the distributive law. 
 Let X, Y be two coinitial vectors and 
 
 S 
 
 On the other hand, let 
 
 m m 
 
SUBMULTIPLE OF A VECTOR 19 
 
 Then, by the distributive law * and by (7), 
 
 so that the preceding equation becomes 
 
 m m m 
 
 Again, if / be another positive integer, we have, by the same 
 distributive law, 
 
 -(X + Y)=-X + -Y, (8) 
 
 m x mm 
 
 where stands for / . . But it can be easily shown that, for 
 m m 
 
 any of our vectors, l( X ) is the same thing as (IK), so that 
 \m ) mm 
 
 in formula (8) can be treated indiscriminately either as Inr 1 or 
 as m~ l l, as is the case with fractional factors in common algebra. 
 The same would be the case for X+Y+Z, and so on. 
 
 Thus the distributive law of multiplication of a vector sum 
 of two or more addends is also valid for any positive fractional 
 factor l/m. This distributive property can also be extended to 
 irrational factors ; but the reader will exempt me from giving 
 here a rigorous formal proof. This being taken for granted, we 
 shall have, for any positive number n, 
 
 n(X + Y+...) = nX + Y+... . (9) 
 
 Negative vectors, defined by X + (-X)=o, will be treated later 
 on. It will be time then to speak of negative scalar factors. 
 
 The above distributive properties, such as (8), followed algebrai- 
 cally from the definition (7) of a submultiple of a vector X, inde- 
 pendently of the actual method one has to adopt in order to draw 
 any such submultiple of a given vector. We might employ for 
 this purpose the construction which has been in use among the 
 projectionists for many years, chiefly in connection with Cayley's 
 theory of distance (cf., for instance, Coolidge, I.e. p. 256 et seq., or 
 W. Killing, Nicht-Euklid. Raumformen, Leipzig, 1885, p. 119). 
 But, from the point of view of this booklet, the following, equivalent, 
 method seems more direct and instructive, and at the same time 
 not less simple. 
 
 * Hitherto valid for integral (positive) factors only. 
 
20 PROJECTIVE VECTOR ALGEBRA 
 
 Let X 1/m be the end-point of the required vector, so that 
 mK' = OX lim . Take, for instance, the case of JX = &X"i. Then, 
 
 by the definition (7), and X being the end-point of X, the four 
 points 
 
 0, X if X, T x (10) 
 
 form a harmonic range, with 0, X and Xi, T% as conjugate pairs. 
 Imagine a ray drawn from up to its terminus T Y ', join T x with T T 
 and project, from any point of this TMine, the harmonic range (10) 
 upon the auxiliary ray. Then, by an elementary theorem of 
 projective geometry, the four points of that ray will again form 
 a harmonic range.* Call Y the point corresponding to X^, and 
 write OY = "Y. Then to X will correspond F 2 , the end-point of the 
 vector 2Y. In short, the range corresponding to (10) will be 
 
 0, Y, Y 2 , T Y . (10') 
 
 In quite the same way, to the range 
 
 0, Xl, X^, *.. Xm-l, X, T x , 
 
 with any integral m, will correspond on the auxiliary Y-line the 
 range 
 
 ^> *l Y z , " * m-li Y mi T r , 
 
 that is to say, itself, the end-points of the vectors Y, 2Y, ... wY, 
 and the terminus T r . This leads at once to the following con- 
 struction of any submultiple, say one-fifth, of a given vector X : 
 
 Through 0, the origin of X, draw any straight OT r , take upon 
 it a vector Y ending in Y and construct in the known way 2Y, 3Y, 
 4Y, 5Y, all beginning in and ending in the points 2, 3, 4, 5 (thus 
 marked in Fig. 9). Draw the straight X*,, crossing the TMine 
 in T. Then the straight TY will cut the ^"-line in X, the end-point 
 of the required vector ^X. At the same time, the rays 7^2, T$, T4 
 will find the end-points of the vectors fX, f X, and iX respectively. 
 Similarly for any other integer m. 
 
 Compare this construction with the familiar method of sub- 
 dividing a segment in common metrical geometry, which consists 
 in laying off, on the auxiliary ray, a chain of m ' rigidly ' equal 
 
 * This theorem can also be demonstrated algebraically from the above 
 principles, as will be shown later on, among the examples. Cf. Section 13. 
 
SUBMULTIPLE OF A VECTOR 
 
 21 
 
 segments Y, joining the last point with X, and drawing parallels 
 to this join through Y and through the remaining auxiliary points. 
 The above is obviously a generalization of the usual method. The 
 rdle of the euclidean parallels is here taken over by the rays drawn 
 
 through a T-point. The relations are exactly of the same nature 
 as those mentioned before in connection with the non-metrical 
 vector addition as compared with the parallelogram construction 
 of a vector sum. 
 
 The proof that the result of the process of subdivision just 
 described is independent of the choice of the auxiliary vector Y 
 may be left to the reader. 
 
 8. The concept of Equality of non-coinitial vectors. Hitherto 
 all our vectors were coinitial. It is now time to pass to vectors 
 having distinct origins. As was already remarked, the vectors 
 of a bundle emanating from a common origin had nothing 
 to do with one another, no two of them being in any sense equal 
 or equivalent to one another, disregarding the trivial case of 
 identical vectors. 
 
 If, however, non-coinitial vectors are contemplated, the possi- 
 bility readily arises of (usefully) defining some pairs of them as 
 equal and others as different. 
 
 Let A. = OA be a vector, with terminus T A on the fixed T-plane, 
 as before, and let 0' be another point, not on the line of A, as in 
 
22 PROJECTIVE VECTOR ALGEBRA 
 
 Fig. 10. Let us consider 0' as the origin of oo s vectors. Is there 
 among them some one codetermined by A, and unique in its kind, 
 worthy of the name of ' equal to A ' ? If so, then we are free to 
 define it as equal to the given vector A. The answer is in the affir- 
 mative, and the form of the definition readily suggests itself. 
 
 FIG. io. 
 
 Let the join 00' cross the T-plane in 7". Draw AT and 0'T A . 
 These will cross in a point A', determining an unique vector O'A', 
 which we will denote by A'. It is this vector I propose to define as 
 equal to A, in symbols, 
 
 A' = A or 0'A' = OA. (n) 
 
 Accepting this definition, which, after all, is but an obvious generali- 
 zation of the usual definition of equality of non-coinitial ' parallel ' 
 vectors, we shall also have to accept the equality of the vectors 
 AA' and 00', as will be shown presently. 
 
 A first consequence of the definition (u) and of what precedes it 
 is a very reassuring one. In fact, remember that the vector E = OA' 
 has been defined, at the outset, as the sum of 00' and OA, i.e. 
 
 But, by the new definition, A = A'. Thus, to respect both defini- 
 tions, we have to accept the equality 
 
 7 a 
 
 OA'-OO' + O'A', (12) 
 
 which reads : the sum of two vectors in chain arrangement * 
 * Origin (0") of second coinciding with end-point of first. 
 
VECTOR EQUALITY 23 
 
 is the vector drawn from the origin of the first to the end-point 
 of the second, exactly as in the familiar rule of ordinary vector 
 algebra. 
 
 We might have started from (12) ; then the definition expressed 
 by (n) would be the result. 
 
 Since OA and AA' is also a chain of vectors, we shall have 
 OA' = OA+AA' = A + AA', and since, on the other hand, 
 
 we must accept also the vector equality 
 
 A A' =00', 
 
 as mentioned before. Thus the quadrilateral OAA'O' of Fig. IO 
 is seen to play the part of the usual parallelogram. 
 
 Let us now consider some further implications of the new defini- 
 tion and of what was deduced in the previous sections. 
 
 First of all, we see (Fig. n) that if the new origin 0' moves, say, 
 along any straight OT', from the old origin up to the T-plane, all 
 
 FIG. n. 
 
 the arrowed intercepts between OT' and AT' will represent vectors 
 A' equal to A. When 0' tends to the T-plane, these vectors 
 dwindle down to nothing. The same construction can at once 
 be applied to points 0' taken beyond 0, where as shown on 
 the figure nothing particular happens. If, however, 0' exceeds 
 the terminus T' t the vectors A'=A begin to slant downwards. 
 But this need not frighten us. The reader will soon convince 
 
24 PROJECTIVE VECTOR ALGEBRA 
 
 himself that this, at first puzzling, behaviour cannot give rise to 
 any self-contradiction of the adopted system of axioms and defini- 
 tions. In short, all the arrowed segments of Fig. II represent, 
 according to the last definition, vectors equal to the given vector 
 A. The whole class of these vectors " equal A " have the terminus 
 T A in common. 
 
 In the second place, if the given vector A happened to be the 
 ' infinite ' vector T A = OT A , then the same definition would require 
 us to consider all the vectors 0'T A as equal among one another 
 and equal to T A . This result, puzzling as it may seem, does not, 
 however, lead to any incompatibilities. We may return to 
 questions of this kind later on. 
 
 The associative and the commutative laws will continue to hold 
 on substituting for any of the originally coinitial vectors an equal 
 vector, with any new origin. In particular, the commutative 
 property for two vectors linked up into a chain, as (Fig. 10) 
 
 OA +AA' = AA' + OA, 
 
 follows immediately from the commutative property of 00' + OA 
 (coinitial vectors) and from the newly defined equality, thus 
 
 OA + AA' = R = 00' + OA = AA' + OA. 
 
 A number of other, more or less direct implications of the definition 
 of vector equality can easily be treated, and may be left to the 
 reader. 
 
 As to the circumstance that the vector O'A', although defined 
 as equal to A = OA, ' dwindles down to nothing ' when 0' reaches 
 the T-plane (or the point 7"), this is in full harmony with the 
 previously stated property A + T' = T'. Further remarks on this 
 behaviour are hardly necessary. The inversion of ' sense ' of the 
 said class of vectors (equal A) beyond the T-plane, as in Fig. 11, 
 need not deter us, no matter whether the contemplated space is 
 euclidean, hyperbolic or elliptic, of the polar variety.* Notice 
 in passing that in the last case the ' inversion of sense ' is par- 
 ticularly well suited to the known property of the (closed) straight 
 lines and (one-sided) planes of a polar elliptic space, according to 
 which a traveller marching off from 0, erect on the plane of Fig. II, 
 towards the left, would arrive at T' from the right (as shown by the 
 
 * The antipodal or riemannian variety being better excluded for reasons 
 explained before. 
 
VECTOR EQUALITY 25 
 
 feathered arrows) and upside down.* Such a traveller would, 
 therefore, perceive no inversion. What we have hitherto called 
 ' beyond T ' and ' beyond ' is in elliptic space but one connected 
 region. In the case of hyperbolic or euclidean space (open straight 
 lines) there is no such natural connection between this-side and 
 that-side of the 7"-plane, and, therefore, no danger of a sudden, 
 discontinuous change of the sense of the vectors in question. 
 For these spaces it is usual in Projective Geometry to regard the 
 straight line as a closed -figure (cf., for instance, Mathews, I.e. 
 p. 12), but this is merely an advantageous artifice based upon 
 certain general conventions about ' elements at infinity.' It is by 
 no means necessary for our purposes. 
 
 But let us continue the subject of the present section, viz. by 
 considering the important sub-class of 
 
 Collinear equal vectors. We already know how to draw a vector, 
 equal to X = OX, from any origin 0' distinct from but not on the 
 line of X. Let us now construct, according to the adopted defini- 
 tion of equality, a vector O'X' equal to X, when 0' is on the line 
 of X, in short, a vector equal to and collinear with X = OX. The 
 original construction (Fig. 10) cannot, of course, be directly applied 
 in this case. But, as on previous occasions, an auxiliary vector 
 Y = 0F (Fig. 12) will help us out of the difficulty. The new origin 
 
 0' being given, say between X and T x , draw A=X+Y by the 
 previous method. Next, construct, from 0' as origin, the vector 
 O'A' or A' equal to A (by drawing 0'T A which crosses YT X in A'). 
 
 * The reader unacquainted with this subject will do well to look it up 
 in Sommerville's fascinating book on Non-Euclidean Geometry, London, 
 Bell, 1914. Chapter III. 
 
26 PROJECTIVE VECTOR ALGEBRA 
 
 This gives A'=X+Y. From this vector subtract Y = Y' = 0'y. 
 The required vector will be O'X' = A' - Y', the difference of two 
 coinitial vectors, which we already know how to construct (Section 6, 
 Fig. 4). In fact, Y'A' is already drawn, up to T x , as required ; 
 and so is T X 0'. Thus, it is enough to draw T r A' which will cut 
 0'T X in the required end-point X' of the vector A'-Y'. This 
 completes the construction and gives 
 
 0'X'==JL'=X==OX. 
 
 In particular, if the new origin 0' coincides with X, the end-point 
 of X, Fig. 12 becomes identical with Fig. 6, i.e. our present X' 
 becomes X z , and OX' becomes 2X, as it should, since we now 
 have two vectors in chain arrangement, OX' ' = OX + XX'. This 
 result justifies our procedure on page 13, where we wrote 
 
 (2X)-X=X, 
 
 although the need for such a justification was hardly felt at 
 that occasion. 
 
 Next, a vector X = OX being given, let us methodically construct 
 the negative of XX' =X, with X (not 0) as origin.* Now, X' 
 (or X^) is easily constructed, being the fourth harmonic to 0, X, T x . 
 In this way the point X' is constructed in Fig. 13. It remains to 
 construct the negative of XX', taking X as origin. This construc- 
 tion is already known from Section 6. We have to draw T B (x) Y', then 
 this will cut the X-line in the required point. But TB(X Y is already 
 drawn (in constructing X'), and it passes through 0. Thus the 
 negative of XX' with X as origin is, in fact, XO. In short symbols, 
 
 -(OX)=-(XX')=XO. (13) 
 
 Here X, are any two points, or at least points not on the T-plane. 
 This simple relation will be found very convenient in handling 
 vector sums with a view of interpreting them geometrically. 
 
 Finally, let us notice that the nil-vector was hitherto used as 
 a special nil-vector, viz. as 00 = 0. It can be expected that every 
 other nil-vector AA will be equivalent to it. This, if desired, can 
 
 * It is naturally interesting to see explicitly whether the negative of OX 
 with origin shifted to X is in fact the reverse, XO, of that vector, and thus 
 to verify the expectation immediately derivable from the chain sum 
 OX+XOOO =0, which, by the definition of -X, is equivalent to 
 XO = - (OX). It will be remembered, however, that this definition relates 
 in the first place to coinitial vectors only. 
 
VECTOR EQUALITY 27 
 
 also be easily proved in a formal way, by means of the associative 
 law for a chain sum. In fact, if 0, A, B are three points, we have 
 
 which proves the statement. Henceforth, speaking of a nil-vector, 
 we need no more specify its origin (which is also its end-point). All 
 
 FIG. 13. 
 
 nil-vectors being equivalent or equal to one another, we can simply 
 denote them by or by 0, the ordinary zero, and write, for example, 
 A+o = A, or AB +BA=O, and so on. 
 
 9. Difference of vectors. Negative scalar factors. We have 
 had recourse to the concept of the difference of two vectors in 
 Section 6, but only incidentally, as a means for constructing the 
 sum of two collinear vectors. In view of the fundamental impor- 
 tance of this concept and of the corresponding operation, the 
 subtraction of two vectors from one another, it seems desirable 
 to deal with this subject somewhat more fully. The special case 
 of the negative of a vector also deserves some attention. 
 
 Let X and Y be two vectors. Then their difference, symbolized by 
 
 D=X-Y, (14) 
 
 is denned as such a vector which, being added to Y, gives X, 
 i.e. such that 
 
 Y + D = D+Y = X. (140) 
 
 We have already noted, in connection with Figs. 4 and 4a, that 
 if, say, X and Y are coinitial in 0, the end-point D of D = OD may 
 fall that-side as well as this-side of the T-plane (or the straight 
 T\TY)- We should have said, more definitively, that D will fall 
 outside the domain of the triangle OT X T Y , and either ' this-side ' or 
 
28 PROJECTIVE VECTOR ALGEBRA 
 
 1 that-side ' of the conventionally fixed T-plane.* But there is 
 more than this. The subtraction of vectors, as defined by (14) 
 and (140), being an inverse operation, it may well happen that 
 there is no such vector OD satisfying the requirements of the 
 problem. Whether this is so or not, will depend upon the properties 
 of the contemplated space as a whole, and upon the given vectors 
 X, Y. 
 
 In fact, to construct D=X-Y we have, by what was said in 
 Section 6, to draw the join YX up to its intersection T D with the 
 TMine of the X, Y plane, and to draw the ray OT D ; then the 
 end-point D of the required vector will be the cross of T r X with 
 OT D . In the case of Fig. 14 both of these crosses, T D and D, are 
 
 FIG. 14. 
 
 actual points (the circumstance that D falls ' this-side ' of T is 
 irrelevant in the present connection). Now, concerning the first 
 of these two points, it may happen that the straight YX does 
 not cut the T-line at all; but this is of no consequence. For, 
 whether T D is merely inaccessible (in the practical sense of the 
 word) or YX does not intersect the TMine at all (and if our space 
 is hyperbolic there is a whole pencil of lines through Y coplanar 
 with T r T x and not intersecting this latter line), the two lines 
 T r T z and YX always send through a perfectly definite straight 
 OR which can be easily constructed. (Cf. Section 10.) But this 
 straight, which is the line of the required D, being thus found, it 
 
 * Or rather of a deliberately limited portion of it. For, should the 
 totality of our space be elliptic (polar), then the T-plane, as any plane, has 
 but one side, somewhat as the familiar leaf of Mobius. 
 
DIFFERENCE OF VECTORS 29 
 
 may happen that the straight T Y X does not intersect it at all. If so, 
 then there is no actual point D and therefore no actual vector X - Y 
 with as origin. If the contemplated space is elliptic, this will 
 never happen, for in such a space every two coplanar straight lines 
 have an actual cross. But if the space is hyperbolic, then there is 
 through X a whole flat-pencil of lines not meeting the straight 
 OR (Lobatschewky's not intersecting lines), that is to say, to a 
 given X there will be an infinity of different values ofY, even 
 on a prescribed line OT r , such that X-Y with as origin is 
 non-existent. The euclidean space lies on the frontier of these 
 two kinds of spaces ; then T T X always intersects OR actually, 
 unless it is parallel to OR, when D becomes ' the point at infinity,' 
 and therefore D, apart from its sense, a definite vector. But it will 
 be remembered, of course, that we are not assuming the parallel 
 axiom which claims the existence of one and but one parallel. 
 
 In short, if X and Y are actually given vectors, and if the space 
 is elliptic, then X-Y is always an actual vector; if parabolic 
 (euclidean), then the end-point of this vector may, at the utmost, 
 become the point at infinity ; and if hyperbolic, then X-Y, always 
 with as origin, may exist or not ; in other words, it may be an 
 actual vector or not, according to the values of X, Y. In the latter 
 case there is no such point as D in the plane X, Y, and therefore 
 in the whole contemplated three-space.* The emphasis put on 
 the specification ' with as origin ' is important. For, even if OD 
 does not exist, we can construct actual vectors equal to X - Y from 
 origins other than 0. Thus, for instance, since 0JV = Y-X is 
 an actual vector, we can take NO as an actual representative of 
 X-Y. This representative will have its origin in N and its 
 end-point in 0. Thus X-Y has always a real meaning. 
 
 Essentially the same remarks apply to the particular case of 
 vectorial difference, in which the minuend is nil. But since this 
 sub-case is particularly well suited for a clear and instructive 
 statement of the relations, it will be useful to treat it in some detail. 
 Let, therefore, X be an actually given vector OX, and let -X, 
 in accordance with the general definition of subtraction, be such a 
 vector that 
 
 (-X)=o. 
 
 * If X-Y, with O as origin, is not an actual vector, Y-X is one. The 
 reader may investigate the validity of this assertion by way of an exercise. 
 
30 PROJECTIVE VECTOR ALGEBRA 
 
 By what has been explained previously, the construction of -X 
 will be as follows : Take an auxiliary vector Y, not collinear 
 with X, construct 05=B=X+Y (Fig. 15), and mark its terminus 
 T B . Then the cross of the straight T B Y with the X-line will be 
 the end-point X_^ of the required vector -X = 0J_ V * This is, 
 in the case of Fig. 15, an actual vector, the said cross being an actual 
 point of the X-line, towards the left of 0. In other cases it may lie 
 to the right of T x , ' beyond ' the T-plane ; at any rate X_^ will fall 
 
 outside the segment OXT X . But it may fall nowhere ; that is to 
 say, for certain X there may be no cross X- at all, i.e. no actual 
 vector - X. This will never be the case if the space is elliptic, but 
 may be the case if it is hyperbolic. 
 
 It is instructive to consider a series of collinear vectors X and 
 to construct to each of these its negative -X, using always the 
 same auxiliary point Y (whose choice does not influence the final 
 result), so that all the lines T B Y form a flat pencil with Y as centre. 
 Then the connection between the existence or non-existence of a 
 - X and the type of space becomes manifest. Let, for instance, 
 a be some standard vector Oa, and let us take the series of vectors 
 
 X = wa, 
 
 where n=i, 2, 3, etc. The end-points of these vectors, with as 
 origin, are shortly denoted in Fig. 16 by I, 2, 3, etc., and the 
 
 * It can be shown that this construction is equivalent to the usual 
 construction of X_^ as the fourth harmonic to O, X, T x , with X. lt X and 
 O, TX as conjugate pairs. Such pairs always separate one another. 
 
NEGATIVE OF A VECTOR 31 
 
 end-points of the corresponding negative vectors, -na, by - 1, 
 - 2, - 3, etc. Between the positive integers n the reader may 
 imagine intercalated fractional ones, and so on. If the archimedean 
 postulate and continuity are assumed, the numbers of the interval 
 n = to +00 just cover the points of the segment OaT X) there being 
 a one-to-one correspondence of points and numbers. And the 
 negative values of n cover at any rate all the points of the X-line 
 outside the segment OaT x , but we certainly are not entitled to 
 assert that all these negative n's are necessary to cover this con- 
 tinuum of points ; some of the negative n's may be superfluous, 
 as will be seen presently. As shown in Fig. 16, where only integral 
 
 -15 
 
 FIG. 16. 
 
 n's are placed, together with their negatives,* the end-points of 
 some of the negative vectors fall to the left of the origin 0, and 
 those of others to the right of the terminus T x . (There is thus a 
 formal connection between these two domains ; these would, from 
 the point of view of the scalar factors or ' indices ' n, form but 
 one domain.) Thus, the points - 1, -2, ... -5 are on the left of 
 ; so would also be - 10, no doubt, if we could give the drawing 
 a greater lateral extension. The position of -II or -12 would 
 look dubious, in the drawing, but -15, -20, and so on are 
 decidedly to the right of, or beyond T x . It is not difficult to see 
 that on approaching T x from that-side T we have n negative and 
 growing in absolute value beyond every limit. Thus, on crossing 
 T x , from left to right, there is a jump of n from +00 to -co . 
 
 * Which are constructed by the above method. The termini T B (in the 
 notation of Fig. 15) corresponding to Y + a, Y + 2a, etc., are, in Fig. 16, 
 marked shortly by (i), (2), etc. 
 
32 PROJECTIVE VECTOR ALGEBRA 
 
 Let us, however, concentrate our attention upon the pencil of lines 
 through the auxiliary point Y. If the space is elliptic, then every 
 line of this pencil intersects the base line in one (and only one) 
 point, and this line as every other closes up, so that there is a 
 continuous succession of negative n values, from o to -oo along 
 the segment (via - i) T x , the supplement of OaT x . Not only 
 does to every one of the points of this supplement-segment 
 correspond some negative value of n, but to every one of these n's 
 corresponds actually some point of this segment. In other words, 
 the segment exhausts all the values of the interval w=o to -oo. 
 
 FIG. i6a. 
 
 This is still the case, if the contemplated space is euclidean. For 
 then the parallel through Y sends us to the infinitely remote 
 point on the left with a certain negative n, say - 12 or - 12-25, an d 
 this same number reappears from the right, growing to -oo on 
 approaching T x from the right. If, however, the space is hyper- 
 bolic, then, if YP and YP' (Fig. i6a) are the two lobatschevskyan 
 parallels * to the base line, YP sends us to the left with a certain 
 negative w-value, say n= -A, and the other parallel YP' takes 
 us to the right with another negative value n= /*< A. And 
 since the region PYQ', P'YQ is filled out with lines not intersecting 
 the base, the interval of values 
 
 -/*<< -A 
 * Or limiting, asymptotic lines. 
 
NEGATIVE OF A VECTOR 33 
 
 will represent no points at all. In short, this will be the super- 
 fluous interval of n, hinted at above. The numerical values of 
 /*, A will depend, of course, on the choice of the ' standard ' a, by 
 means of which we express all vectors X = na.* It is, therefore, 
 better to express the last result without the aid of any such standard 
 vector. This can be done at once in the following manner. Notice 
 that the termini of the two parallels, T and T' (Fig. i6a), play the 
 part of T B of Fig. 15, so that if JL P = OX r be the vector to which 
 YP corresponds, OT will be the line of the vector Xj, + Y = B. 
 Draw, therefore, TO and YT X ; their cross B will be the end-point 
 of X/. + Y, and T r B will find on the base-line the required point X P . 
 Similarly, to find X f , t corresponding to the other parallel, draw 
 OT', crossing YT X in B' t and draw T Y B' ; the latter will cut the 
 base in X P >. In this way we obtain a definite segment XpX P ,,^ 
 and our result can be stated thus : If X = OX is a vector whose 
 end-point X falls within the segment XpX P > t then its negative, 
 -X, with as origin, i.e. as OX_ lt does not exist ; for there is no 
 such point as X_ r And if X falls outside that segment, then -X 
 can be constructed from as an actual vector. 
 
 As in the general case (of vector difference), notice that what 
 does not exist, when X falls within XpX P ,, is the localized vector 
 OX_i. But vectors equal to -X can always be constructed from 
 some origins other than 0, as, for instance, from X as origin ; for, 
 by what has been said before, 
 
 XO=-OX=-X, 
 
 so that, whatever the given X, its negative, -X, always represents 
 a class of actual vectors equal to one another and constructible 
 from various origins other than 0. Similarly for wX, where n is 
 any negative number. 
 
 After what has been said in this and the previous Sections, we 
 shall be justified in employing any real number n, positive or nega- 
 
 * The particular kind of the hyperbolic space being given, and O, Tx 
 being fixed conventionally. 
 
 f The position of the points X P and X P > is obviously independent of the 
 choice of the auxiliary Y ; the segment X P Xf> is a property of the line OT X 
 itself. 
 
 If space is lobatschevskyan, X P and Xp> are distinct points ; if elliptic, 
 there are no such points at all ; and if euclidean, these two points coalesce, 
 and the exceptional segment dwindles down to nothing. 
 P.V.A. c 
 
34 PROJECTIVE VECTOR ALGEBRA 
 
 tive, integral, fractional or irrational, as the scalar factor of a vector. 
 And if this be a vector binomial or polynomial, the distributive 
 property 
 
 w(XY...)=nXY... 
 
 will be generally valid. Moreover we have the associative property 
 of scalar multiplication, 
 
 m (nX) = ( mn] X = wwX, etc. , 
 which implies also, for instance, 
 
 - (wX - wY) = wY - mX, 
 and so on, as in common algebra. 
 
 This settles all questions concerning the algebraical manipulation 
 of vectors in accordance with the adopted definitions of their 
 addition and equality. The resulting Vector Algebra is a generaliza- 
 tion of the common one, without its multiplication of vectors 
 into vectors, whose analogon is not required for the purposes of 
 the present book. 
 
 Before proceeding to show, by a series of geometrical examples, 
 the practical use of the proposed vector algebra, it will be well 
 to remove a certain apparent difficulty which is likely to present 
 itself to some readers in connection with the sum definition given 
 at the very beginning. 
 
 10. Addition of vectors with 'ideal' termini. The definition 
 of the sum of two coinitial vectors X = OX and Y = 0F, given 
 in Section 4, as well as the associated definition of equal non- 
 coinitial vectors, presupposes the possibility of joining X to T Y , 
 and of Y to T Xt where the ' termini ' T x , TY are the crosses of the 
 lines of X and of Y with the T-plane. Suppose, however, that 
 one or both of these lines do not meet the T-plane at all, or at 
 Jeast, that we do not know whether they meet that plane ; in other 
 words, that their intersection with the T-plane is not guaranteed. 
 What then ? At a first sight the definition becomes in such cases 
 meaningless, the vectors in question cannot be added. If our space 
 were either elliptic or euclidean, then every line would either 
 actually intersect the 7"-plane or, in the limiting case, become 
 parallel to it, and in neither case would a difficulty arise. But 
 if the contemplated space is hyperbolic,* then there are, in every 
 
 * Quite apart from metrical properties, and only as concerns the question 
 of intersecting and not intersecting coplanar straights. 
 
IDEM, TERMINI 35 
 
 plane through 0, two distinct asymptotic lines or lobatschevskyan 
 parallels, limiting a whole flat pencil centred at and consisting 
 of straight lines which do not intersect the jT-plane at all. If, 
 therefore, the addend Y, say, happens to be on one of these latter 
 lines, it would seem that the join XT Y , as prescribed by the sum 
 definition, cannot be drawn or must remain indeterminate, and 
 that, therefore, the sum X + Y has in such circumstances no 
 definite meaning. But this is by no means the case. In fact, 
 the axioms of order and connection, mentioned and assumed at 
 the outset, suffice to prove the following fundamental theorem 
 (for its proof see, for example, Schur's quoted work, p. 18) : 
 
 Any plane [as the T-plane] and any straight [OY], or any two 
 coplanar straights determine a bundle of rays, so that through 
 every given point [X] passes a perfectly determined ray of this 
 bundle. v 
 
 In other words, the given plane and straight, say T and OY, 
 whether intersecting or not, are jointly equivalent to a given point, 
 T r , inasmuch as they enable us to draw through every X an unique 
 straight line. It is this equivalence which is meant by saying that 
 a plane and a straight, or two coplanar straights, not actually 
 intersecting to our knowledge, meet in an ' ideal point,' and similarly, 
 that two planes meet in an ' ideal straight line.' The clearest 
 statement of the rdle and the meaning of these old concepts will be 
 found in Schur's book. These ' ideal ' entities have a meaning 
 as real as any given and accessible point or line ; they are shorthands 
 for definite things : A plane and a straight determine a bundle 
 of rays as well as if the centre of this bundle were given, but " if 
 the existence of that centre is [doubtful or] not guaranteed, we call 
 it an ideal point." And the same explanation applies to an ' ideal 
 straight ' as the equivalent of the axis of an axial pencil of planes. 
 The ' ideal ' point or line may even turn out to be actual and 
 accessible beings ; only if we do not know beforehand or do not 
 desire to assume that they are so, we call them ' ideal.' 
 
 Thus, returning to our subject, we can at once extend the previous 
 sum definition to any two vectors JL = OX, Y = 0F, by enunciating 
 it in the following manner : 
 
 Let T x and T Y be the actual or the ideal points (or centres of 
 bundles) determined by the T-plane and by the lines of X or Y 
 respectively. Then T X Y is a definite straight (of the first bundle) 
 and T Y X another definite straight (of the second bundle). These 
 
36 PROJECTIVE VECTOR ALGEBRA 
 
 two lines crossing in a point Z, the vector Z = 0Z is the sum of 
 X and Y. 
 
 The resultant Z may itself be an ideal point, as was in certain 
 cases A"_j of the preceding section. But even such a vector can 
 be added to a further given vector, according to the extended 
 definition. 
 
 This settles the question of adding up any coinitial vectors. 
 The sum of vectors linked up into a chain does not require any 
 elucidations. And the construction through a new origin 0' of 
 a vector equal to any given vector OX is settled by the same 
 remarks as those concerning addition, since it calls only for the 
 ideal point determined by 00' and the J-plane. In short, the 
 required extension of all definitions and theorems will be accom- 
 plished by substituting for 'points T x , T Y , etc.,' the more general 
 expression ' actual or ideal points T x , T Y , etc.' Having, however, 
 given all theseTexplanations, we may drop these cautious specifica- 
 tions and henceforth speak simply of T x as the T-po'mt or the 
 terminus of X. 
 
 The actual method of constructing the straight PI through 
 a given actual point P and the ideal point / determined by a 
 plane and a straight or by two coplanar straights will be found in 
 most of the standard treatises on projective geometry. But it 
 may nevertheless be well to quote here the construction given 
 by Schur (I.e. p. 16), since it seems to be the most legitimate and 
 general one. It will be enough to consider the case in which / is 
 determined by two coplanar straights. Let these be a and b. 
 Then, P being the given point coplanar with a, b, take A and A', 
 any two points on a ; next B, any point on b, and B' any other 
 point on b such, however, that B, A' are not on the same side of 
 AB'. Then AB' and A'B will cross in a point y. On AP take a 
 point j8 not on the same side of B'P with y. Then the straight 
 fiy will cross B'P in a point a. Finally, draw A'/3 and Ba. which 
 will intersect in a point P'. Then P'P will be the required 
 straight, i.e. the straight " drawn from P to the ideal point 
 7 = (a, *)." 
 
 If a, b happen to have an actual cross, the straight P'P is certain 
 to pass through it. If there is no such cross, it is meaningless to 
 ask whether P'P does pass through it or not, but it then remains 
 true that P'P is as definite a straight line as are a, b. In fact, 
 
IDEAL TERMINI 
 
 37 
 
 replacing A, B, etc., by any other auxiliary points, satisfying the 
 above conditions of order or qualitative configuration, we shall 
 always obtain the same final straight through P. The proof of this 
 
 B 
 
 FIG. 17. 
 
 construction will be found in Schur's book. Here it will be enough 
 to notice that his proof contains no allusion whatever to the 
 cross of a, b* whether it exists or not. 
 
 11. On the use of the above vector algebra. The extent 
 of the field of possible applications of the described algebra of 
 vectors is obvious. It can be used to solve by algebraic mani- 
 pulation (and without geometrical reasoning) such problems and 
 to prove such theorems as imply any finite number of the 
 following processes and of these only : the drawing of the join 
 of two points, the laying of a plane through three points or a point 
 and a straight, the finding of crosses or intersections of two straights, 
 
 * It is in this respect different from the usual proof of the familiar 
 construction of the " join of a point with the inaccessible cross X of 
 two straights " (cf. for example, Halsted's Synthetic Projective Geometry, 
 New York, 1906, p. 18), which is but a special case of the above problem. 
 This construction makes no use of the cross X, but the proof of the con- 
 struction does. 
 
38 PROJECTIVE VECTOR ALGEBRA 
 
 of a straight and a plane and of planes with one another. But this 
 is precisely the whole field of projective geometry, which is, no 
 doubt, a vast one. , And that is the reason why we have no need 
 to try to enrich the explained algebra by operations, for the present, 
 such as mutual vector multiplication. The addition with its 
 derivates, the subtraction and the multiplication of vectors by 
 ordinary scalar numbers, is all that is required for the representation 
 of the said processes and eo ipso for the algebraical treatment of 
 that field of problems. 
 
 Such algebraical or ' analytical ' treatment of questions indepen- 
 dent of the axioms of congruence and of parallels has been practised 
 for a long time. But its technical side consisted, as in common 
 metrical or cartesian analytical geometry, in introducing plane- or 
 space-coordinates (defined as the indices of points of a straight 
 obtainable by constructing harmonic ranges). In fact, the very 
 first of our figures (p. 4) illustrating the adopted definition of 
 vector sum will be familiar to those who have studied modern 
 projective geometry, as the figure accompanying the usual definition 
 of such coordinates. If A (Fig. i) be a given point of the plane, and 
 OT X , OT Y two fixed lines (axes), draw T A and T X A crossing the 
 axes in X and Y ; then, if Oa, Ob be two standard segments of these 
 axes, and if OX=x(Oa), OY = y(Ob), the scalar factors x, y, defined 
 as our above n, are the (so-called non-homogeneous) ' coordinates ' 
 of A. The obvious extension to three dimensions may be left 
 to the reader.* In our notation, and assuming the generalized 
 archimedean postulate, every point R of space can be represented 
 as the end-point of the vector B=#a + yb-f0c, where x, y, z are 
 real numbers and a, b, c non-coplanar and arbitrarily fixed vectors 
 (not T-vectors), say coinitial in O t the origin of E. These scalars 
 x, y, z will be the coordinates of B or, as we may say, the 
 ' components ' of the vector E along the three conventional and, 
 generally, artificial axes. 
 
 But our purpose was precisely to avoid any artificial resolution, 
 and, therefore, also the use of coordinates. That is why we 
 have formulated the few definitions and their direct consequences 
 in vector language. The chief point consisted in proving the 
 associative property of the sum of any non-collinear vector 
 
 * Cf. also, for instance, Young and Veblen's Projective Geometry, Vol. I. 
 1910, Art. 69. 
 
39 
 
 addends.* The remainder was but a natural sequel of this pro- 
 perty, requiring at no stage of our reasoning any splitting of vectors. 
 The relation of the proposed vector algebra to the usual coordinate* 
 treatment of projective geometry is the same as that of common, 
 euclidean vector algebra to cartesian analytical geometry. 
 
 In complete harmony with this plan of its structure, if we are 
 te expect advantages of this algebra in applying it to particular 
 problems, we must avoid any splitting of vectors along the 
 axes of an artificial framework. To make this general hint more 
 plain : For the sake of the solution of any problem or the proof 
 of any theorem, let us introduce, besides the T'-plane, only such 
 planes and only such vectors or vector-lines as are given by or 
 inherent in the problem itself. Even the T-plane, which, although 
 fixed for the whole case in question, is after all conventional, 
 should and can be chosen in a manner suitable for, and dictated 
 by the inherent features of, the case in hand ; whenever this is 
 possible (not irrelevant), the result will be a simplification of the 
 required algebraic manipulation. 
 
 The formal handling of problems, as illustrated by the examples 
 given below, will be found to be exactly as in common vector 
 algebra, the only difference being that the scalar (numerical) 
 factors of the vectors have now a different, or rather a more general 
 meaning. Thus, for instance, if X stands for a directed segment 
 OX, the vector JX has its end-point M not half-way between 
 and X, as measured by the rod, but OM and MX are ' equal ' 
 in the more general sense which has been explained before, M 
 being the fourth harmonic to 0, X, T x , and so on. Notwith- 
 standing this, the manipulation with these scalar factors will be 
 exactly as in ordinary vector algebra. We can even, without fear 
 of a misunderstanding, call M the centre or the mid-point of OM t 
 and use similar language in all other cases. This will greatly 
 help to shorten our enunciation of data or results. As a matter of 
 fact, if the !T-plane were laid through a-Centauri and two such other 
 stars, then there would be no appreciable difference between these 
 projective concepts and the ordinary mid-points, etc., not only for 
 
 * To judge from the literature accessible to me the possibility of such a 
 general proof, without having recourse to coordinates, has escaped the 
 notice of projective geometers who treat only the associativity of sums of 
 collinear segments. 
 
40 PROJECTIVE VECTOR ALGEBRA 
 
 terrestrial drawings but also for astronomical constructions 
 embracing our solar system and even reaching a good way beyond 
 its limits. 
 
 12. Vector equations of straights and of planes. The simplest 
 operation, and at the same time one that occurs most often, 
 is the drawing of the join of two given points A, B. It is, 
 therefore, natural to begin with this example, which will also serve 
 the purpose of almost all others. 
 
 Let A, B be the end-points of the vectors A, B, having a con- 
 ventionally fixed origin 0, and let R, the end-point of a vector E, 
 with the same origin, be any point of the straight joining A, B, 
 together with its prolongations. Then, X being a variable real 
 scalar, we shall obviously have (Fig. 18) : 
 
 R-A = A(B-A), (15) 
 
 and this is already a possible form of the required equation. Its 
 only defect, or undesirable feature, is its lack of symmetry, while 
 the rdles of A and B relative to their common join are the same. 
 
 B 
 
 FIG. 1 8. 
 
 But symmetry is readily established. Transfer -A to the right, 
 so that (by the distributive and associative properties) 
 
 Thus^the symmetrical form of (15) will be 
 
 , where x + y-i, 
 
41 
 
 x + y 
 
 where x and y are free scalar variables. The most immediate 
 form is still (15), its meaning being 
 
 and in many cases this will be the most convenient form, notwith- 
 standing its asymmetry. The discussion of the values and the 
 sign of the variable A, in connection with the position of R relative 
 to A, B and the cross of their join with the T-plane, may be left 
 to the reader. 
 
 By a similar reasoning the reader will find that 
 
 *A + yB + (I6) 
 
 x + y + z 
 
 is the equation of the plane passing through the end-points A, B, C 
 of the non-coplanar vectors A, B, C drawn from 0. The discussion 
 of the role of the T-plane in connection with (16) may again be left 
 to the care of the reader as a good exercise. 
 
 13. Miscellaneous Examples. Let L, M, N be three coinitial 
 and coplanar vectors. Then, as we have seen a moment ago, 
 the collinearity of L, M, N, the end-points of these vectors, is 
 expressed by 
 
 N-M = A(M-L), 
 
 where A is some scalar number. To illustrate the utility of this 
 simple vect-algebraical rule, let us consider the following example, 
 concerning the proof of a theorem, of which the equally sounding 
 theorem of common metrical geometry is but a special sub-case. 
 
 Ex. 1. Prove that the mid-points* of the three diagonals of a 
 complete quadrangle (OACB) are collinear. 
 
 Let the quadrangle be given by and the end -points A, B, C 
 {Fig. 19) of the three coplanar vectors A, B and C = aA + /?B, 
 where a, /2 are given numbers. We can say that 
 
 0, A, B, a, ft 
 
 * After what has been said before, it is scarcely necessary to explain 
 that ' mid-point ' or ' bisecting ' point, say L in Fig. 19, is here used in 
 the sense that OL and LC are equal vectors with respect to the chosen 
 T-plane. 
 
42 
 
 PROJECTIVE VECTOR ALGEBRA 
 
 completely determine the quadrangle in question. Now, referring 
 to the figure, 
 
 B' =5 OB' = AB = (i - y) A + y (oA + )8B), 
 
 where y is an unknown scalar. But, since A, B are not collinear 
 {by assumption), A = y/3, and ay-y-n=o, whence 
 
 and, similarly, 
 
 I -a 
 
 A'- 
 
 A ~ 
 
 ,A. 
 
 (17) 
 
 These, by the way, are simple expressions for the required two 
 vertices (or codots) of the quadrangle. Now, L, M, N being 
 
 FIG. 19. 
 
 the ' mid-points ' of the three diagonals OC, AB, A'B', we have, for 
 the vectors L = 0L, etc., 
 
 and, by (17), 
 
 whence 
 and 
 
 2(M-L)=(l-a)A 
 
 2 ' N - M ' - 
 
 , 
 
 ) A 1- (i - 
 
 proving the collinearity of L, M, N, as was required. 
 
MISCELLANEOUS EXAMPLES 
 
 43 
 
 To enunciate fully the theorem, thus proved, in geometrical 
 language, remember that L is the fourth harmonic to O, C, T c , and 
 similarly for M, N. The accompanying figure has been drawn for 
 the case in which the T-plane does not pass through any of the four 
 sides of the quadrangle. The reader may draw various other cases, 
 and will readily convince himself of the general validity of the 
 theorem. 
 
 Ex. 2. Let be the origin of a flat pencil of vectors B, whose 
 end-points R are all collinear (Fig. 20). To each triad 
 0, R, T R construct the fourth harmonic R t , more generally 
 the end-point R n of the vector riR. What is the locus of the 
 R 2 , more generally of the R n ? 
 
 FIG. 20. 
 
 Let A and B be any two vectors of the pencil, and B any third 
 vector of this pencil. Then, by assumption, and by 
 
 cA + yB 
 
 R 
 
 x + y 
 
 Multiply both sides by 2, generally by n ; then, in virtue of the 
 distributive property, and because n(xA.) =x(nA), etc., 
 
 _ *(nA)+y(nB) ,,. 
 
 showing immediately that all the points R n are again collinear ; 
 that is to say, the locus of R z , for instance, is a certain straight 1 2 , 
 and so on, in general, the locus of all points R n is a straight line l n . 
 
44 PROJECTIVE VECTOR ALGEBRA 
 
 This property will, of course, be very familiar to those skilled in 
 projective geometry. But the book is not written for specialists, 
 and this example has been inserted because it shows that the very 
 simple passage from (a) to (b) gives a comparatively intricate 
 geometrical property. 
 
 The above result can also be expressed shortly by saying that 
 if the end-points of a pencil of vectors B are collinear, so also are 
 the end-points of each of them ' magnified ' n times. 
 
 This is a (non-metrical) generalization of the corresponding 
 theorem of euclidean geometry, in which, e.g. the new straight 
 lines / are ' parallel ' to the original straight /, the locus of 7?'s. 
 The latter circumstance has also its analogy in our case. In fact, 
 it will easily be seen that every l n passes through TI, the (actual 
 or ideal) terminus of / on the T'-plane, not drawn in Fig. 20. More- 
 over, such being the case, we see how to find the straight locus 
 of R 2 at once. Let A be one of the vectors of the pencil E. Then 
 it is enough to construct A 2 , the fourth harmonic to 0, A, T A \ 
 the join A 2 Ti will be the required locus (/ 2 ) of the end-points, of 
 2B ; and similarly for R n . That is to say, 0, B, B Zt T B , etc., will 
 again be harmonic ranges. (This property has been used in 
 Section 7 to subdivide a vector.) The expert knows, of course, 
 from the rudiments of projective geometry, that T^OAA^A^), 
 being a harmonic pencil, gives also on any new transversal, OT K , 
 a harmonic range. But if the reader did not know this very 
 fundamental theorem, he could obtain it by the above reason- 
 ing, viz. : If the eq. (a) is true, so also is (b). Of course, 
 all the fundamental projective properties are, so to speak, con- 
 densed in the associative, distributive and other rules of our 
 algebraic manipulations. But such is also the secret of the advan- 
 tages afforded by an algebraical or other symbolism invented for 
 the purpose of describing any other domain of science. 
 
 Ex. 3. An obvious generalization of the above property is that 
 concerning a bundle of vectors whose end-points are coplanar. For, 
 if B is any of the vectors, then the assumed coplanarity of end- 
 points is expressed by (16), where A, B, C are any three non- 
 coplanar vectors picked out from the bundle. Now, multiplying 
 both sides of (16) by n, we have 
 
 x + y+z 
 
MISCELLANEOUS EXAMPLES 45 
 
 which shows that the locus of the points R n is again a plane. It 
 is not difficult to show that this plane passes through the inter- 
 section line of the original plane (locus of R) and the T-plane. 
 Any further remarks concerning the subject of this example may 
 be left to the reader. 
 
 Ex. 4. Let OXS Y be a plane quadrilateral having its opposite 
 sides vectorially equal, i.e. XS = OY = 'Y, and YS = OX="K. Find 
 the properties of the cross M of the diagonals OS = S and X Y - Y - X. 
 
 Let VL = OM. Then, by the rules of addition of a chain of 
 vectors, we have 
 
 FIG. 21. 
 
 where y is thus far an unknown scalar. On the other hand, 
 
 M = *(OS)=*(X+Y), 
 
 where x is another unknown number. Whence 
 (x - y)X + (x + y - i) Y = MM=o, 
 and since X, Y are not collinear, x = y and x + y = l, i.e. 
 
 -- 
 
 that is to say, 
 
46 PROJECTIVE VECTOR ALGEBRA 
 
 It will be easily seen that the metrical euclidean property of 
 mutual ' bisection ' of the two diagonals of a ' parallelogram ' is 
 but a particular, limiting, case of the above relations : 
 
 In our case their meaning is : The point M, the cross of the 
 diagonals, is the fourth harmonic to 0, S, T s , and at the same time 
 to X, Y, T XY (if T XY be the terminus of the straight XY). The 
 reader can, in fact, verify by the usual construction that M is the 
 fourth harmonic to the said triads of points. Vice versa, if 0, S, T a 
 were our original data we could derive from Fig. 21 the way of con- 
 structing the fourth harmonic M (conjugate to T s ), thus : Through 
 S draw two straights / lf 1 2 , and let their termini be 7\, T z ; draw 
 T-f) and T 2 t cutting 1 2 , l in A, B ; then AB will cross OT 8 in the 
 required point M. 
 
 We have already called 2X the double of the vector X. Similarly 
 we can call M = |S one-half of S and XM one-half of Y - X, the 
 other diagonal. Thus the above result can now safely be stated 
 in the following manner : 
 
 The diagonals of a plane quadrilateral, whose opposite sides are 
 vectorially equal* bisect one another. 
 
 Notice that in virtue of this result (and by Ex. 2) the straight 
 T r M bisects OX, and T X M bisects OY. This gives another, very 
 convenient, method of constructing the ' mid-point ' of a given 
 segment. 
 
 It is scarcely necessary to add that if we write (vectorially) 
 AM = x(AB) and MB=x'(AB), where M is any point of the 
 segment AB, then 
 
 For this means only that AM+MB=AB, which was the rule 
 for adding a chain of vectors. 
 
 The above has seemed to throw some instructive light upon the 
 nature of the ordinary parallelogram property, being its generaliza- 
 tion. (Remember that each pair of sides emanates from a common 
 point on the T-plane.) Of a similar character is the following 
 illustration. 
 
 * And these can always be made equal, for any plane quadrilateral, to 
 wit, by taking its appropriate diagonal (join of diagonal points T x , T r ) 
 as the T-line. Thus, ultimately, the theorem covers the general properties 
 of the complete quadrilateral. 
 
MISCELLANEOUS EXAMPLES 
 
 47 
 
 Ex. 5. Let ABC be any triangle. Let A', B', C bisect, in the 
 explained sense of the word, the sides BC, CA, AB respectively. 
 Call AA', BB r , CC' the medians of the triangle in question. Not 
 knowing whether these generalized ' medians ' cross in one and 
 the same point, call M the cross of BB' and CC', M z that of CC' 
 with AA', and M 3 that of AA' and BB'. 
 
 T, 
 
 FIG. 22. 
 
 , or 
 
 Now, 
 
 where y, z are some scalars. Again, AC = -(CA) and 
 
 BA=BC + CA=CA-CB, 
 so that we have, with CA=*X, CB=Y, 
 
 |(s-y)X + (y + |2-i)Y=o, 
 and X, Y being non-collinear, 
 
 y = 2 = |, 
 that is to say, 
 
 CM^ICC, BM^^BB'; 
 
 in exactly the same way we shall find 
 
 whence it follows that 
 in plain language, that 
 
 ^ M t M 3t M t M v are all nil-vectors or, 
 
48 PROJECTIVE VECTOR ALGEBRA 
 
 The three medians cross in the same point M, which, moreover, 
 4 trisects ' each of them. 
 
 This, again, is the projective generalization of the well-known 
 theorem of ordinary geometry. Needless to repeat that in our 
 case ' trisection ' means that the vector CM trebled is, in the defined 
 sense of the word, equal to the double of CC' t and so on ; similarly, 
 ' C bisects AB' means that AC' = C'B t i.e. that AC'BT 9 is a 
 harmonic range, and so forth. Thus, ultimately the above short, 
 and familiarly sounding, theorem is the expression for a relatively 
 complicated projective property belonging to a ruler construction 
 sketched in Fig. 22. As in ordinary geometry, the common cross 
 M of our three medians can be called the centre of the triangle. 
 
 FIG. 22. 
 
 Notice that, unlike the ordinary centre,* this centre is not a 
 fixed particular point of the triangular region ; its position in the 
 plane of the triangle depends upon the choice of the (T-plane and 
 therefore) of the TMine in the plane of the triangle. By appro- 
 priately choosing the TMine any point can be made the centre M of 
 the triangle. Shifting the 7-line towards A will draw M to the 
 corner A ; and if the TMine passes through A, for instance, A itself 
 
 * Which is also the ' mass-centre ' of three equal masses placed at 
 A, B, C. 
 
MISCELLANEOUS EXAMPLES 49 
 
 will be M. If the IT-line is drawn across the triangle, as in 
 Fig. 22a, for example, B' and C will lie outside the triangle, but 
 the three medians will again be concurrent in M, a point outside 
 the triangle. For the above algebraic proof continues to hold. 
 
 It may be a useful exercise for the reader to start from A, B, C, 
 arbitrarily given together with the T-line in the plane ABC, to 
 construct in the known way the fourth harmonics A', B' t C', and 
 to prove, without the aid of our algebraic symbolism, that AA', 
 BB', CC' have a common cross. 
 
 Ex. 6. Similarly to the preceding example, let ABCD be a 
 tetrahedron, and (the T-plane being fixed conventionally) let 
 A', B', etc., be the ' centres ' of BCD, CD A, etc. Then AA', BB', 
 CC', DD' mil be concurrent in a point M, which, moreover, will 
 1 quadrisect ' each of these segments. The algebraic proof may be 
 left to the reader. This is again the non-metrical generalization 
 of a well-known theorem of ordinary geometry. The common 
 cross M of the four lines, which may be called the centre of the 
 tetrahedron, will again migrate together with the T-plane. 
 
 Ex. 7. Let abc, a'b'c be two coplanar triangles perspective from 
 a point 0. Call P lt P 2 , P z , the crosses of be and b'c', etc. In order 
 to obtain a simple algebraic verification of the theorem of Desar- 
 gues, take the centre as origin and draw the TMine through 
 P a , P 2 . Thena' = oa, b'=/fl>, c' = yc, and also 
 
 b'-c' = A(b-c), c'-a'=/t*(c-a). 
 
 Whence b'-a' = /?b-aa = Ab + (fi- A)c-/ua, and therefore A = /3, 
 fjL = a = X, giving b'-a' = A(b-a). Thus also P 8 is on the TMine, 
 i.e. collinear with P,, P 2 . Q.E.D. 
 
 Reasons of paper economy compel us to resist the temptation 
 to add a number of further illustrative examples. We shall, 
 therefore, confine ourselves to only one more special application 
 of the explained Algebra, leaving many other examples (which 
 easily suggest themselves) to the care of the reader. 
 
 14. Pascal's Theorem.* Owing to the great importance of 
 this theorem in the modern discussion of the foundations of geo- 
 
 * This is a short name for a special case of Pascal's more general theorem 
 concerning a hexagon whose corners are on a conic and well-known by the 
 name of ' the mystical hexagram,' discovered by Pascal (1639) at the 
 tender age of sixteen. 
 
 P.V.A. D 
 
50 PROJECTIVE VECTOR ALGEBRA 
 
 metry, it has seemed well to dedicate to it a full Section and to 
 show how readily it can be proved by the proposed vector algebra. 
 Let us, first of all, mention that Schur gave in 1898 (Math. 
 Ann. vol. 51, reproduced in his book, loc. cit. 3) a proof of this 
 theorem based upon the axioms of connection, of order and of 
 congruence (or motion), but without the aid of the archimedean 
 postulate. In 1899 Hilbert (loc. cit.} proved the said theorem 
 (or rather its sub-case, for pairs of parallel lines) without the 
 assistance of the axioms of congruence but with the aid of the 
 archimedean postulate ; Hilbert's proof based upon his own 
 1 algebra of segments ' involving the parallel-axiom is a rather 
 complicated one. This, however, by the way only. Here it is 
 important to notice that Pascal's theorem can be rigorously 
 demonstrated by the axioms of order and connection, here adopted 
 at the beginning, and with the aid of either the (generalized) 
 archimedean or the congruence axioms. Now, the congruence 
 axioms were and are entirely foreign to the present vector algebra, 
 while on the other hand we have, in the later sections at least, 
 assumed the archimedean postulate (it being necessary to 
 assert that if a is a vector, any vector collinear with it can be 
 represented by wa). Let us now definitely adopt this postulate. 
 Then our vector algebra should be sufficient to prove the said 
 theorem. And so in fact it is. 
 
 FIG. 23. 
 
 Let X, Y, Z and X', Y', Z', ordered as in Fig. 23, be six points 
 on two straights crossing in 0. Then Pascal's theorem asserts 
 
PASCAL'S THEOREM 51 
 
 that the three crosses, P t of YZ' and Y'Z, P 2 of ZX' and Z'X, 
 and P 3 of XY' and X'Y are collinear. 
 
 To prove it algebraically, let a = 0a and 9! -Oa' be any two 
 vectors on each of the bearers of the three points.* Then, x, y, z 
 and x', y', z' being some scalar numbers and X standing for OX, etc., 
 we can write 
 
 X = #a, Y = ya, Z = z&, 
 
 and similarly for the dashed vectors. Then, denoting OP l by P t , 
 we have 
 
 Pj = za + a (za -/a') =ja + ( ja - z'a,'), 
 
 where a, ft are some numbers. Since a, a' are not collinear, these 
 numbers are found at once by comparing the coefficients of a, a' in 
 the two expressions for P v Thus, z(i +a) = y(i +/8), and ay' =/te', 
 so that 
 
 or, after a slight rearrangement of terms (the associative as well as 
 the commutative laws being valid), and writing down two similar 
 equations for P 2 , P 3 , 
 
 (yy' - zz')V l =y'z'(y -z)a,' +yz(y f - s')a,, 
 (zz f -xx')V 2 =z'x'(z -*)' +zx(z' -*>, 
 (xx f - yy')P 3 *=x'y'(x - y)a' + xy (x' - y')a, 
 
 which are remarkably simple expressions. To prove the col- 
 linearity of P lt P 2t P 3 , it is enough to show that, say, Pi-P f 
 differs from P 2 - P 3 only by a numerical (scalar) factor. 
 The first two formulae give at once 
 
 (y/ - zz') (zz f - xx') ,_ _ x tl \ / / v 
 
 - a -- - (?! - P 2 ) = [(y - )^ + ( - ^)yy + (x - y)zz'}& 
 
 zz 
 
 + [(y'-z')xx' + (z'-x')yy' + (x'-y'}zz']a,'; 
 
 now, the bracketed expressions are symmetrical in x t y, z, x', y', z' t 
 so that the whole right-hand member of this equation is also equal to 
 
 XX 
 
 * As such a, a' we might have used, say, OX and OX' themselves. But 
 this would spoil the symmetry of our formulae. The best way is always 
 that which (in mathematics or human intercourse) does not artificially 
 privilege an entity having no natural or intrinsic claims ; now, the r61es 
 of the six points are certainly the same. 
 
52 PROJECTIVE VECTOR ALGEBRA 
 
 which proves the asserted collinearity, i.e. the theorem of Pascal, 
 for any configuration of the two triads of points. 
 
 Notice that the particular ordinary case of ' parallel ' pairs 
 YZ' t ZY', etc.,* or better, the case in which P lt P 2 , P 3 are on 
 the (conventional) T-plane, occurs when from our point of view 
 P lf P 2 , P 3 are infinite vectors with respect to a, a' or also Y'Z' t etc. 
 (that is to say, require an infinity of fourth-harmonic con- 
 structions). Now, x, x', etc., being finite numbers, we have 
 P 1= =p 2 = oo when and only when 
 
 yy' - zz' = zz' - xx' ; 
 
 but then also xx' = yy', i.e. P 3 = oo, as in the special theorem of 
 Pascal. The T-plane plays again the part of the plane ' at infinity.' 
 That ' special ' case has nothing particular about it, in our circle 
 of ideas ; for, the T-plane can always be put through P lt P Z) P 3 . 
 
 It will be remembered that the choice of a plane as the T-plane 
 is arbitrary. If the six points X, etc., are fixed (prescribed), so 
 are also, of course, the crosses P lt P z , P 3 . The choice of the 
 T-plane influences only the values of the numbers x, ... z' (after the 
 standard vectors a, a' have been fixed by convention). Now, 
 whatever the straight PjPgPg, we can, if we so desire, lay our 
 T-plane through it. Then we shall have the remarkably simple 
 relations between the six numbers : 
 
 xx' = yy'=zz'. 
 
 This, however, by the way only. The purpose of the present 
 section has been to show that the vector algebra in question gives 
 a simple and perfectly symmetrical proof of Pascal's theorem. 
 Hilbert's proof, which may be looked up in his quoted book, is 
 considerably more intricate. Schur's proof, of course, is much 
 more elegant than Hilbert's, but is based upon the postulates of 
 congruence or motion, which according to our plan were to be 
 strictly avoided. 
 
 15. Vector sum for variable /-line. Conic ranges and pencils. 
 
 Let us consider two fixed coinitial, non-collinear vectors JL = OX 
 and Y = Y. The TMine in their plane being conventionally chosen, 
 we have defined as the sum of these two vectors and denoted by 
 
 OS=S=X+Y 
 
 * When Pascal's theorem asserts the parallelism of the third pair if the 
 first two pairs are parallel. 
 
SUM FOR VARIABLE t-LINE 
 
 53 
 
 the vector whose origin is and whose end-point S is the cross of 
 the straights XT y and YT X (cf. Section 4, Fig. i). If the TMine 
 is kept fixed once and for ever, 5 is an uniquely determined point. 
 Now, it has seemed interesting to inquire into the properties of 
 the locus of 5 when, 0, X, Y being kept fixed, the line of termini 
 takes in turn a one-dimensional manifold of positions, more espe- 
 cially when this line, the /-line, as we will say for distinction, is 
 picked out from a flat pencil of straights with a centre P fixed in 
 the X, Y plane. 
 
 Let this variable vector sum OR (as exemplified by Figs. 24 
 and 240) be denoted by 
 
 R-H+Y]e f 
 
 the suffix reminding us that the sum is taken with respect to a 
 Mine belonging to the flat pencil centred in P, i.e. so that R is 
 
 
 
 A 
 
 \ 
 
 FIG. 24. 
 
 the cross of Xt v with Yt x * The points 0, X, Y as well as the 
 centre P being kept fixed, what is the locus of R ? 
 
 * We need not, for the present, trouble about the fixed T-line. This 
 will be required only when we come to express the properties of the locus 
 of R by means of scalar variables, whose numerical value will, of course, be 
 influenced by the choice of the T-line. As such line we shall introduce 
 later on some convenient straight. 
 
54 PROJECTIVE VECTOR ALGEBRA 
 
 It is not difficult to see that the range represented by the 
 end-point of R will be a genuine range of the second degree, a conic 
 range of points or in more familiar language a conic, embracing 
 
 Fie. 34*. 
 
 as sub-cases the ellipse, the hyperbola, or a pair of straight lines, 
 and the parabola. In fact, to quote the usual wording of the 
 method of generation of such curves as given in every Projective 
 Geometry : 
 
 If two coplanar non-copunctal flat pencils are projective (but 
 not perspective), the crosses of correlated straights form a conic 
 range. 
 
 Now, in our case, X and Y are the (non-coinciding) centres of 
 two coplanar pencils which are projective, and R is, by construction, 
 the cross of correlated straights, the ' correlation ' consisting in 
 joining X and Y with t v and t xt the crosses of the Y-line and of the 
 X-line with a /-straight of the flat pencil centred in P. 
 
 We thus see that the simple vector equation 
 
 B-H + Y] (18) 
 
 represents a conic range, or shortly a conic. 
 
 When the /-line passes through 0, we have obviously R = o, and 
 if it passes through X or through Y, we have R = X and R=Y 
 
CONIC RANGES 55 
 
 respectively. In short, the conic represented by {i 8) passes through 
 the origin and through the end-points of both vector addends, 
 that is, in all through three given points. The construction of a 
 conic, most in use, is based upon five of its points being given, and 
 it is well known that, if all data are to be points of the conic, five 
 points are just necessary and sufficient to determine completely a 
 conic, and that this also exhausts the whole class of conies. Our 
 case is somewhat different, owing to the peculiarity of the con- 
 struction embodied in X + Y. The two coinitial vector addends 
 are equivalent to three points only, viz. 0, X, Y. If nothing 
 more is given we can draw through them oo 2 conies, namely by 
 taking in turn various points of the plane (not 0, X, Y) as the 
 pencil centres P. Thus, to fix our conic entirely we have to give 
 0, X, Y and the centre P. And varying these data we can construct 
 any conic whatever. In fact, if five points of the conic are given, 
 we can use three of them as 0, X, Y ; then join X and Y with the 
 fourth point, say Z, and with the fifth U ; the straights XZ and YZ 
 will cross the Y- and the X-line in a pair of points t v , t xt giving t^ v 
 as one 2-line of the pencil ; similarly XU and YU will give another 
 line of the pencil, and the cross of these two Mines will be the centre 
 P of our pencil, thus reducing the usual construction to that repre- 
 sented by equation (18). 
 
 It is interesting to note the r61e of the supplementary datum, 
 the pencil centre P, in connection with well-established concepts 
 of projective geometry. In fact, it can be readily seen from the 
 construction that the two particular rays PX and PY have, each, 
 only one point in common with the conic, viz. X and Y respectively, 
 in other words, that PX, PY are tangents of the conic, which will 
 also be seen a little later from the developed form of the conic- 
 equation (18). Thus, to give P is equivalent to giving the tangents 
 at X, Y, whose cross is that point. 
 
 In short, the more familiar five points as data are here replaced 
 by three points of the conic and the tangents at two of them which are 
 also known to determine completely a conic. 
 
 At the same time we see, and shall presently verify on the 
 developed form of (18), that P, the centre of the pencil /, is the pole 
 corresponding to XY as polar with respect to the conic. Whence 
 a number of properties can easily be enunciated, relating to the 
 conic and to the above process of its generation. These, however, 
 
56 PROJECTIVE VECTOR ALGEBRA 
 
 must be left to the reader, as exceeding the scope of the present 
 little book. 
 
 On the other hand, it has seemed desirable to dedicate some 
 space to a further algebraical development of the simple vector 
 equation (18). As a matter of fact, this, as it stands, is already 
 the complete equation of a conic. But the reader may desire to 
 see it written in terms of numerical (scalar) variables. This can, 
 with the aid of what was previously explained, be easily done. 
 
 In order to introduce numbers, we require obviously, besides 
 the variable /-line, some fixed line of termini which we will again 
 call the TMine. For this only can enable us, after 0, X, Y (apart 
 from P) have been fixed, to fix the meaning of such beings as 
 2X, etc. 
 
 As such permanent T-line let us choose any straight not passing 
 through 0, X, Y, P, and crossing the two vector lines in T x and T v 
 (Fig. 25). Then, calling L, M the crosses of any t with the X- and 
 
 FIG. 23 
 
 the Y-line, taking for the origin of all our vectors, such as L = 0L, 
 etc., and writing vector sums with respect to T without any suffix, 
 
 where A, /* are some scalar factors. 
 collinear, we have 
 
 Now, L, X and M, Y being 
 
 where the scalars /, m may be considered as numbers defining the 
 
CONIC RANGES 57 
 
 particular /-line ; they will be presently expressed by P as datum 
 and one independent variable parameter. By the last equations, 
 X and ft are easily determined in terms of /, m, remembering that 
 X Y are not collinear vectors. In this manner we obtain 
 
 showing the relation of the variable sum R to the (conventionally) 
 fixed sum of the vectors X, Y, or in terms of the addends them- 
 selves, as required for our purpose, 
 
 /( 
 
 Im-i 
 If the centre P of the /-pencil be prescribed in the form 
 
 so that , rj are given fixed numbers (the coordinates of P), then 
 X+ 7/Y = <r/X + (l -<r)wY, where <r is some scalar number ; whence 
 
 o-/ = , (i-o-)fn = i7, 
 which, for the sake of symmetry, will be better written 
 
 /=-. m = -. where u + v = l. 
 u v 
 
 Thus /, m are reduced to one independent parameter, u or v I - u t 
 and ultimately, by substituting them in (19), 
 
 , l 
 
 
 
 ? - UV (2O) 
 
 a comparatively simple developed form of the equation to a conic. 
 To repeat it, this conic, described by the end-point of R, will pass 
 through the origin and through the end-points of the vectors 
 X, Y, the pole of the secant XY being the end-point of the vector 
 
 For u= the equation (20) gives E=X, and similarly R=Y for 
 v=7j. Thus the points X, Y of the conic correspond to u= and 
 v=rj respectively. The third given point O of the conic will corre- 
 spond to infinite values of M, v. If we desire to split the above 
 equation we may write 
 
 so that x, y will be the coordinates of a point R of the conic with 
 
58 PROJECTIVE VECTOR ALGEBRA 
 
 respect to x, Y as ' axes ' and as ' units ' at the same time ; then 
 the conic will be represented by 
 
 (n-v) w(-u) 
 x^u. t y = -^ - f . u+v-i. 
 gr) -y' f gr)-uv' 
 
 If P is on XY, we have a pair of straight lines (XY itself and a 
 straight through O), if P is inside the triangle OXY, we have a 
 hyperbola, and so on. Let the reader discuss all possible sub-cases 
 by means of the vector equation, paying special attention, perhaps, 
 to the case of collinear P, X, Y, when not only u+v=i t but also 
 +77 = 1, etc., and verifying his conclusions by drawings and by 
 geometrical reasoning, whether necessary or not. 
 
 The above equation of a conic can still be simplified. In fact, 
 whatever the given centre P, we can always take the 7-line so as 
 to make 
 
 Then =?? = I, and the equation (20) becomes 
 
 -* <"> 
 
 where u+ v i, as before. But such being the relation between 
 u and v, the ' numerator ' in (21) is a vector whose end-point, 
 say Q, is on the line XY. In other words, Q is the cross of OR 
 with XY (Fig. 26), and we have the interesting relation 
 
 OQ = (i -uv) (OR}. 
 With Q = OQ the equation of a conic can now be written 
 
 where u + y=i, the pole of the polar XY being the end-point of 
 the vector 
 
 P=X + Y. 
 
 Splitting the equation (21), we have, the coordinates of R, 
 
 u v 
 
 x == \ y ~ . 
 
 l-UV I-UV 
 
 whence x/y=u/v, x+y=(i -uv)~ l , and 
 
 x y 
 
 ; - l - 
 
 * For this purpose we have only to draw XP, YP and to lay the T-line 
 through the crosses of these straights with the T- and X-line respectively. 
 
CONIC RANGES 
 
 59 
 
 introducing these values of u, v, into the last equation, we obtain as 
 the (non-homogeneous) coordinate equation to any conic, 
 
 with O, a point of the conic, as the origin. The point X corresponds 
 to M=I, i.e. to x=i, yo, and similarly Y to y=i, x=o. If we 
 
 introduce homogeneous coordinates through x = , y= ', then the 
 equation becomes ** x * 
 
 (22a) 
 
 or 
 
 Fie. 26. 
 
 If the E-line passes through P, we have Q = AP, which is only 
 possible for u=v=* J, so that Q = P and B = ^Q, that is (cf. Fig. 26) 
 
 OQ:OR:OP = 3:4:6 
 or, the cross- ratio, 
 
 * Which stands for 
 
 90. ^? = iz 6 ~4 
 
 RQ ' OP 3 - 4 ' 6 - o' 
 
60 PROJECTIVE VECTOR ALGEBRA 
 
 Thus, P, R, Q, is a harmonic range, as it should be. In fact, 
 XY being the polar of P, any secant through P should give a 
 harmonic range P, D, E, F. 
 
 Let the reader prove this by means of the equation (21) or (21 a). 
 As a matter of fact, however, this general property of the polar and 
 its pole is already proved by the above case, for any point of a given 
 conic can be made the origin O of our vectors. 
 
 Clearly, every projective property of conic ranges can be proved 
 (or verified) algebraically by means of any of the obtained vector 
 equations, since these represent any conic and have been seen to 
 be equivalent to what is obtained by the usual generation of such 
 point ranges based on five points as data. These remarks will 
 suffice, especially as the whole subject of point-conies has been 
 given here chiefly in the way of an illustration of the use of the 
 proposed vector algebra. 
 
 Let us further consider rapidly some differential properties of the 
 equation of a conic. Such concepts, of course, as ' curvature,' are 
 meaningless (or irrelevant) in non-metrical geometry.* Similarly 
 the idea of a ' normal ' is foreign to us, since we have avoided the 
 congruence axioms relative to angles as well as to segments. But 
 the differential concept of tangent is by no means excluded from 
 our circle of ideas. 
 
 The meaning of the symbols being as above, we may represent 
 any plane curve by the equation 
 
 where / is some scalar function of u, v. If then s be any auxiliary 
 independent variable or parameter, the vector Jl = dR/ds will give 
 us information about the tangent to the curve at any of its points. 
 If, for example, f(u, v) = i t then, X, Y being fixed vectors, 
 
 which expresses the obvious result that the tangent to a straight 
 line XY is along the straight. 
 
 Turning to our vector equation of a conic in the form (21), say, 
 there is no objection in treating the scalar parameters u, v as 
 
 * Unless the ' curvature ' is zero (straight line, or inflexion point of a 
 curve) or infinite (cusp). 
 
CONIC RANGES 61 
 
 continuous and differentiate with respect to an auxiliary variable 
 s, such as the ' time.' * Thus, remembering that X, Y are constant 
 
 vectors and denoting -r by a dot, we have 
 
 Sacrificing the symmetry and taking, say, u as our independent 
 variable s, we have u= I and (since u + v=i) v= - 1, so that 
 
 , . 
 
 From the present point of view the ' size ' of R is of no avail ; 
 what interests us is that the vector R is along or collinear with 
 the tangent to the conic (21) at its point M, v. Thus, dropping the 
 irrelevant (and not vanishing) denominator, the required tangent 
 will be given by the vector 
 
 t=(i-u 2 )X-(i-z; 2 )Y. 
 
 At the point X, for instance, where u = i, z> = 0, we have t= -Y, 
 and since, with our choice of the T-line, P = X + Y, 
 
 verifying that the join PX is tangent to the conic in the differential 
 sense of the word. Similarly for PY. 
 
 The explained method of generating conic ranges based upon the 
 vector binomial may be extended to trinomials and so on. If the 
 fixed vectors X, Y, Z are non-coplanar and if the variable f-plane 
 passes always through a fixed centre P, the locus of the end-point of 
 the vector 
 
 will be a certain surface. As an exercise let the reader develop 
 this equation. If the fixed 7"-plane is so chosen that P=X-f Y + Z, 
 the result should be 
 
 _ _ (u + vw)'K + (v + ivu) Y + (w + uv)Z 
 
 I -t- 2UVW -VW- WU - UV 
 
 * In fact, although we have drawn our conic point by point, using a 
 discrete series ol rays, marking their crosses, etc., yet we can imagine a 
 mechanism consisting of appropriately arranged threads (in tension), by 
 means of which a certain point draws the curve continuously. This by no 
 means requires the use of rigid segments. 
 
62 PROJECTIVE VECTOR ALGEBRA 
 
 where u + v + w=*l. The algebraic and graphic investigation of 
 this locus may be left to the reader. It will be noticed that 
 ttX + 0Y + wZ is the vector whose end-point is the cross of OR with 
 the plane XYZ. Other and more complicated examples of this 
 kind of vector equations can readily be constructed and investi- 
 gated. None of these, however, will be of such fundamental 
 geometric interest as the binomial one covering all point-conies. 
 
 Returning once more to the conic-equation, say, (20), multiply 
 both its members by any scalar a ; then, by the distributive pro- 
 perty of scalar multiplication and putting R'=o-R, 
 
 J(,- P )(.X) + ,(|- B )(.Y) M+B ^ ^ 
 
 fr-uv 
 
 Thus the locus of R', the end-point of R', is again a conic, passing 
 through and X', Y', the end-points of o-X, o-Y, and such (that 
 P=X + >?Y and) that PX' and PY' are tangents. This conic, 
 (20'), may be shortly called the conic (20) ' magnified ' o- times, 
 with fixed. Of course, the character and the appearance of the 
 original conic can be entirely modified by such a ' magnification ' ; 
 this will depend essentially upon the position of the conventionally 
 fixed TMine. The reader will find the careful graphical treatment 
 of such magnifications to be instructive as well as interesting. 
 If OP is varied we have in (20') a simple infinity (oo l ), or a pencil 
 of conies, all passing through 0. If the T-line is drawn across the 
 original conic, i.e. any one of the pencil of conies, cutting it in two 
 points L, M, then (OL, OM playing the part of infinite vectors) 
 the whole pencil of conies will obviously pass through these points 
 as well as through the origin. If L coincides with M, or the T-line 
 is tangent to one of the conies, it will also touch all the conies 
 of the pencil in the same point. If some of the R of the original 
 conic are negative vectors, the corresponding 2R or 2-5 or 3R, 
 etc., according to the position of 0, may already have their 
 end-points beyond the T-plane, although the original drawing 
 has been all on ' this-side ' of T. Thus branches of some of the 
 conies of the pencil will be found to lie ' that-side ' of T. Exercises 
 illustrating these and similar remarks are left to the care of the 
 reader. 
 
 In conclusion, let us dwell a moment upon the line-conic (the 
 plane-dual of the previous point-conic), which is formed by the 
 joins of correlated points of two coplanar protective (but not per- 
 
CONIC PENCILS 63 
 
 spective) point ranges. The joins are the tangents of the ulti- 
 mately resulting conic which, in familiar language, is their envelope. 
 Let a, b be two fixed non-collinear vectors coinitial in 0, and let 
 us take one of the said joins or tangents as our TMine cutting the 
 a and b lines in T a , TI (Fig. 27). Then, if u, v be two numbers 
 
 FIG. 27. 
 
 satisfying the condition M-h v= 1, the whole pencil of the generating 
 joins or, as it is commonly called, the conic pencil, can, with R = OR, 
 be conveniently represented by 
 
 
 24) 
 
 where c is a constant, non-vanishing finite scalar number, defining 
 upon the a- and b-lines some fixed points A and B respectively. 
 If, say, c = 7, then one of our straights will be the join of the points 
 I and 6, another the join of 2 and 5, or 2\ and 4j, or 10 and - 3, 
 and so on, and the curve in question will be the envelope of all 
 these joins, its tangents. For m=c the equation (24) represents 
 the a-line itself, and for n=c, the b-line, so that these two lines 
 will also be tangents to the curve, to wit in A and B, as will be 
 seen presently. To w = oo corresponds n = - oo , so that the 
 TMine is itself a tangent, as required (and even a double tangent 
 inasmuch as it reappears asm=-oo,n=+oo). Thus, the data 
 upon which this construction is based are : three tangents, a, b, T t 
 and two contact-points on two of them, A and B. These data are 
 precisely the dual of our previous point-conic data,* and, as is 
 
 * Viz. three points of the conic and the tangents at two of them. 
 
$4 PROJECTIVE VECTOR ALGEBRA 
 
 well known, completely determine the conic. Moreover, since 
 all of these five data are at our disposal, every line-conic can be 
 represented by the simple manifold of vector equations (24). 
 
 It remains to show that the points, m on a and n on b, which 
 are being joined with one another, are correlated points of pro- 
 jective ranges. Now, the cross ratio of any tetrad of points on the 
 a-line, bearing the numbers, m lt m z , m 3 , w 4 , say, is 
 
 m, - m m z - m t 
 
 = - - - - * 
 
 and since n 1 -n a = m s -m 1 , etc., we see at once that 
 
 (i, 3 M 4) = (*j.w t , m 3 mj, (25) 
 
 which proves the statement. We might as well have taken 
 the formally more general relation n = ^ (aS-/3y=o), which 
 
 would also give (25). But this would, with some elements fixed, 
 entail only a different numbering of the points of the a- and b-lines. 
 As far as the geometrical contents are concerned, the adopted 
 relation of the parameters, tw + w = const, is as general as the last- 
 mentioned one, and it has the advantage of formal simplicity.* 
 
 All questions concerning a line-conic can be treated algebraically 
 with the aid of (24). There is no need of entering upon them here. 
 The discussion of the character of the conic in its dependence on 
 the position of A, B, and so on, may be left to the care of the reader, 
 as a useful exercise. It has seemed advisable, however, to insert 
 at least one illustrative diagram (Fig. 28), about which a few 
 remarks will be made presently. 
 
 If the reader desires to derive from the set (24) of vector equations 
 the coordinate equation of the conic as the envelope, he may 
 proceed in the following manner: Writing R = #a+yb, so that 
 x mu and y = nv will be the coordinates of R, and 
 
 / = * + y_ I=0 , (a) 
 
 J m n 
 
 * The choice of the particular form m + n=c in connection with (24) 
 was suggested by the natural attempt at a generalization of the well-familiar 
 construction of the parabola, given in all elementary text-books of a metrical 
 or semi-metrical character. See, for instance, Dr. K. Doehlemann's small but 
 very good book, Projektive Geometric, Leipzig, 1905 (Sammlung GSschen), 
 P- 139- 
 
the coordinate equation of a generating line, let the reader 
 differentiate it with respect to the parameter m, obtaining 
 
 15f x y dn y x 
 
 "dm 
 
 ,,v 
 
 Then, eliminating according to the well-known prescription the 
 parameter m between (a) and (b), the result will be, 
 
 (x-y) 2 -2c(x+y) + c'* = o, (26) 
 
 the equation of the envelope of the said joins. For y = o, for 
 instance, we have, according to this equation, (#-c) 2 = O, i.e. x = c, 
 
 FIG. 18. 
 
 and we thus see that A is a point of the curve or a ' contact-point ' ; 
 similarly for B (y = c, x = o), as was announced above. It may be 
 worth while to notice that (26) can be further simplified by 
 introducing new coordinates, 
 
 The equation then becomes 
 
 y'* = c( X '-\c). (26') 
 
 The corresponding axes and, at the same time, ' unit ' vectors, 
 replacing a, b, are 
 
 a' = a + b, V = a - b, 
 
 and can be easily constructed. 
 
 Finally, but a few more words on our last illustrative diagram. 
 It is not difficult to see that if c be positive, and the points A, B 
 P.V.A. E 
 
66 PROJECTIVE VECTOR ALGEBRA 
 
 bearing this number are both on the segments OT a , OTi, we have 
 an ellipse, inscribed into and touching the triangle OT a Ti>. Fig. 28, 
 which is precisely of this type, has been drawn by the prescription 
 (24), for c = 7, the ' units ' on OT a , OTi having actually been chosen 
 in an arbitrary manner, and then the scales, 2, 3, etc., set up corre- 
 spondingly in the manner explained in the earlier part of the book. 
 Of course, the " c = J " is no geometrical reality ; we might as well 
 have written c 50, when the scale divisions would have obtained 
 only different numbers. The triangle OT a Ti being drawn and, 
 upon its two sides, A and B marked (as the required contact-points), 
 the conic is herewith uniquely determined, no matter which number 
 c (not zero) is affixed to A and B. Its choice is only a matter 
 of convenience or graphic expediency, while the conic is a real 
 content uniquely determined by the said real, as distinguished 
 from formal, data. 
 
 The familiar construction of the parabola, alluded to a moment 
 ago, is a special case of Fig. 28, as representative of (24), to wit 
 when our TMine is moved away to ' infinity,' in which case the 
 steps 01, 12, etc., become ' rigidly ' equal. The technical expres- 
 sion, used in text-books, according to which the parabola has with 
 ' the line at infinity a (doubly counting) point of contact,' assumes 
 in the presentation given above a clearer meaning. For our 
 conic always touches the TMine which appears as a double tangent 
 (w=oo,w=+oo), no matter whether this line is 'at infinity' 
 or near at hand. The parabola thus viewed is, of course, but a 
 special case of the ellipse. 
 
APPENDIX 
 
 A. On the non-metrical sum of angles of a triangle. The 
 reader will have remarked the absence of any angle concept 
 in what precedes. In fact, no such concept is needed for the 
 treatment of any question that may present itself in the whole 
 field of our Vector Algebra. No use therefore has been made of 
 ' angles ' in the course of the preceding deductions. 
 
 But, although by no means indispensable for the purpose of 
 this book, the concept of an angle, and especially of a non-metrical 
 equality of two angles, has some importance on its own account, 
 sufficient to justify its discussion in an Appendix. 
 
 It is not necessary to enunciate here a formal definition of an 
 angle. Again, the generalization to three dimensions being obvious, 
 it will be enough to speak of angles formed by coplanar rays or 
 sides, in short, of coplanar angles. 
 
 If a, b be any vectors along the sides of an angle aOb, we can 
 denote this angle by (a, b). If <r, r are any positive scalar numbers, 
 (o-a, <rb) will stand for the same angle. The straight angle (familiar 
 from metrics as ' two right angles ') will then be any angle of the 
 type (a, -a), or aOb if the points a, 0, b are collinear. Let us 
 declare all straight angles to be equal among one another, and let 
 us define aOc as the sum of any two covertical and adjacent angles 
 aOb and bOc, provided that Oc falls outside aOb* Then the 
 equality of vertical angles, in short symbols, (a, b) =(-a, -b), will 
 follow in the well-known way. 
 
 Now, let (a, b) and (a', b') be two angles with distinct vertices 
 and 0'. It is then, from the standpoint of the present book, in 
 general meaningless to speak of these two angles as equal or not. 
 Suppose, however, that, apart from irrelevant positive numerical 
 factors, we have, with respect to a conventionally fixed TMine, 
 a' = a and b'b. 
 
 * We shall not require the concept of the sum of any other angles. 
 
 67 
 
68 PROJECTIVE VECTOR ALGEBRA 
 
 Then the idea naturally suggests itself to define such two angles as 
 equal to one another, and to write 
 
 (a', b')=(a,b), 
 
 always under the tacit clause ' with respect to the chosen TMine.' 
 
 Remembering the definition of equality of non-coinitial vectors 
 adopted in Section 8, the above definition will at once be seen to 
 amount to the following one : Two angles, whose vertices lie (in 
 the restricted region) on the same side of the TMine, and whose 
 sides intersect, in pairs, on the T-line, are equal to one another. 
 In short symbols, if T a , T b coincide with T a ,, TV, then (a, b) = (a', V). 
 Thus, for instance, AOT', T A AT', T A A'T in Fig. 10 (Section 8) 
 will all be equal angles ; the opposite angles of the quadrilateral 
 AOO'A' will be equal to one another. It is scarcely necessary to 
 say that this equality definition of angles, puzzling as it may seem 
 at first, is but an obvious generalization of the familiar euclidean 
 equality of angles formed by pairs of parallel sides. And, as has 
 been noted on more than one occasion, the r61e of parallels is here 
 taken over by straight lines pointing to the same T-point. In 
 harmony with what has just been said, and if we agree to con- 
 sider (a, a) as the angle zero, any angle whose vertex is on the 
 T-line, such as that formed by OA and O'A' in Fig. 10, will be 
 zero. 
 
 Besides the explained, particular case of angle equality, we have 
 no means, and at the same time no need, of defining any other 
 pairs of angles as equal or different. Under such circumstances 
 the angle concept, supported only by an equality definition which 
 covers but a special class of pairs of angles, cannot reasonably 
 be expected to yield very much. Yet it has, to my great surprise, 
 turned out to be competent to give one strikingly simple result 
 concerning the sum of angles of a triangle. And, as a matter of 
 fact, this was the only reason which has induced me to mention 
 the otherwise superfluous angle concept in the present book. 
 
 In fact, let ABC, Fig. 29, be any (rectilinear) triangle, and a, (3, y, 
 shorts for CAB, etc., its three angles. Choose any straight 123 
 not traversing the triangle as the T'-line, and prolong the sides 
 BC, AC, AB, obtaining the corresponding termini or T-points, I, 2, 
 and 3. Then iB2, as coterminal with the angle \C2, will be seen to 
 be equal to y, and similarly 2.63 will be equal to a. Thus y, a, /3 
 
ANGLE SUM 69 
 
 are all gathered at B as their common vertex, and we see at a 
 glance that 
 
 a + f3 + y = straight angle. 
 
 In words, the non-metrical sum of the angles of a triangle is equal 
 to a straight angle, with reference to any TMine not traversing the 
 triangle,* and not forming one of its sides. 
 
 The result is thus equally sounding with the familiar euclidean 
 theorem. None the less, the two theorems must not be confounded 
 
 I 
 
 with another, their contents being in reality entirely different. 
 The ordinary, metrical sum of a, ft, y is equal to a straight angle 
 only when space is euclidean ; it is smaller for hyperbolic or 
 lobatschewskyan, and greater for elliptic spaces. Our present, 
 non-metrical angle sum is always equal to a straight angle, 
 independently, that is, of the special nature of the contemplated 
 
 * If the reader redraws Fig. 29, with the T-line across the triangle, so 
 that, caeteris paribus, the point i will fall within the side BC, and 2 within 
 AC, he will readily find that, in order to preserve the above theorem, the 
 sign of the angle 7 = (2, i) is to be reversed in throwing it, so to speak, to 
 the other side of T so as to gather all three angles at B as common vertex. 
 Such a change of sign could well be introduced as a further convention. 
 We prefer, however, to accept the restriction of T-lines not traversing the 
 triangle. The symbol (2, i), just used, is an obvious short for 'terminating 
 in 2 and i. 1 
 
70 PROJECTIVE VECTOR ALGEBRA 
 
 space, be it hyperbolic, parabolic or elliptic. For it is entirely 
 independent not only of Euclid's special axiom of parallels, but 
 also of any notion of angle congruence. On the other hand, it is 
 particularly instructive to compare this non-metrical theorem with 
 Euclid's theorem. In fact, if the contemplated space is euclidean, 
 not only our reasoning but the very drawing (Fig. 29) can be used 
 to find the metrical angle sum, provided that our TMine stands 
 for the line at infinity, when B2 becomes parallel to AC, and so 
 on, as in the classical construction. From this point of view, the 
 above theorem can be considered as a generalization of that 
 embodied in Euclid. It is interesting to note that, although the 
 angles 2.83 and iB2 are, separately, not metrically equal to a and y, 
 yet their sum is, both metrically (in Euclid's case) and non- 
 metrically, equal to a + y. 
 
 If one of the vertices, say C, is on the jT-line, then 7 = 0, and 
 the external angle at B becomes CB$ = CA^, that is to say, equal 
 to a, so that a + ft = straight angle, in accordance with the general 
 theorem. 
 
 Any further explanations are scarcely necessary. But, to 
 close this subject, the following remark does not seem out of place. 
 Suppose that the astronomers are somehow deprived of their 
 divided circles and other sanctioned rigid angle transferers. Then, 
 if one of them, say at Greenwich, directs the axes a and b of two 
 telescopes towards some well-defined pair of ' fixed ' stars I, 2, such 
 as a and ft Ursae Majoris, the ' Pointers ' of the Great Bear, and 
 another astronomer, at Paris or Pulkovo or where not, directs his 
 two telescopes a', b', to the same stars, they will readily accept 
 the two angles (a, b) and (a', V) as equal, without any fear of 
 infringing against the metrical criterion of their equality. Nor 
 would (in the case of the great majority of pairs of stars) any detect- 
 able discrepancy arise even if the vertices of the two angles were 
 separated from one another by the whole diameter of the Earth's 
 orbit. Now, such procedure would be entirely in conformity with 
 the proposed non-metrical definition of angle equality, our co-T 
 or co-terminal angles being here exemplified by what might be 
 called ' co-stellar ' angles. These astronomers would then readily 
 discover the theorem a + /3 + y = straight angle. It would even be 
 immaterial to have a straight as the TMine (or a plane as !T-surface) ; 
 any appropriately selected three stars, collinear or not, could be 
 
LINEAR OPERATORS 71 
 
 utilised as the substitutes of the termini I, 2, 3 of Figure 29. A 
 theorem on the angle sum thus detected and enunciated would 
 certainly be declared, by its fictitious discoverers, to be independent 
 of any parallel-postulate. It is even thinkable that, if such were 
 the historical course of events, the astronomers would prefer the 
 stellar criterion of angle equality to that based on the best available 
 rigid-scale goniometers. Phenomena now associated with the 
 name of ' parallax ' would only come to the knowledge of astro- 
 nomers gradually, revealing themselves by the more or less con- 
 siderable deviation of the metrical angle sum from a straight angle, 
 according to which celestial objects were chosen as termini. This 
 would soon lead to a corresponding classification of these luminous 
 points. 
 
 B. Linear vector operators. Projective collineations. A linear 
 (homogeneous) vector operator C7 can be defined, in much the same 
 way as in ordinary vector algebra, as a continuous distributive 
 operator. Thus, if r and s be any two vectors, 
 
 nr(r + s) = err + crs, 
 
 the sum being taken, on both sides of this equation, with respect 
 to any conventionally fixed T-plane. 
 
 If r be any vector-operand, r'=C7r will be called a homogeneous 
 linear vector function of r, it being tacitly understood that cro = o, 
 if o stands for the nil-vector. But the attribute ' homogeneous ' 
 may henceforth be dropped. We shall also assume that if r is an 
 infinite or T-vector, so is r'. 
 
 It follows at once from the above definition that cr(2r) = 2STr, 
 and so on, more generally 
 
 where o-, T, etc., are any scalar numbers. Whence it follows 
 that an operator CT is completely determined if it is given for any 
 three non-coplanar vectors a, b, c, that is to say, if the three 
 particular vectors 
 
 a' = na, b' = srb, c' = ETC 
 
 are prescribed. For every vector r can be represented as 
 
 r = x& +.yb + zc, 
 where x, y, z are some scalars, the coordinates of r's end-point 
 
72 
 
 with respect to some fixed T-plane and origin 0. The said three 
 vector-data can themselves be expanded, thus 
 
 era = w n a + w 12 b + o > 13 c, 
 sib = o> 21 a + to^b + t 23 c, 
 CTC = <o 31 a + <o 32 b + 33 c, 
 
 where w tK , generally i=<o Kl1 are ordinary numbers. Thus a given 
 vector operator implies, in general, nine independent scalar data. 
 
 The ' product,' or the successive application of two linear vector 
 operators is again some linear vector operator. For if CTj and C7 2 
 be such operators, we have, by the definition of & lt 
 
 tn^r + s) = 
 and by the definition of t? 2 , 
 
 so that CTgCTj is again some operator ST. Notice that, in general, 
 STjCTg will not be the same operator as E^CTj. 
 
 In particular, iterations or successive applications of one and 
 the same operator to, which will be written shortly 
 
 STGT = CT 2 , CTGrar = trr 3 , etc., 
 
 will again be linear vector operators, in general entirely different 
 from one another and from or itself. An operator which leaves 
 any operand r intact will be called an identical operator, or, after 
 Gibbs, an idem} actor. It can be simply denoted by I, so that, for 
 example, CT = i will stand for 
 
 OT = r, for every r. 
 
 The same symbolism can be used for any operator, not necessarily 
 a linear one. 
 
 As in ordinary vector algebra, the principal axes of a linear 
 vector operator T can be defined as those lines or ' directions ' 
 of r, for which 
 
 CTr = a- . r, 
 
 o- being some scalar number, which may be called the corresponding 
 ' principal value ' of ST. It is interesting to note that, notwith- 
 standing the non-metrical character of our vectors and conse- 
 quently also of or, the cardinal property belonging to metrical 
 vector operators continues to hold, inasmuch as our trr can at the 
 
73 
 
 utmost have three principal axes with different principal values, 
 which, moreover, cannot be coplanar. In fact if, say, i and j 
 are two principal axes such that 
 
 5*1 = 0^1, t3j=o-2J, and o-j^o-.j, 
 
 then if any vector r=xi + yj, coplanar with i, j, represented a third 
 axis, we should have OT = cr(;d + yj), and, on the other hand, 
 CTr = o-jtfi + o- 2 jj, which would require o- to be equal to both cr l and <r 2 , 
 contradicting the assumption. Thus, if o-j^ <r 2 , there is no third 
 principal axis coplanar with i, j. (And, if o-j=o- 2 , then every line 
 in that plane is a principal axis with the same principal value.) 
 But it is possible to have a third principal axis non-coplanar with 
 i, j, the first two axes. Let this be k, and let 
 
 trrk = o-gk. 
 
 Then, if r were any fourth principal axis whatever, we could repre- 
 sent it by xi+yj + zk, and if <r were the corresponding principal 
 value, we should have 
 
 <r(xi +yj + zk) = xvj. +yo-.j + zo- 3 k, 
 
 which is only possible for cr = 0^ = 0-2 = 0-3. Thus either <r lt o- 2 , o- & 
 are all different, when no fourth principal axis is possible, or two 
 of these values are equal, when there is a whole flat pencil of 
 principal axes, or finally 0^=0-2 = 0-3, when every line is such an 
 axis. In the latter case, the operator GT degenerates into an 
 ordinary factor <r, a stretcher or ' magnifier,' which has already 
 been mentioned (cf. p. 62), and in particular, for <r=i, an idem- 
 factor. From what has just been said it does not follow, of course, 
 that every operator or has even as many as three real principal 
 axes. It may have but one such axis and two complex ones, 
 notwithstanding that all constituents ta lK are real. Only if 
 w,=w )Cl does the operator or, then called 'symmetrical, 1 always 
 possess three real axes. (The question of their ' orthogonality ' 
 is, of course, entirely foreign to our circle of ideas.) 
 
 The subject of the principal axes of CT has been mentioned 
 here only for the sake of comparison with the familiar and most 
 fundamental behaviour of metrical linear vector operators. But 
 the chief interest of the operator CT, relating to projective geometry, 
 seems to lie in a different direction. 
 
 In fact, if the homogeneous linear function CTr is slightly ampli- 
 
74 PROJECTIVE VECTOR ALGEBRA 
 
 fied, viz. made non-homogeneous by adding to it a constant 
 vector p, it can be used at once to represent any projective collinea- 
 tion of space. To be more explicit, let r represent, through its end- 
 point, any point of space, and r' or flr (with fl as symbol of an 
 operator) the corresponding point. 
 
 Then, if the T-plane * is transformed into itself, any projective 
 collinealion is represented by 
 
 r' = ftr = err + p, 
 
 provided that the determinant of cr does not vanish. This is but 
 a shortened expression of a well-known theorem of projective 
 geometry. (See, for instance, Veblen and Young's work, vol. i. 
 p. 201.) 
 
 The whole operator fl, which may be called a collineator, is, 
 of course, in general not distributive. It consists of the operation 
 CT, as explained before, followed by the addition to all r's of a 
 constant vector p or by a pure translation. In fact, the addition 
 of one and the same vector p to any number of given vectors, is 
 readily seen to be the non-metrical generalization of the usual 
 4 translation ' of the figure determined by the end-points of these 
 vectors. By what has been said in the text of this volume, such 
 a generalized translation of a triangle ABC in its own plane, for 
 example, carries its corners to A', B', C' such that ABC and 
 A'B'C' are triangles in perspective, having a jT-point (the terminus 
 of p) as their point of perspectivity and the jT-line as their line 
 of perspectivity, and similarly for the translation of more com- 
 plicated figures. Such details, however, need not detain us 
 here. The important thing is to note that the projective collinea- 
 tion consists of an operation ra> which is itself distributive, followed 
 by another non-distributive but very simple operation, the pure 
 translation, +p. 
 
 Owing to this circumstance many important questions associated 
 with projective collineations can be very easily treated. In 
 illustration of this we can give here but a few hints. 
 
 Thus, for instance, the successive application of the same pro- 
 jective collineation symbolized by ft* will be given by 
 
 * That is, any plane whatever whose choice completes the conventional 
 framework of reference. 
 
PROJECTIVE COLLINEATIONS 75 
 
 which by the above formula and by the distributivity of GT becomes 
 at once 
 
 This is, manifestly, again a projective collineation with sr 2 as the 
 new linear vector operator, and with the new vector 
 
 as the translation. Similarly 
 
 and so on. 
 
 A collineation is called cyclic, or periodic of period n if, for every r, 
 
 Qr = r, 
 
 n being a positive integer. Thus the equation (with i as idem- 
 factor) of any cyclic collineation will be 
 
 which can also be written 
 
 where the bracketed expression is again a homogeneous linear 
 vector operator. If this is to be satisfied for every r, we must have 
 
 TS n i (idemf actor), 
 
 and if p = 0, we must exclude from all possible roots of this opera- 
 tional equation the particular root TS=I. In short, the required 
 cr will be any root of ET"=I, with the exception of the idemfactor 
 itself. It is scarcely necessary to warn the reader that any such 
 equation as CF l =l stands for nine ordinary equations (in the case 
 of space, and four in the case of a plane), viz. if the constituents 
 of cy are denoted by 
 
 =0, I K, 
 
 and each of these constituents is easily expressible in terms of 
 those of CT itself. 
 
 The simplest and particularly interesting case is that of = 2, 
 when the collineation is involutoric or reflexive, i.e. such that 
 fir = r' and lhr'=r. The equation written above then becomes 
 
76 PROJECTIVE VECTOR ALGEBRA 
 
 Thus, if p=/=o, the necessary condition 
 
 CT= - i 
 
 satisfies also or 2 = I and leaves no other alternative. We thus see 
 that the most general involutoric space-collineation transforming the 
 T-plane into itself, is 
 
 fir = p - r, 
 
 where p is any constant vector. (The assumption p = o would 
 only lead to a sub-case.) 
 
 The only double point of this correspondence, apart from the 
 7-plane, is given by p - r = r, i.e. by r = Jp. Let p, r, r' be the 
 end-points of p, etc., with as origin. Then, in order to obtain 
 the image of any point r, we have, by Section 9, to draw the straight 
 pr crossing the T-plane in T', say ; then, if T be the terminus of r, 
 the straight Tp will cross T'O in the required point r'. A drawing 
 thus executed will be seen to contain at the same time the reciprocal 
 passage from r' to r, in accordance with the involutoric nature of 
 this collineation. 
 
 The investigation of cyclical projective collineations of space or 
 of a plane of period 3, and so forth, may be left to the care of the 
 reader. 
 
INDEX 
 
 Addition, 4 et seq. 
 algebras of segments, i 
 angles, 67-71 
 archimedean axiom, 3 
 associative law, 6 et seq. 
 axes of operator, 72-73 
 axioms of order, etc., 2 
 
 Bisection, 46 
 
 Cayley, 16 
 chain of vectors, 22 
 collinear vectors, 9, 25 
 collineations, 74 
 commutative law, 5 
 conic pencils, 63-66 
 - ranges, 54-62 
 connection, axioms of, 2 
 Coolidge, 12, 19 
 coordinates, projective, 38 
 co-stellar angles, 70 
 cross ratio, 14, 64 
 cyclic collineation, 75 
 
 Desargues' theorem, 2, 8, 49 
 difference of vectors, 27 et seq. 
 
 differentiation , 61 
 
 distributivity, 16, 19, 34 
 Doehlemann, 64 
 
 Elliptic space, 3, 24 
 
 equality of vectors denned, 22 
 
 equation of straight, 40 
 
 of plane, 41 
 of conies, 54 et seq. 
 
 Fourth harmonic, 12 et passim 
 fractional factors, 19 
 
 Gibbs, 72 
 
 Halsted, 37 
 
 harmonic ranges, 17, 60 
 
 Hilbert, i, 50 
 
 hyperbolic space, 29, 32, 34 
 
 Ideal points, 34-37 
 idemfactor, 72 
 
 involutoric collineation, 75-76 
 irrational factors, 19 
 
 Killing, W., 19 
 
 Line conic, 62 
 
 linear vector operators, 71-73 
 
 Lobatchevsky's parallels, 32, 33, 35 
 
 Magnification, 44, 62, 73 
 Mathews, 3, 25 
 medians of a triangle, 48 
 mid-point, 39 
 motion, axioms of, i 
 multiple of a vector, 12, 13 
 
 Negative, of a vector, 29, 31 
 nil- vector, 26-27 
 
 Operators, linear vector-, 71-73 
 order, axioms of, 2 
 
 Parallax, 71 
 
 Pascal's theorem, 49-52 
 
 polar and pole, 55, 60 
 
 principal axes, 72 
 
 projective collineations, 74-76 
 
 ranges, 62, 64 
 prospectivity, i 
 
 Quadrangle, quadrilateral, 41, 45, 46 
 
 Reflexive collineation, 75 
 rigid transferors, 13 
 
 77 
 
78 
 
 Schur, i, 2, 35, 36, 50 
 
 Sommerville, 25 
 
 Staudt, i 
 
 submultiple of a vector, 18-21 
 
 subtraction, 27 
 
 sum of vectors, 5 et passim 
 
 T-plane, 6 
 T- vectors, 15 
 tangents, 55, 61 
 
 INDEX 
 
 termini, 5, 34 
 tetrahedron, 49 
 translation, 74 
 triangle, medians of, 47-49 
 sum of angles, 69 
 
 Veblen and Young, 38, 74 
 
 vector equality, 22 
 
 vector operator, linear, 71 et seq. 
 
 GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. 
 
BELL'S MATHEMATICAL SERIES 
 
 FOR SCHOOLS AND COLLEGES. 
 
 General Editor : WILLIAM P. MILNE, M.A., D.Sc. 
 
 A FIRST COURSE IN THE CALCULUS. By W. P. MILNE, M.A., D.Sc., 
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