Zlbe Uaniverstt? of Cbtca0o FOUNDED BY JOHN D. ROCKEFELLER A SET OF POSTULATES FOR GENERAL PROJECTIVE GEOMETRY A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY MEYER GRUPP GABA PRESS O0 THE NEW ERA PRINTING COMPANY LANCASTER, PA. 1915 A SET OF POSTULATES FOR GENERAL PROJECTIVE GEOMETRY* BY MEYER G. GABA Since Klein promulgated his famous Erlangen Programmet it has been known that the various types of geometry are such tbat each is characterized by a group of transformations. In view of the importance of the concept of transformation in nearly all mathematics and perhaps especially in geometry, geometers may properly seek to develop the various types of geometry in terms of point and transformation. For euclidean geometry this has been done by Pieri. t This paper is devoted to a similar treatment of general projective geometry.~ One would naturally lay such postulates on the system of transformations so as to make the system form the group associated with the geometry. This was the scheme that Pieri used. His postulates make his transformations form the group of motions. In general projective geometry, however, this method is not necessary. If we are given the group of all projective transformations we can deduce the geometry from it but it will be shown in the sequel that we can also do that from a properly chosen semi-group belonging to that group. Our basis, to repeat, is a class of undefined elements called points and a class of undefined functions on point to point | or transformations called collineations. For notation we will use small Roman letters to designate * Read before the American Mathematical Society, April 26, 1913. t F. Klein, Vergleichende Betrachtungen uber neuere geometrische Forschungen, Erlangen, 1872. English translation by M. W. Haskell, B u ll e t i n o f t h e A m e r i c a n M a t h e - matical Society, vol. 2 (1893). t M. Pieri, Della geometria elementare come sistema ipotetico-deduttivo; monografia del punto edelmote, Memorie delle Scienze di Torino (1899). ~ General projective geometry is defined by Veblen and Young as a geometry associated (analytically) with a general number field; that is, its theorems are valid, not alone in the ordinary real and the ordinary complex projective spaces but also in the ordinary rational spaces and in the finite spaces. This paper connects closely with the postulates for general projective geometry given by O. Veblen and J. W. Young in the A m e r i c a n J o u r n a o f M a t h e m a t i c s, vol. 30 (1908), and in their Projective Geometry, Ginn and Company (1910). Il By point to point is meant that to every point p there corresponds a single point p' and no point p' is the correspondent of two distinct points pi and p2. 51 52 M. G. GABA: [January points and small Greek letters for collineations. Thus T(pl, P2, p3) = pl p2, p, 3 means that the collineation r transforms the points pi, p2, ps into pl, p2, p3 respectively. Line will be defined in terms of points and collineations. If we should interpret our undefined collineations as the group of all projective collineations, our defined line will satisfy the Veblen-Young assumptions for their undefined line and the postulates I to VIII that we will soon give are theorems in general projective geometry. This proves the consistency of our postulates. On the other hand, leaving collineation as undefined and using postulates I to VI* we are able to prove as theorems the Veblen-Young assumptions A1, A2, A3, Eo, Ei, E2, ***, E, En,, and P.t This shows that our six postulates are sufficient to establish the general projective geometry of n-dimensions. The undefined collineation will be proven to be a projective collineation which justifies the notation. If we desire that our class of collineations should be the group of all projective collineations we add postulate VII to the preceding six. To the Veblen-Young postulate Ho corresponds our postulate VIII. The independence of our assumptions is proven by the set of independence examples given at the end of this paper. POSTULATE I. There are at least n + 2 distinct points. POSTULATE II. If ri is a collineation and r2 is a collineation then the resultant of operating first with ri and then with r2 (in notation T2 n7) is also a collineation. DEFINITION. A linear set is a class of points sucb that: (a) every collineation that leaves two distinct points of the class invariant leaves the class invariant, (b) every collineation that leaves three distinct points of the class invariant leaves every point of the class invariant. DEFINITION. Points belonging to the same linear set are called collinear. DEFINITION. A linear set that contains at least three distinct points is called a line.$ POSTULATE III. If p, p2, p3 are three distinct collinear points and p', p2, p3 are three distinct collinear points then a collineation exists that transforms pi, p2, p3 into p', p2, p3 respectively. POSTULATE IV. If pl, p2, p3, P4 are four distinct points such that no three * Postulates I to VI explicitly require that our set of collineations form a semi-group but all sets of transformations that we found satisfying I-VI were groups. The question whether they necessarily form a group in the general case has not as yet been proven. t The precise statement of these postulates is given later in this paper. t It will be proved that every pair of distinct points is contained in one and only one line. 1915] POSTULATES FOR PROJECTIVE GEOMETRY 53 are such that each is collinear with the same two distinct points then a collineation exists leaving pi and p2 invariant and interchanging p3 and p4.* DEFINITION. If pi, p2 are two distinct points, a line containing pi and p2 or, in case no line exists containing pi and p2, then the pair of points pi, p2 is called a one-space (P1) containing pi, p2. DEFINITION. If Pk-1 is a (k - l)-space and po is a point not contained in Pk-1, the class Pk - [Pk-1, o] of all points p collinear with the point po and some point of Pk- is called the k-space determined by Pk-1 and po. DEFINITION. k points are called independent of each other if there exists no (k - 2)-space that contains them all. POSTULATE V. If pi, p2, ''*, pn+i are n + 1 distinct points of the same k-space, k < n, then a collineation distinct from the identity exists leaving pi, p2, ', P n+i invariant. t POSTULATE VI. If pl, P2, *, pn+2 are n + 2 distinct points then there exists a k-space, k n, that contains them ail. POSTULATE VII. If pi, p2, * *, pn+2 are n + 2 points of the same n-space such that every n + 1 are independent, and p', p2, ' ", n,+2 are n + 2 points of the same n-space such that every n + 1 are independent then a collineation exists that transforms pi, p2, '., pn+2 into pi, p,,...* pl+2 respectively. DEFINITION. A complete quadrangle is a figure consisting of four distinct coplanar points such that no three are collinear, called its vertices and six distinct lines containing the vertices in pairs called its sides. Two sides having no vertex in common are called opposite and points common to two opposite sides are called diagonal points. POSTULATE VIII. The diagonal points of a complete quadrangle are noncollinear. THEOREM 1. If pi and p2 are distinct points, there is not more than one line containing both pi and p2. Let us assume that two lines P and P' exist such that each contains pl and p2. If the two lines are distinct then at least one of the lines must contain a point not in the other. Let us assume that pS is a point of P' and not of P. Since P is a line it contains in addition to pi and p2 a third point ps. From Postulate III we know that a collineation r exists such that (pi, p2, p ) -= p, p2, p3/ But T leaves two points of P invariant, therefore it leaves P invariant and P3 cannot be transformed into p' which is a point not of P. Since we are led to a contradiction our assumption that pi and p2 are contained in two distinct lines must be false. THEOREM 2. Two distinct lines cannot have more than one common point. * Compare Postulate IV with Theorem 4. Postulate IV is weaker than Theorem 4 as independence example - IV will show. t Postulate V with Theorem 14 shows that all points lie in no Pk, k < n. 54 M. G. GABA: [January THEOREM 3. If three points are such that each is collinear with the same two points, they are collinear and conversely. THEOREM 4. If pi, p2, P3, p4 are four distinct points such that no three are collinear then a collineation r exists such that r (pi, 2, p3, p4 ) = pl, p2, p4, P3 The theorem follows from Postulate IV and Theorem 3. THEOREM 5. If pi, p2 are two distinct points and qi, q2 are two distinct points then a collineation r exists such that r (pi, p2 ) = ql, q2. If pi, p2 are on a line and qi, q2 are cn a line, the theorem follows from Postulate III. If no three of the points pi, p2, ql, q2 are collinear and they are all distinct then by Theorem 4 we know that a collineation ri exists such that i (pl i,, p2, q2) = pi, qi, q2, p2 and a collineation r2 exists such that T2 (pi, qi, q2, p2) = qi, pl, q2, p2. Therefore by Postulate II a collineation 72 ri exists such that 72 1i (pi, p2 ) = ql, 2. There are, notation apart, two possible cases remaining which are: (1) q2 collinear with pi and p2 but qi, q2 on no line; (2) p2 collinear with qi and q2, but pi, p2 on no line. Let us first suppose that a point r exists such that r is non-collinear with every two of the points pi, p2, qi, q2. In the first case, collineations 73 and 74 exist such that 73 (pi, p2, r, qi) = qi, p2, r, pi and r4 ( ql, p2, r, q2 ) = qi, 2, r, p2, and therefore 74 r3 is the required collineation. In the second case, collineations Tr and 76 exist such that r5 (pi, p2, r, q2) = pi, q2, r, p2 and 76(pl, q2, r, qi) = qi, q2, r, pi and in this case T6 T7 is a collineation that transforms pi, P2 into qi, q2 respectively. If no such point r exists then every point is contained in some one of the one-spaces determined by two of the four points pi, p2, qi, q2. Let us consider the case where pi and p2 are on no line and where pi is collinear with ql q2 and let us further suppose that pi is distinct from qi and from q2. All points are in the two-space or plane determined by the line ql q2 and the point p2 is not in the line qi q2. If n = 2 then by Postulate V a collineation not the identity exists having pi, qi, q2 as invariant points and there must therefore be at least one additional point ri on the one-space qi p2 or one additional point si on the one-space q2 p2. If r1 exists then collineations 77 and 78 exist such that 77 (pl, p2, r, q2) = pl q2, ri, p2 and r (qi, q2, pi) = Pi, Q2 qqi If si exists we have collineations T9 and Tro such that T9 (pi, p2, s1, qi) = pi, qi, si, p2 and o0 (ql q2, pI) = q2, PI, ql Hence 78 77 or r10 r9 will be the required collineation according as ri or si exists. IY n > 2 there are by Postulate I at least n + 2 > 4 distinct points; hence the additional point ri or si exists as before, and the argument is completed as in the case n = 2. Let us now suppose tbat pi coincides with qi. Since n > 2, at least four 1915] POSTULATES FOR PROJECTIVE GEOMETRY 55 distinct points exist. Let us assume q3, distinct from qi and q2, exists on the one-space qi q2. A collineation distinct from the identity exists leaving invariant qi, q2, q3 if n = 2 and, if n > 2, qi, q2, q3, and n - 2 other points; therefore a point r1 distinct from q2, p2 must lie on the one-space q2 p2. If the existence of rl had been assumed, then we could have proven in a similar manner that q3 existed and since one or the other must exist, both exist. Since the four points pi = q1, p2, ri, q3 are such that no three are collinear a collineation ri exists such that r11 (pi, p2, r1, q3) = pl, q3, ri, p2 and a collineation 712 exists such that 712 (pi, q2, q3) = ql, q3, q 2. The collineation 712 711 is the collineation that transforms pi p2 into qi q2. For all the other possible cases the proofs are very similar to the preceding and therefore need not be repeated. THEOREM 6. A ine exists. If all linear sets contained but two points there would be but k + 1 points in a k-space. This would make Postulates I and VI contradictory and therefore at least one linear set contains more than two points and hence a line exists. THEOREM 7. Every collineation transforms lines into lines. Let 71 be any collineation that transforms the line P = [p] into a set of points Q = [q]. We are to prove that the set of points Q constitute a linear set. Let 72 be any collineation that leaves two of the q's, say qi and q2 invariant. The points qi and q2 are the transforms under 71 of two points of P which we will call pi and p2. By Theorem 5 there is a collineation 73 that transforms qi, q2 into pi, p2. Then r3 7r2 r (P) = P since the points pl and p2 of P are left invariant. For the same reason 73 71 (P) = P. Therefore 73 (Q) = P, that is to say every p is the transform under 73 of some q and that for every q 3 ( q ) is a p. But 3 r2 ( Q ) = P therefore 72 ( Q ) = Q. Hence any collineation that leaves two points of Q fixed leaves Q invariant. Let T4 be any collineation that leaves three of the q's invariant, say qi, q2, q3, where qi, q2, q3 are the transforms under r7 of pl, p2, p3 respectively. 737471 [P] = P since pi and p2 are left invariant. 73 74 r7(pi, p2, p3) = p, 2, p p. Since pi, p2, P3 and pi, p2, ps are sets of collinear points a collineation 75 exists such that 75 (pl, p2, p3) = Pl P2, p3. Then 75 73 74 7( ( P2, P2, P3) = i, p2, p3 and T5 73 r(pl, P2, P3) = pl, P2, P3. Therefore the collineations 75 73 ri and 75 73 r4 ri leave every point of P invariant. If ri transforms pi into qi, 75 73 must transform qi into pi and hence 74 must leave every point of Q invariant. We have shown that properties (a) and (b) of a linear set hold for Q and since P was a line (containing at least three points) Q is a line. 56 M. G. GABA: [January THEOREM 8. If pi and p2 are distinct points, there is at least one ine containing both pi and p2. We know that at least one line exists from Theorem 6. That line has two points qi and q2. By Theorem 5 a collineation r exists transforming qi, q2 into pi, p2 respectively. The line containing qi, q2 is transformed by r into a line which contains pi, p2. THEOREM 9. Every collineation transforms a k-space into a k-space. THEOREM 10. All points are not on the same line. THEOREM 11. If pi, p2, ps, P4, p5 are five distinct points such that pi, p2, p3 are non-collinear, pi, p2, p4 are collinear and pl, p3, p5 are collinear, then there exists a point p6 such that p2, p3, p6 are collinear and P4, P5, p6 are collinear.* No three of the points p2, p3, p4, p5 are collinear for if they were pi, p2, p3 would be collinear. By Theorem 4 a collineation r exists such that T(p2, p3, p4, P5) = P2, P4, P3, p5. The collineation r transforms the lines ps p5 and p2 p4 into the lines p4 P5 and p2 p3 respectively. The point pi common to the lines P3 p5 and p2 p4 will therefore be transformed into a point p6 common to the lines p4 ps and p2 p3. We have already proven as theorems the Veblen-Young postulates Ai, A2, Aa, Eo, Ei, and E2. The postulate A1 is our Theorem 8; A2 is our Theorem 1; A3 (if pi, p2, p3 are points not all on the same line and p4 and P5 (p4 = p5) are points such that pl, p2, p4 are on a line and pi, p3, p5 are on a line, there is a point p6 such that p2, p3, P6 are on a line and p4, p5, P6 are on a line) is in content equivalent to Theorem 11; Eo (there are at least three points on every line) is true from definition of line; El (there exists at least one line) is Theorem 6; and E2 is Theorem 9. We therefore Iknow that our line satisfies the six preceding postulates that Veblen and Young lay down for their undefined line, hence all theorems that they derive from the six assumptions listed will hold in our geometry. One such theorem is: THEOREM 12. Let the k-space Pk be defined by the point po and the (k - 1)space Pk-l, then (a) There is a k-space on any k + 1 independent points. (b) Every line on two points of Pk has one point in common with Pk-l and is in Pk. (c) Every Pg (g < k) on g + 1 independent points of Pk is in Pk. * Theorem 11 is essentially equivalent to the Veblen-Young postulate As: If p', p2, ps are points not all on the same line and p4 and p5 ( p4 + p5 ) are points such that pl, p2, p4 are on a line and pi, p3, p5 are on a line, there is a point p6 such that p2, p~, p6 are on a line and p4 P, p p are on a line. The form of statement for Theorem 11 was suggested by Professor E. H. Moore as a substitute for As since the latter is redundant in that it includes the obvious cases where p4 is coincident with pi or p2 or where p5 is coincident with pi or p3. 1915] POSTULATES FOR PROJECTIVE GEOMETRY 57 (d) Every Pg (g < k) on g + 1 independent points of Pk has a P-i1 in common with Pk-l provided all g + 1 points are not in Pk-1. (e) Every line P1 on two points of Pk has one point in common with every Pk-, in Pk. (f) If qo and Qk-1 (qo not in Qk-) are any point and any (k - 1)-space respectively of the k-space determined by po and Pk-_, the latter space is the same as that determined by qo and Qk-1. Another important theorem that Veblen and Young prove is: THEOREM 13. On k + 1 independent points there is one and but one k-space. THEOREM 14. If k + 1 points of a Pk-1, such that every k are independent, are left invariant by a collineation r then r leaves every point of Pk-l invariant. This theorem clearly is true for a line or P1. Let us assume that the theorem is true for a Pg-1. In a Pg if every g + 1 of g + 2 points are to be independent, then if pl, * * *, pg determine a Pg-1, pg+l and pg+2 cannot lie in Pg-1, nor can the line pg+l pg+2 contain any of the points pi, - *, pg. The line pg+1 pg+2 has a single point po in common with Pg-1 by Theorem 12 (e). When the g + 2 points are left invariant po, being the intersection of P,-1 and pg+l pg+2, is left invariant. The line Pg-_ and the line pg+~ pg+2 are each therefore left identically invariant. Let p be any point in P, not in Pg-_ nor on the line pg+1 pg+2. The lines ppg+ and ppg+2 each meet the Pg-1 by Theorem 12 (e). When the given g + 2 points are left invariant, these lines and consequently their intersection is left invariant. The theorem being true for a g-space if true for a (g - l)-space and holding for a one-space is therefore true for a k-space. THEOREM 15. All points are not on the same k-space if k < n. It can easily be shown that every k-space has k + 2 points such that every k + 1 are independent. If these k + 2 points and n - k - 1 other points, which exist by Postulate I, are not in the k-space the theorem is true. If these n + 1 points are in the k-space then by Postulate V a collineation distinct from the identity exists leaving these n + 1 points invariant and therefore by Theorem 14 that collineation leaves the k-space identically invariant. Hence not all points can be in the k-space. THEOREM 16. There exist n + 2 points such that every n + 1 are independent. The definitions of perspectivity, projectivity, etc., can now be given exactly as Veblen and Young give them. We will now proceed to identify what we call a collineation with what they call a projective collineation. To do this we will first prove: THEOREM 17. Every central perspective correspondence between points of two lines can be secured by a collineation. Let the perspectivity be defined by pi, p2 having as their correspondents ql, q2. We have by Theorem 4, since no three of pl, p2, ql, q2 are collinear, 58 M. G. GABA: [January that the two collineations ri and 72 exist such that r'(pl, p2, ql, q2) = pl, q2, ql, p2 and 72(pl, q2, ql, P2) = ql, q2, pl, P2, therefore r = r2 ri exists by Postulate II and is such that r (pl, p2, ql, q2) = qî, q2, pi, p2. The lines pl qi and p2 q2 will be left invariant by r and so will their point of intersection o. The lines pl p2 and qi q2 are interchanged by r as well as the lines pi q2 and p2 qi. Let the intersection of pl p2 and qi q2 be r3 and of pi q2 and p2 qi be r4. The collineation r must leave r3 and r4 invariant. There are two cases possible,* the first where o, r3, and r4 are non-collinear and the second where these three points are collinear. Let us first consider case 1. The line r3 r4 meets the lines pi qi and p2 q2 in points ri and r2. The collineation r leaves r1, r2, r3, and r4 invariant; hence every point of the line r3, r4 is left invariant by r. Let p be any point on the line pi p2 and call r the point of intersection of the line op with the line ri r2. The line op is left invariant by r since that collineation leaves two of its points, o and r, invariant. Since the line pi p2 is transformed into the line qi q2 by r, p is transformed by r into the intersection of the lines op and ql q2 which we will call q. For the second case, where o, r3, and r4 are collinear, the points o and r are coincident. The collineation r2 = TT leaves pl, p2, and rs invariant and therefore leaves every point of pl p2 invariant. Hence if q denotes the point of ql q2 into which r transforms a point p of pi p2 then r must transform q into p. Hence the line pq is left invariant by r. Since o, r3, and r4 are collinear the lines pi ql, p2 q2, and r3 r4 are concurrent at o. If the line pq did not pass through o it would intersect the three lines pi qi, p2 q2, and r3 r4 at three distinct points. But this would make the line pq identically invariant under the collineation r and hence p would be invariant and coincide with q, but this is possible only if p is r3 and we assumed that p was any point on the line pl p2 therefore pq passes through o. Therefore the collineation r makes correspond to the points of the line (p) the points of line (q) perspective to (p) with center of perspectivity o. THEOREM 18. If a projective correspondence exists between the points of two Uines, then a collineation exists that transforms the points of the first line into the projectively corresponding points of the second line. A projective correspondence between the points of two lines is the resultant of a sequence of central perspectivities. By Theorem 16, each central perspective correspondence has associated with it a collineation. The resultant of the sequence of collineations corresponding to the sequence of perspectivities that define the projectivity is the required collineation. * If Postulate VIII were assumed, the first case only would arise. 1915] POSTULATES FOR PROJECTIVE GEOMETRY 59 THEOREM 19. If a projectivity leaves each of three distinct points of a lne invariant it leaves every point of the line invariant. THEOREM 20. If P is a P, all points are in P. This theorem follows readily from Postulate VI and Theorem 12. In addition to the Veblen-Young postulates AI, A2, A3, Eo, and E1 we have proven En, which is Theorem 15, E', which is Theorem 19, and P, which is Theorem 18. We have therefore proven all of the postulates that Veblen and Young assume for general projective geometry of n-space and consequently all the theorems that can be derived from their postulates hold in our geometry. We can now prove THEOREM 21. Every collineation is a projective collineation. Let r be any collineation. From Theorem 20 we know that there exist n + 2 points such that every n + 1 are independent. The collineation r will transform these n + 2 points which we will eall pi, p2,, pn+2 into p', p2, *, pn+2 respectively. There exists a projective collineation 7r (from the Veblen-Young geometrywhich is at our disposal) such that 7r (pi, p2, * *, pn+2 ) = Pi, p2,* * * pn+2. The points pi, p2, ~, pn determine an (n - 1)-space which will meet the line pn+l pn+2 in a point po. Let p' denote the intersection of the (n - 1 )-space determined by p, p2, ~ *, p, with the line pn+l p'+2. This point, pO, will correspond to po both by transformation r and transformation. The line pn+l pn+2 having three of its points Pn+l, pn+2, and po transformed into p+1, pl+2, and po both by r and r, will have every one of its points transformed into the same corresponding point of p.+~ p'+2 both by r and r. This is true for every line pi pj (i, j = 1,2,3, * *, n + 2; i + j). Every plane pi pj pk will go into the plane p' pp. ',. and every point in the first plane will have the same correspondent in the second plane both by r and 7r. Let pijc be any point in the plane pi pj pk. Draw the lines pi pij and pj pijk. These lines will intersect the lines pj pk and pi pk in points that we can call ijk and Pik. Transformations r and wr will transform pi, pj, pjk, and pik into the same points, that is into p', p", pj,, and p T,. To pijk will correspond the intersection of Pi pjiz with pj pi, by either r or r. By continuing this process we can prove that to any point p there corresponds by either r or t the same point p'. Hence r and 7r are identical. If we desire the set of collineations from which we start to be the group of all projective collineations we add Postulate VII to the postulates we have used and we have the theorem. THEOREM 22. Every projective collineation is a collineation. 60 M. G. GABA: [January INDEPENDENCE EXAMPLES, n 3- 3 In the following a Roman numeral preceded by a minus sign denotes an example of a system in which the postulate denoted by that numeral is false but all the other postulates of the set I-VIII are true.* - I. Let the class of points consist of a single element and the class of collineations of the identity transformation. - II. Case 1, n odd. Let the class of points consist of 2 (n + 1) elements in (n + 1)/2 sets of four. Let the collineations be all the transformations that permute the points of each set amongst themselves, the identity transformation excepted, together with all the transformations that permute the points of each of all but two sets amongst themselves but leaves one point of each of the two remaining sets invariant and transforms the other three points of each of these sets into the remaining three points of the other set. Each set is a line and there are no other lines. Case 2, n even. Let the class of points consist of 2n + 1 points in n/2 sets of four and one single point. Let the collineations be transformations on the n/2 sets like those of Case 1 leaving the extra point invariant together with the transformations that permute the points of each of all but one of the sets amongst themselves, interchange the single point with one of the points of the remaining set and leaves none of the three other points of that set invariant. Each set is again a line and there are no other lines. - III. Let the class of points be 2 (n + 1 ) elements in n + 1 sets of two. The collineations are the 2f+l transformations on the points leaving each pair invariant. The lines are the n(n + 1)/2 tetrads of points each consisting of two pairs. - IV. Case 1, n odd. Let the class of points consist of 2 (n + 1) elements in (n + 1)/2 sets of four. Let the class of collineations be all the transformations that permute the points of each set or interchange the sets. Each set is a line. Case 2, n even. Let the class of points consist of 2n -H 1 elements in n/2 sets of four and one single point. Let the collineations be like those of case 1 on the sets and leave the extra point invariant. Each set is a line. - V. Ordinary projective geometry of (n - l)-space. - VI. Ordinary projective geometry of (n + )-space. - VII. Let the class of points be the class of all sets of n + 1 rational numbers (X1, X2,.*, Xn+i), the set (0, 0,..., 0) excepted. The sets (X1, X2,..., Xn+l) and (KX1, KX2, *, KXn+1) are understood to be equivalent for all rational values of k distinct from zero. Let the collineations be the class of all linear homogeneous transformations on n + 1 variables, * In - I, Postulates III-VIII are satisfied vacuously. The same is true of Postulates VII and VIII in - I and - III and of Postulate VII in - IV. 1915] POSTULATES FOR PROJECTIVE GEOMETRY 61 having rational coefficients and whose determinants of transformation are (n + 1 )th powers of rational numbers not zero. - VIII. The finite projective geometry of n-space having three points on a line.* * For finite projective geometries see O. Veblen and W. H. Bussey, Finite projective geometries, these T r a n s a c t i o n s, vol. 7 (1906), pp 241-259. VITA Meyer Grupp Gaba was born June 25, 1884, at Syracuse, N. Y. He received his early education in the Chicago public schools, graduating from the Northwest Division High School in 1903. In the autumn of 1903 he entered the University of Chicago, from which institution he received the degree of S.B. in 1907, and of S.M. in 1908. He was instructor in mathematics at the University of Kansas from 1908 to 1910, at the School of Mines of the University of Missouri from 1910 to 1911, and at Dartmouth College from 1911 to 1913. During the summer quarters from 1908 to 1913 and during the year 1913-1914 he was in residence at the University of Chicago. He took courses under Professors E. H. Moore, O. Bolza, H. Maschke, G. A. Bliss, L. E. Dickson, A. C. Lunn, H. E. Slaught, E. J. Wilczynski, H. F. Blichfeldt, G. A. Miller, H. S. White, and J. W. Young in Mathematics and under Professors F. R. Moulton, K. Laves, and W. D. MacMillan in Astronomy, to all of whom he wishes to express his appreciation. He desires especially to thank Professor E. H. Moore, to whose inspiration he owes much of his interest in mathematics, for his helpful criticism in the writing of this thesis.