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A PAPER 
 
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 FOUNDATIONS 
 
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 PROJECTIVE GEOMETRY 
 
 (Read before the Aristotelian Society, Dec. 13, 1897) 
 
 BT 
 
 EDWARD T. DIXON 
 
 (ffamttiljfle 
 DEIGHTON BELL & CO. 
 
 1898 
 
THE FOUNDATIONS 
 
 PROJECTIVE GEOMETRY 
 
A PAPER 
 
 ON THE 
 
 FOUNDATIONS 
 
 OP 
 
 PEOJECTIVE GEOMETRY 
 
 (Read before the Aristotelian Society, Dec. 13, 1897) 
 
 EDWAKD T. DIXON 
 
 OF THE 
 
 UNIVEPSI 
 
 DEIGHTON BELL & CO. 
 1898 
 

 Gift of 
 JMbs: Tberese F.Colin 
 
PREFACE. 
 
 This paper was nominally "read" at a meeting of the 
 Aristotelian Society on December 13th, 1897. It is the ex- 
 cellent custom of that Society to circulate proofs of papers 
 among the members before the meetings, and I was thus able 
 to make the paper longer and more technical than would 
 otherwise have been possible ; but even so the mathematical 
 portion is compressed to the utmost, and a great deal is 
 necessarily left to the good will and intelligence of the reader. 
 
 Since writing the paper I have read up the modem litera- 
 ture of the subject, and I am confirmed in the impression, which 
 I got from Mr. Russell's book, that the conception of " order " as 
 the basis of so-called " Projective Geometry " is essentially new. 
 The only suggestion of this way of looking at the question which 
 I have found is in Pasch's Vorlesungen uher neuere Geometrie. 
 But though he sees that less than four points have no determinate 
 " order," he never extends the conception of order to continuous 
 quantities, nor does he grasp the connection between it and 
 " anharmonic ratio." It seems, therefore, worth while to bring 
 my views before a wider public, for, interesting as the discussion 
 before the Aristotelian Society was, it was chiefly my epistemo- 
 logical views which came in for criticism ; the geometrical points 
 were hardly discussed at all. 
 
 EDWARD T. DIXON. 
 
 January, 1898. 
 
 178634 
 
fc^ THE 
 
 U N i y £ r> r- j J Y 
 
 THE 
 
 FOUNDATIONS OF PROJECTIVE GEOMETRY 
 
 PART I 
 
 I HAD the honour of reading a paper on Natural Realism 
 before this Society last year, the object of which was not so 
 much to convert Idealists to realism (for I held, and hold still, 
 that they are all natural realists except in moments of 
 philosophic extasy), but rather to discover if possible what 
 modern Idealism really is. I failed ; perhaps because my 
 remarks were too general in character. My opponents in 
 argument, while professing to be Idealists, adopted my con- 
 clusions and gave them their own interpretation, so that 
 apparently nothing was left to argue about. I propose 
 therefore to-night to discuss a special case only, our so-called 
 knowledge of space, and to see whether it does not provide a 
 test which shall discriminate between the methods, even if not 
 finally between the opinions, of idealistic and realistic philoso- 
 phers. I shall do so with special reference to a book which 
 has recently appeared on the subject by Mr. Russell, a member 
 of this Society, who is himself, if I mistake not, an Idealist of 
 the modern type. 
 
 Mr. Russell commences his book with the statement that 
 throughout the l7th and 18th centuries Geometry remained, 
 in the war against empiricism, "an impregnable fortress of 
 the idealists," meaning that they successfully maintained the 
 position that in Geometry at least we had obtained certain 
 knowledge about the real world, independently of experience. 
 " None but a madman, they said, would throw doubt on its 
 validity, and none but a fool would deny its objective reference." 
 He implies that modern idealists have receded from this 
 
8 THE FOUNDATIONS OF 
 
 position ; and it is the primary object of this paper to discover 
 whether they really have done so or not. Mr. Russell's book 
 professes to be an exposition of the position modern idealists 
 take up in reference to this question. We shall see whether it 
 differs in any essential from that of the idealists of the last 
 two centuries. 
 
 Before proceeding to consider in detail the constructive 
 portions of Mr. Russell's book, J must notice a point in which 
 his critical method, and that of the school of philosophy to 
 which he belongs, seems to me radically defective; and this 
 is the treatment of contradictions and ' antinomies.' It is 
 true that all Mr. Russell does is to attempt, and generally with 
 success, to explain them away. But this is not the case with 
 all modern philosophers; and that Mr. Russell allows himself 
 ever to attach any importance to so-called 'antinomies,' 
 without at once holding them up to ridicule, shows how much 
 he, like other modern philosophers, is still under the pernicious 
 influence of that arch-sophist Immanuel Kant. Consider for 
 example the 'antinomy of the point' — "There are different 
 points : all points are identical." Surely it is evident at once 
 that this is only a clumsy way of saying, "There are points 
 which differ in position, but are in other respects identical." 
 Mr. Russell's own statement of the antinomy is expressed in 
 rather more specious, or at least more involved, language; but 
 I think he might have explained it away in half the space he 
 actually devotes to it. In his treatment of the second antinomy 
 however he is not so happy. As stated by him, the contra- 
 diction arises solely from an ambiguous use of the term 
 ' relation * ; and so far from explaining this ambiguity away, it 
 remains, and mars much of the reasoning throughout his book. 
 Surely he ought to have seen that when he says spatial figures 
 (he is referring to straight lines and planes) must be regarded 
 as 'relations,' he is using the word in a completely different 
 sense to that which it bears in the assertion that ' a relation is 
 indivisible.' To me it has always seemed that to talk of a 
 straight line as a ' relation,' or still more as ' the ' relation 
 between two points, is an abuse of language. One might 
 indeed fairly say that the fact that one, and only one, straight 
 
PROJECTIVE GEOMETRY. 9 
 
 line can be drawn through two points iudicates the existence 
 of a relation between them ; but the relation is there whether 
 you draw the line or not; it is certainly not the same thing as 
 the line. But if it is desired, or found convenient, to call a 
 straight line ' the relation ' between two points it is of course 
 logically permissible to do so, 'provided that some other term, 
 such as 'relationship,' is used for the kind of abstract con- 
 ception which can not be divided. 
 
 But it is simple waste of time to discuss so-called an- 
 tinomies in detail. To talk of a contradiction in things, 
 whether objective or subjective, is sheer nonsense. To do so is 
 in itself simply to make a contradiction in terms. If you 
 attempt to apply two contradictory terms, or one which has 
 been defined in a self-contradictory manner, to any real thing, 
 it simply won't fit. You can not by doing so affect the nature 
 of the thing, nor find out anything, either positively or 
 negatively, about it. If your language is self-consistent and 
 free from contradictions you may employ its terms as symbols 
 for real things ; and in this way it may assist you to obtain 
 clear conceptions with regard to them. But if it is not self- 
 consistent, it is mere ' gibberish ' ; and it passes my compre- 
 hension to understand how anybody could imagine that terms 
 which were admittedly contradictory could be of any use. If 
 anyone wishes to see the absurdities to which this sort of 
 ' logic ' can lead, let him read Mr. Bradley's " Appearance and 
 Reality." 
 
 Mr. Russell devotes a considerable amount of space in his 
 introduction to the defining of the terms * a prion ' and 
 ' wholly a pHori! I shall show that there is some reason to 
 doubt whether he really does confine himself to the definitions 
 of the terms which he gives at the beginning of his book, 
 when he uses them in the statement of his final conclusions. 
 But at present I wish to discuss the propriety of using the terui 
 a -priori at all. The term is philologically and historically 
 bound up with the view " that certain knowledge independent 
 of experience is possible about the real world," and if you do 
 not want to express this view it is better not to use a word 
 which in the minds of the enormous majority of your readers 
 
10 THE FOUNDATIONS OF 
 
 will certainly imply it. However, Mr. Russell defines the term 
 a 'priori in a new and special sense, in which a priori 
 knowledge, even if ' certain,' and ' independent of experience,' 
 may perhaps not be intended to be knowledge of the real 
 world. Whether he so intends it or not I am not able with 
 certainty to determine from what he says in his introduction ; 
 for he declines to discuss any psychological question. Even if 
 he agreed with Kant, that a priori knowledge is subjective, 
 this would not settle the question, for he gives two discrepant 
 definitions of ' subjective,' one of which, without any apparent 
 justification, he fathers on Psychology, and the other on 
 Physical Science. Nor is it any clearer what he means by 
 ' experience.' He refers to Mr. Bradley for his definition, who 
 certainly includes sensations in the denotation of the term * 
 But Mr. Russell can not seriously mean that he believes in the 
 a priori character of the axioms of projective geometry 
 because without them a toothache would be impossible ! If he 
 had said those axioms were only relatively a priori, as he says 
 of the axioms of metrical geometry, he would only have 
 meant, according to his definition, that a science deduced from, 
 other axioms would not be the science of projective geometry — 
 which is only another way of saying that the axioms do in fact 
 define the terms of the science — they are logically merely 
 definitions. With that I entirely agree, but what more does 
 Mr. Russell mean by ' wholly ' a priori ? That they are true of 
 any form of externality ? But is not this the same thing over 
 again ? — if I admit this, does it not simply define the term 
 ' externality ' ? I might have some other theory of reality, not 
 based on Mr. Russell's axioms, and all he has proved against 
 such a theory would be that I had no right to call it a theory 
 of externality. I have pointed out elsewhere-|* that, unless we 
 define our terms independently but not inconsistently, no 
 proposition can be held to be more than an implicit definition 
 of any doubtful term it may contain. It was of course open 
 to Mr. Russell to have defined the term ' externality ' indepen- 
 
 * See "Appearance and lleality" p. 346. 
 
 t See my Essay on Reasoning, Deighton, Bell & Co. Also an article in 
 Mind, N. S. Vol. II., p. 339. 
 
PROJECTIVE GEOMETRY. 11 
 
 dently and consistently. He might for example have defined 
 it as the conception we actually have, commonly called Space. 
 This would be what I have called a definition by denotation ; 
 and the other terms being defined by connotation, the assertion, 
 that certain axioms must be true of any form of externality, 
 would acquire real import. But this would have involved Mr. 
 Russell in a psychological discussion as to how we actually do 
 conceive space, and would have made the axioms dependent 
 upon subjective experience. What Mr. Russell is trying to do, 
 and it is herein that his idealism appears, is to obtain know- 
 ledge which shall be real, and yet not be " at the mercy of 
 empirical psychology." He wants to know something about 
 the psychological world, if not about the physical, ' a pHori ' ; 
 in the good, old, undiluted sense. We will see how far he 
 succeeds. 
 
 fle commences the detailed discussion of the elements of 
 projective geometry by saying (§ 111) "The two mathe- 
 matically fundamental things in projective geometry are 
 anharmonic ratio and the quadrilateral construction." These 
 things may indeed be essential to the mathematical treatment 
 of the subject, but there are a number of preliminary pro- 
 positions which must be established before we can discuss them, 
 which Mr. Russell tells us nothing about at all, and he hardly 
 tells us anything even about the ' axioms ' upon which he 
 conceives them to be based. He does indeed enumerate these 
 axioms in a subsequent paragraph (§ 122); and I suppose he 
 relies on his readers' knowledge of the ordinary text books to 
 effect the requisite deductions. But in none of the ordinary 
 text books, so far as I am aware, are the preliminary propositions 
 deduced from axioms at all like Mr. Russell's, or even from 
 definitions which profess to be free from spatial implications. 
 The one possible exception with which I am personally 
 acquainted is Veronese's important work on the elements of 
 geometry; but though Mr. Russell quotes from this work he 
 can not have appreciated its importance, or he would have given 
 more attention to those preliminary propositions about which 
 he practically tells us nothing at all. For example, in the 
 diagram on p. 123 of Mr. Russell's book, if the section abed is 
 
12 THE FOUNDATIONS OF 
 
 given, and two points a" c" on the third section, how does Mr. 
 Russell deduce from his axioms that there is a point h" at all ? 
 And how that there is a point d" at all ? 
 
 Mr. Russell's treatment of ' anharmonic ratio ' and the 
 'quadrilateral construction' is just enough to whet the appetite 
 for knowledge without satisfying it. He tells us that two sets 
 of four collinear points each have the same ' anharmonic ratio ' 
 when corresponding points of different sets lie two by two on 
 four straight lines through a single point, or when both sets 
 have this relation to a third set. He adds, in a note, " There is 
 no corresponding property of three points in a line, because 
 they can be projectively transformed into any other three points 
 on the same line." But if this is so he expresses himself very 
 badly when he tells us that " The successive application to any 
 figure of two reciprocal operations of projection and section is 
 regarded as producing a figure projectively indistinguishable 
 from the first." He does indeed make his meaning clearer later 
 on, by saying that " Two sets of points or lines which have the 
 same anharmonic ratio are treated by projective geometry as 
 equivalent," (my italics) that is, two sets oi four points obtained 
 by projection and section are regarded as indistinguishable ; for 
 we know there is no anharmonic ratio between three. But 
 what is meant by the term ' anharmonic ratio ' if it is not used 
 solely on account of its numerical, if not metrical, significance ? 
 In Mr. Russell's exposition it seems merely designed to lead up 
 to the application of analysis to the problems of projective 
 geometry, by means of the quadrilateral construction. Why, 
 for example, does Mr. Russell assign certain numbers to the 
 points A, B, C, D in his quadrilateral construction ? In order, 
 he says, that the differences AB, AG, AD may be in hai'monical 
 progression. Is this merely because we called it an anharmonic 
 ratio ? Or is it merely in order to fit in with preconceived 
 spatial notions ? If Mr. Russell's preconceived notions had 
 been those of, say, spherical geometry, he would not have made 
 the differences a harmonical progression. Does the distinction 
 between the meanings of anharmonic ratio in Euclidian and 
 spherical geometry depend upon nothing of which we can take 
 cognizance without the notion of number ? Or if we did not 
 
PROJECTIVE GEOMETRY. 13 
 
 care to apply numerical analysis to points at all, would anhar- 
 monic ratio be devoid of all significance ? 
 
 To answer this we must attempt to realise what significance 
 projective geometry could have, apart not only from spatial, but 
 even from numerical applications. Mr. Russell does attempt an 
 answer in §§ 117 — 124 in his book. He commences by giving 
 a definition which he admits contains a ' fundamental contra- 
 diction ' ; but, not in the least deterred, he goes on to define 
 'quality in geometrical matters' in the belief that the rest of 
 his definitions will ' follow without difficulty ' ; and in the hope 
 that they will enable him to discover ' what sets of figures are 
 projectively indiscernible.' This he conceives himself to have 
 done in § 119, where he tells us that points "have no intrinsic 
 properties ; but they are distinguished solely by means of their 
 relations. Now the relation between two points is the straight 
 line on which they lie. This gives that identity of quality for 
 all pairs of points on the same straight line which is required 
 by our projective principle. If only two points arc given they 
 can not without the use of quantity be distinguished from any 
 other two points on the same straight line." Very well ; but 
 unfortunately, Mr. Russell sets out to prove this of three points 
 in a line, not of two, and it does not take three points to 
 determine a straight line. Accordingly in § 120 Mr. Russell 
 tries back, and thinks he can now see the reason for what may 
 previously have seemed ' a somewhat arbitrary fact,' namely the 
 necessity of four coUinear points for an anharmonic ratio. I 
 quote the important part of his explanation — " Given one 
 
 point no projective relation to any second point can be 
 
 assigned which shall in any way limit our choice of the second 
 point. Given two points, however, there is such a relation, the 
 third point may be given collinear with the first two. This 
 limits its position to one straight line, but since two points 
 determine nothing but a straight line the third point cannot be 
 further limited." Is this really the case ? Why not limit it by 
 saying it shall be between or outside the two given points ? 
 But I am anticipating. Mr. Russell goes on : " Thus we see why 
 no intrinsic projective relation can be found between three 
 points which shall enable us from two uniquely to determine a 
 
U THE FOUNDATIONS OF 
 
 third. With three given collinear points however we have 
 more given than a straight line." .... Where ? Do three 
 given collinear points determine a plane ? If not, how are 
 you going to perform the quadrilateral construction without 
 assuming something not given ? And if you may assume a 
 plane not given, why might you not do so before, and perform 
 a trilateral or some other construction to determine the third 
 point from the first two ? " and the quadrilateral con- 
 struction enables us uniquely to determine any number of fresh 
 collinear points." Precisely ; anharmonic ratio depends on that 
 beautiful quadrilateral construction, and the quadrilateral con- 
 struction on anharmonic ratio, in a charming, if somewhat 
 vicious, circle. 
 
 The point that Mr. Russell has missed is that if we are 
 given a number of collinear points we are given something 
 more than the relation that they are in a straight line. He 
 says (§ 121), "Let us begin with a collection of points on a 
 straight line. So long as these are considered without reference 
 to any other points or figures they are all qualitatively similar. 
 They can be distinguished by immediate intuition, but when 
 we endeavour, without quantity, to distinguish them con- 
 ceptually, we find the task impossible, since the only qualitative 
 relation of any two of them, the straight line, is the same for 
 any other two." The straight line however is not the only 
 qualitative relation between the points taken together. In fact, 
 as I shall show, the straightness of the line in ' projective 
 geometry ' is not a relation between the points in the line at 
 all, but a relation between them and points outside the line. 
 But there is another relation, or rather a system of relations, 
 between the points themselves which permits us to say they are 
 in a line and not spread over a surface. This system of relations 
 is constituted by the order of the points in the line. If we name 
 only two points, or as I shall show, even if we name three, we can 
 not distinguish any order among them, but if we name four or 
 more we can distinguish an order. And this order is what Mr. 
 Russell calls a ' qualitative,' as distinguished from a quantitative, 
 or even a numerical, relation. It does enable us to perform 
 what Mr. Russell, in the sentence quoted above, describes as " an 
 
PROJECTIVE GEOMETRY. 15 
 
 impossible task." It is not necessary to take into consideration 
 any points external to the line to distinguish this order ; and, 
 unless we do consider such points, there is nothing to tell us 
 whether the line is straight or not — the relation of straightness 
 is a relation to points outside the line, not between the points 
 in it. If Mr. Russell, or geometers generally, had attempted 
 to define a 'line' qualitatively, that is to distinguish not a 
 straight from a curved line, but a line of any kind from a 
 surface, they would probably have been led to a completely 
 different conception of the meaning of projective geometry. 
 It is, in its abstract form, the science of the ordering, oi 
 cataloguing, of continuous groups, and the question whether 
 ' projective ' methods can or can not distinguish one figure from 
 another is the question whether we can, or can not, distinguish 
 the order of the points in one from the order of the points in 
 the other; using this term in an extended sense which is 
 applicable not only to groups of more than one ' dimension,' 
 but also to continuous, as well as to discrete, groups. 
 
16 THE FOUNDATIONS OF 
 
 PART II 
 
 THE ELEMENTS OF A SCIENCE OF CATALOGUING CONTINUOUS 
 
 GROUPS. 
 
 1. I have shown elsewhere* that the ideal method of 
 founding a science is to define the terms by verbal definitions 
 by connotation, the primitive terms employed in the definitions 
 being only such as are ' well understood ' ; and to proceed, in 
 the first place, to a purely symbolic argument. We may always 
 afterwards ascribe to them any real import which the terms of 
 the science will bear, when we have sufficiently investigated 
 the formal consequences of our definitions. If our primitive 
 terms are not * well understood ' we shall sooner or later be led 
 to contradictions or ' antinomies ' ; if that should happen there 
 would be nothing for it but to try back, and make a fresh start. 
 If, however, we come to no contradictions or ' antinomies ' we 
 may feel confident that our primitive terms really were as well 
 understood as we took them to be. This confidence in our 
 primitive terms may be expressed as a (subjective) axiom as 
 soon as we have defined any terms which may be necessary to 
 express the axiom clearly. 
 
 2. By ^passing in review logical units of thought, I mean 
 thinking of them one after the other, or successively. If a 
 collection of units has been ordered or arranged in order, I 
 mean that it has been agreed that we shall pass the units in 
 review in certain ways only, and not in others ; so that after 
 any one unit we shall, in passing in review, (or I may say 
 simply in 'passing) think next of one of a certain number of 
 determined units which may be spoken of as ' contiguous ' to 
 it, and not of any other unit whatever in the collection. Such 
 an ordered collection I shall speak of as a group. I may now 
 express the axiom of the science thus : 
 
 Any number of units may be conceived, discriminated 
 between, and arranged in order. 
 
 * See my Essay on Reasoning, 
 
PROJECTIVE GEOMETRY. 17 
 
 By collating two groups, I mean regarding the order in 
 each as identical, by taking one unit in one group to correspond 
 to each unit in the other, and passing the corresponding units 
 in review together. 
 
 When I say that two units are in different parts of a group 
 with respect to a certain boundary, I mean that the units have 
 been arranged in order, so that in passing in review from the 
 one unit to the other in a different part of the group with 
 respect to the boundary, we must pass through some unit or 
 other which is in the boundary. Thus a boundary is itself a 
 group of units. 
 
 I shall call a group simple if it does not itself form a 
 boundary within itself If we were to collate a group which 
 was not simple with one which was, we should have to collate 
 one or more of its units with two different units in the simple 
 group. 
 
 I call a group uniform if the order proceeds in the same 
 way from every unit in it. If it contains more than two units 
 it must be possible to pass from one to a second throu<^di a 
 third, and therefore it must be possible to pass throuo-h the 
 third from the first to the second, and so on. Hence in any 
 uniform group a single unit can not form a boundary. 
 
 An uniform group of the Jlrst order is one in which any 
 two different units form a boundary; and a simple uniform 
 group of the oixler (l+r), where r is any positive integer, is 
 one in which any simple uniform group of the order r forms a 
 boundary, but one of a lower order does not.* 
 
 Thus sub-groups, as any groups of lower order than the 
 whole group we are considering may be termed, are boundaries 
 in the group of the order next above. As in a group of the 
 first order two units form a boundary, we may call them a oroup 
 of the order zero. 
 
 3. These definitions might be applied to discrete groups, 
 that is to groups of a finite number of distinct units. The 
 
 • Note. — I should like to call attention to the fact that so far as I am aware 
 in no elementary text book (unless my "Foundations of Geometry" comes under 
 that heading) is any proposition analogous to these definitions deduced or deducible 
 from the premises. Euclid simply begs the question in his Uth Book. 
 
 2 
 
18 THE FOUNDATIONS OF 
 
 theory of such groups, however, requires no special elucidation. 
 What wo are here concerned with are continuous groups, which 
 may be thus defined — 
 
 When I call a group continuous, I mean that between any 
 two units in it any finite number of boundaries may be con- 
 ceived. To catalogue such groups it is necessary to have in 
 them certain boundary groups. These boundaries must be 
 defined in some way. We are not concerned to enquire how 
 we recognise individual units, or liow we know whether we can, 
 or can not, pass from one to another without passing through a 
 boundary. All we have to do is to establish a system of cata- 
 loguing which can be applied after these preliminary processes 
 of recognition have been performed. And similarly it is not 
 necessary to say how a boundary is to be recognised, if we lay 
 down as a definition the conditions which such recognition shall 
 imply. The system defined below is not a necessary or the 
 only possible system, but it is a possible one, and that is 
 sufficient for the purpose of the science. 
 
 The definition of the group of order zero might be accepted 
 as a mere verbal definition. But I may as well point out the 
 psychological reason for it. It is that in the only kind of com- 
 plete group which we conceive as a whole, that is without 
 intentionally or through ignorance confusing it with an incom- 
 plete group, two units are required to form a boundary. The 
 geometrical conception of a closed line is obviously a case in 
 point, but the same thing may be exhibited in non-spatial 
 groups. If I regard the numbers 1, 2, 3, 4, .5 as forming an 
 uniform group, I regard 5 as contiguous with 1 ; and conse- 
 quently any single number, say 3, does not by itself form a 
 boundary between two others, say 2 and 4, The subsequent 
 developments of the theory of order among continuous groups 
 may, I think, be held to show that this psychological fact is of 
 universal significance, but for the time being we may regard the 
 definition in question as purely arbitrary. 
 
 4. As I shall define 'unique' boundaries in terms of ' unique' 
 boundaiies of lower orders, the unique boundary of the order 
 zero, which I shall write U^, is defined as two units, either of 
 which uniquely determines the other, and which may be called 
 
PROJECTIVE GEOMETRY. 19 
 
 ' antipodal ' units. The unique boundaries of higher orders may 
 then be defined thus — 
 
 An unique boundary of order r, U^ (where r is an integer 
 higher than zero and lower than n, the order of the whole group 
 we are cataloguing) is uniquely deternained by naming any 
 unique boundary of the order (?•— 1) [Uy_^, and one of the order 
 zero, ( U^ not in the former, in the U^. 
 
 5. It must be carefully observed, that though I define an 
 unique boundary in terms of one of a lower order, there is 
 nothing which I have laid down as yet to define the unique 
 boundary of the lowest order at all. As far as this science is 
 concerned the antipodal relation between units in a JJ^, and the 
 relations which determine the uniqueness of a U^, are arbitrary. 
 They must be determined by methods which do not come into the 
 discussion of the science of continuous groups as such at all. 
 And consequently there is nothing in the science to determine 
 the whole collection as an ' unique ' group of the order n ; we 
 can only say it is a group of the order n. In any particular 
 application of the science our method of determining unique 
 bormdaries may be such as to define uniquely the whole group, 
 but this depends on considerations foreign to the abstract 
 science with which I am now dealing. This alone is sufficient 
 to distinguish my theory from that of Veronese* for he takes 
 the whole group to be defined by the sub-groups which it con- 
 tains ; and this shows, what indeed is otherwise evident, that 
 Veronese's theory is not free from geometrical pre-conceptions ; 
 and it is not therefore surprising to find that its scope is more 
 restricted than that of the theory I am now advancing. 
 
 And in the same way the choice of the ' order ' of the whole 
 group, or that of the highest unique boundary in it, is also 
 determined by considerations foreign to the abstract science of 
 cataloguing. I can illustrate this point at once by an analogy 
 from the case of discrete groups, though of course the analogy 
 can not be carried beyond a certain point. If we wish to be 
 able to find any named book in a library, what we do is to 
 prepare a catalogue. Now a catalogue is simply a group of 
 
 * See GrundzUge der Geometrie, the German translation of Veronese's work, 
 p. 219 (Teubner, Leipzig). And also p. 70 of this paper. 
 
20 THE FOUNDATIONS OF 
 
 units whose order is intuitively known. The simplest example 
 is the group of natural numbers, and we might catalogue the 
 library by simply numbering the books from 1 to as high a 
 number as was necessary. It is not at all necessary that the 
 numbers should be consecutive. If the book I want is number 
 3 and I see numbers 2 and 7 on the shelves, I know my book 
 must be between them — if there is only one book between 
 them either that is the one I want, or the one I want is not in 
 the library. Whether there are or are not books corresponding 
 to numbers 4, 5, 6 does not matter at all. In this way then 
 the library may be catalogued as a (discrete) group of the first 
 order. But I may also number the bookcases, the shelves in 
 each bookcase, and the volumes on each shelf separately. In 
 that case I have catalogued the library as a (discrete) group of 
 the third order. I might catalogue the rooms separately also, 
 and so make it a group of the fourth order. I might use letters 
 of the alphabet instead of numbers, in some or all cases, for we 
 also know intuitively the order of letters of the alphabet. The 
 arithmetical properties of the numbers have nothing to do with 
 it — it is their order only which is maile use of. 
 
 6. I must add a few words of explanation as to what I mean 
 by the unique determination of one group by others. This 
 sort of phraseology is indeed common enough in books on pro- 
 jective geometry and elsewhere, but it needs more precise 
 definition before it can be considered ' well understood.' If I 
 say that a group G is uniquely determined by groups A and B, 
 I mean not only that any group so determined by A and B is 
 the group G and no other, but further that the fact that it was 
 so determined by A and B involves all that can possibly be 
 said about it ; and therefore any conclusions about units in G, 
 which flow from the fact that they are in G, must be deducible 
 from the facts which determine A and B. And further, any- 
 thing that can be said about units in G without referring to the 
 determinations of A, must be deducible from the determinations 
 of B, and vice versa. This is a general principle which applies 
 not only to the particular cases of unique determination which 
 I have already mentioned, in the definition of unique groups, 
 but to other cases where units or unique groups are defined as 
 
PROJECTIVE GEOMETRY. 21 
 
 being common to two or more unique groups, or in other ways 
 which come under the general head of unique determination. 
 The principle will be amply illustrated later on. 
 
 7. If A, B, be two simple, but not necessarily unique, 
 groups of the order (/* — 1) which are boundaries in a group of 
 the order r, and if two units in A are respectively in the two 
 different parts into which the boundary B divides the group, 
 then we can not pass from one of the units in .4 to the other, 
 while passing in review the units of the group A alone, without 
 passing through an unit in the group B. Therefore the units 
 common to the groups A and B form a boundary in A, that is 
 they form a group (or groups) of the order (r — 2). 
 
 8. For the sake of abbreviation I use the symbol Ur to denote 
 an unique boundary, or group, of the order r, and I shall call 
 the whole collection a complete group of the order n, or (r„ 
 though we may use the symbol U^ if it is clearly understood 
 that in this case it is not defined, or determined, by unique 
 groups of lower orders. The unique group of the first order is 
 therefore a closed or complete group, and the lowest unique 
 group is that of the order zero, and consists of two units, each 
 of which determines the other uniquely, and which may be 
 called antipodal units. Since an unit determines a U^ 
 uniquely, we may say a Z7, is determined by a ?7r_, and any 
 unit not in it, instead of saying a U„ not in it. 
 
 Any two different {U^)s, or two different units which are 
 not antipodal, {i.e. are not in a UJ determine a U^. Therefore 
 any three ( ?7Js, or any three units, not in a t/,, determine a U^ 
 and a U^ not in it; and therefore a U„. Similarly any four 
 which are not in a U^ determine a U^. And generally, any 
 (r+ 1) units, or {Ug)s, which are not in a f7,._j, determine a f/^. 
 and any (p + 1) units, or U^s, which are not in a Up_^, if they 
 are in a U^, determine a Uj, which is wholly in that Ur. Thus 
 if any two unique gi'oups have a common unit, they have 
 common an unique group of units. In particular, if any unit 
 is in an unique group, so is its antipodal. 
 
 9. If a Up and a U^, A and B, have common a Us, C, and 
 no other units, then they determine a Up + q^s (which contains 
 both of them), uniquely. For if we take (s + 1) (UJs in C, 
 
22 THE FOUNDATIONS OF 
 
 which are not in a Ug-i, and {p-s) more {U^s in A, but not 
 in G, and therefore not in B, and which, with the (s + 1) {U^s 
 chosen in C, are not in one Uj,_^\ then in all we have (p+ 1) 
 (trjs in J., which determine A. If we now chose a (I7J in J^ 
 which is not in C, it determines with A a Up^^ ; and with C a 
 ZJs+i, which is common to B and the Up_^_^. If we chose 
 another U^ in 5 not in this t/^ + i, it will determine with the 
 Up^^, containing A a Up_^_^,B,Vidi with the f/^ + i common to the 
 Up^^ and B, a Z7s+2, which is common to the U^^^ and B. And 
 so we go on, till we have chosen {q —s) (U^)s in B, and have a 
 Up + g-s which contains A, and a Ug+g-s, that is a ^7,, common 
 to the Up^g-s and 5, which is necessarily B itself. 
 
 Conversely if a U^ and a f/^ are both in one ?7„ they will have 
 at least a ?7^+,_r common, ii (p-^q) is greater than r. For if a 
 group of order s contained all the (11^)8 they had common, the 
 Up and the U^ would determine a Up + g.g- But they cannot 
 determine a group higher than r, since they are contained in a 
 group of order r. Hence r must be at least as great as 
 (p + q — s), or s must be at least as great as (p + q — r). If 
 p-\-q = r they must have at least a U^ common ; if it is less 
 than r however, they may have no common units at all. 
 
 We may say that a U^ and a U^ determine an unique group 
 common to them, in the same sense as that they determine an 
 unique group which contains them. In the one case the group 
 determined consists of the whole of the common units, and is 
 not any group of lower order contained in it, and in the other 
 case the group determined is the lowest order of group which 
 contains both of them, and not any higher group. 
 
 10. Perhaps a modern philosopher would be inclined to see 
 an ' antinomy ' in the fact that though all we can say about an 
 unit is that it is or is not in the same part of the group as 
 another unit, with respect to a given boundary, yet I speak of 
 units as being determined by being 'in' certain boundaries. 
 The apparent contradiction is however easily explained away 
 if we remember that a boundary is itself nothing but a group 
 of units, which may be reciprocally defined by the units which 
 are in different parts of the whole group, with respect to the 
 boundary, just as well as the units may be by the boundary. 
 
PROJECTIVE GEOMETRY. 23 
 
 This is most readily seen in the cnse of a complete group of 
 the first order. If a and c are in different parts of such a 
 group with respect to J and (^ as a boundary, we may just as 
 well define the boundary {hd) by means of a and c, as the unit 
 a by means of the boundary {bd) and the unit c. Which is the 
 one to be defined, and which the one already understood, 
 depends on considerations outside the scope of a symbolic 
 science ; it must be determined for any particular application 
 of the science on grounds with which formal logic has nothing 
 to do. Thus we may always use units to catalogue boundaries, 
 as well as boundaries to catalogue units ; and the whole process 
 of cataloguing consists as a matter of fact of a combination of 
 the two processes. We may say of a that it is ' in ' the 
 boundary ahcd, and not in the same part of it as c with respect 
 to (hd) ; but this is only a first step in the cataloguing of a. 
 We "must go on to say {ahcd) is a boundary which is ' between ' 
 two units e and /, in a group of the second order, just as we 
 might have said h was in a boundary of the order zero ' between ' 
 a and c. 
 
 The assertion that a boundary is * between ' two units is in 
 fact merely the logical converse of the assertion that two units 
 are separated by a boundary. The boundary has in this case 
 become a logical unit of thought, though we reserve to ourselves 
 the right of subsequently analysing it as a group of subordinate 
 boundaries or of ultimate units. 
 
 11. If therefore we wish to catalogue a group of the order 
 n, we may say of any unit in it (1) it is in an unique boundary 
 of the order {n—l) which is 'between' two known units; (2) 
 in this boundary gi'oup it is in an unique boundary of the 
 order (w— 2) which is between two known units... and so on ; till 
 we come to {n) it is in an unique boundaiy of the lowest order, 
 zero, which is between two known units in a ?7j ; and lastly, 
 it is, or is not, a known one of the two units in this group. 
 
 Now since in any continuous group we may take any 
 number of boundaries between any two given units, the 
 determination of a boundary in this manner can never be more 
 than approximate. Though theoretically sufficient, in practice 
 any system of cataloguing of continuous groups can never be 
 
24 THE FOUNDATIONS OF 
 
 more than practically sufficient for the purpose in hand, what- 
 ever that may be. But the degree of approximation reached 
 will not depend on the order of the group ; that only determines 
 the number of approximate determinations. The degree of 
 accuracy can be made to depend on the accuracy with which 
 we can catalogue the units of a group of the first order, 
 as I shall proceed to prove. At present I wish to call 
 attention to the fact that to catalogue a group of the wth 
 order we have first to catalogue n groups of unique boundaries, 
 considered as logical units of thought for the time being. And, 
 while doing so, any proposition which is proved generally as 
 applicable to the cataloguing of units, applies i'pso facto to the 
 cataloguing of unique boundaries. 
 
 12. Without two distinct units we can have no boundary. 
 This is obviously true in the case of the only kind of group we 
 can clearly conceive as a whole, which is 'totus teres atque 
 rotundus ' ; of which a closed curve is an example. But 
 perhaps it is not quite so obvious that it is equally true in any 
 method whatever of cataloguing a group. If so, however, it is 
 only because we confuse our minds by introducing the term 
 ' infinity ' into the discussion. Logically an unit or point ' at 
 infinity' is not different from any other unit or point; it is 
 only because we look at it from a different point of view that 
 it appears to us different. If, for extraneous reasons, we have 
 adopted this point of view, we must not let ourselves be led 
 into making statements which would obviously be erroneous if 
 we changed our point of view. That is why in this discussion 
 I have adopted the point of view from which the group can be 
 conceived ' totus teres atque rotundus.' If we "project points to 
 infinity " we are apt to lose sight of them, but if it is logically 
 done it can not alter the logical relations between them. Con- 
 sequently it is always true that it takes at least two distinct 
 units of thought to form a boundary. And there is no sense in 
 talking of a boundary unless it separates two units. Therefore it 
 requires at least four units in order that we may recognise any 
 order among them. This proposition is very important, and it 
 applies to all systems of cataloguing alike. 
 
 13. If in any complete group of the first order we are given 
 
PROJECTIVE GEOMETRY. 25 
 
 two units which are uot antipodal, we are at the same time 
 given two more, namely their antipodals. And as the antipodals 
 are uniquely determined by the units, we are given the order 
 of the four units. It may be shown that an unit and its 
 antipodal are in different parts of the group with respect to 
 any other unit and its antipodal, considered as a boundary. 
 Let aa be two antipodal units in a complete group of the first 
 order (which may or may not be ' unique,' if it is a sub-group 
 in a higher group). Let us pass in review the units in one of 
 the parts into which [aa) divides the group, calling the unit 
 under review b, and its antipodal h'. If a^a^a^... be a succession 
 of the units passed under review, and a^a^a^... their antipodals, 
 then as we successively collate b with a^a/i^... we collate b' with 
 a^'a^a^'... ; and until b passes the boundary [aa), b' can not do so. 
 Therefore b, b', are always in the same, or always in different, 
 parts of the group with respect to the boundary (aa). But if 
 h' were in the same part as b, then (b'a) would form a boundary, 
 with respect to which b and a would initially be in different 
 parts of the group. And b reaches a without passing through 
 a ; it would therefore have to pass through b' ; which, however, 
 it can not do, since an unit and its antipodal are by definition 
 distinct. Hence b, b', are always in dif event parts of the group 
 Avith respect to (aa). Further, we see that if b passes any 
 unique boundary (a^a^'), V also passes the same unique boundary. 
 Hence the order of units is the same as the order of their 
 antipodals. 
 
 14. If besides {aa) {bb') we are given a fifth unit, c, we can 
 say of it that it is in one of the four parts into which the two 
 {U^s divide the whole group. We may for example say that it 
 is not in the same part of the group with respect to the 
 boundary {ab) as either of the antipodals of a and b, or we 
 may make a similar statement with respect to {ba), [ah') or 
 [b'a). 
 
 In the same way in a complete group of the second order, 
 if we have a complete group of units which forms a boundary 
 in the group of the second order, and is such that the antipodals 
 of the units are all in one part of the group with respect to 
 this boundary, then the group of the first order, with its 
 
26 THE FOUNDATIONS OF 
 
 antipodal group, divides the group of the second order into three 
 parts, and we can say of any unit not in either group, either that it 
 is not in the same part with respect to one of them as the anti- 
 podal group, or that it is in the same part, or that it is not in the 
 same part, with respect to the antipodal group as the original 
 group was. And similarly with groups of orders higher than 
 the second. 
 
 15. Now I have said that in any system of cataloguing we 
 can say of two units that they are separated by a boundary, 
 which must itself consist of at least two units ; and reciprocally 
 we may speak of a group of at least two units which forms a 
 boundary as 'between' two units. Bat we can not, without 
 further definition, speak of a single unit as ' between ' two others. 
 In a complete group, however, a single unit uniquely determines 
 an unique boundary, and we might speak of a single unit as 
 between two others, meaning that the U^ determined by it was 
 between them. 
 
 I intend, however, further to limit the use of this expression. 
 If I say unit a is between units b and c in a C/,, I mean not 
 only that the ( U^ [aa] is between b and c, but that neither is 
 the [Ug) [hb') between a and c, nor the U^ [cc) between a and b. 
 This is the same thing as saying that a is in a different part of 
 the C/j, with respect to (5c) as boundary, from the antipodals of 
 b and c. Thus, in this special sense, we can, in a complete 
 group, speak of one unit as ' between ' two others, but it must 
 clearly be understood that this is only possible by an implicit 
 reference to their antipodals, and that if there were no uniquely 
 related pairs of units the expression would be meaningless. In 
 the case of a bounded group we might indeed give another 
 meaning to it by saying an unit was between two others if it 
 was in a different part of the group from the boundary of the 
 whole group, with respect to the boundary formed by the named 
 units within the group. But in that case, though not referring 
 to antipodal units, we should be referring implicitly to a 
 boundary assumed arbitrarily in the group, the units in which 
 would be specially related to all the other units in the group. 
 
 In the same way if a group of units with the group of 
 antipodal units form two boundaries which divide a group of 
 
PROJECTIVE GEOMETRY. 27 
 
 order higher than the first into three parts, we may speak of 
 units as * between ' the two groups or ' within ' either of them. 
 
 If, however, we take {U^s as our units of thought, and 
 consider a [U^ as a group of the first order of such logical 
 units, we shall require two of them to form a boundary ; a 
 single U^ is not a boundary for ( f/Js. And conversely we can 
 not say that one ( U^ is ' between ' two others in a Uy Nor, 
 in a U^, will a f/, be a boundary for [U^s\ and we can not 
 therefore say that a [U^] is between two [U^s; though it may 
 be between two units, one selected from each. But two {U^s 
 form a boundary for a U^ in a t/^ ; and therefore we may say 
 that two [U^)s are between two [U^s, or that two [U^s are 
 between two ( U^s. And the group of ( U^s which forms the two 
 antipodal groups of units which divide a U^ into three distinct 
 groups of units, forms a boundary for {U^)s, and divides the U^, 
 considered as a group of [U^s, into two distinct parts. We 
 may therefore say of a U^ that it is within or between the 
 groups of units, according as the units in it are so, and though 
 it sounds unfamiliar, we may retain the same expression and 
 say a C^ is within or between the single group of ( U^s formed 
 by the two groups of units. Such a group of {U^s, which is 
 a boundary in a U^ for [U^s, I shall call a 5,. A group of 
 units which contains the antipodal of every unit in it, of which 
 a Z7j is a special case, I shall call an A group. Similarly among 
 higher orders of groups we may distinguish B from A groups, 
 according as they are, or are not, boundaries for ( U^s. It will 
 be observed that a U^ is always an A„ though all [Ar)s are not 
 
 16. If we pass in review the units in any complete group, 
 and in so doing pass any boundary, we must repass that 
 boundary before we can return to the unit from which we set 
 out. Therefore on the whole we must pass that boundary an 
 even number of times. For example, either an A^ or a B^ in a 
 U^ must have an even number of U^s common with a B^ in it. 
 And any complete group of units in it has an even number of 
 units common with any other complete group. But to U^s an 
 A^ is not a boundary in a U.,. Hence a complete group of Uji 
 may have an odd number of U^s common with an A^. If the 
 
28 TBE FOUNDATIONS OF 
 
 group is itself an A^ it will always have an odd number; for if 
 we trace an unit in one A^, the other A^ is a boundar}' for units 
 and the antipodal of the unit under review must be in the 
 opposite part of the U^ with respect to this boundary. There- 
 fore as we trace the unit under review from a given unit to its 
 antipodal, it must pass the boundary an odd number of times. 
 We shall at the same time have traced the U^ from the initial 
 U^ back to it again, and the Z7„ will therefore pass the boundary 
 an odd number of times, and we shall have passed in review 
 the whole group of (f/Js. Similarly an A^ has an odd number 
 of U^8 common with any A group, an even number with any B 
 group if they are boundaries for units in a group of higher order. 
 
 17. A group of {Up)s determined by a (C^.J, called the 
 origin of projection, and the various ( JJ^s in a U^, which has 
 no U^ in the origin, is called a 'pencil of ( C^)s (each of which 
 is called a ray of the pencil) of the order g^. The Z7, is called 
 the initial section of the pencil, and any other U^ in the Up^ 
 containing the whole pencil, which has no U^ in the origin, (and 
 which therefore has one U^ common with each ray) is called a 
 section of the pencil. 
 
 There are only two sorts of pencils which I intend to 
 discuss — pencils of [U^s of various orders, and pencils of various 
 unique boundaries of the first order. By means of one or other 
 of these we can reduce the question of cataloguing of units in 
 any group to that of cataloguing of units, or {U^s, in a U^. 
 This may be done as follows : — 
 
 Take any two ( U^s, Op O^, and the Z7, determined by them. 
 We may describe this as taking Oj as origin, and 0^ as section, 
 of a pencil of ( Z7Js of the order zero. Next take another Z/^, Og, 
 not in this pencil ; and take a pencil of ( U^s with origin 0^ and 
 section [Ofi^, that is the U^ {Ofifi^. Then choose a fourth 
 U^, 0^, outside the pencil, and a pencil of the first order with 
 (Ofi^) as origin, and (O3OJ as section. Choose 0^, outside this 
 pencil, and a pencil of the first order with [Ofifi^ as origin, 
 and {Ofi^ as section ; and so on. In the end we have chosen 
 [n + 1), {U^8, and (w — 1), [U^s as sections of various pencils of 
 the first order. But we may equally well follow the process in 
 this way. Take an origin Oj, and a U^ {Ofi^ through it. 
 
PROJECTIVE GEOMETRY. 39 
 
 Through 0^ take any f/", [Ofi^ as section of a pencil of the 
 first order of {U^s, with 0^ as origin. Through {0.fl^ take 
 any U^ {Ofifi^ as section of a pencil of the second order of 
 ( U^)s ; and so on, till in the end we have a U,^ determined by 
 the n {U^)s [0J)^...0„^,^, and a pencil of the order (w — 2) of 
 ( Z7j)s, with Oj as origin. We want names for these two ways of 
 collating, and an obvious analogy suggests that we should call 
 them the Cartesian and Polar methods respectively. But the 
 methods are not logically distinct. In the polar method we 
 have to commence by collating together the rays of a pencil 
 of {U^s of the order (n — 2). In the cartesian method we have 
 however to collate together the sections of the various pencils of 
 (Z7Js of the first order, and they form a pencil of (Z/Js of the 
 order (w — 2). In the polar way of looking at it we must collate 
 together the rays of each pencil of ( U^s, though these need not 
 necessarily be collated with the rays of the next pencil. In the 
 cartesian way of looking at it we must collate together the 
 sections of the various pencils of any given order, but the sections 
 of pencils of different orders need not be collated together ; they 
 may, if we like, be catalogued independently on ditierent 
 principles; just as in a library we may catalogue the book- 
 cases by roman figures, the shelves in each book -case by 
 letters of the alphabet, and the books in each shelf by arable 
 numerals. The next question to be investigated therefore 
 is, how to collate together the rays of a pencil of {U^s of 
 any order. 
 
 18. I have now to prove a most important proposition, 
 namely, that if we pass in review a continuous series of sec- 
 tions of any pencil of {U^s, each ray of which therefore has a 
 single U^ common with each section, and if we trace the unit 
 common with the sections in each ray, from the unit in the 
 initial to the unit in the final section, then the orders of the 
 units so traced will be the same in the initial and final sections. 
 I shall call this process collation by projection. It is sufficient 
 for my present purpose to consider only the case of pencils of 
 the first order, where the section therefore is a U^; for as we 
 have seen, the cataloguing of the whole collection may be made 
 to depend on the collation and cataloguing of ( U^)s. The proof 
 
30 THE FOUNDATIONS OF 
 
 however is general in its nature, and obviously extends to pro- 
 jection by pencils of U^s of all orders. 
 
 1 have to show that in passing in review sections of a pencil 
 of the first order of (C^)«, either no unit in the section under 
 review passes any boundary formed in the section by other 
 units, or else that every unit passes every boundary so formed, 
 in which case the order, though we may regard it as reversed, 
 remains the same as before, in so far as it is determined by 
 boundaries collated with each other. For the principle of 
 boundaries does not allow us to distinguish between ' passing 
 in review ' forwards or backwards. 
 
 Let oo' denote the origin of projection and abed ... a'b'dd' ... 
 the units in the section under review ; oao'a', oho'h', and so on, 
 denoting always the same rays, respectively. Let suffixes to 
 the letters denote corresponding units in particular sections, 
 the suffix zero those in the initial section, the suffix n those in 
 the final one. 
 
 Now hh' are always units in the ray o\o'b^', and this is a 
 boundary to units in the U^. Hence any other unit in the 
 section undej" review, say a, can not pass the boundary [bb') in 
 the section, unless it passes through the boundary ob^o'b^ in the 
 U^. Now a is always in the ray oa^o'a^, and the only units 
 common to this ray and the ray ob^o'b^ are the units o and o'. 
 Therefore a cannot pass ihe boundary [bh') in the section under 
 review unless it passes through o or o', which it can only do if 
 the section itself passes over oo, the origin of projection. Con- 
 sequently if the section does not pass over the origin, no unit 
 in it can pass the boundary formed by any two other units, and 
 therefore the order of the units collated by projection in the 
 initial and final sections must be the same. 
 
 Next suppose the section does pass over the origin. The 
 initial and final sections must have a U^ common, for they are 
 ( U^s in a U^. Let the one denoted by [a^a^) in the initial section 
 be this common f/^. And first let us suppose that, in passing 
 from the initial to the final sections, the section under review 
 always has the U^ («o^o') ^^ ^^' ^^'^ further let b be an unit 
 which passes through o as we trace it from b^ to Z>„. 
 
 Now those units in the ray a^oa^'o' which are in the same 
 
PnOJECTIVE GEOMETRY. 31 
 
 part of it with respect to the boundary {a^d^), as o is, and those 
 in the initial section aj>^a^h^ which are in the same part of it 
 with respect to the same boundary [a^a^), as 6 is, form a closed 
 group of units which divides the whole U^ into two distinct 
 parts, and we may say that as h passes along a ray of the pencil 
 from h^ to o it is always xuithin this group, and h' is always 
 within the antipodal group. As a^, a^ are units in this boundary 
 group, any unit in the section under review in the same part of 
 it with respect to {ajci^), as h is, will also be within the group. 
 Now if any one of the units in the section under review, besides 
 a^a^, were to pass through the initial section, they would all do 
 so together. And the units in the same part of the section 
 under review, with respect to [o/i^], as h is, must all pass into 
 or out of the boundary group together. Consequently, if one of 
 them, b, passes out through o, they have all got to pass out, and 
 they cannot do so otherwise than through o. That is to say, all 
 of them, with the possible exception of a and a, pass through o, 
 and their antipodals through o', as the section under review 
 passes over the origin of projection. And as they do so, each of 
 them will cross each of the boundaries in the U^ formed by the 
 rays corresponding to the others, and will therefore cross each 
 of tlie boundaries in the section under review formed by any 
 two other units. Thus the order of all the units in the section 
 under review, with the possible exception of a and a, will 
 remain the same. 
 
 Now 'pHimi facie we are unable to say whether a and a 
 pass through the origin or not. For though we can certainly 
 say they do not do so unless the section under review passes over 
 the origin, yet, if it does pass over we have no means of tracing a or 
 a, since the section and the ray a^o a^o' are then identical. But 
 for the same reason we cannot say positively that they do not 
 do so. We might therefore, merely for the sake of uniformity, 
 say that when the section passes over the origin, a and a pass 
 through it, and therefore a„ is identical with a^ and a„' with a„. 
 And it can be shown that in fact we must say so, for the sake 
 not only of uniformity, but of consistency. For there is no 
 reason why the section under review should always pass through 
 (a^ «(,'), and if it does not happen to pass through it at the 
 
32 THE FOUNDATIONS OF 
 
 moment it passes over the origin of projection, it will never 
 coincide with the ray [a^o a^'o'), and in that case without doubt 
 a and a' will pass through o or o'. Hence we are bound to say, 
 that if in projection the section under review passes over the 
 origin, then all the units, as we trace them from the initial to 
 the final section, pass through the origin; and consequently 
 their order in the section under review remains unaltered. It 
 is to be noted that if the section under review does not pass 
 over the origin, a^a^ are each collated with themselves. If, 
 however, the section does pass over the origin, each is collated 
 with its antipodal. If, however, we are not concerned to distin- 
 guish antipodals, or if we are only discussing {U^)s, then we 
 may say that the unit of thought common to the initial and final 
 sections is, in a single projection, always collated with itself. 
 
 19. In the course of this discussion I have successively 
 'collated' h^ with 5,, b^, b^, ... up to &„, and while passing the 
 sections in review I called the unit under review alwa5's b. 
 The symbols with suffixes always meant particular units, a 
 symbol without suffix meant sometimes one sometimes another 
 unit. I may say that I identified the units with the symbols 
 with suffixes, but I only collated them, temporarily, with the 
 symbols without suffixes. When the symbols are ' identified ' 
 with the units, to collate two symbols is to collate the two units, 
 and to ' identify ' two symbols is to identify the two units. For 
 example, if I say that \ is collated with 5„, I mean that the 
 units with which these are respectively identified are collated 
 together. If I say that a^ is identified with a„, I mean that the 
 units are identified. On the other hand, if the symbols are 
 only 'collated' with the units, to identify two symbols only 
 collates the units. For example, b was collated first with b^ 
 and then with b„. The 'b' in each case was identified with the 
 ' 6 ' in the other, but the result was only to collate b^^ with &„, 
 not to identify them. It will be seen therefore that the use of 
 symbols to denote units is simply an example of what I have 
 called 'collation,' and that what I call 'identification' with 
 symbols is really only a collation which it is agreed shall be 
 permanently maintained during the discussion in hand. 
 
 If our symbols have among themselves a well-understood 
 
PROJECTIVE GEOMETRY. 33 
 
 order, to collate or identify them with units is to ascribe that 
 order to the units themselves. This collation with a well- 
 understood order is what I call cataloguing, as distinguished 
 from simple collation. How this cataloguing can be effected 
 in continuous groups it is the object of this paper to enquire, 
 but among discrete groups the process is sufficiently familiar 
 not to require elucidation. For the present purpose it is 
 sufficient to take as an example of a discrete group, whose 
 order is ' well understood,' the twenty-five letters of the 
 alphabet. In order that the group may be a ' complete ' 
 group, I include in it the same letters repeated, but with 
 accents to distinguish them, and regard any letter and the 
 same letter accented as antipodals. If we regard the letters 
 with suffixes in the above discussion as cataloguing the groups 
 of the first order in this way, then we always collate a„ with a^, 
 but whether we identify it with a^ or with a^, will depend upon 
 whether the section under review does or does not pass 
 over the origin as we trace it from the initial to the final 
 section. 
 
 This point being settled, no generality will be lost if in 
 future I assume (i) that the section under review always passes 
 through the U^[ao%), and (ii) that any other unit in it, by 
 which, in addition to {%%), it is completely defined, and by which 
 it may be said to be catalogued, passes only through units in 
 one ray which are ' between ' the units in the initial and final 
 sections which are to be collated together, and passes through 
 each of them only once. This convention will determine whether 
 the section passes through the origin or not, according as the 
 origin is or is not ' between ' the units in any one ray which are 
 to be collated together. It is therefore always possible to collate 
 together the units in two given sections which have a common 
 ?7q by a single projection ; the section under review passing 
 through, or not passing through, the origin, whichever we please ; 
 and the units in the final section collated with the units in the 
 initial section in the one case, will be the antipodals of those 
 collated with them in the othei. Hence we infer that the order 
 of units is the same as that of their antipodals (which however 
 we knew already). 
 
 3 
 
34 THE FOUNDATIONS OF 
 
 20. Suppose we wish to collate together certain units in the 
 initial, and the same number in the final section, selected 
 arbitrarily. We may suppose the units in the initial section to 
 be catalogued by the symbols a^h^... and so on, and a^l>„... will 
 denote the units we collate with them respectively by projection. 
 To use these symbols to denote the arbitrarily selected units 
 would be to ascribe to them a certain order, which we must not 
 do, as they are to be selected arbitrarily. I shall therefore use 
 capital letters for the units of the final section, and shall conceive 
 these letters not to have any intrinsic order, so that they do not 
 catalogue the units which they denote. To catalogue them we 
 have to collate them with the already catalogued units in the 
 initial ray. Now we can say of c^ that it is ' between * a^ and d^, 
 and that it is not 'between' a^ and 6^. Hence it must necessarily 
 be impossible to collate a^ with A, h^ with B, and c^ with C, if C 
 is 'between' A and £; and similarly it must be impossible to 
 collate a^ with A, c^ with G, and d^ with D, if G is not between 
 A and D. Hence it can not always be possible to collate 
 together arbitrarily three units in a complete group with three. 
 But I shall show that it is always possible to collate two with 
 two by a single projection, if neither of them are in the ?7^ 
 common to the initial and final sections, and by more than one 
 projection in any case, and further, that it is always possible to 
 collate two out of three with two out of three and the third with 
 the given third, or with its antipodal. Thus in any case it is 
 always possible arbitrarily to collate three U^s in a U^ with three 
 others in a U^, by one or more projections. 
 
 Let {a^a^), as before, be the U^^ in the initial section which 
 is also in the final one. Then by a single projection we can not 
 collate ttg with any unit A, unless this unit is also in the initial 
 section ; but if it is, we can effect the collation, whether A is 
 identified with a^ or a^, by tracing the section under review 
 either through, or not through, the origin ; which origin may be 
 taken anywhere in the U^ containing the sections, except in tlie 
 sections themselves. 
 
 Suppose further we wish to collate h^ with B in the final 
 section. We can do so by choosing any origin in the U^ {l\J^]- 
 If neither of the units in the origin is ' between ' h. and B we 
 
PROJECTIVE GEOMETRY. 35 
 
 shall at the same time collate a^ with A. If one of them is 
 'between' h^ and B we shall collate a^ with A. 
 
 If further we wish to collate c^ with C, the origin must also 
 be in the U^ {CqG), and it is therefore definitely determined. If 
 one of its units is between b^ and B, and also between c^ and G, 
 then we shall collate h^ with B, c^ with C, and so we shall if 
 neither unit is between either h^ and B, or c^ and G. But in the 
 first case we collate a^ with A, and in the second a^ with A. If 
 one unit of the origin is between J„ and B, and neither unit 
 between c^ and C, we collate h^^ with 5, but c^' with C. And 
 similarly we can discuss the remaining possibilities. It is clear 
 enough from these illustrations how it comes about that we can 
 not always project three units on to three, but can always pro- 
 ject two out of the three on to two, and the third on to the 
 third oi: its antipodal, when one pair is in the U^ common to 
 the initial and final sections. 
 
 If no pair is in the common C(,, or if the two sections have 
 no common U^, we shall require two projections to effect the 
 collation. By one projection on to a section determined by, 
 say, a^ and B, we can collate a^ with a,, which is identified with 
 it in the intermediate section, b^ with J,, which is identified with 
 B or B', and c^ with Cj. Then by a second projection we can 
 collate rtj, h^, c^ with A ov A',B or B', C or C' as the case may be. 
 
 If the initial and final sections have two ( U^)s common, that 
 is, if they are coincident, and if no pair of units to be collated 
 are identical or antipodal, it will require three projections. But 
 in the end we can always collate by projection two units out of 
 three with two, and the third with the third or its antipodal. 
 If we do not care to discriminate between antipodal units, 
 or if we are only concerned with C^s, we may say that we can 
 always collate three with three. 
 
 21. To anyone who thoroughly understood the expression 
 ' unique determination,' as I employ it in this science, this would 
 be a foregone conclusion. It would be sufficient to say, "No 
 order is distinguishable among three units of thought however 
 we may grou]^ them ; and since projection is simply an unique 
 method of collation, that is of comparing orders, it must be 
 possible by projection to collate together any three (U^.s, and 
 
36 THE FOUNDATIONS OF 
 
 any three others, two and two. But since, in a complete group, 
 the selection of any three units carries with it if so facto the selec- 
 tion of three antipodals, there is a certain order distinguishable 
 among them, namely by the relations with their antipodals. 
 Hence it cannot always be possible to project three units on to 
 three, unless we agree to overlook the distinction between 
 antipodals." This, I say, is a logically sufficient proof; but 
 from a philosophical point of view it is interesting to see how 
 the definitions work themselves out. In the next proposition I 
 shall however adopt at once the simple logical method, and leave 
 the detailed investigation to be worked out by others. The 
 logical principle may be enunciated thus. If a number of units 
 a, /8, 7 . . . are respectively determined by units a, b, c ... by 
 means of relations which determine them uniquely with reference 
 to certain other arbitrary factors, then any statement which can 
 be made about a, yS, T--- must be deducible from statements 
 made about a,b,c..., or about the arbitrary factors, or both. 
 And further, any statement which can be made about a, /8, 7 . . . 
 independently of any determination of the arbitrary factors 
 must be deducible from a similar statement made about a,h,c ... 
 and vice versa. 
 
 22. Now suppose the [U^s abc... in a U^, A, are projected 
 by a pencil of U^s with any origin, and that any other section 
 of this pencil is again similarly projected from some other 
 origin, and so on p times successively. And next let A be 
 projected again by a succession of q different projections, by 
 pencils of ( f/,)s. If the corresponding pairs of rays in the pih. 
 and gth pencils have any common [U^s let them, or as many 
 of them as there may be, be denoted by greek letters corres- 
 ponding to the latin letters in the group A. And let the group 
 of ( U^s so determined be called the group X. 
 
 If the group X consists of a single U^, a, corresponding to 
 a in ^, then there is nothing that can be said about a by itself, 
 without reference to the origins of projection, nor is there 
 anything that can be said about any other U^ in A with 
 reference to a, without reference to the origins of projection. 
 Even if X consists of two [U^s alone, a and /S, there is nothing 
 which can be said about them by themselves, nor is there 
 
PROJECTIVE GEOMETRY. 37 
 
 anything that can be said about any other U^ in A with 
 reference to a and b alone, without reference to the origins of 
 projection. There is therefore no reason why the group X 
 should not consist of two (CJs alone. 
 
 But if there is any third U^, 7, in X corresponding to c in 
 A, then we can make a statement about c ; namely that it is in 
 a group of the first order in which [ah) forms a boundary. 
 Hence 7 must be in a corresponding group of the first order, 
 namely, X, in which (a/3) forms a boundary. And of any other 
 ( f/g) c? in ^ we can say that it is, or is not, in the same part of 
 A with respect to the boundary [ah) as c is. There must there- 
 fore be in Jl" a corresponding U^, 8, of which we can say that 
 it is, or is not, as the case may be, in the same part of X with 
 respect to (a/3) as 7 is. That is, if X contains as many as three 
 {U)s it must be a group of the first order and contain a U^ 
 corresponding to each U^'m A. We can not, however, say that 
 X must be an unique group of the first order, for as I have 
 pointed out there is no sense in calling A unique, except in 
 reference to units in a wider group which contains it. Hence 
 whether X is unique or not depends on units outside A, and 
 may be affected by our choice of origins of projection. If, 
 however, the ^^th and gth pencils are not in a U^, they have a 
 U^ common if they have more than one U^ common. There- 
 fore if three pairs of rays have common [U^s, X can only be 
 this common U^ ; and as every pair of rays have a common U^ 
 in this f/j it is a common section of the /?th and gth pencils. 
 
 If, however, the f)th and 5th pencils are in one U^, X may 
 be a group of the first order, but not an unique group. In that 
 case let e be an unit (not U^ in A'' corresponding to an unit e 
 in A. Then in the U^ containing X the U^ [a^) forms a 
 boundary for units, and we can therefore say of e that it is, 
 or is not, in the same part of the U^ with respect to (a/3) as 
 another given unit is, or else that it is in the boundary (a/3). 
 We can correspondingly say that e is, or is not, in the same part 
 of A, with respect to [ah) as a boundary, as a given unit is ; but 
 we can not say it is in the boundary [ah). Hence if one other 
 unit e in A is not in the U^ (a/3), there can not be any other 
 unit in X which is in it, and conversely if e is in (a/S), every 
 
38 THE FOUNDATIONS OF 
 
 other uuit in A' must be in it also. That is to say, if three 
 units, no two of which are antipodal, or if three ( U^s in X are 
 in a U^, then X is that U^, and is a common section of the pih 
 and qih. pencils, in which every pair of corresponding rays deter- 
 mines a f/g. 
 
 Now X must be either an A^ or a B^, that is, either it 
 consists of a single group of units containing the antipodals also 
 of each unit, or it consists of two distinct groups of units, one of 
 which contains all the antipodals of units in the other. For if 
 some and not all of the units had antipodals in one part of the 
 group, we should be able to say something about these units 
 which we could not say of the others, without reference to any 
 units outside X, and we could not say anything correspondingly 
 about the units in A. It is easily seen that if X is a 5^ as we 
 pass in review the units in A, the corresponding units in X will 
 all be in one part of the group till we come to an unit uniquely 
 related in some way to the origins of projection. This will be 
 the unit in X, which is in the U^ determined by the pth and 
 qth. origins. 
 
 Since an A^ must have an odd number of {U^)s common 
 with any other A^, if X has only two f/^^s common with any t/j 
 (which is a special case of an A^), it must itself be a i?j and not 
 another A^. This particular kind of B group, which is further 
 specialised by the connotation that it can only have two {UJs 
 common with any U^, I shall call a C group of the first order, 
 or C,. 
 
 23. We may in the same way investigate the nature of the 
 group X in the case of successive projections of the units of a 
 U^, A, by pencils of {U^)s. If there are only one or two {U^)s 
 in X, then as before we can say nothing about them ; nor can 
 we do so if there are only three, unless the three {U^^)s in A, 
 to which they correspond, are in one U^, which would be the 
 case just discussed. For if there are three iUg)s a, /3, 7, in X 
 corresponding to a, h, c, in A, which are not in one U^, we are 
 unable to make any statement in respect to a fourth U^, d, in A 
 with reference to a,b, c; for a U^ is 7wt a boundary for ( U^)s 
 in a t/j, and we can not therefore say that d is, or is not, in a 
 different part of the U^ from a, h, or c. But any two of the 
 
U N I V E ] 
 
 PROJECTIVE GEOMETRY. 39 
 
 (trjs [ab], {be), {ca) form a boundary in A, and therefore if 
 there is a fourth U^, h, in A'^ corresponding to d in A, we can 
 say of any fifth C/^, e in ^, that it is, or is not, in the same part 
 of the U^ as d is, with respect to one or other of these boundaries. 
 Hence in this case it follows that X is a group of the second 
 order, in Avhich (a/8), (/Sv), (7a) determine boundaries. This 
 group must be a U^, and a common section of the pth and ^'th 
 pencils. For a, /3, 7 can not all three be in one U^ containing 
 thejMth and qih. origins of projection, since a, b, c were not in 
 one t/j ; and if S were not in the U^ determined by a, /9, 7, then, 
 as before, we could show that no other U^ in X, besides a, /3, 7, 
 could be in this U.^. But if X is a group of the second order, 
 it must have at least a group of the first order common with 
 the i/g determined by (ot, /3, 7). Hence, if there are four {11^)8 
 in X, which correspond to four in A, which are not in one (/,, 
 then they are in a f/^ ; which is a common section of the pth 
 and qih pencils. And similar results may be deduced in the 
 case of successive projections by pencils of ( f/Js of higher orders. 
 24. These results are very important. They might perhaps 
 have been deduced by methods more familiar to students of 
 projective geometry, though as a matter of fact I am not aware 
 that any exactly corresponding propositions are to be found in 
 the ordinary text books of the subject. It would probably be 
 possible to deduce them indirectly from the proofs of the 
 uniqueness of the quadrilateral construction, though the sig- 
 nificance of these proofs appears to have been misunder- 
 stood by most writers on the subject. But from the 
 philosophical point of view the proofs I have given are much 
 more interesting, as they enable us to trace the true meaning 
 of the results. The reason that we can not project any four 
 collinear points on to any other four is to be traced, not to the 
 fact that it takes two points to define a straight line, as Mr. 
 Hussell supposes, but to the fact that it takes two units, or 
 (i7p)s, to form a boundary which shall separate two other units, 
 or {U^)s, and consequently we can make a definite statement 
 about the order of four units, or (f/^js, but not about that of 
 any smaller number. It will, however, make this clearer to put 
 the results in another way. We may say that there exists a 
 
40 TtlE FOUNDATIONS OF 
 
 definite relation among any number of units, or [U^^s, in a U^ 
 greater than three, which remains unaltered by projection. 
 Similarly there exists a definite relation among any number of 
 units or [U^s in a U^, greater than four, unless all or all but 
 one of them are in a U^, which is unaltered by projection. 
 Now I have already shown that in its ordinary sense, with which 
 we are familiar in dealing with discrete groups of units, the 
 ' order ' of units, or {U^s, is unaltered by projection ; and further, 
 that this is so even in the extended sense in which I have 
 already used the word, that, namely, in which it is defined by 
 the relations of units or ( U^s to boundaries, and which applies 
 to groups of a higher order than the first. This new relation 
 which we have discovered, which is unaltered by projection, is 
 in fact nothing but the 'order' of the units or {U^s in a new 
 sense, or rather in an extension of the old sense which can be 
 applied to continuous, as well as to discrete, groups. It is 
 because no ' order ' is distinguishable in a collection of less than 
 four units, even if we regard it as a group of the first order, 
 that we can always project three {U^s in a U^ on to any 
 three other [U^s in a U^. And it is because an order is 
 recognisable among three units, or more than three ( U^s, that 
 we can not always project three units in a ' complete ' group on 
 to three, or more than three {U^s on to more than three; and 
 that in either case, when we have projected three on to three, 
 the remainder of the units in the initial group of the first 
 order are uniquely projected on to, or collated with, the units 
 in the final group. We shall become more familiar with this 
 extended sense of the term ' order ' as we proceed, but it is 
 already clear that it is a sense which is capable of exact 
 definition. It is no use ' rhapsodising ' loosely about ' qualita- 
 tive identity ' or calling a relation which, ex hypothesi is non- 
 metrical, ' anharmonic ratio.' If we can not strictly define our 
 terms they serve at best only to disguise our ignorance, and it 
 is more than probable that their employment will, sooner or 
 later, lead us into positive fallacies, of the kind known to 
 logicians as ' fallacies of the ambiguous middle.' 
 
 25. If in a complete group we collate the rays of a pencil of 
 {U^)s by means of the units in each in the section, we shall 
 
PROJECTIVE GEOMETRY. 41 
 
 collate each ray twice over, with each of the two antipodal 
 units. To evade this difficulty we must collate together the 
 half-rays as divided by the origin of the pencil, collating 
 together the units in the origin, and two other units in each 
 half-ray. This would require the arbitrary selection of two 
 units in each half-ray, besides the origin ; but by the method I 
 am about to describe it is sufficient to chose two units in the 
 whole ray, one in each half of it, to collate a single ray, and 
 only two more, besides the origin, in one other ray of a pencil 
 of the first order, to collate the whole of the rays of the pencil ; 
 and, generally, 2« units besides the origin, to collate a pencil of 
 the order (w — 2). 
 
 I shall call this process of collation of half-rays in a pencil 
 reversion ; and in reference to the process I shall speak of the 
 origin as the 'pivot of reversion, and the section of the pencil as 
 the equator of the reversion. It will appear that reversion is 
 only a specially determined kind of projection, it involves no 
 new principle. 
 
 26. The simplest case of reversion is where we are concerned 
 only with two {U^s in a f/",. We have seen that if the origin 
 of projection is not in the U^ and the final section coincides with 
 the initial one, every unit is projected into its antipodal. But 
 if we take the origin of projection in the initial sectiou, the 
 collation of all units in it, with the exception of the origin, and 
 the units in the U^ which is supposed to remain always in the 
 section under review, is indeterminate. The unit in the origin 
 ma,y however be traced along any one of the rays of the pencil 
 to its antipodal. At the same time, since the section does not 
 pass through the origin, the units in the U^^ through which the 
 section under review always passes, are collated with themselves, 
 not each with its antipodal. Hence we collate a, b, a, h' with 
 a, U , a , h respectively ; that is, we can always regard the order 
 a, b, a , b' as identical with the order a, b', a', b, if we regard 
 aa, bb' as pairs of antipodal units. If these four units con- 
 stituted the whole collection, and we catalogued the collection as 
 a complete group by the letters a, b, a , b' this antipodal relation 
 would ipso facto be established, and what we have proved only 
 shows that it makes no difference to the ' order ' of the letters 
 
42 THE FOUNDATIONS OF 
 
 whether we read them forwards or backwards. If however we 
 catalogued the group as aa'bh' , then [ab], [ah') would be the 
 pairs of antipodal units, and in that case we could not collate 
 a, b, a, b' respectively with a, b' , a, b. The term 'singular 
 reversion' will do to denote this special case of projection. 
 
 27. I go on to the more general case of the reversion of the 
 whole of the units in each of the two halves of a single ray. 
 Let PP' be the pivot of the reversion and QQ the equator. 
 That is to say, P is to be collated with P, and P' with P', in each 
 half ray, but Q in one is to be collated with Q in the other, and 
 vice versa. Take any U^ through PP', and in it choose any two 
 origins of projection (0^ C/), [0^ 0^). To fix our ideas let 0^, 
 Og, be in the same part of the U^ with respect to [PP'), and let 
 Og be ' between ' 0^ and P. 
 
 Let D be an unit in the U^ between 0^ and 0^. If we pro- 
 ject the units of [PQ) on to a U^, [QDQ), from one origin, and 
 then project them back on to PQ from the other origin, P and 
 P' will always be collated with themselves, and Q will always be 
 collated with Q and vice versa, for in one case the section will 
 pass over the origin of projection, and in the other not. The 
 other units in PQ will however in general be projected differently, 
 according as we project first by 0^ then by 0^, or first by 0^ and 
 then by 0^. If however any one of them is projected on to the 
 same unit in either case, then, as we have seen, they will all be 
 projected on to the same unit in either case. It is easily seen 
 that any unit between P and Q will be projected on to an unit 
 between P and Q, and that the ' order ' of the units from P to 
 P' via Q will be the same as the order of the projected units 
 from P to P' via Q. Thus we shall have uniquely ' reverted ' 
 the half ray PQP' on to the half ray PQ'P'. It only remains 
 to be shown that an intermediate section QDQ' can always be 
 determined which will effect this, after we have chosen the 
 origins 0^ 0^ ; and that our choice of the origins 0^ and 0^, so 
 long as they are in a ?7j with P, does not affect the result. 
 
 It will simplify the discussion to neglect for the moment the 
 distinction between antipodal units, and regard unaccounted 
 letters as denoting [U^)s instead of units. In this case it is 
 evident that the distinction between pivot and equator dis- 
 
PROJECTIVE GEOMETRY. 43 
 
 appears ; either may be regarded as pivot and the other as 
 equator — the reversion will be identical in either case. In order 
 to make my treatment of the question more readily comparable 
 with Mr. Russell's I will adopt the notation he employs in § 113 
 of his book, merely translating his paragraph, as literally as I 
 can, into my own language. 
 
 * Given any three ( U^s A, B, 1) in one U^, the quadrilateral 
 construction finds the f/„ C into which A may be reverted with 
 B,D,as pivot and equator,' and (as I shall show) it also huds the t/„, 
 G which, together with A as pivot and equator, reverts B into IJ, 
 and further, shows us how to determine the intermediate section, 
 by means of which all the {Ug)s in a U^ may be reverted with 
 respect to a given pivot and equator. However, to continue the 
 translation, ' Take any U^, 0, not in the f/,, ABD, and which 
 determines [U^s with B and D. Take any U^ through A, having 
 (f/Js P and Q common with the [U^)s {OD), [OB), respectively. 
 Let R be the U^ common to the ((7,)s {DQ), [BF), and C the 
 U„ common to the {U,)s [OR) and [ABD). Then C is the i/„ 
 required. 
 
 * To prove this, let T be the U^ common to the {U^)s {JJRQ) 
 and {(JA), and S the (f/„) common to the {U^]s [AR), {OB). 
 Then by projection with origin R and section [OB), we collate 
 A, B, G, D with *S', F, 0, D; and by a second projection, witli 
 origin A and section DQ, wc collate them with R, Q, T, D. But 
 again by a third projection with origin and section [ABD) wo 
 collate them with C, B, A, D. Thus the order ABCD is the 
 same as the order GBAD.' 
 
 28. Now at first sight this conclusion seems simply banal. TIjc 
 two orders are evidently the same, considered as discrete orders 
 of letters, except that one is written backwards. The point is, 
 however, that it is not only the discrete order of the four Us 
 which is the same, but the continuous order of the iuhole of the 
 [U^)s in ABD, which have been projected at the same time. 
 The simplest way to see this is to regard C as pivot, A as equator, 
 and [AQP) as intermediate section, 0, R being the origins of 
 projection. In this case, projecting first by as origin, if L be 
 the U^ common to [OC) and the intermediate section, we collate 
 A, B, G, D with A, Q, L, P, and then with R as origin we 
 
44 THE FOUNDATIONS OF 
 
 collate them with A, B, C, B. If we project by R first, we 
 collate A, B, C\ D with A, F, L, Q, and then with A, D, G, B, 
 obtaining the same collation in each case. 
 
 If any reverted order be projected, it projects into a reverted 
 order, taking corresponding units or ( TJ^s as pivot and equator. 
 Hence it can easily be shown that G reverts into L with 0, R, 
 as pivot and equator. Consequently, to return to the former 
 discussion, if we wish to revert a U^, PQF'Q with respect to 
 FP' as pivot and QQ' as equator, taking 0^ 0^ as origins of pro- 
 jection, to find D we perform the ' quadrilateral construction,' so 
 that F reverts into I) with 0^ 0^ as pivot and equator. 
 
 But now let us discriminate between the units in each U^. 
 If the letters in Mr. Russell's discussion represent only units, 
 instead of ( Ug)s, we can not say that the order ABGD is the same 
 as GBAD ; for, wherever the antipodals may come in, they will 
 serve to discriminate between the orders, unless {AG), [BD] are 
 themselves pairs of antipodal units, which would be the case of 
 singular reversion already discussed. It is, however, obvious that 
 in one of the projections he makes (the first in his diagram) the 
 section passes over the origin of projection, and in the other two 
 it does not. Hence, what he has shown in the case of his 
 diagram is that the order ABGD is the same as GBAD', if we 
 take the letters to denote units in a complete group. Thus we 
 discriminate between the pivot B and the equator D of the 
 reversion. Looking at the reversion in the other way, as obtained 
 by only two projections from R and from 0, in either order, in 
 one projection the section passes over the origin (namely when 
 R is the origin, in Mr. Russell's diagram) and in the other not. 
 Hence A reverts into A', and G is the pivot and A the equator 
 of the reversion. 
 
 29. We may determine a reversion by naming the pivot, 
 say G, and two units B and D, which shall revert into each 
 other. We have in this case clearly named three units in the 
 initial and three in the final sections, considering the reversion 
 as a projection. But if we name the pivot and equator we have 
 also named three units in each, namely G, A, and A' in one, 
 G, A', and A in the other. For in reversion, not only do we 
 collate B in the initial with D in the final section, but also D in 
 
PROJECTIVE GEOMETRY. 45 
 
 the initial with B in the final. Similarly, not only do we collate 
 A in the initial with A' in the final, but A' in the initial with 
 A in the final section. The case where the equator is named is 
 a limiting case of the other, but it does not really by itself 
 suffice to determine the reversion, for as we have seen in 
 singular reversion the pivot and equator are named, but the 
 collation of the other units remains indeterminate. The 
 additional assumption implied in naming the pivot and equator 
 of a definite reversion is however merely that the other units 
 shall be reverted somehow ; if they are reverted at all they are 
 reverted in an unique manner. 
 
 We have seen that, so long as we are concerned only with 
 (t/g)s, the pivot and equator need not, or rather can not, be 
 distinguished. But even if we are dealing with units, the 
 construction I gave may be used whether P is regarded as pivot 
 or Q. P was made pivot by the fact that we chose that the 
 section should cross the origin in one projection and not in the 
 other. If it is made to cross it both times, or neither time, 
 Q will be collated with Q, and P with P', so that Q will be the 
 pivot and P the equator. 
 
 We have seen also that if we consider only four units 
 A, B, G, D in Mr. Russell's construction, we may regard either 
 A and B as pivot and equator, or C and D. The results as 
 regards other units in the U^ will however be totally different 
 in the two cases. Reversion of a U^ may be called a function 
 of two U^s, symmetrical with respect to them, or a function of 
 two units, though not symmetrical with respect to them ; but 
 it is not a function of four, rather than of the infinite number 
 contained in a whole U^. 
 
 30. But if we are careful not to attach to it any ' metrical ' 
 or spatial significance, we may usefully employ the term 
 ' symmetrical ' in another way with respect to reversion. We 
 may say that in the case of complete groups it defines an order 
 .symmetrical with respect to the pivot, and in which the units 
 of the equator occupy symmetrical places. In the case of a 
 group of [U^s of the first order, the order defined is symmetrical 
 with respect to either pivot or equator, but iu a pencil of {U^]s 
 the symmetry with respect to the equator, though it still exists, 
 
46 THE FOUNDATIONS OF 
 
 is uot of the same nature as that with respect to the pivot, for 
 the equator is a U^ or higher group, and the pivot a [U^. 
 
 31. I return to the question of collation of boundaries. We 
 have to arrange a system by which the units in the various half 
 rays of each pencil of [U^s shall be collated in the simplest 
 manner, that is, with the fewest arbitrary assumptions. It will 
 be sufficient to show how the units in a pencil of the first order 
 of half U^s may be collated with each other, and with the section 
 of the pencil. We are already given two U^fi in each ray, 
 namely the origin and the U^ in the equator. It will obviously 
 be simplest to collate together the U^ in each ray which 
 constitutes at once the origin and the dividing boundary of 
 each half ray. As the unit in the section of the pencil belonging 
 to each half ray is used to collate the half rays, these units must 
 all be collated together, and therefore lue inust revert the units 
 in each ray with respect to the origin as pivot and the section 
 as equator by an unique reversion, which will completely collate 
 the units of the two halves of any one ray. If (Oj 0/), [0^ 0^), 
 (O3 O3') be the first three arbitrarily selected JJ^s out of the (/i+ 1) 
 which determine the system of pencils used to collate the whole 
 group, (Oj 0/) will be the first origin and (0^ O3 0,/ 0,') the first 
 section ; (0,^ O./) will be the second origin, and ( 0^ 0^) the U^ with 
 one of whose units the whole pencil of U^s will be collated in 
 the next stage of the process. Hence, to collate the two halves 
 of the section with each other we similarly revert it uniquely 
 with respect to ( 0^ 0,/) as pivot and ( 0^ OJ) as equator. To collate 
 together the section and the rays of the pencil, if we wish to do 
 so, we may collate together the pivots of reversion and the 
 equators in each case ; thus 0^ in the ray is collated with 0^ in 
 the section, and 0^ in the ray with 0^ in the section. Or we 
 may catalogue the rays and the sections independently. To 
 determine the collation completely, we have, however, to collate 
 together some third unit in each ray and in each section. I 
 will show how these third units may be determined for all the 
 rays of a pencil of the first order by choosing arbitrarily units 
 in two of them, or an unit in one and an unit in the section. 
 
 Select any two units, P^ between 0^, 0^, and P^ between 
 Oj, O3, in two rays of the pencil. Let R be the unit common 
 
PROJECTIVE GEOMETRY. 47 
 
 to (O^P^) and (O^PJ, and between the units named in each. 
 Let Qj be the unit common to [O^R), [0^0^ between 0^, 0^, 
 and P2 the unit common to (P, QJ, and 0^ 0^ between 0^ and 
 0,'. Next, revert the U^ ( 0^ OJ with respect to Pj as pivot, so 
 that (9, reverts into 0^ ; and let any unit X between 0^ and Pj 
 revert into an unit Y between P^ and 0^. And let Z be the 
 unit common to p^A' and P^F, between the units named in 
 each case. 
 
 Now if we pass in review a series of units in the place of 
 A'^, we get a series of units in the place of Z. And these units 
 are common to corresponding rays of two pencils with origins 
 P^, j).^, and with sections which have themselves been projected 
 from one and the same f/,. For Y was obtained from X by 
 reversion, which is only a special case of projection. Therefore 
 the locus of Z is either a f/, or a (7,. It is easy to show that it 
 is the latter, and that it passes through P^, P,^, and /?,/, which is 
 the unit into which P^ reverts with 0, as pivot and as 
 equator. For Z is within the boundary (\0.,P,^, since it is 
 between P^ and F, and F is between 0.^ and 0,. And if we 
 trace X to 0, at the same time we trace F to 0^ and therefore 
 Z to P^. And if we trace X to P, we trace also F and Z to P,. 
 
 The locus of Z will have one unit common with every half 
 ray in the pencil with origin Oj, and section [OJJ^). If we 
 collate together all these units, and the units in the origin 
 and section of the pencil, we shall completely determine the 
 collation of each half ray. And it can easily be shown that 
 the loci of all units collated together are (CJs, no two of which 
 will have a common ^7^. We may imagine the collation to bo 
 effected by passing in review the rays of the pencil as sections 
 of the group of (CJs, and tracing the units to be collated 
 together along these (CJs. This is what I shall call collation 
 of the rays of the pencil by reversion. 
 
 32. To determine the collation, besides the units , 0., O , 
 we chose two more arbitrary units P^, P^. We might instead 
 have chosen the units P^, Q^, arbitrarily ; that is, one in the 
 initial ray and one in the initial section. If in the next stacre 
 of the process we choose a fresh arbitrary unit P^ in {0 0), it 
 will determine a fresh unit in every section. If on the other 
 
48 THE FOUNDATIONS OF 
 
 hand we choose Q^ and another arbitrary unit Q^ in 0^0^, it will 
 determine the arbitrary unit in every ray of the pencil with 
 origin 0^. Thus in the end we have only to choose [n] arbitrary 
 units (PjPj,...P„), or [P^Q.Q^ ... Q„_,) besides the (?2 + l) units 
 {0^0^ .. 0„^^), to determine the collation completely. If we 
 look at the whole collation from the polar point of view, we 
 should naturally choose the units {P^P^... P^^) arbitrarily. If 
 on the other hand we look at it from the cartesian point of 
 view, it would seem most natural to begin by selecting 
 (PjQj... Q,j_i). But it is of course logically indifferent which 
 we do. If we choose [P^Qi ... Q,t_i) arbitrarily in the initial 
 sections, we are choosing the corresponding units in all the other 
 sections by collation by reversion. In fact, the sections in the 
 cartesian way of looking at it are really rays of pencils, and 
 the rays in the polar way of looking at it their sections. 
 
 83. There is one property of unique boundaries which, 
 though not very directly connected with the present discussion, 
 deserves notice, not only on account of its intrinsic beauty, but 
 because of the prominent place it occupies in most works on 
 Projective Geometry ; and this is what is known as the Principle 
 of Duality. We have already seen that we may say that a U^ 
 and a Ug determine a Us common to them in the same sense 
 in which they determine a Up + g^s containing them. And 
 generally, we may say a U^^ is determined by n ( U^^)s just as 
 n{U^)s determine a U^^. To every proposition about {U^)s 
 will correspond one about ( t/„_Js, and so on. It must be noticed, 
 however, that the principle of duality applies primarily to the 
 whole considered as a group of ( U^)s, and not considered as a 
 complete group of units. Now I have several times already 
 spoken of considering the group as a group of {U^.,)s instead 
 of one of units, and the time has come to consider more 
 exactly what this means. 
 
 34. I will take first the simplest case, that of a group cata- 
 logued as a complete group of the first order of units, each unit 
 of which determines another, which together with it forms a f7„. 
 Now we have seen that in cataloguing such a group of units by 
 means of unique boundaries we in the first place, as it were, run 
 to earth a U^, and then dig out of it the particular unit we want. 
 
PROJECTIVE GEOMETRY. 49 
 
 But suppose we stopped at the U^, and so far from attempting 
 to dig out a particular unit we regarded it logically as an unit in 
 itself. In other words, suppose we started by cataloguing the 
 complete group ivithout establishing any unique relation between 
 pairs of units, but retained all the rest of my investigation, in so 
 far as it does not discriminate between the units in a ?7j as it 
 stands. This would indeed logically have been the best way of 
 starting the science, only it would have involved a re-investiga- 
 tion when we wished to get at what I have called the ' complete ' 
 method of cataloguing, whereas by the plan I have adopted, the 
 system in which there is no antipodal relation, which I will call 
 the Monistic system, is really being investigated all the time 
 alongside of the complete system. We have only to say that 
 the (CJs are the units of our collection, and our conclusions 
 about them apply at once. 
 
 35. In the monistic system, if we pass in review the units, 
 that is {U^]s, in a U^, we come back to the one with which we 
 started, and so we do in exactly the same way if we pass in 
 review the rays of a pencil of the first order of unique groups of 
 any order whatever. In a fT, it requires two units, i.e. [11^)8, to 
 form a boundary, and we cannot say that one U^ is ' between ' 
 two others. This introduces a complication into the method of 
 cataloguing, for we can no longer identify an f/j, or any unique 
 group, even approximately, by saying it is ' between ' two units ; 
 these being, on the M system, {UJs. We can only say it is, or 
 is not, in the same part of the group with respect to these two 
 {Ug)s as a third named (Z7J is. It would be interesting to go 
 on and investigate how this might most simply be done, but to 
 do so would be beyond the scope of the present paper. I will 
 merely call attention to a few more salient peculiarities of the 
 M system. A single unit, (?7J, may be finally determined by 
 two {U^)s in a U^, or by n (t/'„_Js in the whole collection. An 
 A^ does not divide a U,^, nor. an A„_^ the whole collection, into 
 two parts, for it is not a boundary to the units of the M system. 
 But a B group of the same order is a boundary, and divides the 
 group into two parts which may be distinguished from each 
 other by the fact that in one we can not have a whole A group 
 of any order, whereas in the other we may. Any A^ has an odd 
 
 4 
 
50 THE FOUNDATIONS OF 
 
 number of units common with any A,._^, but an even number 
 common with any B,._^ in a U,.. Hence an A^ must have at 
 least one unit common with any A,_^ if both are in one U,., but it 
 need not have any at all common with a jB^_j. We can always 
 collate by projection any three units, i.e. {U^)s, in a U^ with any 
 other three. The principle of duality applies without any 
 reservations. 
 
 86. In a complete group on the other hand, that is one 
 catalogued on what I shall call the U system, if we pass in 
 review the units of a U^ we reach first the antipodal of 
 the unit from which we started, and then, in order, the anti- 
 podals of all the units already passed in review, till we come 
 back again to the initial unit. If, however, we pass in review 
 the rays of a pencil of the first order, the case is different, for 
 these rays, considered as units of thought, are catalogued 
 monistically. To make the catalogue one on the U system we 
 have to take half rays, divided by the origin of the pencil. In 
 a i7j two units, or one U^, form a boundary. Hence we can 
 say a t/o is ' between ' two units. We can also, in the special 
 sense I have defined, say one unit is ' between ' two units, if 
 these are not themselves antipodal. Similarly, any unique 
 group of a higher order, being a boundary, can be said to be 
 * between ' two units. A B group is a double group of units, the 
 two parts being antipodal to each other, and it divides the 
 group in which it forms a boundary into three parts, and it may 
 be said that an unit is ' within ' one or other of the antipodal 
 groups, or 'between ' them. An A^ has an odd number of (UJs 
 common with any A group, and an even number common with 
 any B group, as in the M system. We cannot always collate 
 three units in a U^ with three others, but we can always collate 
 two and two, and the third with the third or its antipodal. 
 The principle of duality applies to unique boundary groups, 
 that is a U^ corresponds to a U„_^, and so on, not an unit with 
 a U^,. 
 
 37. We have now to consider certain other ways of cata- 
 loguing groups. There is one in particular which combines 
 certain of the advantages of both the M and U systems, in that 
 it is not necessary to establish an antipodal relation between 
 
PROJECTIVE GEOMETRY. 51 
 
 every pair of units, and on the other hand we are still able to 
 speak of an nnit or an unique group as ' between ' two units. 
 But to explain this I must go back to my original definitions 
 and consider what is actually done when we establish the anti- 
 podal relation between two units. Suppose I have a collection of 
 four units, and arrange them as a group of the first order, abed, 
 d and a being considered contiguous. Here h and d form a 
 boundary which separates a and c. That is to say, if I pass in 
 review from a to c I must pass through h or d. But why not 
 pass straight from a to c ? There is no unit to be passed through, 
 no boundary. The answer is that I have agreed, by cataloguing 
 the collection as a group of the first order, not to do so. I may 
 say I have put a harrier between a and c, and also between 
 h and d, in addition to the boundary formed by {bd) in the one 
 case and [ac] in the other. Why, or how, I put the barrier it 
 is not for formal reasoning to enquire. Now, there being an 
 even number of units in my group, I may, if I please, just as I 
 erected a barrier, forge a connecting link between a and c, 
 h and d, and call them respectively antipodal units. If, however, 
 my collection consisted of an odd number of units, I could not 
 in this way establish an unique relation between pairs of units. 
 I am at present unable to say how far this analogy may be 
 carried in the case of continuous groups, or whether we ought 
 not to say that though cataloguing on the U system undoubtedly 
 implies an even number (however great) of units, yet cataloguing 
 on the M system does not necessarily imply an odd number. 
 But however this may be, the number of units in the U system 
 being even we may divide it by two, and consider a collection 
 of either an even or odd number of units to be catalogued as 
 one half of a f/" group of any given order. I shall call this the 
 semi-complete, or 8 system of cataloguing. 
 
 38. The simplest way of explaining it is to take the polar 
 way of looking at a catalogued JJ group of the order w, and 
 divide it in two by the section, or as I may call it the equator, 
 of the pencil of {U^s with origin [0^0^) ; and consider the S 
 group to be formed of those units only which are in the equator 
 or in the same part of the U group as 0^ is. It will then be 
 seen that in the 8 group there is an antipodal relation only 
 
f>2 THE FOUNDATIONS OF 
 
 between units in the equator, which will form an infinitesimally 
 small portion of the whole continuous group, and which may for 
 many purposes be neglected altogether. The conclusions we 
 reached in the discussion of the U system may now be applied 
 to the S system, by making certain provisos and reservations. 
 In the first place we observe that whenever we have ' run to 
 earth ' a Z/^ in the catalogue of the U group, if that U^ is not 
 in the equator, only one of its units is in the >S^ group ; and 
 therefore the trouble of 'digging one out' is saved us. If we 
 can afford to neglect units in the equator this means an 
 immense simplification of our catalogue. But it must not be 
 supposed that the units of the S group are simply units in the 
 U group. They are units which are defined as one or other of 
 the units of a U^ of the U group. The importance of this 
 distinction will appear if we pass in review the units in the S 
 group corresponding to U^s in a TJ^ of the U group ; say a ray 
 of the pencil of U^s, through 0^. As the U^ under review 
 passes towards the equator, the unit in the S group passes to 
 the equator, and when the U^ is in the equator the unit in the 
 S group is indeterminate. But the moment the U^ has passed 
 through the equator, the unit in the S group reappears as the 
 antipodal in the U group of the unit we were tracing before. 
 Hence, though in one sense the unit of the S group corresponds to 
 the unit in the U group, in another and more important sense it 
 corresponds to the U^. For the equator is not a boundary in 
 the (S^ group. And further, though we can say that one unit in 
 a ?7, is between two others (in the same sense that we can in 
 the U system), yet we can not say that a U^ is a boundary in 
 a U^, any more than we can in the M system, or that a single 
 unit is a boundary in a U^. We can say that a Z7j in a U^ is 
 ' between ' two units, but not that it divides the U^ into two 
 distinct parts. Again, though, as in the U system, we can say 
 that one unit is between two others, in accordance with our 
 convention, yet ive can always collate by projection any three 
 units in a U^ with any other three ; for whether we take the 
 third unit or the antipodal of the third in the U system, we 
 take in the S whichever of them is in the half of the TJ system 
 we are using as a catalogue. B groups in the S system will 
 
PROJECTIVE GEOMETRY. 53 
 
 appear in various forms. Either one or portions of both the 
 antipodal groups may be within it. C groups would be 
 particularly interesting to discuss, but I refrain. If parts of 
 both antipodal groups are within the S group the distinction of 
 ' within ' and ' between ' the groups may be maintained If not, 
 we may say ' within ' and ' without.' In either case they are 
 boundary groups, while A groups are not. Two units always 
 determine a U^ unless both are in the equator. Two U^s in a 
 U^ have always one unit common, and may have two, if they 
 are in the equator. Two U^s in a U^ always have a U^ common, 
 but it may be a t/", in the equator. I need not say more, as 
 these consequences are sufficiently obvious and familiar to us 
 all. 
 
 39. There remains one, or perhaps I should say two, more 
 systems of cataloguing to be discussed. Instead of taking one 
 half of the U group to catalogue a collection I might take less 
 than half, namely, a portion defined by a boundary group in 
 the U group. I might take either one of the portions within 
 one of the antipodal groups, (I shall call this the H system) or 
 the portion between the antipodal groups (I shall call this the / 
 system). It will be seen that there is a very close logical 
 relationship between the two systems, though for practical 
 purposes one may be much more convenient than the other. 
 The boundary in the U group would naturally, if not quite 
 necessarily, be taken to be one of the C groups which we used 
 in collating the pencil of {U^s with origin 0,. Thus the H 
 group would be symmetrical with respect to the origin, and the 
 I group with respect to the section, of the first pencil of ( U^s. 
 Any U^ would either lie wholly in the I group, or there would 
 be only a single portion of it in the H group, since a U^ can 
 not have more than two ( U^s common with a C group. (If we 
 were to take any other B group as boundary we might have the 
 J7j determined by two units in an H group divided into two 
 distinct parts within the H group). The logical units in the H 
 group would be single units of the U group ; in the I group 
 on the other hand we might take either units or Uji, that is we 
 might catalogue the section of the pencil on the M or U, or >S' 
 systems, the only U^s in the I s^^stem which have as yet been 
 
54 THE FOUNDATIONS OF 
 
 catalogued being the rays of the first pencil. It is, however, 
 not worth while for my present purpose to enquire further into 
 the / system of cataloguing. I shall treat it merely as the cor- 
 relative of the H system. It will be found that in most 
 respects the peculiarities of the H system correspond with 
 those of the S, and those of the / with those of the U. We 
 may in fact regard an S group as a limiting case of an H group, 
 when the 1 group has shrunk up to nothing more than the 
 equator ; and similarly an U group as a limiting case of an 1 
 group, when the H group has shrunk up to the origin. 
 
 40. If we pass in review the units of a U^ in the H system, 
 starting from an unit within the boundary, we shall sooner or 
 later come to the boundary and have to stop. There is no 
 unique relation between the terminal units in the U^ as there 
 is in the S system, so we can not in any sense look upon them 
 as contiguous elements. I may mention that this and the other 
 conclusions I am about to draw are not merely due to the way 
 we are looking at the question, namely, considering the H group 
 as part of a complete U group. They can be proved inde- 
 pendently, from considerations affecting units in the H group 
 itself alone. 
 
 In the H system, as in the S and U systems, we can say of 
 one unit in a U^ that it is ' between ' two others. But more 
 than this, we can say that (together with the terminal units, or 
 one of them), it forms a ' boundary.' Similarly in a TJ„ a U,._^, 
 (together with the boundary group), forms a boundary. It divides 
 the whole group into two distinct parts. 
 
 We may have the whole of one of the groups of units in a 
 B group of ( U^s within the H group, or we may have only a 
 part of it, or we may have part of each of the two groups of 
 units. If, however, parts of both groups are in the H group, 
 there is no longer any essential distinction between the three 
 parts into which the H group is divided. It is only by taking 
 into consideration parts of the B group without the boundary 
 that we can distinguish between the part ' between ' and the 
 parts ' within ' the two groups. 
 
 Within the // group an A^ may have an even number of 
 units common with another A group, or none at all. And with 
 
PROJECTIVE GEOMETRY. 55 
 
 a B group it may have an odd number common, for the remaining 
 common units may be outside the H group. 
 
 We cannot always collate by projection any three units in a 
 i/j with any three, unless we are at liberty to take origins of 
 projection outside the boundary of the H group. Nor can it be 
 said that the order of units in two sections of the same pencil 
 of [U^s is identical, unless we neglect the boundary, for the 
 boundary will not in general come in between corresponding 
 units in the two sections. If we ignore units beyond the 
 boundary, there may be no units to collate with certain of the 
 units of the initial section. 
 
 The principle of duality applies in a yet more restricted 
 manner than in the U and S systems. In a U^ two U^s do 
 not always determine an unit or U^ at all in the H and / 
 systems; for they may determine one beyond the boundary, 
 which 'would not count. There is, however, another kind of 
 principle of duality which exists reciprocally between the H 
 and I systems. It may be shown by the ordinary methods of 
 projective geometry, that to each unit in the H group as ' pole ' 
 there corresponds a ( U^^ wholly in the / group as ' polar.' If 
 we allow ourselves to discuss the / system as an ' imaginary ' 
 complement of the // system, we get in this way an * imaginary ' 
 principle of duality which may serve to obscure the anomalies 
 of the H method of cataloguing. 
 
 41. It appears from this discussion that the five systems are 
 not logically correlative. The most simple and logically perfect 
 is the M system, in which the U^ is closed, but no unique 
 relation is postulated between any two of its units, or between 
 any one and the group as a whole. The U and S systems 
 come next in order of simplicity, and the anomalies in the one 
 are in a sense complementary to those in the other. In the U 
 system, like the M, the U^ is a ' closed ' group, whereas in the 
 /S' system it is ' open.' In the U system we postulate an ' anti- 
 podal ' relation between each unit and a certain other unit ; in 
 the S system this is only done in the case of units in ' the 
 equator.' The H and / systems are also complementary to 
 each other, as the *S and U are, the U^ in the H system being 
 open, and in the / system sometimes open and sometimes closed. 
 
56 TEE FOUNDATIONS OF 
 
 But the H, I systems are also complementary in another sense. 
 It is only by arbitrarily shutting our eyes to units ' beyond the 
 boundary' that they can be considered separately at all ; whereas 
 in the U and S systems this is not the case. We may use 
 abusive epithets towards units outside the pale, and call them 
 * imaginary,' but in any complete logical discussion of either the 
 H or the / system they will intrude themselves, nevertheless. 
 
 42. In the course of the discussion so far I have been 
 representing units by letters of the alphabet or numbers, and in 
 certain cases I have assumed that these symbols themselves had 
 a well understood order. But the order in question has never 
 been more than a discrete order, the catalogue which in reality 
 I have employed has been the time conception. And that is 
 why I have not attempted to catalogue groups of more than the 
 first order, except by collation of unique groups as pencils of the 
 first order. The time conception is the first and best understood 
 conception of a continuous group, and ' passing in review ' the 
 units in a group of the first order is, in effect, nothing but 
 collating them with units of the time series, used as a catalogue. 
 Until the group of numbers became ' well understood,' as a con- 
 tinuous group, it was not available as a catalogue for continuous 
 groups; and accordingly in primitive attempts at cataloguing 
 such groups it is not employed. I cannot say historically when 
 it first occurred to anybody to employ it so ; but the first attempt 
 is popularly associated with the name of Descartes. He however 
 did not regard the question from quite so abstract a point of 
 view as I have done, and it will be worth while to go over the 
 ground again in relation to my theory. 
 
 The advantage of cataloguing groups of the first order by 
 means of numbers is that they constitute a group which we can 
 ' carry in our heads,' so that we instinctively recognise its ' order ' 
 in the vulgar, discrete, sense of the word. But before we can 
 apply it to the cataloguing of continuous groups we must under- 
 stand its order in the extended meaning of the term. We must 
 be able to 'project' the number series, that is, to apply such 
 transformations to each of its units as will enable us to collate 
 any three with any other tlirce, but which shall then determine 
 uniquely what fourth is collated with any given fourth. 
 
PROJECTIVE GEOMETRY. 57 
 
 43. It is evident that whatever transformation we apply 
 must not alter the order of the units in the discrete sense, though 
 it may be that not every possible transformation which fulfils 
 this preliminary condition would in other respects be suitable. 
 The most general transformation of this kind would in mathe- 
 matical language be described as ' taking a one- valued function ' 
 of each of the numbers. But a few preliminary considerations 
 will enable us to divide the business of ' taking ' this function 
 into stages, and so simplify the matter. 
 
 In the first place, if we call an unit a, in a group of the first 
 order, '1'; J, '2'; and c, '3,' then, if we want to call i, ' 1 ' instead 
 of a, we may call c, ' 2/ and a, ' 0.' This is obvious in the case of 
 discrete groups. But further, it is generally obvious that we 
 may always call any one unit by any number we please, by add- 
 ing or subtracting from each number in the series an appropriate 
 constant. The only units whose numbers will remain the same 
 in spite of this treatment, will be the units we have called oc 
 and — X . 
 
 Or else if we wish to call a * 2 ' instead of ' 1 ,' we may do so 
 by multiplying each unit in the series by 2, instead of adding 
 one to each unit. In this way we may call any unit by any new 
 number, except that the units numbered zero and + oc will 
 remain unaltered. By these two methods we can change the 
 numbers of any two units for any other two, and so we have 
 only to determine in addition a 'function' to change the number 
 of a third in an assigned manner. 
 
 44. To do this, let us proceed thus. Suppose the series of 
 numbers to be represented by the algebraical symbols a, /S, 7, S, e, 
 and so on ; the symbols being ' collated ' with the numbers, but 
 not employed to catalogue them alphabetically. 
 
 First subtract a constant number, say a, from each. Then 
 take a one-valued function F, which I may call the ' projective 
 function,' of each of the differences. Next subtract again a 
 constant number, this time F{^~a), from each. And lastly 
 multiply the number so derived in each case by a constant 
 number, h. The derived series of numbers is 
 
 h{F{o]-F{0-o^)]; 0; h{F{y~a)-F{^-a)}; 
 h[F{B-a)-F{l3-a]}, etc. 
 
^a THE FOUNDATIONS OF 
 
 I may call this process projecting the series of numbers, for it 
 collates the units in the original series with the corresponding 
 ones in the derived series. 
 
 Now project the series all over again, using a diffeient 
 function, F' instead of F, and a different constant, h' instead of 
 h, if we prefer to do so ; putting any three numbers a'/SV which 
 it is desired arbitrarily to collate with a yS 7 in their places, 
 respectively, in the process. They will be collated through the 
 two derived series as an ' intermediate section,' if we can identify 
 the numbers derived from them respectively. 
 
 The numbers derived from /3, /S' are already identical, both 
 being zero. The numbers derived from 7, 7' can be made so 
 by choosing the ratio h : h' suitably. And whatever ratio is 
 selected the numbers derived from a, a will be identical if F{0) 
 and F' (0) are both infinite. And now, if S' is the number 
 collated with 8, it is determined by the equation 
 
 F{8-a)-F{^-a) _ F'{B'-a.')~F'{0'-a') 
 F{ry-a.)-F{^-a)~ F'{Y-a')-F'{/3'-a')' 
 
 The only conditions which, so far, limit the functions i^'and F\ 
 are that they shall be one-valued, and that they shall be infinite 
 for zero value of the argument. They need not necessarily be 
 the same function. 
 
 This ratio, which I may call k, is the relation between four 
 numbers, in a continuous series, which remains unaltered by 
 projection. It defines, in fact, their ' order ' in the continuous 
 sense. With a', ^', and y constantly collated with a, /3, 7, any 
 value of k determines two units B and 8', which are collated 
 together. 
 
 The particular projective function, F, to be adopted in any 
 case, will depend on the system of cataloguing. But where we 
 are dealing with rays of pencils of iU^)s catalogued all on one 
 principle, as in reversion, it will be the same for all. 
 
 45. Now suppose that the (continuous) order of the units 
 a, /3, 7, 8 is the same as that of a, /8, S, 7 (it cannot, of course, 
 be the order in which I have written the symbols). That is, 
 suppose we revert 7 into 8 with respect to (a, /S) as pivot and 
 equator. Then 
 
PROJECTIVE GEOMETRY. 59 
 
 F[h-a)-F{^-a) _ F {ri-a.)-F[^-a) 
 F{y-a)-F{i3-a]~F{B-a.)-F{0-a)' 
 .: [F{S-a)-F{8-a)Y={F{j-a)-F{^-a)Y. 
 
 Therefore either F{y-a)= F{B-a) (i), 
 
 or F[8-a) + F{y-a) = 2F{^-a) (ii). 
 
 Now equation (i) would represent a ' singular ' reversion, for 
 only three units enter into it; and singular reversion is only 
 possible if we discriminate between antipodals. In this case 
 therefore 7 and S must be antipodal, namely the units in the 
 equator of the reversion, and equation (ii) would not be true 
 unless 13 were identical with one or other of these units. In 
 the alternative, that is if 7, 8 are not antipodals in the U system, 
 or in any case in any other system, reversion is represented by 
 equation (ii). We may say that ordinary reversion is always 
 represented by (ii), and that in the U system, where alone it is 
 generally possible, singular reversion is represented by (i). The 
 former case is given by k = — 1, the latter by /^ = + 1. 
 
 46. We may now investigate the series of numbers appropriate 
 to the various systems of cataloguing. In every case it must be 
 remembered that any one unit, or U^, in a f/,, may be catalogued 
 by any number. There is no reason why more than one number 
 should not be assigned to each unit, or U^, but to each number 
 there must unambiguously be assigned one unit, or U^, as the 
 case may be. 
 
 In the M system there is no distinction between any one 
 unit of thought ( UJ and another, and if we pass in review the 
 whole group, starting from one numbered, say zero, we shall in 
 the end come back to the same U^ ; and this time suppose we 
 number it s. If we pass the whole group in review again, when 
 we return to the initial unit we shall number it 2s, and so on. 
 If we start with any other U^, numbered a, when we come back 
 to it it will be numbered (a + s), when we get to it a second 
 time (a + 2.s), and so on. Hence (a+ns), where n is any positive 
 or negative integer, must be collated with the same U^ as a. 
 If now we perform the process of projection upon the series of 
 numbers so obtained, since (a + ns) represents the same U^ as a, 
 F [a -\- ns) must represent the same U^ as F (a). And, moreover, 
 the function of a plus anything but an integral multiple of s. 
 
60 THE FOUNDATIONS OF 
 
 must not represent the same U^ as function of a. Hence the 
 projective function, which in this case we may denote by M, 
 must be a periodic function with the single period s. Either 
 the tangent or cotangent would answer this condition, but since 
 it must be infinite for zero value of the argument, it must be 
 the latter. Hence 
 
 M[<T) = cot-a. 
 
 ^ ' s 
 
 Similar considerations show that in the case of the U system 
 the projective function must also be periodic. But in this case 
 we have the anomaly that we can not always project three units 
 on to any other three — there is an ambiguity as to whether it 
 is in any case the unit we want or its antipodal, on to which 
 we have projected a given unit. In the cataloguing by numbers 
 the corresponding ambiguity must be that after projection we 
 do not distinguish between a fourth unit and its antipodal. 
 Hence in this case it may be that the projective functions for 
 an unit and its antipodal are identical. This we find to be the 
 case. For if t be the whole cycle, so that [a + nt] is collated 
 with the same unit as a, and if 7 be the antipodal of a, as we 
 pass in review the units from 7 to (a + 1) the antipodal of the 
 unit under review passes from a to 7. Hence 
 
 a + c — 7 = 7 — a or 7 — a=-. 
 
 And consequently the equation representing singular reversion 
 shows us that 
 
 v{e) = jj{6^^^ 
 
 and as in the case of the M system we deduce 
 
 CA(^) = coty6>. 
 
 In the S) system the only two antipodal units in a JJ^ are those 
 in the equator. We must therefore be able to revert one of 
 these into the other with any pivot, by singular reversion. 
 That is, if we call them 7, S, we must have 
 
 ^(7-a)=ASf(8-a) 
 for all values of a. Further, there must be no units in the 
 series of numbers beyond 7 and 8, in either direction. Hence 
 
PROJECTIVE GEOMETRY, 61 
 
 7 and S must respectively be positive and negative infinity; 
 and as in addition *Si (o) = cc , we may put 
 
 where r is any constant number. 
 
 In the HI systems, as in the case of the S, we can have no 
 units in the series of numbers beyond those collated with the 
 boundary — at least any numbers which there are must for some 
 reason be held not to count. The easiest way of finding the 
 appropriate projective function is to deduce it from that of the 
 *S system. The boundary in either case is a C group whose 
 units, on the >S' system, are all collated together. Hence on the 
 S system they all have the same number. As in the S function 
 we have an arbitrary constant at our disposal it does not matter 
 what number we choose, so for simplicity I take unity. Hence, 
 if 4> b^ the number collated on the H system, -^ that on the / 
 system, with the unit collated with p on the S system, we must 
 have <^ and y^ both infinite when p is unity, and when p is zero 
 (^ must be zero, when p is infinite >/r must be zero. These con- 
 ditions are fulfilled by putting 
 
 0=plog^, .^ = ^log^, 
 
 p = tanh -^ , p = coth -^ , 
 
 V q 
 
 H {(f>) = coth -<f), I (i/r) = tanh - 1^. 
 
 Now in reality either of these systems of numbers catalogue 
 the whole of the U gioup, for the / part is catalogued by the 
 imaginary numbers on the H system, and the H part by 
 imaginary numbers on the /. For symbolic purposes therefore 
 there can be no possible objection to the HI systems. It is 
 only when we try to give them real import that any difficulty 
 arises. 
 
 To transform from the S to the U or M systems as we have 
 done for the H and / it is clearly only necessary to write 
 
 p = tan ^ 6 = tan - a. 
 '^ t 8 
 
 47. On any system of cataloguing it is convenient, though 
 
63 THE FOUNDATIONS OF 
 
 not necessary, to assign the numbers so that the fundamental 
 reversions in the process of cataloguing shall be purely arith- 
 metical reversions, that is, that they shall be effected with 
 reference to zero as a pivot by a mere change of sign. By this 
 convention if yS be the unit collated with the equator in the U 
 
 system, and /3 — - that collated with its antipodal, also in the 
 
 equator, 
 
 And if 7 be the unit collated with Pj, or y^, and we move the 
 zero of the series of numbers to Pj by substracting 7 from each, 
 we must have 
 
 ^-7 = 7, .. 7 = 2^8- 
 In the M system, by similar reasoning, the {JJ^s on the 
 equator will be collated with + - those in the G^ with + -. In 
 
 the S> system those in the equator, as we have already seen, are 
 collated with + cc , and those in the G^ by this convention will 
 also be collated with + cc , that is, they will not be distinguished 
 from the units in the equator. In the H system the units in 
 the equator will be collated with the 'imaginary' number iir, 
 which will be collated with the origin in the 7 system. From 
 the point of view of pure logic 'imaginary' numbers are just 
 as good as ' real ' ones ; we might use them just as effectively 
 for cataloguing a library (for the use of mathematicians onlyj. 
 To the mathematician who employs numbers to catalogue a 
 collection on the H system, the / system will always be present 
 in imagination ; he will call it ' imaginary.' It is only when we 
 try to give real import to the term that ' imaginary ' comes to 
 mean ' not real.' 
 
 48. I have now laid the foundations of the science of the 
 Cataloguing of Continuous Groups; but before proceeding to 
 consider its applications I wish to impress upon you very 
 strongly that it is not geometry at all. It is not only not 
 metrical, and not spatial, but it is not even necessarily numeri- 
 cal I have shown how groups may be catalogued by the aid 
 
PROJECTIVE GEOMETRY. 63 
 
 of the number series, but I have also shown how they may be 
 catalogued without it. Geometry, and also Arithmetic, are 
 developments in special directions, or rather special applications, 
 of a far more general or abstract theory, which Mr. Kempe calls 
 the theory of Mathematical Form, and which includes the 
 theory of cataloguing of both continuous and discrete groups, 
 with or without the aid of numbers. It is almost inevitable in 
 thinking of the theory of continuous groups, at least of those of 
 lower order than the fourth, that we should attach more or less 
 spatial import to the terms. If, however, I have succeeded in 
 my exposition, it will be clear that this spatial import is not 
 necessary to the logical validity of the arguments employed. 
 The ' experience ' upon which they depend is not experience of 
 * externality,' except in the somewhat ambiguous sense in which 
 Time is spoken of as a ' form of externality.' 
 
 49. I may further point out that I do not, of course, claim 
 any originality for the propositions I have proved, considered as 
 propositions in ' projective geometry.' The object of the exposi- 
 tion has rather been to show that they are not essentially 
 geometrical propositions at all. I accept Mr. Russell's authority 
 for saying the philosophical principles of Projective Geometry 
 required further elucidation, though I cannot accept his exposi- 
 tion as a further elucidation of them. 
 
64 THE FOUNDATIONS OF 
 
 PART III 
 
 But if it is clearly realised that the science of the cataloguing 
 of continuous groups is not per se geometry, I may go on to 
 show how it is related to the latter science. And this can, I 
 think, be done without prejudging the question whether space 
 ' is ' or ' is not ' Euclidian. 
 
 Mr. Russell, after revelling in contradictions in a manner of 
 which only a modern philosopher could fail to see the humorous 
 side, arrives at the conclusion that we must " give every geo- 
 metrical proposition a certain reference to matter in general," 
 and thus distinguish a point from a position. He might have 
 found this conclusion much more simply expressed in my 
 Foundations of Geometry* I gave as the implicit definition 
 of position, the two assertions " (a) A position may be conceived 
 to be indicated by a portion of matter, called a point, which is 
 so small that for the purpose in hand variations of position 
 within it may be neglected ; (6) But a position is not the same 
 thing as a point, for a point may be conceived to move, that is 
 to change its position, whereas to talk of a position as moving, 
 is a contradiction in terms." 
 
 I carried out the same idea by distinguishing between a 
 (material) line and a ' path,' between an angle, the concrete 
 measure, and an ' inclination,' the abstract relation. I propose 
 to adopt similar distinctions here ; for I could not, like a modern 
 philosopher, go on, after admitting that my definitions contained 
 " a fundamental contradiction." I must however point out that 
 I do not, and did not in my former treatise, use the term 
 ' matter ' in an objective sense, any more than, I presume, 
 Mr. Russell does in the passage I have quoted. The geometry 
 I was discussing then, and which I am discussing now, is purely 
 subjective, or ideal, geometry. 
 
 It is obvious that the group of positions in a line is a group 
 of the first order, which we may, and habitually do, collate with 
 
 * Deighton Bell and Co., 1891, 
 
PROJECTIVE GEOMETRY. 65 
 
 the time series directly. I imagine myself as a material point 
 passing along the line. The line may be conceived as open or 
 closed, and taken by itself I might, if it is closed, catalogue its 
 units on the M or U systems, or if open, on the H or S. So"'iar 
 it is obvious that the choice of system is arbitrary. But further, 
 if it is regarded solely by itself, there is no logical reason for 
 regarding it as open rather than closed. The distinction lies in 
 the fact that, in conceiving the line * in space,' I have already 
 catalogued its points by means of my conception of space as a 
 catalogue. This would prevent my passing in review from one 
 end of a terminated line to the other ; such a line must there- 
 fore be catalogued on the H system, if it is to be catalogued by 
 itself The conception of an unterminated but open line, if we 
 do conceive such, would, as we shall see presently, indicate that 
 we had adopted the S system of cataloguing in our conception 
 of space. The conception of a straight line has undoubtedly 
 the property that ' in general ' it is determined by two points, 
 that is, we conceive the positions in a straight line as a U^ of 
 positions, and those positions are the logical units of our 
 catalogue ; though, if it is an M catalogue, they are what I have 
 called (?7Js. Further, it will be easily admitted that the group 
 of positions in a surface is a group of the second order, and the 
 group of positions in a volume one of the third order. In a 
 closed or bounded surface, a closed line, or one terminating in 
 the boundary, divides it into two distinct parts, and in a 
 bounded volume a surface terminated in the boundary divides 
 it into two parts. And as we conceive three points not in a 
 straight line as in general determining a plane uniquely, the 
 positions in a plane form a U^. As, however, we do not con- 
 ceive any positions outside a group of the third order, as this 
 group is our whole collection of positions, there is 'prima fade 
 no point in calling our whole space a U^ rather than a G^. 
 
 Now, how do we ' collate ' our conceptual space ? We do 
 so by ' the method of superposition.' That is, we pass in review 
 the series of positions occupied by points in a line or surface, 
 which is supposed, in virtue of the axiom of congruence, " Es 
 existiren in sick feste Korijer" to remain identical with itself. 
 We do not, however, conceive the points to pass along straight 
 
 5 
 
66 THE FOUNDATIONS OF 
 
 lines which meet in a point — we do not collate by projection. 
 The line or figure in superposition moves, that is, changes its 
 positions, by either ' rotation ' or ' translation.' But if our space 
 is catalogued on the U or M systems these two movements are 
 identical in nature, if on the S system the only distinction is that 
 in translation the axis of the rotation is in the equator, and on 
 the H or I system that it is beyond the boundary, so that trans- 
 lation is an ' imaginary ' rotation. We may say, therefore, that 
 superposition is effected by one or more rotations, and in one 
 rotation the axis of rotation 'remains fixed,' i.e., is always collated 
 with itself, and the positions through which the other points of 
 the figure pass are C^s. In fact, collation by superposition is 
 collation by reversion. The locus of each point is a group of 
 positions of the first order, such that no three positions in it 
 are in a straight line. The further properties of a circle — 
 which distinguish it from other conies — are due to the assump- 
 tion that something more than the order of the points in a line 
 remains identical as we move it about. Of this I shall say 
 more presently. 
 
 Now our conceptions of space are essentially egoistic, / am 
 am the hub of my own universe ; and it is in reality always in 
 relation to a position conceived to be occupied by myself that 
 I conceive space. And with reference to that position I 
 conceive it to be perfectly symmetrical. This leads at once to 
 the rejection of the J system as incompatible with our con- 
 ception of space; for from that point of view, as we have seen, 
 space would be regarded as symmetrical with respect to a plane, 
 but not with respect to a point. We look at space primarily 
 from the polar, not the cartesian, point of view. Now from the 
 polar point of view we may, if we please, look upon one of the 
 C groups, the locus of P^ say, as the section, instead of the 
 equator. That is to say, we may catalogue the rays of the 
 pencil of (Z7Js by the units or {U^)s in them in this Cj. In 
 the case of space in the same way we may catalogue straight 
 lines radiating into space in all directions from any origin by 
 the positions of points in a sphere with that origin as centre. 
 In this way we may obtain at once a catalogue of the rays, 
 without begging the question whether the rays themselves are 
 

 PROJECTIVE GEOMETRY. 67 
 
 catalogued on the M, U, S, or H system. It is clear that the 
 two points on one ray are related as antipodals, and that the 
 catalogue of the ' celestial sphere ' is on the U system. 
 
 The logical consequences which would flow from the 
 adoption of the U, S and H systems have been sufficiently 
 worked out by the meta-geometers, and they have further been 
 illustrated by actually cataloguing groups of points on various 
 surfaces in space, with or without the assumption of an axiom 
 equivalent to the axiom of congruence. In the illustrations of 
 two dimensional spaces by surfaces of constant curvature such 
 an axiom is assumed, though in the case of surfaces of negative 
 curvature it is not assumed in quite the natural sense. In 
 illustrations of three dimensional hyperbolic spaces the axiom 
 is, however, frankly abandoned. Nevertheless, Mr. Russell 
 regards these illustrations as necessary to " prove the possibility " 
 of U and H spaces, and I shall therefore give a similar illus- 
 tration of an M space ; after which Mr. Russell must hold it to 
 be just as " possible " as the others. 
 
 Mr. Russell gives an example of what he holds to constitute 
 an objection to M geometry, the geometry of Klein's ' elliptic 
 space.* I will reproduce what he says in a form slightly altered 
 to suit my present purpose. Suppose I take a triangle, one 
 corner of which is one yard in front of me, one corner one yard 
 to the right, and the third one yard to the left ; and I move 
 this triangle, with tlie first corner and the middle point of the 
 opposite side remaining in a fixed straight line, ' round the 
 universe ' ; so that the corners to the right and left of me return 
 to where they were before. If ' the universe ' were a pencil of 
 rays grouped on the M .system, the result would be that the first 
 time the triangle came round its vertex would be behind me, 
 and I should have to move it round a second time in order to 
 get it in front of me again. Mr. Russell says this "seems like 
 an action of empty space ; " and though I can not quite follow 
 his reason for rejecting this appearance, I quite agree with him 
 that it is not so. After what I have said about the M system 
 the analogy I am about to propose is obvious. Instead of the 
 triangle as we have conceived it, push the triangle a couple of 
 yards upwards into the air ; and for each corner substitute the 
 
68 THE FOUNDATIONS OF 
 
 straight line determined by my position and the corner ; for each 
 side substitute the plane determined by me and the side. Both 
 lines and planes so substituted are to be continued upwards and 
 downwards through my position, indefinitely. Now in order 
 that the lines representing the corners of the base may return 
 to their initial situation, it is only necessary that the plane con- 
 taining them should be related through two right angles. But 
 when this is done, the line representing the vertex will not have 
 returned to its initial situation. To bring all three into con- 
 gruence at once, we must rotate through four right angles. 
 
 And generally, just. as the points on a sphere illustrate the U 
 system, so the diameters of the sphere illustrate the M system 
 of cataloguing. 
 
 It might thoughtlessly be objected to this illustration that 
 after all it was not a genuine illustration of M space, since lines 
 are not points. Now if the illustration is to be regarded as 
 showing that it is possible that we do catalogue space on the M 
 system, this objection is perfectly valid. Only, there are similar 
 objections to all the illustrations which have been given of U 
 and H spaces. The great circles on a sphere are not straight 
 lines, any more than the diameters of it are points. The 
 geodesies of a surface of constant negative curvature are not 
 rigid lines, though they are rigid against bending in the surface. 
 We have, in both cases, substituted new conceptions and called 
 them by old names. The same is even more obvious when, for 
 example, Poincard substitutes for ' distance ' the logarithm of a 
 certain anharmonic ratio. All these illustrations illustrate the 
 logical processes involved in meta-geometry, but they do not in 
 the least degree illustrate the psychological processes which 
 would be required to conceive space as non-Euclidian. The 
 question whether the space we conceive is Euclidian or not, is 
 not a logical question at all, but a psychological one. It can be 
 determined with certainty ; not indeed with the verbal certainty 
 of formal logic, but with the absolute certainty of direct intuition. 
 I have already attacked this question of subjective geometry in 
 my book on the Foundations of Geometry, and from the point 
 of view which I there took up I have nothing particular to add. 
 I showed there, upon what I still believe to be perfectly sound 
 
PROJECTIVE GEOMETRY. 69 
 
 logical and psychological principles, that our subjective geometry 
 is Euclidian, and no other. It may, however, throw some light 
 upon what T still consider to be psychologically the fundamental 
 elements of our space-conception (position, and direction, con- 
 ceived as independent of each other), to bring the views I then 
 expressed into relation with the theory of the cataloguing of 
 continuous groups, 
 
 I have already pointed out that in the latter theory collation 
 by reversion corresponds logically to the method of superposition, 
 but that psychologically the axiom of mobility, or congruence, 
 implies more than this. The theory of cataloguing by reversion 
 would allow us to collate together points on an ellipse just as 
 well as points on a circle, in the rays of a pencil of straight lines 
 of the first order ; it would not even be necessary that the origin 
 should be the centre of the ellipse. The conception of distance 
 as a relation between the positions of points in a straight line, 
 which is unaltered when the straight line is moved so that the 
 points occupy new positions, is psychologically a different con- 
 ception from that of the order of the points. And consequently, 
 if we conceive this sort of relation between the positions of space, 
 space is no longer any kind of continuous group. I have already 
 shown that it follows at once, that it is not an / group. If this 
 were the only new psychological relation assumed between the 
 positions of ' empty space,' it would remain indeterminate 
 whether the conception was that of an M, U, S, or H group. ?\ 
 But as a matter of fact we conceive another relation between 
 two positions besides distance, namely direction. It is commonly 
 said that this conception is not independent of the other, that 
 no meaning can be attached to the statement that the direction 
 from A to B is the same as the direction from C to D, unless all 
 four points are in one straight line ; or else, unless we assume 
 space to be Euclidian. This argument is logically sound, but 
 psychologically it is putting the cart before the horse. We do 
 attach a meaning to the statement, and that proves that our 
 space-conception is Euclidian, and no other. I need not repeat 
 what I have said elsewhere on this point, but will consider more 
 particularly some of the psychological absurdities to which the 
 statement that our conceptual space is non-Euclidian would 
 
70 THE FOUNDATIONS OF 
 
 lead us. I must first, however, point out one important philo- 
 sophical consequence of the psychological import of the two 
 conceptions of distance and direction, as relations between two 
 positions. It is that these elementary conceptions define not 
 only the form in which we conceive particular figures, but also 
 the form in which we conceive ' empty space ' as a whole. They 
 are independent, not only of each other, but of what Mr. Russell 
 calls the axiom of dimensions. Just as we might conceivably 
 have a different conception of direction, such, for example, as 
 would be applicable to an H or U group, so we might conceivably 
 have a different conception of the number of dimensions of our 
 ' empty space,' without our other elementary space-conceptions 
 being changed. We might in this way conceive space as. four- 
 dimensional, and I believe that, psychologically, this change 
 would be much more easily effected than any other. In my 
 Foundations of Geometry, I even went so far as to express the 
 opinion that we might come to conceive a space of four dimen- 
 sions in the course of a single life-time. But I do not think it 
 would be possible for most men, probably not for any, to obtain 
 a new conception of direction in the same sense that they have 
 the Euclidian conception ; and philosophers in general seem to 
 be agreed (though they do not express their agreement in this 
 form), that it would be impossible to alter our conception of 
 distance. It is admitted that meta-geometrical spaces are 
 * Euclidian in their smallest parts,' and that therefore the 
 Euclidian conception of space would be applicable to small 
 figures, even if not to large ones. I am convinced that if we 
 should find the Euclidian conception which we have of space 
 inapplicable to large objective figures, the easiest way out of 
 the difficulty would be to try if we could not conceive material 
 space as a boundary in an * empty ' Euclidian space of four 
 dimensions. However, this sort of speculation has little practical 
 value, and I return to more important considerations. 
 
 The most obvious of them is found in the answer to the 
 question — Do we conceive a straight line to be open, or closed ? 
 The answer to this is indeed so obvious that very little attention 
 has been given to Riemann's spherical space, and still less to 
 elliptical, or monistic, in comparison with ' hyperbolic ' space. 
 
PROJECTIVE GEOMETRY. 71 
 
 It was perceived that, logically, spherical space was as valid as 
 hyperbolic — that the one was just as a priori possible as the 
 other. That, however, spherical space has never been considered 
 to be a practical possibility — that, for example, no astronomer, 
 as far as I am aware, has attempted to measure large triangles 
 for excess, but only for defect, is due to an instinctive recognition 
 of the importance of psychological considerations, combined with 
 an erroneous ascription to them of objective validity. It has 
 indeed been urged that we really conceive only a small part of 
 a straight line, just as, objectively, we can measure only a small 
 part of it; and that beyond the part we conceive it might return 
 and close on itself The answer, of course, is that the straight 
 line in our conceptional space is what we conceive it to be, and 
 nothing else. If I imagine a straight line which is not closed, 
 to be produced and then to become closed, I have altered my 
 original conception, and the produced line is no longer * straight ' 
 in the sense in which it was before. Moreover, I conceive shape 
 to be independent of size, and to say a closed line is infinitely 
 great does not alter my conception of its shape in the least 
 As, however, this may be held to beg the question at issue, for 
 it has been noted that Euclid's disputed axioms might be 
 deduced from the postulate that shape is independent of size, 
 I will waive this point, and examine some of the further 
 consequences of assuming that a straight line may be closed. 
 If I conceive two straight lines in a plane, each of them 
 perpendicular to a third, it may be shown by the method of 
 superposition that the figure is symmetrical with respect to the 
 third line ; that is, that the other two must intersect in both 
 directions from the third line, or in neither. Now, as far as I 
 conceive the two straight lines, they certainly do not intersect — 
 I should not call them 'straight' if they did; and to say that 
 the straight lines which I conceive intersect somewhere else, 
 where I do not conceive them as intersecting, is to talk nonsense. 
 Subjectively, therefore, I hold that Euclid was perfectly correct 
 in saying that parallel straight lines do not intersect at all. But 
 as long as we are admittedly only talking symbolically it is quite 
 permissible to talk of parallel straight lines as intersecting ' at 
 infinity ' ; that being a locality which it is understood that we 
 
72 THE FOUNDATIONS OF 
 
 do uot conceive at all. In symbolic reasoning cc is no better, 
 and no worse, than any other symbol, if it is consistently used. 
 But as soon as we attempt to give any real subjective import to 
 such an expression as 'an infinite distance' or 'points at infinity' 
 we find we are unable to do so ; they remain purely symbolic 
 terms. This might be a serious drawback to our space-conception 
 were it not compensated for by the fact that we conceive shape 
 to be independent of size, and that consequently we may discuss 
 what happens at home without troubling ourselves about what 
 might or might not be said to happen 'at infinity.' It is as 
 easy to conceive the motion of the solar system through space, 
 as that of a vortex ring constituting an atom of matter, for we 
 may discuss the shape of figures independently of their sizes. 
 But we can not in any true sense be said to conceive ' points at 
 infinity'; and though to talk of such points is not logically 
 false, " non est geomietria." Geometry used to be regarded as 
 the most perfect of sciences, because its terms could all be ' well 
 understood ' ; and old fashioned geometers took care so to 
 exhibit their proofs that this should always remain so. The 
 modern habit of employing purely symbolic arguments, and still 
 calling the science geometry, is calculated to lead to serious 
 logical misconceptions — if indeed it does not indicate that they 
 exist already. I know some people who believe that the 
 * asymptotes of an ellipse,' or the ' circular points at infinity ' 
 really exist somewhere, only that our drawing boards are not 
 quite big enough to hold them. It ought to be clearly under- 
 stood, that when we begin to talk about what happens ' at 
 infinity ' we have left the ground of pure geometry and have 
 launched forth into abstract analysis. 
 
 However, the way of abstract analysis is a good way of 
 acquiring knowledge, if it is properly employed, and if we are 
 careful not to use our terms inconsistently. It is perfectly 
 legitimate to say, symbolically, that two straight lines with a 
 common perpendicular intersect ' at infinity ' — if we say that 
 they intersect in both directions from the common perpendicular. 
 But it is not permissible to speak of them as intersecting on a 
 straight line, or plane, 'at infinity,' as is so often done. For, 
 either we must say that the intersections in the two directions 
 
PROJECTIVE GEOMETRY. 73 
 
 from the common perpendicular are one and the same point ' at 
 infinity,' or that they are two different points. If we say 
 they are the same point, then the straight lines are regarded 
 as limiting cases of closed curves, and if we transfer our 
 attention to the point ' at infinity ' where they intersect, we see 
 at once that we are regarding space as a limiting case of one of 
 Klein's elliptic spaces, or as I should put it, that we are cata- 
 loguing positions on the M system, with infinity as the period 
 of our projective function. In this case the locus of the inter- 
 sections of straight lines in a plane which have a common 
 perpendicular within a finite distance of the origin, would be 
 the infinite plane itself, and not a straight line in that plane, 
 'at infinity.' Similarly the locus of the intersections of such 
 lines in space would be infinite space itself, and not a plane 'at 
 infinity.' If on the other hand we say that two straight lines 
 which' have a common perpendicular within a cognisible distance 
 of us, i.e., the parallel straight lines of Euclidian geometry, 
 intersect in two distinct ' points at infinity,' these two points 
 are uniquely related as antipodal points, and their locus is, not 
 a plane in the Euclidian sense of the term, for the points in a 
 plane have no antipodal points, but what may appropriately be 
 called a great sphere ' at infinity.' To use the same term for 
 this great sphere and an Euclidian plane leads to the 
 ' antinomies.' In this way of looking at it we regard Euclidian 
 space as the limit, not of a Reimann's or spherical space, but of 
 what I have called an S group of the third order ; and it has 
 this great advantage — that it does not involve saying that straight 
 lines are closed line?, and therefore it constitutes the best sym- 
 bolic representation of our space-conception. We do as a 
 matter of fact conceive, not a boundary to our space, but a 
 ' celestial sphere ' ever so far off, beyond which we do not 
 trouble ourselves to conceive anything. The points in which 
 we conceive any straight line to reach this celestial great sphere 
 are antipodal, and not identical, points. We can not, as I have 
 said, be truly said to conceive this great sphere, but we can, 
 quite naturally and easily, talk about it ; and so avoid certain 
 latent antinomies which surely we must all have suspected in 
 the ' plane at infinity.' 
 
74 THE FOUNDATIONS OF 
 
 It is worth while in this connection to point out an arith- 
 metical error Avhich is sometimes made, from a fancied 
 geometrical analogy. It is sometimes thought that positive 
 and negative infinity are arithmetically indistinguishable. We 
 can not indeed say that the difference of two infinites is zero, 
 unless we know by what process the limit was reached in each 
 case. But even if we distinguish the two infinities by suffixes 
 and write 
 
 we can add to both sides of the equation identically the same 
 infinity as the second, and so obtain, unambiguously, the result — 
 
 + cc, + x^ = 0, 
 that is, the sum of two positive quantities equated to zero; 
 which is a contradiction in terms. 
 
 We may of course collate + cc and — a; with the same unit 
 in a group of the first order, just as well as we may collate 
 + TT and — TT. But this is arithmetically equivalent only to 
 equating a function of infinity with the same function of minus 
 infinity, and it only shows that the function is a periodic one. 
 The geometrical equivalent of this is the fact that I have 
 already noted, that if we say two straight lines which have a 
 common perpendicular intersect in one point only at infinity, 
 then the straight line is a limiting case of a closed curve. It is 
 of course only if we regard the straight line as open that the 
 distance measured along it between two positions is unam- 
 biguously determined, or that it can be unambiguously 
 represented by the difference of the numbers collated with the 
 positions. It is therefore only in this case that we can regard 
 the positions and numbers as identified, and not merely collated. 
 
 In my Foundations of Geometry, Part III., I have shown 
 that if we take the axiom of mobility in its natural sense, that 
 is, that neither the size nor the shape of geometrical figures 
 depends on their position in space, no mere extension of the 
 number of dimensions would allow us to conceive space as an 
 hyperbolic (though we might conceive it as a spherical) boundary 
 group in a higher group. The ' possibility ' of hyperbolic space 
 in this sense depends upon our ignoring changes of shape which 
 do not affect ' sizes,' as determined solely by measurements in 
 
PROJECTIVE GEOMETRY. 75 
 
 space. Though this view is of course logically permissible, I 
 can not admit that it represents our actual conception of ' free 
 mobility.' I, however, also quoted a ' proof of Euclid's Xlth 
 Axiom,' by a M. Vincent, which deserves more notice than it 
 has hitherto received. I will briefly reproduce it here. I have 
 already shown that we must say that two straight lines in a 
 plane which have a common perpendicular either intersect each 
 other on both sides of that perpendicular, or on neither. The 
 proof is based on the latter assumption, and proceeds as follows. 
 Two straight lines at right angles in a plane divide the whole 
 plane into four equal quadrants. If we successively cut off from 
 one of these quadrants infinite ' half-bands,' by straight lines 
 through equi-distant points along one of the arms of the 
 quadrant, and at right angles to it, then these half-bands can 
 be shown by the method of superposition to be all equal to 
 each other. But the remaining portion of the quadrant can be 
 shown, by superposition, to be still equal to the original 
 (quadrant. Hence, we must say, that the area of a quadrant 
 is infinitely great in comparison with that of any half-band, or of 
 any finite number of half-bauds. But a sector containing any 
 finite angle is numerically comparable with a quadrant, and 
 therefore infinitely greater than any half-band. Hence, if from 
 the vertex of the original quadrant we draw a straight line 
 making any finite angle with the perpendicular line, it must 
 intersect the opposite side of any half-band. 
 
 Now the only objection I have ever heard urged against this 
 proof is to the effect that it is a kind of sacrilege to talk about 
 infinities in this way. No doubt it is very dangerous for incom- 
 petent persons to have anything to do with infinities, as it would 
 be for anyone but a trained metaphysician to attempt any 
 further determination of the absolute. But trained mathe- 
 maticians are accustomed to deal with infinities every day, and 
 are quite competent to determine whether this argument is 
 sound or not. There need, however, be no doubt about the 
 question, for it is easily shown that it is not really inconsistent 
 with Lobatchewsky's geometry at all. It depends on the 
 assumption, which is psychologically justifiable, that space is 
 ' unbounded ' ; and meta-geometers admit that hyperbolic space 
 
76 THE FOUNDATIONS OF 
 
 is not unbounded. The importance of M. Vincent's proof is 
 that it brings home to us the fact that this peculiarity of hyper- 
 bolic space has a psychological, as well as a logical, significance. 
 Lobatchewsky may push his ' boundary ' to infinity, but he can- 
 not thereby push it out of sight altogether ; its influence is still 
 felt at home. We cannot regard space as hyperbolic, and 
 continue to assert the axiom of mobility in its ordinary sense 
 (even if we agree to neglect alterations of shape, such as occur 
 in figures moved about on a surface of constant negative 
 curvature, merely because they are not accompanied by altera- 
 tions in the dimensions of the figures as measured along geodesies 
 of space). 
 
 There is one last point, before I leave the subject of conceptual 
 space, which I particularly commend to the consideration of 
 Mr, Russell. It is that neither the principles of projective 
 geometry, nor the principle of duality, as enunciated by him, 
 are true without reservations of any one of the systems of 
 geometry which he regards as 'possible.' They, strictly speaking, 
 apply only to the M system, to Klein's elliptic space. In the 
 S system, or in Euclidian geometry, the reservations are so 
 slight that they may perhaps fairly be considered as merely 
 verbal. But still there are exceptions, though it will be 
 sufficient to point out the more important anomalies in the 
 U and H systems, and leave those in the S to be inferred 
 from them. In the U system it is not true that any three 
 units in a U^ can be collated with any other three, or in 
 geometrical language, that any three collinear points can be 
 projected on to any other three. We can, in general, only 
 project two on to two, and the third on to the third or its anti- 
 podal. And in the H system it is not true that the order of 
 units in a U^ is unaltered by projection ; unless we take into 
 account units beyond the boundary. That is, if we project 
 points in one 'straight line' in hyperbolic space on to another 
 straight line, there may, in the second, be no points at all 
 corresponding to certain points in the first. It cannot therefore 
 be said that the second section is ' projectively indistinguishable ' 
 from the first, unless we choose to recognise ' imaginary ' points 
 in the second section. But, if we do that, the raison d'etre of 
 
PROJECTIVE GEOMETRY. 77 
 
 hyperbolic geometry is removed. We are only calling things 
 by different names, and not really conceiving anything different 
 from Euclidian space at all, 
 
 I submit that the fact that Mr. Russell has overlooked these 
 exceptions to the applicability of projective geometry to meta- 
 geometrical spaces, is due to the fact that he never really 
 conceived them as such ; that, in fact, his argument, so far as it 
 is valid, is purely symbolic ; and, in so far as it is invalid, it is 
 .so because it is tainted with spatial, and more particularly with 
 Euclidian-spatial, prejudices. 
 
 I may mention, for the benefit of those who have not made 
 a study of projective geometry, that the above is not the only 
 geometrical application of the theory of continuous groups. In 
 order that the positions of points in them may be regarded as 
 forming {U^s it is not necessary that lines should be 'straight,' 
 as long as they are drawn according to some rule which enables 
 us to consider space as a pencil of such lines with any origin, 
 the lines being in any case determined uniquely by the origin 
 of the pencil and one point in its section. The familiar illus- 
 tration is the system of geodesies on a surface of constant 
 curvature. The geodesies on any other surface would not do, 
 for here, although any particular geodesies drawn might not have 
 more than one or two common points, yet two points would not 
 uniquely determine any one of them. The difficulty in the case 
 of non-congruent surfaces is that we have no general rule by 
 which unique groups of positions can be determined. But it 
 may easily be shown that it is theoretically always possible to 
 determine various systems of them. If we stretch an elastic 
 membrane tightly over any congruent surface, say a plane or a 
 sphere, and draw on it projective diagrams, in straight lines or 
 great circles, and then remove the membrane and stretch it 
 again over any other surface whatever, say an ellipsoid or an 
 egg, the projective properties of the figures remain, and the 
 lines, or as many of them as have been drawn, are unique 
 boundaries. Similarly we might imagine a three-dimensional 
 diagram, drawn on a lump of elastic jelly, and we might then 
 screw up the jelly, or stretch or compress it, into any .shape 
 whatever, so long as we did not tear it anywhere, and our pro- 
 
78 THE FOUNDATIONS OF 
 
 jective diagram would lose none of its properties, for the order 
 of the points would be unaltered. It would of course be easy, 
 without making use of an elastic jelly, to draw such projective 
 diagrams ' free hand,' provided only that no two lines in the 
 diagram had more than one common point, and that the 
 diagram was only to contain a finite number of lines. The 
 necessity for some rule, by which to draw the lines, would only 
 arise if we wished to draw an infinite number, that is a con- 
 tinuous series, of them. In the case of surfaces of constant 
 curvature we have such a rule, if we make the lines all 
 geodesies. But it would be an error to suppose that the theory 
 of continuous groups applies only to surfaces of constant 
 curvature, or only to geodesic lines upon them. 
 
 As in our space-conception we have a well-understood 
 example of a continuous group of the third order, catalogued 
 by unique boundaries, we are able to use it as a catalogue for 
 other continuous groups of the second and third order, and so 
 to arrive at an intuitive perception of the relations among units 
 of such groups. This is what we do when we assist our 
 imagination by 'graphic ' representations of statistical and other 
 results. In the same way, if we admit the objective validity of 
 our space conception, we may employ ' graphic ' methods to solve 
 statistical and other problems which involve two or more 
 variables, whether these variables are in themselves spatial 
 quantities or not. It is worthy of note that the word 'graphic' 
 has acquired almost the same connotation as ' well-understood,' 
 simply because we regard our space conception, not without 
 truth, as ' well-understood.' 
 
 The questions which remain to be discussed are these — Can 
 we consistently catalogue the positions of material points in the 
 way I have described, collating them on the assumption that 
 there is an invariable relation of distance between the positions 
 of any two points in a rigid body, and that there is a relation 
 of direction between any two positions ? and if not, would it be 
 necessary to alter our space theory, or might not the adjustment 
 be made more simply by altering some of our physical hypotheses? 
 
 To these (]uestions I now proceed, but if ever I use ordinary 
 language, and say that material space ' is ' or ' is not ' Euclidian 
 
PROJECTIVE GEOMETRY. 79 
 
 it must be understood that I do not intend thereby to institute 
 any comparison between material and conceptual space. To 
 perceive a material body is not to compare it with our conception 
 of it, nor is it only to believe that it exists ; but it is to intuitively 
 catalogue its points by means of our space-conception, and to 
 believe that method of cataloguing will be satisfactory. To say 
 the material body occupied a material space which was 'like' 
 our conceptual space, would lead to ' antinomies,' and therefore 
 I avoid doing so, whenever I do not forget myself. 
 
 Now the fact that we have an instinctive conception of space, 
 and that geometry is psychologically distinguishable from such 
 hypothetical theories as, for example, the wave theory of light, 
 proves that for the ordinary purposes of every day life the 
 ordinary theory of Euclidian geometry is, and has for generations 
 been, good enough. But more than this ; since the most far- 
 reaching and exact calculations of geodesy and astronomy have 
 been made on the Euclidian hypothesis, and have turned out 
 satisfactorily, the theory has already been tested more severely 
 than probably any other ; and from the point of view of Natural 
 Science there is nothing more to be done than to accept it fully 
 and frankly. 
 
 But from the point of view of philo.sophy it is not so much 
 the fact of the certainty which is of interest, as the epistemo- 
 logical grounds upon which it is based. I wish to show that its 
 certainty does not differ in hind, however much it may differ in 
 degree, from that of any other hypothesis of physical science. 
 
 Mr. Russell makes a great point of the superior certainty of 
 the 'axiom of dimensions' over that of, say, the law of gravitation. 
 The supposed superiority is however based on a misconception. 
 The certainty that conceptual space has only three dimensions 
 depends of course on utterly diflferent grounds from that of the 
 law of gravitation, but the certainty that our three-dimensional 
 conception is objectively sufficient, is only the same in kind, 
 even if superior in degree to that of any physical ' law.' 
 Astronomers used anxiously to watch any new comet, to see 
 whether it 'obeyed the law of gravitation' or not; and if they 
 have now given up doing so, it is because they find it easier to 
 explain any apparent discrepancies by hypotheses about meteor 
 
80 THE FOUNDATIONS OF 
 
 streams, and what not. In precisely the same way, Zollner 
 anxiously watched the medium Slade, and came to the conclusion 
 that his strips of hide did not 'obey the law of three dimensions,' 
 though perhaps other people might find simpler hypotheses to 
 account for the results. Zollner was however logically justified 
 in assuming that the fact that a three-dimensional space 
 hypothesis had been found sufficient to explain all previously 
 observed phenomena, does not prove that it will account for the 
 next phenomenon to be observed, any more than the fact that 
 the law of gravitation explains the movements of the planets, 
 proves that it will also explain the movements of the next new 
 comet. There is, moreover, no sense in saying that material space 
 ' is ' or 'is not ' three-dimensional, rather than n dimensional. 
 All we can say is that the assumption of three independent 
 directions has hitherto sufficed to catalogue points in space 
 satisfactorily, and that you may safely ' bet your bottom dollar ' 
 that it will continue to do so. 
 
 Besides the ' axiom of dimensions,' Mr. Russell recognises 
 two others, which he calls the 'axiom of mobility' and the 
 ' axiom of distance,' and he compares the latter with the ' axiom 
 of the straight line ' in projective geometry. I cannot however 
 help thinking this classification implies some confusion of 
 thought ; for it is the axiom of mobility which he says gives us 
 our criterion of spatial equality, and therefore our measure of 
 distance, and which moreover serves to define a straight line in 
 the sense in which that term is used in projective geometry. 
 The term ' axiom of mobility ' seems to me unfortunate in 
 another way — the axiom is not to the effect that it does not 
 require force to move figures about in space ; that would be a 
 dynamical, not a geometrical, axiom. It may be easy or difficult 
 to move figures in space ; all that the geometrical axiom asserts 
 is that when moved they do not, ipso facto, change their dimen- 
 sions. But whatever we call the axiom, the effect of it is to 
 enable us not only to compare objective distances, but to con- 
 struct objective straight lines and planes, independently of any 
 physical hypotheses about the nature of light. The assumption 
 of this ' axiom ' is equivalent, in my theory of cataloguing of 
 groups, to the two assumptions (1) that we can catalogue 
 
PROJECTIVE GEOMETRY. 81 
 
 positions of material points as a continuous group by means of 
 unique boundaries (straight lines and planes), and (2) that we 
 can collate the positions by reversion in the way I have described 
 above, by assuming an invariable relation of ' distance ' between 
 the positions of any two points in a rigid body. This second 
 assumption, therefore, defines our method of cataloguing as either 
 M, U, S, or H. The further axiom or hypothesis required to 
 define ' material space as Euclidian ' may fitly be called the 
 hypothesis of direction, namely that there is a relation of 
 direction as well as of distance between the positions of any 
 two points of a material body ; or we may call it the axiom of 
 the independence of shape and size. It is this latter hypothesis, 
 or axiom, which meta-geometers have seen fit to doubt, but it is 
 a logical error to suppose that no physical experiments could 
 throw, a doubt upon the axiom of congruence, or that it can be 
 known, independently of experience, that a space hypothesis 
 built upon that axiom must fit the facts. It may be that under 
 any conceivable circumstances it would be simpler to reconcile 
 our explanations of the facts by altering some other hypothesis, 
 rather than this one, but this is a matter of practical convenience, 
 not of logical necessity. 
 
 I can best show that this is so by proposing a single experi- 
 ment to test the rival theories of objective geometry, which 
 will at the same time test the axiom of congruence. Mr. Russell 
 has pointed out that even if we found that large stellar triangles 
 bad a measurable ' defect,' we should probably attempt to explain 
 it by modifying our hypotheses as to the propagation of light 
 through interstellar space. It would require a much more 
 fundamental re-construction of our physical conceptions to 
 explain away the results of the experiment I am going to 
 propose; if they were not in accordance with the predictions 
 of Euclidian geometry. Whitworth's method of manufacturing 
 true planes, or ' surface plates,' depends solely on the objective 
 axiom or hypothesis of congruence ; and as the intersection of 
 two such true planes is a straight line, it enables us to construct 
 a series of material points occupying positions which form an 
 ' unique group of the first order ' — if we assume it to be possible 
 to catalogue the positions of material points by the method of 
 
 6 
 
82 TEE FOUNDAJIONS OF 
 
 unique boundaries at all. Suppose then that I construct three 
 such material ' straight edges,' each one yard long, and arrange 
 them to form a triangle ABC. I then, by the same method, 
 construct another straight edge, say one foot long, and by it 
 measure off AD, AE along AB and AC, each equal to one foot, 
 and place my one-foot rule with one end at D, and try whether 
 I can at the same time place the other end, E', at E. No 
 a 'priori reasoning can predict with certainty whether E' will 
 fall upon E, or whether it will fall short, or over. The hypothesis 
 of congruence, if objectively valid, tells me that whichever way 
 it falls in one part of the universe it will fall the same way in 
 any other, or at any other time. If, therefore, it fell short in one 
 place and over in another, or short one day and over the next, 
 we might indeed try to explain this by altering physical 
 hypotheses, but it might conceivably be so difficult to frame 
 a physical hypothesis to account for all the experiments, that 
 we should in the end abandon the axiom, ' Es existiren in sich 
 feste Korper,' as objectively valueless. But meta-geometers 
 draw a yet more far-reaching conclusion from this axiom. They 
 can, namely, calculate the exact amount by which E' will fall 
 short of, or over, £* in the case of triangles of any linear dimensions 
 whatever, if they are given the amount it falls short, or over, in 
 one particular case. This is a perfectly legitimate deduction 
 from the axiom, or hypothesis, of congruence, and it shows that 
 we have a yet more searching test of the validity of this 
 hypothesis, in that from the measurement of one particular 
 CMC we could predict the measurement in any other. The 
 objective application of the axiom of congruence is therefore 
 empirical — in exactly the same sense, if not in quite the same 
 degree, as the ' law of gravitation ' is. And this is where, I 
 believe, that Mr. Kussell's idealistic bias has misled him. 
 
 He calls the axioms of projective geometry ' wholly a priori^ 
 by which he means something more than that they are neces- 
 sary merely in order to establish the conclusions of a particular 
 science. He says that they riiust be true of 'any form of 
 externality,' and that ' without them experience would be 
 impossible.' Whatever qualifying or minimising explanations 
 may be found in the rest of the book I am sure his readers 
 
PR0JEC7IVE GEOMETRY. 83 
 
 will take this to mean that, in his view, we may apply either 
 Euclidian or meta-geometrical calculations to practical space- 
 measurements, with something more than empirical confidence. 
 Indeed, he says distinctly that the empirical element in geometry 
 "arises out of the alternatives of Euclidian and non-Euclidian 
 space " ; by which of course he understands Euclidian, spherical, 
 or hyperbolic space. Now I have shown how these alternatives 
 may be tested, not only as against each other, but as against non- 
 congruent geometries, without any reference to light-theories. 
 The result of one such test can not, even according to Mr. Russell, 
 be predicted on a 'priori grounds. But the test once made with 
 sufficient accuracy, the empirical element, according to Mr. 
 Russell, is decided once for all. We could predict from one 
 experiment exactly how far E' would fall short of, or over, E, in the 
 case of any triangles, whenever or wherever measured ; the 
 only possibility of failure of our prediction lying in the pos- 
 sibilities of inaccuracy in our first experiment. Does Mr. 
 Russell really believe that it is, on a p^'iori grounds, 
 impossible that we should have such an experience as that, say, 
 in England E' should always fall short of E, and in Germany 
 always fall over? Or is it impossible that the distance EE' 
 should be any other function of the linear magnitudes of the 
 triangles than those assigned by spherical or hyperbolic 
 geometry ? 
 
 There is of course one way by which Mr. Russell, or any 
 idealist, might seek to evade this question. He might say that 
 as a philosopher he had nothing to do with actual physical 
 measurements, that he recognised no reality beyond subjective 
 reality, and never intended his theories to have what I call 
 objective validity. But, if so, Mr. Russell has failed to achieve 
 the object he set himself in the Introduction to his book. He 
 is still on the horns of the dilemma that ' none but a madman 
 would throw a doubt on the validity of geometry, and none but 
 a fool would deny its objective reference.' He has still to 
 explain why practical men rely on the validity of Euclidian 
 geometry, and, if he thinks them fools for doing so, why it is 
 that predictions made on the strength of it always turn out true, 
 within the limits of experimental error. 
 
84 F0UNDA7I0NS OF PROJECTIVE GEOMETRY. 
 
 This is the test by which I proposed at the commencement 
 of this paper to distinguish between realistic and idealistic 
 theories of philosophy. It may be that the distinction is in 
 reality only verbal : Mr. Muirhead, for example, when I read my 
 paper on Natural Realism before this Society, called me an 
 idealist, and I called him a realist in reply. It may even be 
 that when idealistic philosophers talk about the absolute they 
 really mean something, which, if only I could learn their 
 language, I should recognise as something I have long been 
 familiar with. But even modern philosophers are ordinary 
 human beings in private life, and if this is the explanation of 
 their theories they must know enough plain English to be able 
 to translate them into ordinary language. There are of course 
 plenty of philosophers who do use plain language, such as Locke, 
 Mill, or even Herbert Spencer. And if T do not agree with 
 their views it is easy to trace where, in my opinion, they have 
 gone wrong. Though I may hold their theories to be erroneous, 
 they are perfectly intelligible. But with Idealism in its modern 
 form this is not the case. Modern philosophers talk the 
 language of genuine Idealism ; but instead of frankly abandon- 
 ing its errors they alter the meanings of its terms, much, or 
 little, according to the force of the criticisms directed against 
 them. The consequence is that to attack them is like punching 
 a bolster ; you may make a dent, but soon after your back is 
 turned it will bulge out to its original shape again. If anyone 
 here can follow me in my theory, and agree in the conclusions I 
 have reached, I shall hail him with pleasure as a natural realist ; 
 only in that case I must really beg him finally to sever any 
 connection he may have had with a 'priori philosophy. If on 
 the other hand he thinks there is a logical necessity which 
 makes it impossible that triangles should have an excess in 
 England and a defect in Germany — then I shall know what to 
 think of him, as a philosopher. 
 
 EDWARD T. DIXON. 
 
 Caubbidoe, 
 
 August, 1897. _ 
 
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